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Chapter 12
GENERAL EQUILIBRIUM AND
WELFARE
Copyright 2005 by South-Western, a division of Thomson Learning. All rights reserved.
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Perfectly Competitive
Price System We will assume that all markets are
perfectly competitive
there is some large number of homogeneousgoods in the economy both consumption goods and factors of
production
each good has an equilibrium price there are no transaction or transportation
costs
individuals and firms have perfect information
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Law of One Price A homogeneous good trades at thesame price no matter who buys it or
who sells it
if one good traded at two different prices,
demanders would rush to buy the good
where it was cheaper and firms would try
to sell their output where the price washigher
these actions would tend to equalize the price
of the good
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Assumptions of Perfect
Competition There are a large number of people
buying any one good
each person takes all prices as given andseeks to maximize utility given his budget
constraint
There are a large number of firmsproducing each good
each firm takes all prices as given and
attempts to maximize profits
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General Equilibrium Assume that there are only two goods,x
and y
All individuals are assumed to haveidentical preferences
represented by an indifference map
The production possibility curve can beused to show how outputs and inputs are
related
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Edgeworth Box Diagram Construction of the production possibility
curve forxand ystarts with the
assumption that the amounts of kand l
are fixed
An Edgeworth box shows every possible
way the existing kand lmight be used to
producexand y any point in the box represents a fully
employed allocation of the available
resources toxand y
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Edgeworth Box Diagram We will use isoquant maps for the two
goods
the isoquant map for goodxuses Oxas theorigin
the isoquant map for good yuses Oyas the
origin
The efficient allocations will occur where
the isoquants are tangent to one another
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Edgeworth Box Diagram
Ox
Oy
Total Labor
T
otalCapital
x2
x1
y1
y2
A
PointAis inefficient because, by moving along y1, we can increase
xfromx1tox2while holding yconstant
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Edgeworth Box Diagram
Ox
Oy
Total Labor
T
otalCapital
x2
x1
y1
y2
A
We could also increase yfrom y1to y2while holdingxconstant
by moving alongx1
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Production Possibility Frontier
The locus of efficient points shows the
maximum output of ythat can be
produced for any level ofx we can use this information to construct a
production possibility frontier
shows the alternative outputs ofxand ythat
can be produced with the fixed capital and
labor inputs that are employed efficiently
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Rate of Product Transformation
The rate of product transformation (RPT)
between two outputs is the negative of
the slope of the production possibilityfrontier
frontierypossibilit
productionofslope)for(of yxRPT
)(along)for(of yxOOdx
dyyxRPT
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Rate of Product Transformation
The rate of product transformation shows
howxcan be technically traded for y
while continuing to keep the availableproductive inputs efficiently employed
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Shape of the Production
Possibility Frontier
The production possibility frontier shown
earlier exhibited an increasing RPT this concave shape will characterize most
production situations
RPTis equal to the ratio of MCxto MC
y
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Shape of the Production
Possibility Frontier Suppose that the costs of any output
combination are C(x,y)
along the production possibility frontier,
C(x,y) is constant
We can write the total differential of the
cost function as
0
dy
y
Cdx
x
CdC
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Shape of the Production
Possibility Frontier Rewriting, we get
y
xyx
MCMC
yCxCOO
dxdyRPT
//)(along
The RPTis a measure of the relative
marginal costs of the two goods
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Shape of the Production
Possibility Frontier As production ofxrises and production
of yfalls, the ratio of MCxto MC
yrises
this occurs if both goods are produced
under diminishing returns
increasing the production ofxraises MCx, while
reducing the production of ylowers MCy this could also occur if some inputs were
more suited forxproduction than for y
production
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Shape of the Production
Possibility Frontier But we have assumed that inputs are
homogeneous
We need an explanation that allows
homogeneous inputs and constant
returns to scale
The production possibility frontier will be
concave if goodsxand yuse inputs in
different proportions
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Opportunity Cost
The production possibility frontier
demonstrates that there are many
possible efficient combinations of twogoods
Producing more of one good
necessitates lowering the production ofthe other good
this is what economists mean by opportunity
cost
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Opportunity Cost
The opportunity cost of one more unit of
xis the reduction in ythat this entails
Thus, the opportunity cost is bestmeasured as the RPT(ofxfor y) at the
prevailing point on the production
possibility frontier this opportunity cost rises as morexis
produced
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Concavity of the Production
