lecture notes general equilibrium theory: ss205fede/lecture_notes/echenique_ge.pdf · the ideas and...

LECTURE NOTESGENERAL EQUILIBRIUM THEORY: SS205

FEDERICO ECHENIQUECALTECH

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Contents

0. Disclaimer 4

1. Preliminary definitions 5

1.1. Binary relations 5

1.2. Preferences in Euclidean space 5

2. Consumer Theory 6

2.1. Digression: upper hemi continuity 7

2.2. Properties of demand 7

3. Economies 8

3.1. Exchange economies 8

3.2. Economies with production 11

4. Welfare Theorems 13

4.1. First Welfare Theorem 13

4.2. Second Welfare Theorem 14

5. Scitovsky Contours and cost-benefit analysis 20

6. Excess demand functions 22

6.1. Notation 22

6.2. Aggregate excess demand in an exchange economy 22

6.3. Aggregate excess demand 25

7. Existence of competitive equilibria 26

7.1. The Negishi approach 28

8. Uniqueness 32

9. Representative Consumer 34

9.1. Samuelsonian Aggregation 37

9.2. Eisenberg’s Theorem 39

10. Determinacy 39

GENERAL EQUILIBRIUM THEORY 3

10.1. Digression: Implicit Function Theorem 40

10.2. Regular and Critical Economies 41

10.3. Digression: Measure Zero Sets and Transversality 44

10.4. Genericity of regular economies 45

11. Observable Consequences of Competitive Equilibrium 46

11.1. Digression on Afriat’s Theorem 46

11.2. Sonnenschein-Mantel-Debreu Theorem: Anything goes 47

11.3. Brown and Matzkin: Testable Restrictions OnCompetitve Equilibrium 48

12. The Core 49

12.1. Pareto Optimality, The Core and Walrasian Equiilbria 51

12.2. Debreu-Scarf Core Convergence Theorem 51

13. Partial equilibrium 58

13.1. Aggregate demand and welfare 60

13.2. Production 61

13.3. Public goods 62

13.4. Lindahl equilibrium 63

14. Two-sided matching model with transferable utility 64

14.1. Pseudomarktes 67

15. General equilibrium under uncertainty 70

15.1. Two-state, two-agent economy 71

15.2. Pari-mutuel betting 72

15.3. Arrow-Debreu and Radner equilibria. 77

16. Asset Pricing 78

16.1. Digression: Farkas Lemma 82

16.2. State prices 82

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16.3. Application: Put-call parity 84

16.4. Market incompleteness. 85

16.5. The Capital Asset Pricing Model (CAPM) 86

16.6. Consumption CAPM 88

16.7. Lucas Tree Model 89

16.8. Risk-neutral consumer and the martingale property 92

16.9. Finite state space 92

16.10. Logarithmic utility 93

16.11. CRRA utility 93

17. Large economies and approximate equilibrium 96

17.1. The Shapley-Folkman theorem 96

17.2. Approximate equilibrium 100

17.3. Core convergence revisited 103

References 107

0. Disclaimer

The ideas and organization in these lecture notes owe a lot to ChrisShannon’s Econ 201(a) class at UC Berkeley, as well as various text-books: notably Mas-Colell et al. (1995), and to a lesser extent Bewley(2009). Over the years, when I’ve had time in class, I’ve added somemore advanced material. Usually in the last week of the quarter. I’venever covered all the material here in a single quarter.

The current write-up started from Alejandro Robinson’s notes fromthe class I taught in the Winter of 2015. I have added material, andrewritten the notes each time I have taught the class since. They area work in progress, so please let me know of any problems you find.


1. Preliminary definitions

1.1. Binary relations. Let X be a set. A binary relation on X is asubset of X×X. If B is a binary relation on X, and x, y ∈ X we writex B y to denote that (x, y) ∈ B. A binary relation B is complete ifx B y or y B x (or both) for any x, y ∈ X. It is transitive if for anyx, y, z ∈ X

x B y and y B z imply x B z.

A binary relation is a weak order if it is complete and transitive. Ineconomics we use the term preference relation (or rational pref-erence relation) for weak orders.

A preference relation is denoted by �. Associated to any preferencerelation � are two more binary relation. The first captures indifference:x ∼ y is x � y and y � x. The second captures strict preference: x � yif x � y and it is not the case that x ∼ y.

