94
Industrial Organization Lecture Notes Sérgio O. Parreiras Fall, 2017
Industrial OrganizationLecture Notes
Seacutergio O Parreiras
Fall 2017
Outline
Mathematical ToolboxIntermediate Microeconomic Theory RevisionPerfect CompetitionMonopolyOligopoly
Mathematical Toolbox
How to use the toolbox1 Casually read it once so you can classify your
understanding of the topics in three categories masteredfamiliar but not mastered and never seen before
2 Read it again but skip the mastered topics3 In your second reading make sure to have pen and paper
at hand and Mathematica open and running4 Work the learning-by-doing exercises using pen and paper
and verifymdashusing Mathematicamdashif your answers arecorrect
5 When you have problems with Mathematicamdashas you willfor suremdashrefer to the crash tutorial and Mathematicarsquoshelp documentation As last resort email me your nb file
Mathematical ToolboxMatrices
A matrix is just a convenient way of displaying information An by m matrix A is composed of n times m entires The entry Aij isdisplayed in the ith row and jth column
Example of a 3 by 3 matrix
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
Mathematical ToolboxMatrices
A matrix is just a convenient way of displaying information An by m matrix A is composed of n times m entires The entry Aij isdisplayed in the ith row and jth column
Example of a 3 by 3 matrix
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
Mathematical Toolbox Matrix Multiplication
a11 a12 a1p
a21 a22 a2p
an1 an2 anp
A n rows p columns
b11 b12 b1q
b21 b22 b2q
bp1 bp2 bpq
B p rows q columns
c11 c12 c1q
c21 c22 c2q
cn1 cn2 cnq
a 21times
b 12a 22
timesb 22
a 2ptimes
b p2
+
+ hellip+
C = A times B n rows q columns
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationMathematica
In Mathematica to enter the matrices
A =
(1 0 3
5 4 7
)and B =
1 2
3 4
7 0
we type
A =1 0 3 5 4 7
B =1 2 3 4 7 0 shift+enter
To multiply the matrices type AB shift+enter
A times B =
(1 middot 1 + 0 middot 3 + 3 middot 7 1 middot 2 + 0 middot 4 + 3 middot 05 middot 1 + 4 middot 3 + 7 middot 7 5 middot 2 + 4 middot 4 + 7 middot 0
)
Mathematical ToolboxPartial Derivatives
Often we wish to evaluate the marginal impact of ONE givenvariable on some function of several variables
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
We refer to My as1 The marginal change f with respect to y2 The partial derivative of f wrt y3 The slope of f wrt y
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Outline
Mathematical ToolboxIntermediate Microeconomic Theory RevisionPerfect CompetitionMonopolyOligopoly
Mathematical Toolbox
How to use the toolbox1 Casually read it once so you can classify your
understanding of the topics in three categories masteredfamiliar but not mastered and never seen before
2 Read it again but skip the mastered topics3 In your second reading make sure to have pen and paper
at hand and Mathematica open and running4 Work the learning-by-doing exercises using pen and paper
and verifymdashusing Mathematicamdashif your answers arecorrect
5 When you have problems with Mathematicamdashas you willfor suremdashrefer to the crash tutorial and Mathematicarsquoshelp documentation As last resort email me your nb file
Mathematical ToolboxMatrices
A matrix is just a convenient way of displaying information An by m matrix A is composed of n times m entires The entry Aij isdisplayed in the ith row and jth column
Example of a 3 by 3 matrix
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
Mathematical ToolboxMatrices
A matrix is just a convenient way of displaying information An by m matrix A is composed of n times m entires The entry Aij isdisplayed in the ith row and jth column
Example of a 3 by 3 matrix
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
Mathematical Toolbox Matrix Multiplication
a11 a12 a1p
a21 a22 a2p
an1 an2 anp
A n rows p columns
b11 b12 b1q
b21 b22 b2q
bp1 bp2 bpq
B p rows q columns
c11 c12 c1q
c21 c22 c2q
cn1 cn2 cnq
a 21times
b 12a 22
timesb 22
a 2ptimes
b p2
+
+ hellip+
C = A times B n rows q columns
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationMathematica
In Mathematica to enter the matrices
A =
(1 0 3
5 4 7
)and B =
1 2
3 4
7 0
we type
A =1 0 3 5 4 7
B =1 2 3 4 7 0 shift+enter
To multiply the matrices type AB shift+enter
A times B =
(1 middot 1 + 0 middot 3 + 3 middot 7 1 middot 2 + 0 middot 4 + 3 middot 05 middot 1 + 4 middot 3 + 7 middot 7 5 middot 2 + 4 middot 4 + 7 middot 0
)
Mathematical ToolboxPartial Derivatives
Often we wish to evaluate the marginal impact of ONE givenvariable on some function of several variables
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
We refer to My as1 The marginal change f with respect to y2 The partial derivative of f wrt y3 The slope of f wrt y
