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Intermediate Microeconomic Theory BUndergraduate Lecture Notes

by Oz Shy

University of HaifaMarch 2001June 2001File=microb12 Revised=2001/06/04 21:05

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Contents

Remarks vi

1 Cost of Production 11.1 Cost Functions 1

1.2 Single-factor case: A demonstration 11.3 Cost minimization and long-run cost functions 11.4 Properties of Cost Functions 21.5 Optimal plant size 4

2 Profit Maximization 52.1 Choosing profit-maximizing output 52.2 Choosing profit-maximizing factor employment 5

3 Long-Run Supply and the Competitive Industry 63.1 Assumptions and Goals 63.2 A Numerical Example 6

4 The Monopoly 84.1 Demand Characterization 84.2 The Simple Monopoly 84.3 Discriminating Monopoly 94.4 The Cartel 114.5 Multiplant Monopoly 12

5 Oligopoly: Competition Among Few Firms 135.1 Noncooperative Game Theory: Nash Equilibrium 135.2 The Cournot Market Structure: Quantity Competition 135.3 Stackelberg Equilibrium: Sequential Moves 14

5.4 Bertrand Market Structure (price game) 155.5 Dominant Firm 15

6 Welfare Economics 166.1 Pure Exchange Economy: Basic Definitions 166.2 Contract Curves 176.3 Efficient production 176.4 Integrated economy (Full efficiency) 19

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CONTENTS v

7 Competitive Equilibrium and the Neoclassical Welfare Theorems 207.1 Competitive equilibrium 207.2 The First-Welfare Theorem (the essence of capitalism) 21

7.3 The Second-Welfare Theorem 227.4 Monopoly in Edgeworth Box 22

8 Externalities 238.1 Consumption Externalities 238.2 Production Externalities 24

9 Public Goods 299.1 Definition 299.2 Samuelsons Efficiency Condition 299.3 The Tragedy of the Commons 30

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Remarks

Notes prepared during the 1st semester at the University of Haifa, March 2001 to June 2001(Tash-Nach)

For a Syllabus see a separate handout in Hebrew (summarized by the present Table of Con-tent)

Texts:1. Blumental, Levhari, Ofer, & Sheshinski. 1971. Price Theory. Academon Press.

2. Varian H. 1987. Intermediate Microeconomics. W.W. Norton

3. Shy, O. 1986. Industrial Organization: Theory & Applications. Cambridge, Mass.: TheMIT Press

Lecture is 3 45 minutes (given nonstop once a week

Department of Economics, University of Haifa (June 4, 2001)

ozshy@econ.haifa.ac.il

http://econ.haifa.ac.il/ozshy

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Topic 1

Cost of Production

1.1 Cost Functions

Let, W wage rate, R rental on capital T C(W,R,y) maps factor-rental prices to $s Emphasize duality: cost function can derived from a production function, and vise versa. Marginal Cost: MC(y)def= TC(y)/y

Average Cost: AC(y)def= TC(y)/y

1.2 Single-factor case: A demonstration

How to derive the cost function from a production function y = ? Let, W wage rate and > 0.

1. Input-requirement function: = y1/

2. cost means payment to factors: T C(W, y) = W = W y1/.

3. Note: return to scale (see Figure 1.1):

() > iff > 1

1.3 Cost minimization and long-run cost functions

Given W and R, find and k that minimize cost of producing y0 units of output.

min,k

W + Rk s.t. f(, k) y0

Discuss corner vs. interior solutions. If min, kmin > 0,

MPMPk

=W

R

The second equation is y0 = f(min, kmin) to get LRTC: T C(W,R,y)

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Cost of Production 2

lll

Q Q Q

Q = l

T C = wQ1

0 < < 1 = 1 > 1

Q Q Q

AC = wQ1

Q = l

Q = l

T C(Q) = wQ1

T C = wQ1

AC = wQ1

AC = wQ1

Figure 1.1: Duality between cost- and production functions

Example: find LRTC for y = k1 (CRS).

k =1

W

R

yielding conditional demand functions

(W,R,y) =

1 R

W

1y

k(W,R,y) =

1 a

W

R

y

yielding

LRTC(W,R,y) = W + Rk =

1 1

R1

W

+1

R

W1

y

Note: MC(y) =AC(y) is constant

1.4 Properties of Cost Functions

1.4.1 Relation between T C, AC, M C

As an example, consider the total cost function given by T C(Q) = F + cQ2, F, c 0. This costfunction is illustrated on the left part of Figure 1.2. We refer to F as the fixed cost parameter,

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Cost of Production 3

Q Q

M C(Q)

AC(Q)

T C(Q)

Fc

2

cFF

Figure 1.2: Total, average, and marginal cost functions

since the fixed cost is independent of the output level.It is straightforward to calculate that AC(Q) = F/Q + cQ and that M C(Q) = 2cQ. The

average and marginal cost functions are drawn on the right part of Figure 1.2. The M C(Q) curveis linear and rising with Q, and has a slope of 2c. The AC(Q) curve is falling with Q as long asthe output level is sufficiently small (Q

F/c). Thus, in this example the cost per unit of output reaches a minimum at an outputlevel Q =

F/c.