1.2. Preferences in Euclidean space. A subset A ⊆ Rn is convexif λx+ (1− λ)y ∈ A for any x, y ∈ A and any λ ∈ (0, 1).

We use the following notational conventions: For vectors x, y ∈ Rn,x ≤ y means that xi ≤ yi for all i = 1, . . . , n; x < y means that x ≤ yand x 6= y; and x� y means that xi < yi for all i = 1, . . . , n.

When X ⊆ Rn and � is a preference relation over X we have thefollowing definitions: � is

• locally nonsatiated if for any x ∈ X and ε > 0 there is y ∈ Xsuch that ‖x− y‖ < ε and y � x.• weakly monotone if x ≤ y implies y � x and x � y impliesy � x• strongly monotone if x < y implies y � x.

Define the following sets: Let U(x) = {y ∈ X : y � x} be the uppercontour set of � at x and let L(x) = {y ∈ X : x � y} be the lowercontour set of � at x.

The preference relation � is

• convex if U(x) is a convex set for all x ∈ X;• strictly convex if

λy + (1− λ)y′ � x

6 ECHENIQUE

for any y, y′ ∈ U(x) and λ ∈ (0, 1), for any x ∈ X;• and continuous if U(x) and L(x) are closed sets (in X) for allx ∈ X.

The following theorem is due to Debreu.

Theorem 1. If a preference relation � is continuous on X ⊆ Rn thenthere exists u : X → R such that x � y iff u(x) ≥ u(y).

The function u is a utility representation for �.

2. Consumer Theory

A consumer is a pair (X,�), where X is a set termed consump-tion space , and � is a preference relation on X. As we shall see, weshall require a bit more information when we place the consumer in aneconomy. The set X represents all the possible consumption bundlesthat the consumer can choose. The preference � is harder to interpret,but the standard view in economics is that � is simply a description ofthe consumer’s choice behavior: � specifies what the consumer choosesfrom each pair of alternative bundles in X.

We assume throughout that X = RL+. This means that there are Lgoods, and that the consumer can choose to consume these goods in any(continuous) quantity. As we shall see, the notion of “good” is quiteflexible, and accommodates goods that differ in the time when theyare consumed, or the uncertain events upon which they are delivered(Debreu, 1987).

The consumer chooses from a budget set. Let p ∈ RL+ and W ≥ 0.The set

B(p,W ) = {y ∈ RL+ : p · y ≤ W}is the budget set for a consumer when prices are p and income is W .

Given a preference relation �, the optimal choices of the consumer are:x∗(p,W ) = {x ∈ B(p,W ) : x � y for all y ∈ B(p,W )}.

The mapping from (p,W ) into x∗(p,W ) is the consumer’s demandcorrespondence.

The theory of the consumer predicts that demand (choices made frombudget sets) are optimal choices according to an underlying (rational)preference relation.


The following results are simple observations that you should prove onyour own (or find in MWG).

Observation 2. x ∈ x∗(p,W ) if and only if y � x implies that p ·y > W .

2.1. Digression: upper hemi continuity. a correspondence is afunction φ with domain A and range 2B for some set B, such that φ(a)is a nonempty subset of B for each a. We denote a correspondence byφ : A→ B.

A correspondence φ : A ⊆ Rn → B ⊆ Rm has closed graph if{(x, y) ∈ A×B : y ∈ φ(x)} is a closed subset of A×B.

Let φ : A ⊆ Rn → B ⊆ Rm, where B is closed. We say that φ isupper hemicontinuous (uhc) if it has closed graph and the imageof compact sets are bounded.

Note: this is a practical way of understanding uhc. It is how we useit in this class. But you can find a general definition, for example, inAliprantis and Border (2006).

2.2. Properties of demand.

Proposition 3. If � is locally nonsatiated and continuous then x∗(p,W )is nonempty and satisfies that p · x = W for all x ∈ x∗(p,W ), for all pand W . Moreover, the demand correspondence (p,W ) 7→ x∗(p,W ) isupper hemicontinuous.

Proposition 4. If � is locally nonsatiated, continuous and convex,then x∗(p,W ) is nonempty, compact, and convex, for all p and W .Moreover the demand correspondence (p,W ) 7→ x∗(p,W ) is upper hemi-continuous.

