Chapter 5 General EquilibriumEconomics 5113 Microeconomic Theory

Kam Yu

Winter 2019

Chapter 5General

Equilibrium

Kam Yu

Introduction

ExchangeEconomy

Barter

Markets

WelfareTheorems

ProductionEconomy

Firms

Consumers

Equilibrium

Welfare

Outline

1 Introduction

2 Exchange EconomyBarterMarketsWelfare Theorems

3 Production EconomyFirmsConsumersEquilibriumWelfare

Chapter 5General

Equilibrium

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Introduction

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Barter

Markets

WelfareTheorems

ProductionEconomy

Firms

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Equilibrium

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Invisible Hand Theory

• Adam Smith (1776) asserts that free trade amongself-interested agents in a society results in the greatestsocial welfare.

• In competitive markets, incentives in gain from trade guidethem towards the equilibrium like an invisible hand.

• Our goal is to show that such an equilibrium (fixed point)exists, under the neoclassical framework.

• There are other questions, such as the uniqueness of theequilibrium, the dynamic process in reaching it, stability,and evolution.

• Also, complexity theory suggests that the concept ofequilibrium is less important than the process itself.

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A Simple Exchange Economy

• We start with an economy with two consumers. There isno production.

• Each is endowed by nature with two goods. Consumer 1’sendowment is e1 = (e11 , e

12). Consumer 2 e2 = (e21 , e

22).

• The two guys are free to exchange what they have (bartertrade).

• Question: Is there any gain from trade?

• We illustrate the problem with an Edgeworth box, withwidth e11 + e21 and height e12 + e22 .

• The endowment point e = (e1, e2) is a point in the box.

• Notice that each point in the box has four coordinates.

• Each point x = (x1, x2) in the box represents an allocationof the two goods among the consumers resulting from thetrade.

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Edgeworth Box

196 CHAPTER 5

5.1 EQUILIBRIUM IN EXCHANGE

Here we explore the basic economic problem of distribution in a very simple society with-out organised markets. Our objective is to describe what outcomes might arise through aprocess of voluntary exchange. By examining the outcomes of this process, we can estab-lish a benchmark against which the equilibria achieved under competitive market systemscan be compared.

The society we consider is very stark. First, there is no production. Commoditiesexist, but for now we do not ask how they came to be. Instead, we merely assume eachconsumer is ‘endowed’ by nature with a certain amount of a finite number of consumablegoods. Each consumer has preferences over the available commodity bundles, and eachcares only about his or her individual well-being. Agents may consume their endowmentof commodities or may engage in barter exchange with others. We admit the institution ofprivate ownership into this society and assume that the principle of voluntary, non-coercivetrade is respected. In the absence of coercion, and because consumers are self-interested,voluntary exchange is the only means by which commodities may be redistributed from theinitial distribution. In such a setting, what outcomes might we expect to arise? Or, rephras-ing the question, where might this system come to rest through the process of voluntaryexchange? We shall refer to such rest points as barter equilibria.

To simplify matters, suppose there are only two consumers in this society, con-sumer 1 and consumer 2, and only two goods, x1 and x2. Let e1 ≡ (e11, e

12) denote the

non-negative endowment of the two goods owned by consumer 1, and e2 ≡ (e21, e22) the

endowment of consumer 2. The total amount of each good available in this society thencan be summarised by the vector e1 + e2 = (e11 + e21, e

12 + e22). (From now on, superscripts

will be used to denote consumers and subscripts to denote goods.)The essential aspects of this economy can be analysed with the ingenious

Edgeworth box, familiar from intermediate theory courses. In Fig. 5.1, units of x1are measured along each horizontal side and units of x2 along each vertical side. Thesouth-west corner is consumer 1’s origin and the north-east corner consumer 2’s origin.

e11 e1

2

e21 e2

2

e22

x22

!

!

"

e11x1

1

x12 e1

2

x21

e21

02

01

(x 1, x 2)

e (e1, e2)

Figure 5.1. The Edgeworth box.

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Gain From Trade• We now impose the indifference curves of the two

consumers in the box.• The line CC is the locus of the tangent points of the two

sets of curves, which is called the contract curve.• Convince yourself that only the points in the lens-shaped

area enclosed by the the two indifference curves through ewill make both of them better off.

• Consider the point B. Since it is not on the contractcurve, there exists two indifference curves crossing eachother at B, creating another lens. Therefore there are stillrooms for improvement.

• Only points on the contract curve inside the original lenscan be equilibria (line cc).

• Once the two consumers agree to any such point on cc ,say point D, there is no further gain from trade.