We now demonstrate an easy method for finding the output level that minimizes the averagecost.

Proposition 1.1 If Qmin > 0 minimizes AC(Q), then AC(Qmin) = M C(Qmin).

Proof. At the output level Qmin, the slope of the AC(Q) function must be zero. Hence,

0 =AC(Qmin)

Q=

TC(Qmin)Qmin

Q

=M C(Qmin)Qmin T C(Qmin)

(Qmin)2.

Hence,

M C(Qmin) =T C(Qmin)

Qmin= AC(Qmin).

We now return to our example illustrated in Figure 1.2, where T C(Q) = F+cQ2. Proposition 1.1states that in order to find the output level that minimizes the cost per unit, all that we need todo is extract Qmin from the equation AC(Qmin) = M C(Qmin). In our example,

AC(Qmin) =F

Qmin+ cQmin = 2cQmin = M C(Qmin).

Hence, Qmin =

F/c, and AC(Qmin) = M C(Qmin) = 2

cF.Do it in general (graphically only)

1.4.2 Another useful condition

W

MP= M C =

R

MPk

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Cost of Production 4

Proof.dT C(y)

d=

dT C(f(, k))

d=

T C(y)

y

y

= MC(y) MP

1.5 Optimal plant size

Choosing the level of fixed costs (fixed factors) k denotes plant size (say k squared meters) k(y) the optimal size plant given output level y Short run: SRTC(y, k)

SRAC(y, k) = SRTC(y, k)/y LRTC(y) = SRTC(y, k(y)) Result: LRTC(y) SRTC(y, k) Plot envelope long run optimal plant AC, and SRACs for given values of k.

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Topic 2

Profit Maximization

Profit definition: = TRTC Two methods:

1. choose the profit-maximizing output, y, using T C(y)

2. choose the profit-maximizing factor employment using f(, k)

2.1 Choosing profit-maximizing output

maxy

(y) = T R(y) T C(y) = pyy T C(y)

If y > 0, py = MC(y)

Condition needed: py AT C(y)Second order MC is declining with y.Draw figures.

2.2 Choosing profit-maximizing factor employment

= TR TC = pyf(, k) W Rk

yielding W = pyMP = VMPL and R = MPk = VMPKSOC:

f < 0, fkk < 0, and ffkk (fk)2 > 0

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Topic 3

Long-Run Supply and the Competitive Industry

3.1 Assumptions and Goals

Each firm a competitive (price taker), thus faces a perfectly-elastic demand curve Long-run means free entry and exit Short-run, the number of firms is fixed (no entry or exit) Identical firms (not necessary, but will be assumed here)

The purpose of this analysis is

1. to calculate the long-run number of firms and aggregate output

2. to calculate the short run output level for a given number of firms

3.2 A Numerical Example

3.2.1 Data

Industry faces the (inverse) demand curve: p = 10000/Q

n = 100 identical firms: TC(qi) = 50 + (qi)2/2

3.2.2 Long-run equilibrium

MC = q, AC =50

q+

q

2.

MC = AC = qi = 10 and minq

AC(10) = 10.

Hence, the long-run industry supply is perfectly elastic at p = 10.Intersecting demand and supply yields

p =

10000

Q = 10 = Q = 1000 = qi =Q

n =

1000

100 = 10.

3.2.3 The effect of a rise in fixed cost

Suppose that the government imposes a license fee of 15. We look for the new long-run equilibrium.

TC(qi) = 65 +(qi)

2

2.

MC = q, AC =65

q+

q

2.

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Long-Run Supply and the Competitive Industry 7

MC = AC = qi =

130 = minq

AC(q) =65

130+

130

2=

130,

which is the industrys long-run supply curve.

p =10000

Q=

130 = Q = 10000130

= n = Qqi

=10000

130= 76.92.

3.2.4 The effect of a rise in unit cost

Suppose that the government imposes a per-unit tax of 4.5. Calculate the short-run equilibrium(i.e., n = 100):

TC(qi) = 50 +(qi)

2

2+ 4.5qi.

MC = q+ 4.5, AC =50

q

+q

2

+ 4.5.

Now,

p =10000

100q=

100

q= q+ 4.5 = MC = q = 8 = p = 12.5.

In the long run, the number of firms will decline.

q = 10 = p = MC = 10 + 4.5 = 14.5 = 1000010n

= n = 100014.5

= 68.96.

3.2.5 The effect of a demand shock

Suppose that the Ministry of Health declared the product to be unhealthy. Formally, the demand

drops to p = 6400/Q. In the short run, n = 100. So, solve

p =6400

100q= q = MC = q = 8, = Q = 800, = p = 6400

800= 8 < 10 = minAC.

Therefore, in the long run, the number of firms must decline.

p = q = 10 = 10 = p = 6400Q

= Q = 640, = n = Qq

=640

10= 64.