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical Toolbox
How to use the toolbox1 Casually read it once so you can classify your
understanding of the topics in three categories masteredfamiliar but not mastered and never seen before
2 Read it again but skip the mastered topics3 In your second reading make sure to have pen and paper
at hand and Mathematica open and running4 Work the learning-by-doing exercises using pen and paper
and verifymdashusing Mathematicamdashif your answers arecorrect
5 When you have problems with Mathematicamdashas you willfor suremdashrefer to the crash tutorial and Mathematicarsquoshelp documentation As last resort email me your nb file
Mathematical ToolboxMatrices
A matrix is just a convenient way of displaying information An by m matrix A is composed of n times m entires The entry Aij isdisplayed in the ith row and jth column
Example of a 3 by 3 matrix
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
Mathematical ToolboxMatrices
A matrix is just a convenient way of displaying information An by m matrix A is composed of n times m entires The entry Aij isdisplayed in the ith row and jth column
Example of a 3 by 3 matrix
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
Mathematical Toolbox Matrix Multiplication
a11 a12 a1p
a21 a22 a2p
an1 an2 anp
A n rows p columns
b11 b12 b1q
b21 b22 b2q
bp1 bp2 bpq
B p rows q columns
c11 c12 c1q
c21 c22 c2q
cn1 cn2 cnq
a 21times
b 12a 22
timesb 22
a 2ptimes
b p2
+
+ hellip+
C = A times B n rows q columns
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationMathematica
In Mathematica to enter the matrices
A =
(1 0 3
5 4 7
)and B =
1 2
3 4
7 0
we type
A =1 0 3 5 4 7
B =1 2 3 4 7 0 shift+enter
To multiply the matrices type AB shift+enter
A times B =
(1 middot 1 + 0 middot 3 + 3 middot 7 1 middot 2 + 0 middot 4 + 3 middot 05 middot 1 + 4 middot 3 + 7 middot 7 5 middot 2 + 4 middot 4 + 7 middot 0
)
Mathematical ToolboxPartial Derivatives
Often we wish to evaluate the marginal impact of ONE givenvariable on some function of several variables
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
We refer to My as1 The marginal change f with respect to y2 The partial derivative of f wrt y3 The slope of f wrt y
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolboxMatrices
A matrix is just a convenient way of displaying information An by m matrix A is composed of n times m entires The entry Aij isdisplayed in the ith row and jth column
Example of a 3 by 3 matrix
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
Mathematical ToolboxMatrices
A matrix is just a convenient way of displaying information An by m matrix A is composed of n times m entires The entry Aij isdisplayed in the ith row and jth column
Example of a 3 by 3 matrix
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
Mathematical Toolbox Matrix Multiplication
a11 a12 a1p
a21 a22 a2p
an1 an2 anp
A n rows p columns
b11 b12 b1q
b21 b22 b2q
bp1 bp2 bpq
B p rows q columns
c11 c12 c1q
c21 c22 c2q
cn1 cn2 cnq
a 21times
b 12a 22
timesb 22
a 2ptimes
b p2
+
+ hellip+
C = A times B n rows q columns
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationMathematica
In Mathematica to enter the matrices
A =
(1 0 3
5 4 7
)and B =
1 2
3 4
7 0
we type
A =1 0 3 5 4 7
B =1 2 3 4 7 0 shift+enter
To multiply the matrices type AB shift+enter
A times B =
(1 middot 1 + 0 middot 3 + 3 middot 7 1 middot 2 + 0 middot 4 + 3 middot 05 middot 1 + 4 middot 3 + 7 middot 7 5 middot 2 + 4 middot 4 + 7 middot 0
)
Mathematical ToolboxPartial Derivatives
Often we wish to evaluate the marginal impact of ONE givenvariable on some function of several variables
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
We refer to My as1 The marginal change f with respect to y2 The partial derivative of f wrt y3 The slope of f wrt y
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolboxMatrices
A matrix is just a convenient way of displaying information An by m matrix A is composed of n times m entires The entry Aij isdisplayed in the ith row and jth column
Example of a 3 by 3 matrix
A =
A11 A12 A13
A21 A22 A23
A31 A32 A33
Mathematical Toolbox Matrix Multiplication
a11 a12 a1p
a21 a22 a2p
an1 an2 anp
A n rows p columns
b11 b12 b1q
b21 b22 b2q
bp1 bp2 bpq
B p rows q columns
c11 c12 c1q
c21 c22 c2q
cn1 cn2 cnq
a 21times
b 12a 22
timesb 22
a 2ptimes
b p2
+
+ hellip+
C = A times B n rows q columns
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationMathematica
In Mathematica to enter the matrices
A =
(1 0 3
5 4 7
)and B =
1 2
3 4
7 0
we type
A =1 0 3 5 4 7
B =1 2 3 4 7 0 shift+enter
To multiply the matrices type AB shift+enter
A times B =
(1 middot 1 + 0 middot 3 + 3 middot 7 1 middot 2 + 0 middot 4 + 3 middot 05 middot 1 + 4 middot 3 + 7 middot 7 5 middot 2 + 4 middot 4 + 7 middot 0
)