Possibility Frontier Suppose that the production ofxand y
depends only on labor and the production
functions are5.0)( xxfx ll
5.0)( yyfy ll
If labor supply is fixed at 100, then
lx+ ly= 100
The production possibility frontier is
x2+ y2= 100 forx,y0
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Concavity of the Production
Possibility Frontier The RPTcan be calculated by taking the
total differential:
y
x
y
x
dx
dyRPTydyxdx
2
)2(or022
The slope of the production possibility
frontier increases asxoutput increases the frontier is concave
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Determination of
Equilibrium Prices
We can use the production possibility
frontier along with a set of indifferencecurves to show how equilibrium prices
are determined
the indifference curves represent
individuals preferences for the two goods
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Determination of
Equilibrium Prices
Quantity ofx
Quantity of y
y1
x1
U1
U2
U3
y
x
p
pslope
C
C
The price ofxwill rise andthe price of ywill fall
x1
y1
There is excess demand forxand
excess supply of y
excess
supply
excess demand
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Determination of
Equilibrium Prices
Quantity ofx
Quantity of y
y1
x1
U1
U2
U3
y
x
p
pslope
C
C
x1
y1
The equilibrium output will
bex1* and y1*y1*
x1*
The equilibrium prices will
bepx* andpy*
C*
C*
**slope
y
x
pp
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Comparative Statics Analysis
The equilibrium price ratio will tend to
persist until either preferences or
production technologies change If preferences were to shift toward good
x,px/pywould rise and morexand less
ywould be produced we would move in a clockwise direction
along the production possibility frontier
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Comparative Statics Analysis
Technical progress in the production of
goodxwill shift the production
possibility curve outward this will lower the relative price ofx
morexwill be consumed
ifxis a normal good
the effect on yis ambiguous
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Technical Progress in the
Production of x
Quantity ofx
Quantity of y
U1
U2
U3
x1*
The relative price ofxwill fall
Morexwill be consumed
x2*
Technical progress in the production
ofxwill shift the production possibility
curve out
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General Equilibrium Pricing Suppose that the production possibility
frontier can be represented by
x2+ y2= 100
Suppose also that the communitys
preferences can be represented by
U(x,y) =x0.5y0.5
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General Equilibrium Pricing Profit-maximizing firms will equate RPT
and the ratio ofpx/py
y
x
pp
yxRPT
Utility maximization requires that
y
x
p
p
x
yMRS
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General Equilibrium Pricing Equilibrium requires that firms and
individuals face the same price ratio
MRSxy
pp
yxRPT
y
x
or
x* = y*
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The Corn Laws Debate High tariffs on grain imports were
imposed by the British government after
the Napoleonic wars
Economists debated the effects of these
corn laws between 1829 and 1845
what effect would the elimination of these
tariffs have on factor prices?
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The Corn Laws Debate
Quantity of Grain (x)
Quantity ofmanufactured
goods (y)
U1
U2
x0
If the corn laws completely prevented
trade, output would bex0and y0
y0
The equilibrium prices will be
px* andpy*
*
*slope
y
x
p
p
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The Corn Laws Debate
Quantity of Grain (x)
Quantity ofmanufactured
goods (y)
x0
U1
U2
y0
Removal of the corn laws will change
the prices topx andpy
'
'slope
y
x
p
p
Output will bex1 and y1
x1
y1
y1
x1
Individuals will demandx1and y1
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The Corn Laws Debate
Quantity of Grain (x)
Quantity ofmanufactured
goods (y)
y1
x0
U1
U2
x1
y0
'
'slope
y
x
p
p
x1
y1
Grain imports will bex1x1
imports of grain
These imports will be financed by
the export of manufactured goods
equal to y1 y1exportsof
goods
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The Corn Laws Debate
We can use an Edgeworth box diagram
to see the effects of tariff reduction on
the use of labor and capital If the corn laws were repealed, there
would be an increase in the production
of manufactured goods and a decline in
the production of grain
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The Corn Laws Debate
Ox
Oy
Total Labor
TotalCapital
A repeal of the corn laws would result in a movement fromp3to
p1where more yand lessxis produced
x2x1
x4
x3
y1
y2
y3
y4
p4
p3
p2
p1
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The Corn Laws Debate
If we assume that grain production is
relatively capital intensive, the movement
fromp3top1causes the ratio of ktoltorise in both industries
the relative price of capital will fall
the relative price of labor will rise
The repeal of the corn laws will be
harmful to capital owners and helpful to
laborers
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Political Support for
Trade Policies Trade policies may affect the relative
incomes of various factors of production
In the United States, exports tend to beintensive in their use of skilled labor
whereas imports tend to be intensive in
their use of unskilled labor free trade policies will result in rising relative
wages for skilled workers and in falling
relative wages for unskilled workers
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Existence of General
Equilibrium Prices Beginning with 19th century investigations
by Leon Walras, economists have
examined whether there exists a set ofprices that equilibrates all markets
simultaneously
if this set of prices exists, how can it befound?