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Topic 4

The Monopoly

4.1 Demand Characterization

Faces the entire market demand curve Characterize TR(Q), MR(Q), for the inverse demand curve p = a bQ Relate max TR to elasticity If the demand function is linear, p = a bQ, then the marginal-revenue function is also

linear, has the same intercept as the demand, but has twice the (negative) slope. Formally,M R(Q) = a 2bQ.

M R(Q) = p(Q)

1 +

1

p(Q)

.

Proof.

MR(Q) dTR(Q)dQ

=d[p(Q)Q]

dQ= p + Q

dp(Q)

dQ

= p1 + Qp

1

dQ(p)dp

= p 1 + 1p(Q)

.

4.2 The Simple Monopoly

Solve for the simply monopoly

maxQ

= TR(Q) TC(Q) = MR(Q) = MC(Q) provided that pm minMC(Qm).

Example: Figure 4.1 illustrates the monopoly solution for the case where TC(Q) = F + cQ2

, anda linear demand function given by p(Q) = a bQ. MR(Q) = a 2bQ. Hence, if Qm > 0, then Qmsolves

MR(Q) = a 2bQm = 2cQm = MC(Q)implying that

Qm =a

2(b + c)and hence pm = a bQm = a(b + 2c)

2(b + c).

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The Monopoly 9

Q

M C(Q)

AC(Q)

Q

M C(Q)

AC(Q)

Fc

D

M R(Q)

pm

QmM R(Q)

D

Figure 4.1: The monopolys profit maximizing output

Consequently,

(Qm) TR(Qm) TC(Qm)

=a2(b + 2c)

4(b + c)2 F c

a

2(b + c)

2=

a2

4(b + c) F.

Altogether, the monopolys profit-maximizing output is given by

Qm =

a

2(b + c)if F a2

4(b + c)0 otherwise .

4.3 Discriminating Monopoly

Selling to different markets (different demand curves) How to enforce anti-arbitrage measures (e.g., student discounts, senior citizens, hours of

operation, late editions (book publishers))

In some cases, it is profitable not to sell on some markets

Figure 4.2 illustrates the demand schedules in the two markets (market 1 and market 2).

MR is the horizontal sum of MR1 + MR2

If it is profitable to serve both markets, then solution is found from MR = MC(q1 + q2) Find q1 and q2 from MC(q1 + q2) = MR1(q1) = MR2(q2) Find p1 and p2 from each market demand curve It is NOT clear that it is profitable to serve market 1 (must be checked!)

Example: Two segmented markets: q1 = 2 p1, and q2 = 4 p2. Marginal cost is c = 1.

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The Monopoly 10

M R2M R1

M C(q1 + q2)

pp2p1

D1 D2

D1 + D2

Qmqm2qm1

pm2

pm1

q1

QM R

q2

Figure 4.2: Monopoly discriminating between two markets

1. In market 1, p1 = 2 q1. Hence, MR1(q1) = 2 2q1. Equating MR1(q1) = c = 1 yieldsq1 = 0.5. Hence, p1 = 1.5.In market 2, p2 = 4 q2. Hence, MR2(q2) = 4 2q2. Equating MR2(q2) = c = 1 yieldsq2 = 1.5. Hence, p2 = 2.5.

2. 1 = (p1 c)q1 = (0.5)2 = 0.25, and 2 = (p2 c)q2 = (1.5)2 = 2.25. Summing up, themonopolys profit under price discrimination is = 2.5.

3. Suppose now that price discrimination is infeasible (markets are open). There are two cases tobe considered: (i) The monopoly sets a uniform price p 2 thereby selling only in market 2,or (ii) setting p < 2, thereby selling a strictly positive amount in both markets. Let usconsider these two cases:

(a) Ifp 2, then q1 = 0. Therefore, in this case the monopoly will set q2 maximize its profitin market 2 only. By subquestion 1 above, = 2 = 2.25.

(b) Here, if p < 2, q1 > 0 and q2 > 0. Therefore, aggregate demand is given by Q(p) =q1 + q2 = 2 p + 4p = 6 2p, or p(Q) = 3 0.5Q. Hence, M R(Q) = 3 Q. EquatingM R(Q) = c = 1 yields Q = 2, hence, p = 2. Hence, in this case = (p c)2 = 2 < 2.25.

Altogether, the monopoly will set a uniform price of p = 2.5 and will sell Q = 1.5 units inmarket 2 only.1

Finally, to find the relationship between the price charged in each market and the demand

elasticities, pm1 (1 + 1/1) = pm2 (1 + 1/2).

Hence, pm2 > pm1 if 2 > 1, (or |2| < |1|, recalling that elasticity is a negative number). Hence,

a discriminating monopoly selling a strictly positive amount in each market will charge a higherprice at the market with the less elastic demand.

1Note that consumers in market 1 are better off under price discrimination than without it, since under no

discrimination no output is purchased in market 1. Given that the price in market 2 is the same under price

discrimination and without it, we can conclude that in this example, price discrimination is Pareto superior to

nonprice discrimination, since both consumer surplus and the monopoly profit are higher under price discrimination.