• The set of points on the segment cc is called the core ofthe economy, and are potential equilibria.

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Equilibrium in Two-PersonExchange

GENERAL EQUILIBRIUM 197

Increasing amounts of x1 for consumer 1 are measured rightwards from 01 along the bot-tom side, and increasing amounts of x1 for consumer 2 are measured leftwards from 02along the top side. Similarly, x2 for consumer 1 is measured vertically up from 01 on theleft, and for consumer 2, vertically down on the right. The box is constructed so that itswidth measures the total endowment of x1 and its height the total endowment of x2.

Notice carefully that each point in the box has four coordinates – two indicatingsome amount of each good for consumer 1 and two indicating some amount of each goodfor consumer 2. Because the dimensions of the box are fixed by the total endowments, eachset of four coordinates represents some division of the total amount of each good betweenthe two consumers. For example, the point labelled e denotes the pair of initial endowmentse1 and e2. Every other point in the box represents some other way the totals can be allocatedbetween the consumers, and every possible allocation of the totals between the consumersis represented by some point in the box. The box therefore provides a complete picture ofevery feasible distribution of existing commodities between consumers.

To complete the description of the two-person exchange economy, suppose eachconsumer has preferences represented by a usual, convex indifference map. In Fig. 5.2,consumer 1’s indifference map increases north-easterly, and consumer 2’s increases south-westerly. One indifference curve for each consumer passes through every point in the box.The line labelled CC is the subset of allocations where the consumers’ indifference curvesthrough the point are tangent to each other, and it is called the contract curve. At anypoint off the contract curve, the consumers’ indifference curves through that point mustcut each other.

Given initial endowments at e, which allocations will be barter equilibria in thisexchange economy? Obviously, the first requirement is that the allocations be somewhere,

x1

x2

01

02

A

B

D

C

C

c

c

e

Figure 5.2. Equilibrium in two-person exchange.

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Definitions

• Consider a set of consumers I = {1, 2, . . . , I}. Eachconsumer has a rational preference relation %i on n goods.

• Endowment of consumer i is ei = (e i1, ei2, . . . , e

in).

• An exchange economy is defined as

E ={

(%i , ei ) : i ∈ I}.

• The economy’s endowment is e = (e1, e2, . . . , eI ).

• An allocation of bundles among the consumers is writtenas x = (x1, x2, . . . , xI ).

• Given the endowment e, the set of feasible allocations is

F (e) =

{x :∑i∈I

xi =∑i∈I

ei

}.

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Pareto-Efficient Allocations

• An allocation x ∈ F (e) is Pareto efficient if there is noother allocation y ∈ F (e) such that yi %i xi for allconsumers with at least one strict preference.

• On the contrary, if such an allocation y exists, it is called aPareto improvement over x.

• A Pareto-efficient allocation is called an exchangeeconomy equilibrium.

• Since exchanges are voluntary, any consumer can block afeasible allocation if she does not like it. It follows that anexchange economy equilibrium must be Pareto efficient.

• A subset of consumers S ⊆ I is called a coalition.

• The Pareto-efficiency condition should be satisfied by anycoalition. That is, gain from trade for each consumer isexhaustive.

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Core of an Economy

• Let S ⊆ I be a coalition. We say that S blocks anallocation x ∈ F (e) if there exists y ∈ F (e) such that,within the coalition S ,

1 y is feasible,∑

i∈S yi =∑

i∈S ei ,2 y is a Pareto improvement over x in S .

• Again since trade is voluntary, any coalition can block anallocation.

• Consequence: An exchange economy equilibrium is afeasible allocation x ∈ F (e) which is not blocked by anycoalition.

• The set of all equilibria is called the core of the economy,C (e).

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Market System

• In the above barter trade economy, consumers have tocommunicate and negotiate with each others.

• An alternative narrative is to imagine a central plannerwho has information of all consumers’ preferences andtheir endowments.

• The total transaction cost of reaching an equilibrium inboth cases can be very high.

• We now consider an economic institution that is moredecentralized. Every consumer only knows her ownpreference structure, and she make decisions on tradebased on a set of market prices, which are publiclyobservable.

• In a competitive exchange economy, the consumers areprice takers. Their individual actions have no impact onthe market prices.

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A Competitive Market

• Assume that a price vector p� 0 for the n goods andservices is observable by every consumer.

• Each consumer’s preferences are represented by acontinuous, strongly increasing, and strictly quasi-concaveutility function U i (xi ).

• The utility maximization problem for each consumer i ∈ Iis

maxxi

U i (xi ) subject to pTxi ≤ pTei ,

where pTei is the real income of the consumer.