Mathematical ToolboxPartial Derivatives
Often we wish to evaluate the marginal impact of ONE givenvariable on some function of several variables
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
We refer to My as1 The marginal change f with respect to y2 The partial derivative of f wrt y3 The slope of f wrt y
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical Toolbox Matrix Multiplication
a11 a12 a1p
a21 a22 a2p
an1 an2 anp
A n rows p columns
b11 b12 b1q
b21 b22 b2q
bp1 bp2 bpq
B p rows q columns
c11 c12 c1q
c21 c22 c2q
cn1 cn2 cnq
a 21times
b 12a 22
timesb 22
a 2ptimes
b p2
+
+ hellip+
C = A times B n rows q columns
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationMathematica
In Mathematica to enter the matrices
A =
(1 0 3
5 4 7
)and B =
1 2
3 4
7 0
we type
A =1 0 3 5 4 7
B =1 2 3 4 7 0 shift+enter
To multiply the matrices type AB shift+enter
A times B =
(1 middot 1 + 0 middot 3 + 3 middot 7 1 middot 2 + 0 middot 4 + 3 middot 05 middot 1 + 4 middot 3 + 7 middot 7 5 middot 2 + 4 middot 4 + 7 middot 0
)
Mathematical ToolboxPartial Derivatives
Often we wish to evaluate the marginal impact of ONE givenvariable on some function of several variables
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
We refer to My as1 The marginal change f with respect to y2 The partial derivative of f wrt y3 The slope of f wrt y
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationMathematica
In Mathematica to enter the matrices
A =
(1 0 3
5 4 7
)and B =
1 2
3 4
7 0
we type
A =1 0 3 5 4 7
B =1 2 3 4 7 0 shift+enter
To multiply the matrices type AB shift+enter
A times B =
(1 middot 1 + 0 middot 3 + 3 middot 7 1 middot 2 + 0 middot 4 + 3 middot 05 middot 1 + 4 middot 3 + 7 middot 7 5 middot 2 + 4 middot 4 + 7 middot 0
)
Mathematical ToolboxPartial Derivatives
Often we wish to evaluate the marginal impact of ONE givenvariable on some function of several variables
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
We refer to My as1 The marginal change f with respect to y2 The partial derivative of f wrt y3 The slope of f wrt y
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Matrix MultiplicationExamples
A1timesn = (p1 p2 pn) and Bntimes1 =
u(x1)u(x2)
u(xn)
C1times1 = A times B = p1u(x1) + p2u(x2) + + pnu(xn)
Matrix MultiplicationMathematica
In Mathematica to enter the matrices
A =
(1 0 3
5 4 7
)and B =
1 2
3 4
7 0
we type
A =1 0 3 5 4 7
B =1 2 3 4 7 0 shift+enter
To multiply the matrices type AB shift+enter
A times B =
(1 middot 1 + 0 middot 3 + 3 middot 7 1 middot 2 + 0 middot 4 + 3 middot 05 middot 1 + 4 middot 3 + 7 middot 7 5 middot 2 + 4 middot 4 + 7 middot 0
)
Mathematical ToolboxPartial Derivatives
Often we wish to evaluate the marginal impact of ONE givenvariable on some function of several variables
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
We refer to My as1 The marginal change f with respect to y2 The partial derivative of f wrt y3 The slope of f wrt y
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Matrix MultiplicationMathematica
In Mathematica to enter the matrices
A =
(1 0 3
5 4 7
)and B =
1 2
3 4
7 0
we type
A =1 0 3 5 4 7
B =1 2 3 4 7 0 shift+enter
To multiply the matrices type AB shift+enter
A times B =
(1 middot 1 + 0 middot 3 + 3 middot 7 1 middot 2 + 0 middot 4 + 3 middot 05 middot 1 + 4 middot 3 + 7 middot 7 5 middot 2 + 4 middot 4 + 7 middot 0
)
Mathematical ToolboxPartial Derivatives
Often we wish to evaluate the marginal impact of ONE givenvariable on some function of several variables
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
We refer to My as1 The marginal change f with respect to y2 The partial derivative of f wrt y3 The slope of f wrt y
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolboxPartial Derivatives
Often we wish to evaluate the marginal impact of ONE givenvariable on some function of several variables
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
We refer to My as1 The marginal change f with respect to y2 The partial derivative of f wrt y3 The slope of f wrt y
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Partial DerivativesHow to Compute
Myf = part
partyf(x y z) = lim∆rarr0
f(x y +∆ z)minus f(x y z)∆
To compute the partial derivative with respect a givenvariablemdash y in the above examplemdash we use the exact samerules of derivation you learn in calculus with one variable
What about the other variables We treat all the othervariables that are not of interest (x and z in the example above)as constants
In Mathematica we use the command D to compute partialderivatives For example we use D[f[x y z] y] to compute MyfSee the crash tutorial for additional examples
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Partial DerivativesExamples
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Partial DerivativesLearning-by-doing exercises