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The Monopoly 11

4.4 The Cartel

Contract among N competing firms

Agreement on price or quantity quota (our focus) Examples: OPEC, IATA

The objective of the cartel is to choose q1, q2, . . . , q N to

maxq1,q2,...,qN

(q1, q2, . . . , q N) Ni=1

i(qi) (4.1)

= a bN

i=1

qiN

i=1

qiN

i=1

T Ci(qi).

The cartel has to solve for N quantities, so, after some manipulations, the N first-order conditionsare given by

0 =

qj= a 2b

Ni=1

qi M Cj(qj) = M R(Q) M Cj(qj), j = 1, 2, . . . , N . (4.2)

4.4.0.1 A simple cartel example

10 firms, each has TC(qi) = 200 + 2(qi)2

Market demand: p = 140 Q Solve for the cartels output level, market price, and profit

MR(Q) = 140 2Q = 140 2 10 q = 4q = MC(Q) = q = 356

Hence,

Q = 10 q = 1753

= p = 140 Q = 2453

.

Hence,

i = 245

3 35

6 200 235

6 2

=625

3. = = 10i = 6250

3 208.

4.4.0.2 A more general example

Our calculations will rely on TC(qi) = F + c(qi)2

Hence, MC(qi) = 2cqi and AC(qi) = F/qi + cqi Industry market demand: p = a bQ

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The Monopoly 12

Since all plants have identical cost functions, we search for a symmetric equilibrium q1 = q2 = . . . =qN q. Hence,

a

2bN q = 2cq implying that q =

a

2(bN + c)

. (4.3)

The total cartels output and the market price are given by

Q = N q =N a

2(bN + c)and p = a bQ = a(bN + 2c)

2(bN + c). (4.4)

4.5 Multiplant Monopoly

Same as a cartel, but can adjust N (the number of producers/plants), since all under thesame ownership.

i.e., MR(Q) = MC(qi) for all i = 1, . . . , N

Choose qi that minimizes AC(qi)

4.5.0.3 The simple multiplant-monopoly example

Variable (controlled) number of firms, each has TC(qi) = 200 + 2(qi)2

Market demand: p = 140 Q Solve for # firms, output levels, market price, and profit Key issue: Here the monopoly adjusts output by changing the number of plants. In contrast,

a cartel adjusts output by putting production quotas on member firms.

MC = 4qi, AC =200

qi+ 2qi = arg min AC(qi) = 10, min AC = 40

Now,MR = 140 2N q = 140 20N = 40 = MC = N = 5

Hence,Q = 50, = p = 90 = = 90 50 5 400 = 2500

4.5.0.4 The more general example

Hence, q

i= F/c

Solve MR(Q) = MC(qi) Hence, qi = a/[2(bN + c)] Altogether,

F/c = a/[2(bN + c)], Hence,

N =a

c

2b

F c

b

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Topic 5

Oligopoly: Competition Among Few Firms

5.1 Noncooperative Game Theory: Nash Equilibrium

See Shy (1996), Chapter 2: pp.1215, 1820.

5.2 The Cournot Market Structure: Quantity Competition

5.2.1 Example of Cournot equilibrium

Market demand: Q = 3200

1600p, Hence,

p = 2 Q1600

= 2 q1 + q21600

2 firms, firm 1 has a cost advantage:

TC1(q1) = 0.25q1 TC2(q2) = 0.5q2

Solve for the Cournot output levels, market price, and profit levels

Firm 1 solves:

maxq1 1 =

3200

q1

q21600 q1 0.25q1 (5.1)

yielding a best-response function given by

q1(q2) = 1400 12

q2

Firm 1 solves:

maxq2

2 =3200 q1 q2

1600q2 0.5q2 (5.2)

yielding a best-response function given by

q2(q1) = 12001

2 q1 (5.3)

Solving the two best-response functions yield

q1 =3200

3q2 =

2000

3= Q = 5200

3 1733 = p = 11

12

Hence,

1 =6400

9 711 > 278 2500

9= 2

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Oligopoly: Competition Among Few Firms 14

5.2.2 General Cournot Theory

See Shy (1996), Chapter 6: pp.98103.

5.3 Stackelberg Equilibrium: Sequential Moves

5.3.1 Example

Two-stage game (two periods) Suppose now that firm 1 sets q1 (stage I) before firm 2 sets q2 (stage II) Firm 1 is called a leader Firm 2 is called a follower (choosing q2 by taking q1 as given)

Solving the game backwards starting in the 2nd stage Stage II: Firm 2 takes q1 as given and chooses q2 to solve maxq2 2 which is essentially the

same as (5.2),

yielding firm 2s best response function: (5.3) Stage I: Firm 1, knowing that firm 2 reacts according to (5.3) Thus, substitute (5.3) into (5.1), firm 1 solves

maxq1

1 =3200 q1 q2(q1)

1600q1 0.25q1 = q1(3200 q1)

3200(5.4)

yielding

q1 = 1600 >3200

3= q2 = 400 < 2000

3= Q = 2000 = p = 3200 Q

1600=

3

46400

92 = 100 must hold.