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Existence of General
Equilibrium Prices Suppose that there are ngoods in fixed
supply in this economy
let Si(i=1,,n) be the total supply of good iavailable
letpi(i=1,n) be the price of good i
The total demand for good idepends onall prices
Di (p1,,pn) for i=1,,n
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Existence of General
Equilibrium Prices We will write this demand function as
dependent on the whole set of prices (P)
Di (P)
Walras problem: Does there exist an
equilibrium set of prices such that
Di (P*) = Si
for all values of i?
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Excess Demand Functions
The excess demand function for any
good iat any set of prices (P) is defined
to beEDi (P) = Di (P)Si
This means that the equilibrium
condition can be rewritten asEDi (P*) = Di (P*)Si= 0
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Excess Demand Functions
Demand functions are homogeneous of
degree zero
this implies that we can only establishequilibrium relative prices in a Walrasian-
type model
Walras also assumed that demand
functions are continuous
small changes in price lead to small changes
in quantity demanded
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Walras Law
A final observation that Walras made
was that the nexcess demand equations
are not independent of one another Walras lawshows that the total value of
excess demand is zero at any set of
prices
n
i
ii PEDP1
0)(
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Walras Law
Walras law holds for any set of prices
(not just equilibrium prices)
There can be neither excess demand forall goods together nor excess supply
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Walras Proof of the Existence
of Equilibrium Prices The market equilibrium conditions
provide (n-1) independent equations in
(n-1) unknown relative prices can we solve the system for an equilibrium
condition?
the equations are not necessarily linear all prices must be nonnegative
To attack these difficulties, Walras set up
a complicated proof
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Walras Proof of the Existence
of Equilibrium Prices Start with an arbitrary set of prices
Holding the other n-1 prices constant,
find the equilibrium price for good 1 (p1)
Holdingp1 and the other n-2 prices
constant, solve for the equilibrium price
of good 2 (p2) in changingp2from its initial position top2,
the price calculated for good 1 does not
need to remain an equilibrium price
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Walras Proof of the Existence
of Equilibrium Prices The importance of Walras proof is its
ability to demonstrate the simultaneous
nature of the problem of findingequilibrium prices
Because it is cumbersome, it is not
generally used today More recent work uses some relatively
simple tools from advanced mathematics
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Brouwers Fixed-Point Theorem
A mapping is continuous if points that are
close to each other are mapped into other
points that are close to each other The Brouwer fixed-point theorem considers
mappings defined on certain kinds of sets
closed (they contain their boundaries)
bounded (none of their dimensions is infinitely
large)
convex (they have no holes in them)
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Proof of the Existence of
Equilibrium Prices These new prices will retain their original
relative values and will sum to 1
1'1
n
i
ip
j
i
j
i
p
p
p
p
'
'
These new prices will sum to 1
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Proof of the Existence of
Equilibrium Prices We will assume that the feasible set of
prices (S) is composed of all
nonnegative numbers that sum to 1 Sis the set to which we will apply Brouwers
theorem
Sis closed, bounded, and convex we will need to define a continuous mapping
of Sinto itself
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Free Goods
Equilibrium does not really require that
excess demand be zero for every market
Goods may exist for which the marketsare in equilibrium where supply exceeds
demand (negative excess demand)
it is necessary for the prices of these goods
to be equal to zero
free goods
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Mapping the Set of Prices
Into Itself In order to achieve equilibrium, prices of
goods in excess demand should be
raised, whereas those in excess supply
should have their prices lowered
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Mapping the Set of Prices
Into Itself We define the mapping F(P) for any
normalized set of prices (P), such that
the ith component of F(P) is given by
Fi(P) =pi+ EDi (P)
The mapping performs the necessary
task of appropriately raising or lowering
prices
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Mapping the Set of Prices
Into Itself Two problems exist with this mapping
First, nothing ensures that the prices will
be nonnegative the mapping must be redefined to be
Fi(P) = Max [pi+ EDi (P),0]
the new prices defined by the mapping must
be positive or zero
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Mapping the Set of Prices
Into Itself Second, the recalculated prices are not