Compute the marginal utilities MUX and MUY for the followingutility functions
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
If h(x) = f(g(x)) then
hprime(x) = fprime(g(x)) middot gprime(x)
Mathematical Tool Box
The Chain Rule
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Chain RuleLearning-by-doing exercises
1 For each of the composite functions below tell us What arethe corresponding f and g and compute hprimea) h(x) =
radic2x
b) h(x) = minus exp(minusρ middot x)c) h(x) = (4 + xσ) 1
σ
2 Use the Chain Rule to obtain the marginal utilities MX andMY of the utility function
u(x y) = minus2
3exp(minusx)minus 1
3exp(minusy)
3 If k(x) = f(g(h(x))) is a composition of three functionsapply the chain rule twice to compute kprime(x)
4 Consider f(x y) and g(x) compute the total derivative off(x g(x)) with respect to x using the Chain Rule and partialderivatives
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolboxTaylorrsquos Approximation
Consider a function of one variable defined on the real linef R rarr R If f is differentiable we write the first order Taylorapproximation
f(x + h)minus f(x) ≃ f prime (x) middot h
The approximation works well only if |h| is ldquosmallrdquo
For a function of two variables and h = (h1 h2) we have asimilar expression
f(x + h1 y + h2)minus f(x y) ≃ part
partxf(x y) middot h1 +part
partyf(x y) middot h2
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
∆U = U(x +∆xy +∆y)minusU(xy)
≃ MUx middot∆x +MUy middot∆y
Mathematical Toolbox
Marginal Utility amp Taylorrsquos Approximation
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Taylorrsquos ApproximationLearning-by-doing exercises
Using Taylorrsquos approximation for each of the utility functionsbelow compute the change in utility when the consumer movesfrom consuming the basket (100 100) to consuming the basket(105 99)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint What is the value of ∆X What is the value of ∆YWhat is the value of MUX and MUY when the basket (100 100)
is consumed
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolboxImplicit Functions
Let f(x y) be a real-valued function of two variables and let g(x)be a real-valued function of one-variable with the followingproperty
If we set y = g(x) f remains constant as we change x
that is f(x g(x)) = c for all x where c is a constant
We refer to the function g as an implicit function since g isimplicitly defined by the equation f(x g(x)) = c
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Implicit FunctionsLearning-by-doing exercises
For the utility curves below find the equation of theindifference curve that gives utility c
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)Hint set u(x y) = c and solve for y this is your g function
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolboxImplicit Function Theorem
If f(x g(x)) = c for all x where c is a constant
Then gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolsThe Implicit Function Theorem
Proof Taking the total derivative of f with respect to x
ddxf(x g(x)) = part
partxf(x g(x)) middot part
partxx + part
partyf(x g(x)) middot part
partxg(x)
=part
partxf(x g(x)) + part
partyf(x g(x)) middot gprime(x)
Since ddxf(x g(x)) = 0 rArr gprime(x) = minus
part
partx f(x g(x))part
party f(x g(x))
As f is constant along g(x) we also call g an iso-curve of f
Indifference curves and iso-cost curves are examples ofiso-curves that you should be familiar
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
The Implicit Function TheoremLearning-by-doing exercises
For the utility curves below a) find the marginal rate ofsubstitution b) in the MRS replace y by the implicit function gyou found in the previous learning-by-doing exercise andsimplify the expression c) compute gprime(x) for the implicitfunctions of the previous exercise d) compare the results youfound in items (b) and (c)
1 u(x y) = 14x + 3
4y2 u(x y) = 1
2
radicx + 1
2
radicy3 u(x y) = 1
3 ln(x) + 23 ln(y)
4 u(x y) = minus23 exp(minusx)minus 1
3 exp(minusy)
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolboxInterior Solutions
Maximizing a function of one variable defined on the real linef R rarr R
Maximization Problem maxxisinR
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolboxInterior and Corner Solutions
Maximizing a function of one variable defined on an intervalf [a b] rarr R As before
Maximization Problem maxbgexgea
f(x) (P)
First order condition fprime(x) = 0 (FOC)Second order condition fprimeprime(x) le 0 (SOC)
Any point x satisfying FOC and SOC is a candidate for aninterior solution and now
x = a is a candidate for a corner solution if fprime(a) le 0x = b is a candidate for a corner solution if fprime(b) ge 0
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolboxConcavity and Convexity