• The solution is the ordinary demand function

xi∗ = d i (p,pTei ),

which is unique and continuous in p.

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Excess Demand

• For any price p� 0, define the excess demand functionfor consumer i as

z i (p) = d i (p,pTei )− ei .

• The aggregate excess demand function for the wholeeconomy is

z(p) =∑i∈I

(d i (p,pTei )− ei

).

• Properties of z :1 z is continuous in p,2 z is homogenous of degree zero, that is, for any λ > 0,

z(λp) = z(p).3 Walras’ law: pTz(p) = 0.

• Walras’ law implies that if n − 1 markets are inequilibrium, the remaining one also is.

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Proof of Walras’ Law

Consumer i ’s budget constraint implies that∑nk=1 pk(x i∗k − e ik) = 0. Summing up all consumers, we have

0 =I∑

i=1

n∑k=1

pk(x i∗k − e ik)

=n∑

k=1

I∑i=1

pk(x i∗k − e ik)

=n∑

k=1

pk

(I∑

i=1

(x i∗k − e ik)

)

=n∑

k=1

pkzk(p)

= pTz(p).

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Walrasian Equilibrium

• A price vector p∗ ∈ Rn++ is a Walrasian equilibrium if

z(p∗) = 0.

• Theorem: If every consumer’s utility function iscontinuous, strongly increasing, and strictly quasi-concave,and

∑Ii=1 ei � 0, then there exists at least one price

vector p∗ ∈ Rn++ that is a Walrasian equilibrium.

• See JR 207–211 for a proof of the above theorem.

• Suppose p∗ is a Walrasian equilibrium. The resulting x∗ iscalled a Walrasian equilibrium allocation (WEA).

• Given the endowment e, the set of WEAs is denoted byW (e).

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Walrasian Equilibrium Allocations

Theorem: Every Walrasian equilibrium allocation of anendowment e is in the core. That is, W (e) ⊆ C (e).

Proof : Suppose on the contrary that x∗ ∈W (e) butx∗ /∈ C (e). Then there exists an allocation y ∈ F (e) and acoalition S such that∑

i∈Syi =

∑i∈S

ei , (1)

U i (yi ) ≥ U i (x∗) ∀ i ∈ S , (2)

with at least one strict inequality. Equality (1) implies that

(p∗)T

(∑i∈S

yi

)= (p∗)T

(∑i∈S

ei

). (3)

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Inequality (2) implies that

(p∗)Tyi ≥ (p∗)Txi∗ = (p∗)Tei ∀ i ∈ S

with at least one strict inequality. Summing over all i ∈ S gives

(p∗)T

(∑i∈S

yi

)> (p∗)T

(∑i∈S

ei

),

which contradicts equation (3).

Consequence of the theorem: Every WEA is Pareto efficient.This is called the First Welfare Theorem of an exchangeeconomy.

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Spreading the Wealth

• The idea of Pareto efficiency is not concerned with incomedistribution. An allocation x ∈W (e) may not be sociallyor politically desirable.

• Suppose that x is a socially desirable Pareto-efficientallocation. Is it possible to redistribute endowment from eto e∗ such that x ∈W (e∗)?

• The answer is yes. In fact we can set x = e∗. Thenx ∈W (x).

• This is called the Second Welfare Theorem of anexchange economy.

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Proof of SWTSet e∗ = x and let x ∈W (x). We want to show that x = x. Bydefinition, a Walrasian equilibrium allocation is utilitymaximizing. That is, for all i ∈ I,

U i (x) ≥ U i (x).

Since x is Pareto efficient, no one can be better off withoutmaking at least one consumer worse off. So equality holds forall i ∈ I.

For each consumer, if xi 6= xi , then by strict convexity ofpreferences any convex combination of the two bundles isstrictly preferred to them and is affordable. This contradictsthe assumption that xi is utility maximizing. Therefore xi = xi

for all i ∈ I.

Question: Why are the convex combinations of the two bundlesaffordable?

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Social transfers in kind fromgeneral governments, Canada

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Household actual finalconsumption, Canada

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The Production Sector

• The economy has a set of firms J = {1, 2, . . . , J}, eachwith a net output vector yj .

• Each firm’s technology is represented by a production setY j ⊆ Rn with the following properties:

1 0 ∈ Y j .2 Y j is a compact set. ( No free disposal)3 Y j is strongly convex: For all y1 6= y2 ∈ Y j and α ∈ (0, 1),

there exists a y ∈ Y j such that y > αy1 + (1− α)y2.

• All firms are price takers, which observe a common pricevector p� 0.