(b) A feasible allocation is said to be Pareto Optimal if there does not exist an allocation whichis Pareto superior to it.

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Welfare Economics 17

6.2 Contract Curves

Draw the contract curve for different preferences

Prove that on any interior allocation on the contract curve,

RCSA =MUAXMUAY

=MUBXMUBY

= RCSB

Example (interior contract curve): UA = xA yA endowed with (3, 2) and UB = xB yBendowed with (1, 6). Solution:

yAxA

=yBxB

=8 yA4 xA = yA = 2xA

Example (non-interior contract curve, perfect substitutes): UA = xA+ 2yA, UB = 2xB + 2yB,with x = 4 and y = 8. Note: Only the aggregate endowment matters for drawing the contractcurve. Solution: 2 sides: xA = 0 and yA + yB = 8; and yA = 8 and xA + yA = 4.

Example (perfect complements and Cobb-Douglas): UA = xA yA endowed with (0, 10), andUB = min{xB, yB} endowed with (20, 5). Solution:

yA =

0 if xA 55 + xA if 5 xA 20

Example (perfect substitutes and Cobb-Douglas): UA = xA + yA endowed with (60, 10); and

UB = xB yB endowed with (20, 30). Solution:

yA =

0 if xA 4040 + xA if 40 xA 80

Difficult example (both consumes have perfect complements preferences): UA = min{xA, yA}and UB = min{xB, yB} with aggregate endowment of x = 20 and y = 10.

As above but UA = min{2xA, yA} and UB = min{xB , yB} As above buy UA = xA + yA and UB = xB + yB. Solution: The contract curve is the entire

box.

6.3 Efficient production

6.3.1 Basic conditions

2 factors (labor & capital), aggregate endowment L and K 2 goods produced by x = f(LX, KX) and y = g(LY, KY) Monotonicity implies full employment where LX+ LY = L and KX+ KY = K

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Welfare Economics 18

Draw Edgeworth Box in the factor space and iso-quants Interior efficient production allocations (contract curve) satisfy

RTSXdef

=MPXLMPXK

=MPYLMPYK

def

= RTSY

If industry X contains more than one firm (say 2 firms) than efficient production allocationimplies that

MPX,IL = MPX,IIL and MP

X,IK = MP

X,IIK

6.3.2 Capital and labor intensities in production

Definition 6.4 The production of X is said to be capital-intensive relative to Y if

KX

LX>

KY

LY

Draw contract curves reflecting unique and reversible intensities. Example: X = (LX)1/3(KX)2/3 and Y = (LY)1/2(KY)1/2

MPXLMPXK

=1

2

KXLX

=KYLY

=MPYLMPYK

= KXLX

= 2KYLY

>KYLY

implying that the production of X is capital-intensive relative to Y

To find the contract curve:KX

LX= 2

KY

LY>

KY

LY=

KX =

2KLX

L + LX LXLK

6.3.3 Production efficiency and the production-possibility curve

If the contract curve is a 1 : 1 function (i.e., not a set-valued function) then each allocationon the contract curve is associated with a unique point on the PPF

Each allocation in the box which is not on the contract curve is associated with point insidethe PPF

Example: x = LX

KX and y =

LY

KY

RTSXdef

=MPXL

MPXK

=KX

LX=

K KXL LX

=MPYL

MPYK

def

= RTSY =

KX =

K

L

LX

which is the contract curve. Now,

x =

LX

K

LLX = LX

K

L= LX = x

L

K

y =

L LX

K KL

LX = (L LX

K

L) = LX = L

L

Ky

Altogether,

y =

KL x

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Welfare Economics 19

How to calculate the slope of the PPF: dy/dx? Claim: RPTXY (rate of product transformation)

RPTXYdef

=dy

dx=

MPYLMPXL

andMPYKMPXK

Proof.

MPXL =dx

dLand MPYL =

dy

dL= dy

dx=

dydLdxdL

=MPYKMPXK

6.4 Integrated economy (Full efficiency)

We look at interior conditions (although, we have already discussed cases in which allocations are

not on the interior of Edgeworth Box.

consumption:

RCSA =MUAXMUAY

=MUBXMUBY

= RCSB

Production:

RTSXdef

=MPXLMPXK

=MPYLMPYK

def

= RTSY

Industry:MPX,IL = MP

X,IIL and MP

X,IK = MP

X,IIK

Consumption-production: (representative indifference curve is tangent to the PPF)

RCSXY = RPTXY

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Topic 7

Competitive Equilibrium and the Neoclassical Welfare

Theorems

7.1 Competitive equilibrium

7.1.1 Definition

All agents: Consumers, producers, and factors of production are price takers. Competitive equilib-rium is a price vector (pX, pY,W,R) such that

(a) given these prices firms choose how much to produce, xp and yp to maximize profits