Consider any function f Rk rarr R
Definition f is concave if and only if for all α isin [0 1] and anytwo points x y isin Rk we have
f (α x + (1minus α) y) ge α f(x) + (1minus α) f(y)
Another definition We say that f is convex if minusf is concave
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Mathematical ToolBoxGlobal Maxima
Proposition Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (ie x is a global maximum)
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Consumer ChoiceECON 410 Revision
List of ingredients
U R rarr R utility function
x = (x1 x2 xn) basket of goods
p = (p1 p2 pn) price list
I consumerrsquos income
Consumerrsquos goal
maxx
subject to
xge0pmiddotxleI
U(x)
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Consumer ChoiceLearning by Doing Exercise
Let ε gt 0 be a fixed positive number1 How many dollars does the consumer save when heshe
reduces consumption of good i by ε units2 How many units of good j can the consumer buy with the
saved amount you found in (1)3 Use Taylorrsquos approximation to estimate the change in
utility as a function of MUi MUj ε pi and pj
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Consumer ChoiceECON 410 Revision
Assume the the optimal basket xlowast has positive amounts of eachgood Then it must be the case that
MUipi
=MUj
pj
for any two goods i and j and
p middot x = I
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
410 Review
Work in groups of two or three to answer the followingAssume two goods such that MU1 = 3 and MU2 = 10 for Anna
1 If the utility of Annarsquos endowment is 2 but she gives up 1
unit of good 1 in exchange for 12 units of good 2 what isher utility after the trade
2 If p1 = 10 and p2 = 20 should Anna buy more or less ofgood 1 and of good 2
3 If p1 = 10 and p2 = 40 should Anna buy more or less ofgood 1 and of good 2
4 If p1 = 1 what is the maximum value of p2 so that Annawould be willing to not reduce her consumption of good 2
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Consumer ChoiceLearning By Doing Exercises
1 Let ∆x1 be the (monthly) change in consumption of good 1and ∆x2 the change in consumption of good 1 Thechanges for the other goods is zero If prices and (monthly)income do not change and the consumer always spend theentirety of the income every month show or argue thatp1 middot∆x1 = minusp2 middot∆x2
2 Using the Taylorrsquos formula for estimating the change inutility ∆U = MU1 middot∆x1 + MU2 middot∆x2 Show that ifMU1p1 gt MU2p2 and ∆x1 gt 0 with p1 middot∆x1 = minusp2 middot∆x2then ∆U gt 0 Hint First try to show the result usingnumerical values for MU1MU2 p1 and p2 and set∆x1 = +1 Next do the same but replace the numericalvalue for the economic variables
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Lecture 1What IO is about
Trusts and cartels became pervasive in the US economy in theXIX century watch this clip But with the passage of theSherman Antitrust Act (1890) and the Clayton Antitrust Act(1904) read Chapter 1 of section 15 of the US code
However it is not easy to answer if a dominant firm isexercising monopoly power and restricting competition Listento this podcast on Google
A major topic of Industrial Organization (IO) is marketstructure That is characterizing and identifying differentmarket structures competitive markets monopoly oligopolyetc
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Perfect CompetitionDefinition
Competitive Behavior
A buyer or a seller is said to be competitive if heshe believesthat the market price is given and that hisher actions do notinfluence the market price
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Perfect CompetitionNon-Increasing Returns to Scale
Two firms i = 1 2 with costs TCi(qi) = ci middot qiLinear inverse demand p = a minus b middot Q = a minus b middot (q1 + q2)
Competitive Equilibrium
The triplet (plowast qlowast1 qlowast2) is a competitive equilibrium if1 give plowast qlowasti solves
maxqi
πi(qi) = plowast middot qi minus TCi(qi)
2 plowast = a minus b middot (qlowast1 + qlowast2)
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Individual Supply FunctionsConstant Returns to Scale
The supply function of firm i = 1 2 is
qi(p) =
+infin if p gt ci
[0+infin] if p = ci
0 if p lt ci
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
The Competitive Equilibrium
If a gt c2 ge c1 the unique competitive equilibrium price isp = c1 and
1 if c1 lt c2 then q2 = 0 and q1 = (a minus c1)b2 if c1 = c2 then Q = q1 + q2 = (a minus c1)b and q1 q2 ge 0
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Returns to ScaleDefinition
Assume we increase all inputs by the same ratio λ gt 0 So thattotal costs increase by λ If production increases by
1 a ratio of λ2 more than a ratio of λ3 less than a ration of λ