necessarily normalized
they will not sum to 1
it will be simple to normalize such that
n
i
i PF1
1)(
we will assume that this normalization has
been done
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Application of Brouwers
Theorem Thus, Fsatisfies the conditions of the
Brouwer fixed-point theorem
it is a continuous mapping of the set Sintoitself
There exists a point (P*) that is mapped
into itself For this point,
pi* = Max [pi* + EDi (P*),0] for all i
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Application of Brouwers
Theorem This says that P* is an equilibrium set of
prices
forpi* > 0,pi* =pi* + EDi (P*)
EDi (P*) = 0
Forpi* = 0,
pi* + EDi (P*) 0
EDi (P*) 0
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A General Equilibrium with
Three Goods The economy of Oz is composed only of
three precious metals: (1) silver, (2)
gold, and (3) platinum there are 10 (thousand) ounces of each
metal available
The demands for gold and platinum are
1121
3
1
22
p
p
p
pD 182
1
3
1
23
p
p
p
pD
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A General Equilibrium with
Three Goods Equilibrium in the gold and platinum
markets requires that demand equal
supply in both markets simultaneously
101121
3
1
2 p
p
p
p
101821
3
1
2 p
p
p
p
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A General Equilibrium with
Three Goods This system of simultaneous equations
can be solved as
p2/p1= 2 p3/p1= 3
In equilibrium:
gold will have a price twice that of silver
platinum will have a price three times thatof silver
the price of platinum will be 1.5 times thatof gold
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Smiths Invisible Hand
Hypothesis Adam Smith believed that the
competitive market system provided a
powerful invisible hand that ensuredresources would find their way to where
they were most valued
Reliance on the economic self-interestof individuals and firms would result in a
desirable social outcome
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Smiths Invisible Hand
Hypothesis Smiths insights gave rise to modern
welfare economics
The First Theorem of Welfare
Economics suggests that there is an
exact correspondence between the
efficient allocation of resources and thecompetitive pricing of these resources
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Efficiency in Production
An allocation of resources is efficient in
production (or technically efficient) if no
further reallocation would permit more of
one good to be produced without
necessarily reducing the output of some
other good
Technical efficiency is a precondition for
Pareto efficiency but does not guarantee
Pareto efficiency
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Efficient Choice of Inputs for a
Single Firm A single firm with fixed inputs of labor
and capital will have allocated these
resources efficiently if they are fullyemployed and if the RTSbetween
capital and labor is the same for every
output the firm produces
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Efficient Choice of Inputs for a
Single Firm Assume that the firm produces two
goods (xand y) and that the available
levels of capital and labor are kand l The production function forxis given by
x= f(kx,lx)
If we assume full employment, theproduction function for yis
y= g(ky,ly) = g(k-kx,l
-lx)
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Efficient Choice of Inputs for a
Single Firm Technical efficiency requires thatx
output be as large as possible for any
value of y(y) Setting up the Lagrangian and solving for
the first-order conditions:
L= f
(kx,lx) + [yg
(k
-
kx,l
-
lx)]L/kx= fk+ gk= 0
L/lx= fl+ gl= 0
L/= yg
(k
-
kx,l
-
lx) = 0
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Efficient Choice of Inputs for a
Single Firm From the first two conditions, we can see
that
ll gg
ff kk
This implies that
RTSx(kfor l) = RTSy(kfor l)
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Efficient Allocation of
Resources among Firms Resources should be allocated to those
firms where they can be most efficiently
used
the marginal physical product of any
resource in the production of a particular
good should be the same across all firmsthat produce the good
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Efficient Allocation of
Resources among Firms Suppose that there are two firms
producingxand their production
functions are
f1(k1,l1)
f2(k2,l2)
Assume that the total supplies of capital
and labor are kand l
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Efficient Allocation of
Resources among Firms The allocational problem is to maximize
x= f1(k1,l1) + f2(k2,l2)
subject to the constraintsk1+ k2= k
l1+ l2= l
Substituting, the maximization problembecomes
x= f1(k1,l1) + f2(k-k1,l
-l1)
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Efficient Allocation of
Resources among Firms First-order conditions for a maximum
are
02
2
1
1
1
2
1
1
1
k
f
k
f
k
f
k
f
k
x
02
2
1
1
1
2
1
1
1
lllll
ffffx
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Efficient Allocation of
Resources among Firms These first-order conditions can be
rewritten as
2
2
1
1
k
f
k
f
2
2
1
1
ll
ff
The marginal physical product of each