(b) given these prices firms choose hiring of factors of production, LX, KX, LY, KY, to maximizeprofits

(c) Consumers, given goods prices, and their income from owning factors, choose how much toconsumer by maximizing utility subject to their budget constrains, and choose (aggregate)consumption xc, yc

(d) Equilibrium in all markets: xc = xp, yc = yp, LX+ LY = L, and KX+ KY = K

7.1.2 Example: Pure exchange economy with 2 Cobb-Douglas

Consumer A: UA = (xA)2

yA endowed with (2, 6) Consumer B: UB = xByB endowed with (4, 2) Solving for consumers demand function yield

xA =2

3

IApX

yA =1

3

IApY

xB =1

2

IBpX

yB =1

2

IBpY

Consumers incomes are generated from selling their endowments:IA = 2pX+ 6pY IB = 4pX+ 2pY

Market equilibrium conditions: xA + xB = 2 + 4 = 6 and yA + yB = 6 + 2 = 8 Substituting incomes into equilibrium condition of market for X yield

2

3

2 + 6

pYpX

+

1

2

4 + 2

pYpX

= 6

Substituting incomes into the equilibrium condition of market Y yield1

3

2

pXpY

+ 6

+

1

2

4

pXpY

+ 2

= 8

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Competitive Equilibrium and the Neoclassical Welfare Theorems 21

Numeraire: Since only the price ratio pX/pY can be calculated, we can set pX def= 1 as anumeraire, so only pY remains to be solved!

Walras Law: Let there be n perfectly competitive markets. Then, if n 1 markets clear, theremaining market must also clear.

In our case, n = 2, hence, if the market for X clears, the market for Y must clear. Proof for Walras Law: Implied by having all agents satisfying their budget constraints. solving the market equilibrium for Y yields

pXpY

=15

8= xA = 52

15, xB =

38

15, yA =

13

4, yB =

19

4

7.2 The First-Welfare Theorem (the essence of capitalism)

7.2.1 The Theorem

Suppose that: (1) here are no externalities, (2) Complete markets: i.e., no missing markets Then,a competitive equilibrium allocation is Pareto efficient.Incomplete proof:

Consumers:

RCSA =MUAXMUAY

=pXpY

and RCSB =MUBXMUBY

=pXpY

Producers (input markets):

pXMPXL = W pXMP

XK = R pYMP

YL = W pYMP

YK = R

hence,

RTSX =MPXLMPXK

=W

R=

MPYLMPYK

= RTSY

Producers & Consumers (output markets): Profit maximization implies:

RPT =MPYLMPXL

=MPYKMPXK

=pXpY

= RCS

(draw the economys iso-profit line tangent to the PPF, and to the indifference curve)

7.2.2 Why this theorem is so useful? an example of a single-person economy

Goods: a product X, and leisure L (prices: p and w) Consumer: U = x L Firm: t = employment level. x = t Production possibility curve: x = 24 L

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Competitive Equilibrium and the Neoclassical Welfare Theorems 22

Pareto Optimum implies

RPT =

dy

dx=

1

224 L= RCS =

MUL

MUX=

x

L

Hence,1

2

24 L =x

L=

24 L

L= L = 16, x =

8 = 2

2

which also a competitive equilibrium allocation

Now, suppose that we did not have the First-Welfare Theorem! Profit maximization

maxt

= px

wt = p

t

wt =

td = 1

2

p

w

To find the labor supply we need to calculate the firms profit

= p

t wt = 14

p2

w

Consumers income I = 24w + Leisure demand (hence labor supply)

L =24w +

2w=

ts = 24

L =24w

2w= 12

1

8

p2

w2

Labor market equilibrium

ts = 12 18

p2

w2=

1

2

p

w

= td = p

w= 2

8 = t = 8 = x =

8

7.3 The Second-Welfare Theorem

For every Pareto-optimal allocation there exist an initial endowment for which this Pareto-optimal allocation is a competitive equilibrium

Demonstrate using an Edgeworth Box Explain the importance: Possibility of achieving any PO allocation via the competitive mech-

anism

7.4 Monopoly in Edgeworth Box

Demonstrate.

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Topic 8

Externalities

8.1 Consumption Externalities

8.1.1 Definition

General formulation: UA(xA, yA, xB , yB) and UB(xA, yA, xB, yB) Define and explain the difference between external economies and external diseconomies A competitive equilibrium allocation need not be Pareto efficient. For example, UA(xA, yA)

(selfish) but UB

(xB , yB, xA) (externality)

Pareto optimality requiresMUAXAMUAYA

=MUBXB MUBXA

MUBYB

Proof. Pareto optimal allocations are found solving

max UB(xB, yB, xA) s.t. UA(xA, yA) = U

A0 , xA + xB = x, yA + yB = y.

DefineL() def= UB(x xA, yB, xA) + U

A0 UA(xA, y yB) .

The first-order conditions are given by

0 =dL

dxA= MUBxB + MUBxA MUAxA and 0 =

dL

dyB= MUByB + MU

AyA

which yield the result.