We say that the firmrsquos technology has 1 constant returns to scale
Q rArr ATC(Q) = TC(Q)Q stays constant2 increasing returns to scale
Q rArr ATC(Q) = TC(Q)Q 3 decreasing returns to scale
Q rArr ATC(Q) = TC(Q)Q
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Competitive EquilibriumIncreasing Returns to Scale
TC(q) = F + c middot q
Suppose that a gt c then if the firmsrsquo technology exhibitincreasing returns to scale (decreasing average cost) acompetitive equilibrium does not exist
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Social Welfare
DefinitionGiven a market price p and N firms in the industry we definethe social welfare by
W(p) = CS(p) +Nsum
i=1
πi(p)
That is the social welfare is the consumer surplus (area underthe demand curve) plus the firmrsquos profits
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Social Welfare
Recall (ECON 410) that the consumer surplus CS(p) at amarket price is p0 measures how much consumers would bewilling to pay for the quantity that they demand at this priceminus the actual amount they pay
In turn the willingness to pay to consume Q(p0) units ismeasured by the area under the demand curve
p
Q
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Social Welfarecontinued
The social welfare is
W(p) = CS(p) +Nsum
i=1
πi(p)
That os the social welfare is the consumer surplus (area underthe demand curve minus payments to the firms) plus the firmrsquosprofits Moreover since the sum of the firms profits isp middot Q minus
sumNi=1 TCi(qi) where Q =
sumNi=1 when we add the
consumer surplus (which subtracts the payments to the firms)to the profits (which includes the payments to the firms) theterm p middot Q (the payment to the firmsthe firmsrsquo revenue) iscancelled Thus we can also write
W(p) = area under the demand curve at Q(p)minusNsum
i=1
TCi(qi)︸ ︷︷ ︸total cost
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Social WelfareAn Example
Assume a linear inverse demand p = a minus b middot Q and linear costsTCi(qi) = c middot qi Thus the total costs are simplysumN
i=1 TCi(qi) = c middotsumN
i=1 qi = c middot Q Since fixed costs are zero (in thisexample) total cost is just the area under the marginal cost curveThe are under the demand curve is(aminusp0)middotQ0
2 +p0 middot Q0= (a + p0)Q02 = (2a minus bQ0)Q02Total costs are c middot Q0
p
Q
a
c
p = a minus b middot Q
p0 = a minus b middot Q0
Q0
W(p0)
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Social WelfareThe competitive equilibrium maximizes the social welfare
In the example
W =(2a minus bQ)Q2minus cQpart
partQW =(a minus bQ)minus c = p minus c = 0 rArr p = c
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Perfect CompetitionDiscussion
1 In a competitive market can we have lsquofewrsquo firms or firmswith different technologies
2 In a competitive equilibrium how the profits of the firmswith the lsquobetterrsquo technology (ie lower marginal andaverage costs) differ from the profits of the firms with thelsquoworsersquo technology
3 What are the long-run incentives that firms in acompetitive market face
4 Listen to the podcast ldquoHow stuff gets cheaprdquo
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Monopoly
Unlike competitive firms the monopolist firm takes intoaccount its actions affects the market price
maxQ
πi(Q) = p(Q) middot Q minus TC(Q)
part
partQπ = pprime(Q) middot Q + p(Q)︸ ︷︷ ︸MR
minusMC(Q) = 0
Notice that
MC(Qlowast) = p(Qlowast) + pprime(Qlowast) middot (Qlowast) lt p
The monopoly equilibrium price is higher than the marginalcost
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Monopolywith two markets
maxQ1Q2
πi(Q) = p1(Q1) middot Q1 + p2(Q2) middot Q2 minus TC(Q1 + Q2)
part
partQ1π = pprime1(Q) middot Q1 + p1(Q1)︸ ︷︷ ︸
MR1
minusMC(Q1 + Q2) = 0
part
partQ2π = pprime2(Q) middot Q2 + p2(Q2)︸ ︷︷ ︸
MR2
minusMC(Q1 + Q2) = 0
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
MonopolyPodcasts by Planet Money NPR
1 What A 16th Century Guild Teaches Us AboutCompetition
2 Mavericks Monopolies And Beer3 Why Itrsquos Illegal To Braid Hair Without A License
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Cartel
There are N firms in the industry which form a cartel whoseaim is to maximize their joint profits We assume a linearinverse demand p = a minus b middot Q and quadratic costsTCi(qi) = F+ c middot qi in which F and c are positive constants (thefixed cost and marginal cost)
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
CartelEquilibrium
The cartelrsquos profit is
π = (a minus b(q1 + + qN)︸ ︷︷ ︸p
) middot (q1 + + qN)︸ ︷︷ ︸Q︸ ︷︷ ︸
Revenue
minus
TC1(q1) + TCN(qN)︸ ︷︷ ︸Sum of total costs
To find the quantities that max its profits we must solve thefirst-order conditions
part
partqiπ = minusbQ + (a minus bQ)︸ ︷︷ ︸
MR
minusMCi(qi) = 0 i = 1 N