input should be equal across the two
firms
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Efficient Choice of Output
by Firms The Lagrangian for this problem is
L=x1+x2+ [y* - f1(x1) - f2(x2)]
and yields the first-order condition:
f1/x1= f2/x2
The rate of product transformation
(RPT) should be the same for all firms
producing these goods
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Efficient Choice of Output
by Firms
Trucks Trucks
Cars Cars
Firm A Firm B
50 50
100 1001
2
RPT 1
1RPT
FirmAis relatively efficient at producing cars, while Firm B
is relatively efficient at producing trucks
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Theory of Comparative
Advantage The theory of comparative advantage
was first proposed by Ricardo
countries should specialize in producingthose goods of which they are relatively
more efficient producers
these countries should then trade with the rest
of the world to obtain needed commodities
if countries do specialize this way, total
world production will be greater
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Efficiency in Product Mix
Technical efficiency is not a sufficient
condition for Pareto efficiency
demand must also be brought into thepicture
In order to ensure Pareto efficiency, we
must be able to tie individualspreferences and production possibilities
together
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Efficiency in Product Mix
The condition necessary to ensure that
the right goods are produced is
MRS= RPT the psychological rate of trade-off between
the two goods in peoples preferences must
be equal to the rate at which they can be
traded off in production
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Efficiency in Product Mix
Output ofx
Output of y Suppose that we have a one-person (Robinson
Crusoe) economy and PPrepresents the
combinations ofxand ythat can be produced
P
P
Any point on PPrepresents a
point of technical efficiency
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Efficiency in Product Mix
Assume that there are only two goods
(xand y) and one individual in society
(Robinson Crusoe) Crusoes utility function is
U= U(x,y)
The production possibility frontier is
T(x,y) = 0
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Efficiency in Product Mix
Crusoes problem is to maximize his
utility subject to the production
constraint
Setting up the Lagrangian yields
L= U(x,y) +
[T(x,y)]
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Efficiency in Product Mix
First-order conditions for an interior
maximum are
0
xT
xU
xL
0
y
T
y
U
y
L
0),(
yxT
L
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Efficiency in Product Mix
Combining the first two, we get
yT
xT
yU
xU
/
/
/
/
or
)for()(along)for( yxRPTTdxdyyxMRS
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Efficiency in Production
In minimizing costs, a firm will equate
the RTSbetween any two inputs (kand
l) to the ratio of their competitive prices(w/v)
this is true for all outputs the firm produces
RTSwill be equal across all outputs
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Efficiency in Production
A profit-maximizing firm will hire
additional units of an input (l) up to the
point at which its marginal contributionto revenues is equal to the marginal
cost of hiring the input (w)
pxfl= w
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Efficiency in Production
If this is true for every firm, then with a
competitive labor market
pxfl1
= w=pxfl2
fl1= f
l2
Every firm that producesxhas identical
marginal productivities of every input inthe production ofx
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Efficiency in Production
Recall that the RPT(ofxfor y) is equal
to MCx /MCy
In perfect competition, each profit-maximizing firm will produce the output
level for which marginal cost is equal to
price
Sincepx= MCxandpy= MCyfor every
firm, RTS= MCx /MCy=px/py
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Efficiency in Production
Thus, the profit-maximizing decisions
of many firms can achieve technical
efficiency in production without any
central direction
Competitive market prices act as
signals to unify the multitude of
decisions that firms make into one
coherent, efficient pattern
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Efficiency in Product Mix
The price ratios quoted to consumers
are the same ratios the market presents
to firms
This implies that the MRSshared by all
individuals will be equal to the RPT
shared by all the firms
An efficient mix of goods will therefore
be produced
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Efficiency in Product Mix
Output ofx
Output of y
P
P
U0
x* and y* represent the efficient output mix
x*
y*
Only with a price ratio of
px*/py* will supply and
demand be in equilibrium
*
*slope
y
x
p
p
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Laissez-Faire Policies
The correspondence between
competitive equilibrium and Pareto
efficiency provides some support for the
laissez-faire position taken by many
economists
government intervention may only result in
a loss of Pareto efficiency
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Externalities
An externality occurs when there are