However, in a competitive equilibrium

MUAXAMUAYA

=pXpY

=MUBXBMUBYB

8.1.2 The Roomates Example

Two roommates: Tom and Jerry Two goods: Cookies and Music UT(cT, mT) = cT + mT and UJ(cJ, mT) = cJ (mT)2/2 Endowment: each with 30 cookies, and 24 hours No communication allowed: cT = cJ = 30 and mT = 24 hours

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Externalities 24

Pareto Optimality: mT = 1. Why?

max UJ = cJ

(mT)2

2

s.t U0T = cT + mT = 60

cJ+ mT

Since cT = 60 cJ, hence cJ = mT

maxmT

60 + mT U0T

(mT)2

2

= mT = 1

Suppose property rights belong to music lovers. Since M RST = MUTcT/MUTmT = 1, J canbribe T to reduce the playing time length from 24 to 1 by giving him 23 cookies.

8.1.3 Competitive Marriage

Married couple: consumers A and B

3 goods: F = fish, E = Elvis music, and V = Vivaldi music Utility functions:

UA = fA + 120 ln eA 60vB and UB = fB + 120 ln vB 60eA

Market prices: pf = 2, pE = 6, pv = 3 Incomes: IA = IB = 100 Calculate a competitive equilibrium

Calculate the Pareto-optimal allocations8.2 Production Externalities

8.2.1 Definition

Production externality occurs when the output level of one firm affects the production function(cost function) of a different firm.

Examples: Pollution, infrastructure, training Distinguish from pecuniary (not real) externality which stems from equilibrium price effects

8.2.2 The Farmers Example

Suppose that a honey farm is located next to an apple orchard, and each acts as a competitivefirm. Let the amount of apples produced be measured by A and the amount of honey by H. Thecost functions of the two firms are given by:

CH(H) =H2

100, and CA(A) =

A2

100H.

The fixed market price of honey is pH = $2 and the fixed market price of apples is pA = $3.Answer the following questions:

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Externalities 25

(a) Compute the equilibrium amount of honey and the number of apples produced under theseprices. Calculate the profit of each firm and the sum of the firms profits in this equilibrium.Honey producer chooses He to maxHH = 2H

H2/100 yielding He = 100.

Apple producer takes He = 100 as given and chooses Ae to maxA A = 3A A2/100 + 100yielding Ae = 150. Hence, eH = 100,

eA = 325, and

eH+

eA = 425.

(b) Suppose that the honey and apple firms merged. What would be the profit-maximizing amountof apples and honey produced. Calculate the profit of the merged firms. Will the merged firmincrease or decrease the production of honey compared with the case where the honey produceris independent? Explain!The merged firm chooses Hm and Am to

maxH0A0

H+ A = 2H H2

100+ 3A A

2

100+ H

Hence, Hm = 150 and Am = 150. That is, due to the positive externality, the merged firmincreases the production of honey (which reduces the cost of producing apples). Thus, H+A =450 > 425 = eH+

eA.

(c) Suppose now that merger is illegal, so each firm is forced to be independent. Is there anytax/subsidy on one of the producer that will bring the economy to produce the optimal amountof honey and apples?Suppose that the government pays the honey producer a subsidy of s per-unit of honey pro-duced. We now calculate the subsidy that induces the honey producer to produce H = 150.With the subsidy, the honey producer solves

maxH

H = 2H + sH H2100

, yielding s = H

50 2 = 150

50 2 = 1.

8.2.3 The Airport Example

Airport choosing the number of landings x Housing developer choosing # houses to build, y

A = 48x x2 and D = 60y y2 xy

8.2.3.1 No regulations, no communication

Airport: 0 = 48 2x = x = 24.Developer: 0 = 60 2y x = y = 18.Profit levels: A = 576; D = 324;

def

= A + D = 900.

8.2.3.2 Pareto- (industry) optimal allocation

maxx,y

= 48x x2 + 60y y2 xy

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Externalities 26

0 =

x= 48 2x y

0 =

y= 60

2y

x

Hence, y = 24 and x = 12.

A = 432; D = 576; = 1008

8.2.3.3 Strict prohibition: developer has all property right (bargaining not allowed)

Hence, no planes, x = 0, A = 0.

D = 60y y2 0 = 0 = 60 2y = y = 30 = D = 302 = 900 < .

Hence, this mechanism does not support PO.

8.2.3.4 Airport is liable for all damages (bad mechanism!)

Airport must pay xy to developer (compensation for the damage).

maxx

A = 48x x2 xy = 48x x2 30x = 0 = 18 2x = x = 9

D = 60y y2 xy + xy = 0 = 60 2y = y = 30Hence,

A = 81; D = 900; = 981 <

This mechanism also does not support PO since the developer will over-produce to increase thecompensation.

8.2.3.5 Coase equilibrium (bargaining after property rights are assigned to the airport): developerbribes the airport

The developer decides on x subject to leaving A = 576.Level of the bribe = 576 (48x x2).

maxx,y

D = 60y y2 xy [576 (48x x2)]

0 =D

y= 60 2y x

0 =D

x= y + 48 2x

Hence, y = 24 = y and x = 12 = x. Pareto-allocation!!!