Since the marginal revenue is the same for all firms in eq their marginal cost which equalsthe MR must also be equal But then the individuals quantities must also be equal SoQ = N middot q where the value of q solves
minusbNq + (a minus bNq) minus cq = 0
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
A crash review of Game Theory
Consider N firms indexed by i = 1 2 N and let πi denote theprofit of firm i Let si denote a strategy of firm i (si could bethe quantity that firm i is producing or the price it charges oreven the location choice) Fix the strategy that the others arechoosing sminusi = (s1 siminus1 si+1 sN) We define the bestresponse function of firm i to sminusi and we write BRi(sminusi) as thestrategy or strategies that maximize the payoff of firm i whenthe others follow the strategy sminusi
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
A crash review of Game Theory
Notice that the best response is a function its value depend onthe choice of the othersrsquo firms Of course when choices aresimultaneous the firm does not observe the strategy choices ofthe rival So you should think of the best response as the planthe firm intended to carry if the firm believes that the otherswill play a given strategy sminusi
We read the subscript minusi as ldquonot irdquo That is sminusi is a list ofstrategies (one for each firm) of the firms that are distinct fromfirm i
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
A crash review of Game TheoryNash equilibrium
DefinitionAn equilibrium (a Nash eq) is a list of strategies (one for eachfirm) such that each firm is best responding to the choice of theothers
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Cournot OligopolyCompetition in Quantities
We consider an industry that produces an homogeneous goodwhose market demand is p = a minus b middot Q There are N firm andeach firm has a total cost function TCi(qi) (notice that differentfirms might have different cost functions)
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Cournot OligopolyCompetition in Quantities
We assume non-increasing returns to scale so that the marginalcost is increasing with quantity In this case to find thebest-response function of firm i we set the marginal profit equalto zero and solve for qi
πi =(a minus b middot (q1 + + qN)qi minus TCi(qi)
part
partqiπi =a minus b middot (q1 + + qiminus1 + 2qi + qi+1 minus MCi(qi) = 0
BRi(qminusi) solves the above equation
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Finding a best response
Consider the case of three firms with TCi(qi) = c middot qi fori = 1 2 3 (in this case all firms have the same cost function)To find the best-response function of firm 1
π1 =(a minus b middot (q1 + q2 + q3) middot q1 minus c middot q1part
partq1π1 =a minus b middot (2q1 + q2 + q3)minus c = 0
q1 = BR1(q2 q3) =a minus c minus bq2 minus bq3
2b solves the above equation
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Finding a Nash eq
Continuing the example above to find the Nash eq we mustsolve the system of equations below which is equivalent toevery firm be best responding
q1 =a minus c minus bq2 minus bq3
2b
q2 =a minus c minus bq1 minus bq3
2b
q3 =a minus c minus bq1 minus bq2
2b
Exercise Check that q1 = q2 = q3 =a minus c4b is a Nash eq
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Bertrand OligopolyCompetition in prices
In the Cournot games the best response was single valued butin general the best response is set-valued We may have several(or even infinite) number of best responses to sminusi It is evenpossible the best response to sminusi may fail to exist This is thecase when firms compete in prices
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Bertrand CompetitionDuopoly
There are no fixed costs and the marginal cost is constant c
The market demand is P = a minus bQ but firms compete in prices
The firm charging the lowest price captures the entire market
If both firms charge the same they split the market
Profits are
π1(p1 p2) =
(p1 minus c) middot
(a minus p1
b
)if p1 lt p2
(p1 minus c) middot(
aminusp1b2
)if p1 = p2 and
0 if p1 gt p2
For player 2 π2(p1 p2) = π1(p2 p1)
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Bertrand OligopolyEquilibrium
Suppose that p2 gt p1 then π1 = (p1 minus c) middot(
a minus p1b
)and thus
part
partp1π1 =
a + c minus 2p1b gt 0
That is as long as firm 2 is charging a higher price and(a + c)2 gt p1 firm 1 has incentives to raise its own price up tothe point in which p1 = (a + c)2 But if p1 gt c firm 2 has anincentive to charge a lower price and capture the entire marketThus in an eq it must be that p1 = p2 le cExercise verify that p1 = p2 = c is a Nash equilibrium
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
We consider a demand system for two products given by