• The profit function of firm j is

πj(p) = maxy

{pTy : y ∈ Y j

}.

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Net Supply and Profit

If the production set Y j satisfies the three assumptions above,then

1 the net supply function yj∗ = s j(p) is unique andcontinuous,

2 the profit function πj(p) is well-defined and continuous,

3 s j(p) is homogeneous of degree zero,

4 πj(p) is linearly homogeneous.

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Aggregate Production Set

Let Y =∑

j∈J Y j . Then Y has the following properties:

1 If all Y j satisfy the three assumptions, so does Y .(Exercise: Prove this.)

2 For all p� 0 and y ∈ Y , pTy ≥ pTy if and only if for allj ∈ J , there exists a yj ∈ Y j with y =

∑j∈J yj such that

pTyj ≥ pTyj for all yj ∈ Y j .

In words, the aggregate net output vector y ∈ Y is profitmaximizing if and only if the individual firms’ profitmaximizing vectors yj add up to y.

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Proof of Aggregate ProfitMaximization

• Suppose that y is profit maximizing for the aggregatedfirm. That is, pTy ≥ pTy for all y ∈ Y .

• By the definition of Y , y =∑

j∈J yj for some yj in each

of Y j .

• For each j ∈ J , we claim that yj is profit maximizing.Otherwise there is a profit maximizing yj ∈ Y j that willadd up to a different aggregated y with pTy > pTy.

• Proof of the converse is left as an exercise.

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On the Demand Side

• The set of consumers, I, have rational preferencestructure as before.

• Their endowments are e = (e1, e2, . . . , eI ).

• The firms are owned by the consumers. Let θij beconsumer i ’s share of ownership in firm j . It follows that0 ≤ θij ≤ 1 for all i ∈ I and j ∈ J , and

∑i∈I θ

ij = 1 forall j ∈ J .

• Each consumer make her own decision:

maxxi

U i (xi )

subject to pTxi ≤ mi (p),

mi (p) = pTei +∑j∈J

θijπj(p).

• The demand function d i (p,mi (p)) is unique andcontinuous.

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The Production Economy

• A production economy is defined as

E ={

(U i , ei , θij ,Y j) : i ∈ I, j ∈ J}.

• The excess demand function for all the n goods andservices is defined as

z(p) =∑i∈I

d i (p,mi (p))−∑j∈J

s j(p)−∑i∈I

ei .

• The function z also satisfies the three properties:

1 continuous,2 homogeneous of degree zero,3 pTz(p) = 0.

• A price vector p∗ � 0 is call a Walrasian equilibrium of aproduction economy if z(p∗) = 0.

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Walrasian Equilibrium

Reading Assignments:

1 Proof of the existence of the Walrasian equilibrium: JR225–226.

2 An example of a simple general equilibrium model:• Two goods (leisure and coconut)• One consumer (Robinson Crusoe)• One firm (Crusoe and Co., with θ11 = 1)• Question: How does this model compare with the Ramsey

dynamic model in macroeconomics?

Exercise: Prove the three properties of the excess demandfunction for a production economy.

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Always in Equilibrium?

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Walrasian Equilibrium Allocations

Let p∗ be a Walrasian equilibrium. Then the Walrasianequilibrium allocation is defined as

(x∗, y∗) =(

x1∗, x2∗, . . . , xI∗; y1∗, y2∗, . . . , yJ∗).

Properties:

1 Consumption efficiency: All consumers maximize utility,with xi∗ = d i (p∗,mi (p∗)) for all i ∈ I.

2 Production efficiency: All firm maximize profit, withyj∗ = s j(p∗) for all j ∈ J .

3 Product mix efficiency: All markets clear, with z(p∗) = 0.

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Definitions

1 Feasible allocation: Given endowmentse = (e1, e2, . . . , eI ), an allocation

(x, y) =(

x1, x2, . . . , xI ; y1, y2, . . . , yJ)

is feasible if for all i ∈ I and j ∈ J , xi ∈ Rn+, yj ∈ Y j ,

and ∑i∈I

xi =∑i∈I

ei +∑j∈J

yj .

2 An allocation (x, y) is Pareto efficient if• it is feasible,• there is no other feasible allocation (x, y) such that

U i (xi ) ≥ U i (xi ) for all i ∈ I with at least one strictinequality.

Note: By definition any WEA is feasible.

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First Welfare Theorem of aProduction Economy

Every Walrasian equilibrium allocation is Pareto efficient.