interactions among firms and individuals
that are not adequately reflected in
market prices With externalities, market prices no
longer reflect all of a goods costs of
production there is a divergence between private and
social marginal cost
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Public Goods
Public goods have two properties that
make them unsuitable for production in
markets
they are nonrival
additional people can consume the benefits of
these goods at zero cost
they are nonexclusive extra individuals cannot be precluded from
consuming the good
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Imperfect Information
If economic actors are uncertain about
prices or if markets cannot reach
equilibrium, there is no reason to expect
that the efficiency property ofcompetitive pricing will be retained
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Distribution
Although the First Theorem of Welfare
Economics ensures that competitive
markets will achieve efficient allocations,
there are no guarantees that these
allocations will exhibit desirable
distributions of welfare among individuals
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DistributionO
J
OS
Total Y
Total X
UJ4
UJ3
UJ2
UJ1
US4
US3
US2
US1
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DistributionO
J
OS
UJ4
UJ3
UJ2
UJ1
US4
US3
US2
US1
A
Any trade in this area is
an improvement over A
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Contract Curve
In an exchange economy, all efficient
allocations lie along a contract curve
points off the curve are necessarily
inefficient individuals can be made better off by moving to
the curve
Along the contract curve, individualspreferences are rivals
one may be made better off only by making
the other worse off
C C
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Contract CurveO
J
OS
UJ4
UJ3
UJ2
UJ1
US4
US3
US2
US1
A
Contract curve
Exchange with Initial
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Exchange with Initial
Endowments Suppose that the two individuals
possess different quantities of the two
goods at the start it is possible that the two individuals could
both benefit from trade if the initial
allocations were inefficient
Exchange with Initial
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Exchange with Initial
Endowments Neither person would engage in a trade
that would leave him worse off
Only a portion of the contract curveshows allocations that may result from
voluntary exchange
Exchange with Initial
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Exchange with Initial
Endowments OJ
OS
UJA
USA
A
Suppose thatArepresents
the initial endowments
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Exchange with Initial
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Exchange with Initial
Endowments OJ
OS
UJA
USA
A
Only allocations between M1
and M2will be acceptable to
both
M1
M2
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The Distributional Dilemma
If the initial endowments are skewed in
favor of some economic actors, the
Pareto efficient allocations promised by
the competitive price system will also
tend to favor those actors
voluntary transactions cannot overcome
large differences in initial endowments
some sort of transfers will be needed to
attain more equal results
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The Distributional Dilemma
These thoughts lead to the Second
Theorem of Welfare Economics
any desired distribution of welfare among
individuals in an economy can be achieved
in an efficient manner through competitive
pricing if initial endowments are adjusted
appropriately
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Important Points to Note:
Preferences and production
technologies provide the building
blocks upon which all general
equilibrium models are based
one particularly simple version of such a
model uses individual preferences for two
goods together with a concave productionpossibility frontier for those two goods
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Important Points to Note:
Competitive markets can establish
equilibrium prices by making marginal
adjustments in prices in response to
information about the demand and
supply for individual goods
Walras law ties markets together so that
such a solution is assured (in most cases)
I P i N
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Important Points to Note:
Competitive prices will result in a
Pareto-efficient allocation of resources
this is the First Theorem of Welfare
Economics
I t t P i t t N t
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Important Points to Note:
Factors that will interfere with
competitive markets abilities to
achieve efficiency include
market power
externalities
existence of public goods
imperfect information
I t t P i t t N t
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Important Points to Note:
Competitive markets need not yield
equitable distributions of resources,
especially when initial endowments are
very skewed
in theory any desired distribution can be
attained through competitive markets
accompanied by lump-sum transfers there are many practical problems in