p1 = αminus βq1 minus γq2p2 = αminus βq2 minus γq1
We define δ = γ2β2 and refer to it as a measure of productdifferentiation The smaller δ is the more differentiated are theproducts The larger δ is the more homogeneous are theproducts Notice that since we assume γ lt β the measure δ liesin the interval (0 1)
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Oligopoly with differentiated productstextbook 71-713 pp 135ndash141
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 Write the profits of the firms as a function of the quantitiesq1 q2 and the demand parameters α β and γ
2 Equate the marginal profit of each firm to zero and solvefor the best-response functions
3 Find the Nash equilibrium4 Show that profits decrease as γ gt 0 increases5 Explain why product differentiation increases the
equilibrium profitsSolution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
First firms choose locations A and B on the interval [0L]Second firms chose their prices pA and pB Consumers areuniformly distributed on the interval [0L] Assume that A le BThe utility of a consumer who is located at point x is
ui =
minuspA minus τ middot |A minus x| if buys from firm A
minuspB minus τ middot |B minus x| if buys from firm B
Here τ gt 0 measures the consumerrsquos transportation costs
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Assume A le B and that there is a consumer located atA le xlowast(pA pB) le B who is indifferent between buying from Aor B Then
minuspA minus τ(xlowast minus A) = minuspB minus τ(B minus xlowast) rArrxlowast = (A + B)2 + (pB minus pA)(2τ)
Learning by doing exercise assume that firms compete choosingquantities and face zero costs of production and
1 In the above case explain why the demand for A is qA = xlowast
and the demand for B is qB = L minus xlowast2 Write the profit of firm A3 Compute the best-response (price) function of firm A to pB4 Do the same for firm B and then solve for the eq prices
Solution (Mathematica nb file)
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Oligopoly with Differentiated (by location) productstextbook 73-731 pp 149ndash152
Once we obtain the equilibrium prices pA and pB which shouldbe functions of the locations We analyze the choice of locationPlug the eq prices you found previously and then computepartpartAπA and show it is postive That is firm A wants to movecloser to the location of firm B
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
MergersMeasures of Industry Concentration
Label firms so that q1 ge q2 ge q3 ge etcDefinitions
si = 100 qiQ is the market share of firm i
I4 = s1 + s2 + s3 + s4 is the market share of the top fourfirmsIHH =
sumNi=1 s2i is the index of market concentration
note that 0 lt IHH le 10000
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
MergersHorizontal Mergers
Merger increases joint profits but Firms outside the mergermay profit from the mergerThe merger reduces consumersrsquo surplus but the social surplusmight increase (if the merger has strong cost saving efficiencyproperties)
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
MergersVertical Mergers textbook 822 pp 176-179
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Research and Development
A basic modelV value of innovationα probability of discovering a successful innovationI research and development costs (RampD investment)
With one firm invest if and only the expected benefits arelarger than the costs
αV minus I ge 0
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Research and Development
With two firms i=12 if both invest profits are
πi =
V minus I only firm i is successful
V2minus I both firms are successful
0 firm i fails
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Research and Development
With two firms i=121 If both invest expected profits are
π1 = π2 = α2(V2) + (1minus α)αV minus I2 If firm 1 invests but firm 2 does not expected profits are
π1 = αV minus I and π2 = 0
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Research and Development
Depending on V α and I there are several possible cases toconsider
1 If αV minus I lt 0 no firms invest2 If α2(V2) + (1minus α)αV minus I gt 0 both firms invest3 If α2(V2) + (1minus α)αV minus I lt 0 lt αV minus I one firm invests
while the other does not
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
Research and DevelopmentIs investment efficient
The social surplus is the sum of profits (for simplicity weassume zero consumer surplus) if both firms investW = (1minus (1minus α)2)V minus 2I and if only one firm investsW = αV minus I
1 If α2(V2) + (1minus α)αV minus I gt 0 but(1minus (1minus α)2)V minus 2I lt αV minus I there is over investment
2 Ifα2(V2) + (1minus α)αV minus I lt 0 lt αV minus I
but (1minus (1minusα)2)Vminus2I gt αVminus I there is under investment
