chapter 12 of economics

8/11/2019 chapter 12 of economics

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Combining the first two, we get

yT

xT

yU

xU

/

/

/

/

or

)for()(along)for( yxRPTTdxdyyxMRS


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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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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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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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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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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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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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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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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



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123


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



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124


Endowments OJ

OS

UJA

USA

A

Suppose thatArepresents

the initial endowments


125/133



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126


Endowments OJ

OS

UJA

USA

A

Only allocations between M1

and M2will be acceptable to

both

M1

M2


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127

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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128

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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129

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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130


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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131


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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132


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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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

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chapter 12 of economics

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