Coase Theorem: if agents can bargain, then optimality is restored regardless of property rightsassignment.

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Externalities 27

8.2.3.6 Coase equilibrium (bargaining property rights are assigned to the Developer): Airport bribesthe developer to allow him to increase x

Airport has to leave the developer with at least D

900 (see Subsection 8.2.3.4).The compensation is 900 (60y y2 xy). Hence,maxx,y

A = 48x x2 [900 (60y y2 xy)].Yielding the same PO allocation with different profit levels.Remark: Why property rights are needed? Answer: to define the reservation payoffs of the playersbefore bargaining starts.

8.2.3.7 Optimal tax

It is essential that the tax will be levied on the externality-producing activity directly!!!

i.e., do not tax the food sold in airport restaurants, since consumers will bring their own food,but will not reduce travel. A tax on gas, will lead to refueling abroad. Exception: when the taxed item is a perfect complement to the externality activity. However, even if you tax the activity only, flights may switch to airports nearby in a different

country.

Which tax on airport will restore optimality? i.e., x = 12, hence, y = 24.

A = 48x x2 tx = 0 = 48 2x t = t = 24The rest follows.

8.2.4 Upstream-downstream Example

Steel producers cost function: cs(s, x), where s and x are quantity of steel and pollutionproduced

Fishermans cost function: cf(f, x) A competitive steel factory solves

maxs,x

s = ps s cs(s, x) = ps = dcsds

and 0 =dcsdx

A competitive fisherman solves

maxf

f = pf f cf(f, x) = pf = dcfdf

Socially optimal outcome is found by maximizing joint profit

maxf,s,x

def

= s + f = 0 = dcsdx

+dcfdx

= dcsdx

= dcfdx

Draw figure (see Varian) Numerical example (see Varian)

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Externalities 28

8.2.5 Advertising Externality

In a shopping mall there are 2 stores. Advertising of one stores brings more customers toboth stores

a1 advertising of store 1, a2 advertising of store 2 Profit functions: 1 = (60 + x2)x1 2(x1)2 and 2 = (105 + x1)x2 2(x2)2

Solve for non-cooperative and collusion equilibria

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Topic 9

Public Goods

9.1 Definition

A good in which the consumption of one consumer does not exclude the consumption of otherconsumers.

Examples: Broadcasting (radio & TV) Problematic examples: Bridges, roads (because of congestion)

9.2 Samuelsons Efficiency Condition

9.2.1 Efficiency

Let X be public good and y be private good. Let F(x, y) = 0 be the economys production possibility curve Efficient allocation must satisfy

RCSA + RCSB =MUAXMUAYA

+MUBXMUBYB

=FxFy

= RPT

As opposed to having X private good where optimality requires that each consumer equatesRCS to RPT.

9.2.2 Example: Transportation and Cobb-Douglas preferences

2 goods: F = fish (private) and T = train (transportation infrastructure)

N people in town: Ui def= (fi)2 T Production possibility frontier: F2 + 3T2 = 1800

Pareto-optimal provision of T is found from:

N MRSi = N MUiT

MUifi= N fi

2T= RPT = 3

T

F= T = 30

9.2.3 Example: Fireworks

In Nahalal (a small town in Northern Israel) there are N > 1 people. Every year they have afireworks show on the Israels Independence Day. All consumers are identical, each consumer iconsumers only 2 goods: a private good xi and a public good F (fireworks). The utility of each

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Public Goods 30

consumer i is given by ui(xi, F) = xi +

F. Let wi denote the (fixed) income of consumer i.Suppose that the price of the private good is normalized to px = $1, and that the cost of producingfireworks is c(F) = F2.

(a) If fireworks is privately provided by a single competitive firm, calculate the total amount offireworks they have on independence day. Does the amount of fireworks increase or decreasewith the population size N? Explain!Let pf be the price of one firework. Suppose that consumer i buys fi units of fireworks. Then,fi solves

maxfi

ui = wi pffi +

fi +j=i

fj.

The first-order condition yieldspf = 1/(2

fi +

j=i fj). The producer equatesp

f = M C(F) =

2F. By symmetry, fi = f for all i, hence, F = N f. Altogether,

Fe =1

42

3

and fe =1

42

3 N

Thus, although the free-rider effect decreases the amount of fireworks purchased by each con-sumer when population increases, the aggregate level stays the same.

(b) What is the socially optimal amount of fireworks? Does it increase or decrease with N? Explain!The social planner chooses a firework level F that solves the Samuelson condition given by

i

1

2

F= N

1

2

F= MC(F) = 2F, hence, F =

N

4 2

3

and f =1

42

3 N1

3

.

Here, the social planner increases the aggregate level when N increases to capture the benefitfrom an increased positive externality.

9.3 The Tragedy of the Commons

See a fishing example in Shy (1996), Ch.17, pp.448451.