Proof : Suppose on the contrary that (x, y) is a WEA but notPareto efficient. Since (x, y) is feasible,∑

i∈Ixi =

∑i∈I

ei +∑j∈J

yj . (4)

But (x, y) is not Pareto efficient. Thus there exists anallocation (x, y) such that

U i (xi ) ≥ U i (xi )

for all i ∈ I with at least one strict inequality.

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Since (x, y) is a WEA, all consumers maximize utility so that xi

must be unaffordable for some consumers, that is,

(p∗)Txi ≥ (p∗)Txi

for all i ∈ I with at least one strict inequality. Summing up allconsumer gives

(p∗)T∑i∈I

xi > (p∗)T∑i∈I

xi .

By equation (4) and the fact that (x, y) is feasible,

(p∗)T

∑i∈I

ei +∑j∈J

yj

> (p∗)T

∑i∈I

ei +∑j∈J

yj

.

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The last inequality implies that

(p∗)T∑j∈J

yj > (p∗)T∑j∈J

yj ,

meaning(p∗)Tyj > (p∗)Tyj

for some firm j and yj ∈ Y j , which is a contradiction since yj isalready the profit maximizing bundle.

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Helping the Poor

• Given a production economy

E ={

(U i , ei , θij ,Y j) : i ∈ I, j ∈ J},

such that for some y ∈ Y ,

y +∑i∈I

ei � 0.

For example, we are not endowed with any humanartifacts but we can make them with other inputs.

• Suppose that (x, y) is a socially desirable Pareto-efficientallocation.

• The government wants to achieve this goal by a taxsystem that transfer incomes between consumers.

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Second Welfare Theorem

There exist income transfers, T1,T2, . . . ,TI , satisfying∑i∈I Ti = 0, and a Walrasian equilibrium p such that

1 For all i ∈ I, xi is the solution to the utility maximizationproblem

maxxi

U i (xi )

subject to pTxi ≤ mi (p) + Ti .

2 For all j ∈ J , yj is the solution to the profit maximizationproblem

maxyj

pTyj

subject to yj ∈ Y j .

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Proof of SWT

For each firm j ∈ J , define Y j = Y j − {yj}.Exercise: Convince yourself that

1 0 ∈ Y j ,

2 Y j is compact,

3 Y j is strongly convex.

Consider the economy

E ={(

U i , xi , θij , Y j)

: i ∈ I, j ∈ J}.

Notice that we have replace the endowment ei with xi and theproduction set Y j with Y j . A Walrasian equilibrium p existsfor E . Let the Walrasian equilibrium allocation be (x, y). Wewant to show that x = x.

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Equilibrium

Welfare

Since 0 ∈ Y j for all j ∈ J , πj(p) ≥ 0. Therefore xi is at leastas good as the endowment xi , that is, for all i ∈ I,

U i (xi ) ≥ U i (xi ) (5)

Since yj ∈ Y j , by definition yj = yj − yj for some yj ∈ Y j .Since (x, y) is feasible in E ,∑

i∈Ixi =

∑i∈I

xi +∑j∈J

yj

=∑i∈I

xi +∑j∈J

(yj − yj

)

=

∑i∈I

xi −∑j∈J

yj

+∑j∈J

yj

=∑i∈I

ei +∑j∈J

yj

Chapter 5General

Equilibrium

Kam Yu

Introduction

ExchangeEconomy

Barter

Markets

WelfareTheorems

ProductionEconomy

Firms

Consumers

Equilibrium

Welfare

This means that (x, y) is feasible in E . We now show that(x, y) = (x, y). Since x is feasible in E , the inequality in (5)must be equality for all i ∈ I since x is Pareto efficient. Bystrictly convexity of U i , we have x = x. In other words, (x, y) isa WEA with p in E . This means that

xi = arg maxxi

U i (xi ) : pTxi ≤ pTxi +∑j∈J

θijπj(p)

.

Therefore profit income for each consumer is zero, whichmeans that πj(p) = 0 for all j ∈ J . It follows that yj = 0 andtherefore yj = y so that (x, y) = (x, y). Finally, the transfersare

Ti = pTxi −mi (p), i = 1, 2, . . . , I .

Chapter 5General

Equilibrium

Kam Yu

Introduction

ExchangeEconomy

Barter

Markets

WelfareTheorems

ProductionEconomy

Firms

Consumers

Equilibrium

Welfare

Discussions

• The theory does not tell us what allocations are sociallydesirable.

• The government cannot directly observe U i and Y j . Onlyei and θij are potentially observable.

• Common targets in redistribution:

1 health care insurance and health care provision,2 education,3 criminal justice,4 basic income.

• Issues in welfare economics are related to social ethics.

• In the long run, political and economic institutions evolvewith technology.