Proposition 5. If � is locally nonsatiated, continuous and strictlyconvex then x∗(p,W ) is a singleton for all p and W . Moreover thedemand correspondence (p,W ) 7→ x∗(p,W ) is continuous as a function.

Consider a consumer with convex, continuous, and monotone prefer-ences �. Fix a bundle x in Rn. What do we need to do if we wantthis consumer to demand x? Consider the upper contour set U(x).Given our assumptions on preferences, this set is in the hypothesis ofthe supporting hyperplane theorem.1 Consider the picture in Figure 1.

1You should be familiar with the supporting hyperplane theorem, but you willin any case prove it as part of your homework.

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xU(x)

x1

x2

Figure 1. The “second welfare theorem for a single consumer.”

A supporting hyperplane at U(x) gives us prices p and income W forwhich x is a demanded bundle. The reason is that any bundle that isstrictly preferred to x must be in the interior of U(x), and thereforenot affordable to a consumer with such income, and facing such prices.Observe that we can use the monotonicity of preferences to ensure thatthe prices obtained are positive.

The observation in Figure 1 can be called the “second welfare theoremfor an individual consumer,” for reasons that will be clear in a fewlectures.

3. Economies

3.1. Exchange economies. Our first model of an economy is meantto capture the motivations behind pure economic exchange. I desireoranges but own only apples. You own oranges but no apples. Byexchanging goods, we satisfy our needs for the fruits that we lack: a“coincidence of wants” (or “mutual needs”). Prices capture the rate ofexchange between goods; how many oranges should I receive for eachapple that I give you.

The model primitive is a description of a collection of consumers. Eachconsumer is described by a preference relation and a consumptionspace, as in the theory of the consumer. We take consumption spaceto be RL+. An important feature of the theory is to account for whoowns and sells the goods that the consumers buy. We also need toaccount for where consumers’ incomes come from. Each consumer will


be endowed with non-negative quantities of each good. These arethe goods that are sold in market, at the prevailing market prices, andconsumers’ incomes come from selling the goods that they are endowedwith at market prices. Thus, a consumer will be described (fixing con-sumption space to be RL+, by a preference and an endowment vectorin RL+.

An economy then is a description of a collection of consumers. For-mally, an exchange economy is a tuple

E = ((�1, ω1), (�2, ω2), . . . , (�I , ωI)),denoted by (�i, ωi)Ii=1, in which each �i is a preference relation overRL+ and each ωi is a vector in R

L+.

Let E = (%i, ωi)Ii=1 be an exchange economy.

• the aggregate endowment in E is

ω̄ =I∑i=1

ωi;

• An allocation≤ in E is a collection of consumption bundles(xi)

Ii=1, where xi ∈ RL+, i = 1, . . . , I, such that

I∑i=1

xi ≤ ω̄

• an allocation= in E is an allocation≤(xi)Ii=1 such that∑I

i=1 xi =ω̄.

Definition 6 (Pareto Optimality). An allocation≤(xi)Ii=1 is Pareto

optimal in E if there is no allocation≤(yi)Ii=1 such that yi %i xi forevery i = 1, . . . , I, and yh �h xh for some h ∈ {1, . . . , I}.

Pareto optimality is a basic notion of economic efficiency. In a Paretooptimal allocation, one cannot make some agent better off withoutmaking some other agent worse off.

Definition 7 (Walrasian Equilibrium). A Walrasian equilibriumin E is a pair (x, p) such that x = (xi)Ii=1 ∈ RIL+ , and p ∈ RL+ (a pricevector), s.t.:

(i) for every i = 1, . . . , I, xi ∈ B(p, p · ωi), and x′i ∈ B(p, p · ωi)⇒xi %i x′i (all consumers optimize when choosing xi at prices p);

(ii)∑I

i=1 xi =∑I

i=1 ωi (demand equals supply).

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ω

B(p, p · ω1)

B(p, p · ω2)

ω̄1

ω̄2

Figure 2. The Edgeworth box

When there are two goods (L = 2), an exchange economy with twoconsumers can be conveniently represented using an Edgeworth box .Consider the following figure. The two agents’ preferences are repre-sented by indifference curves in the figure. The allocation correspond-ing to the red dot is not Pareto optimal: any point in the orange setdescribes an allocation that is better for both consumers. Note thateach point in the box describes an allocation=,

The Pareto optimal allocations are those for which, if we fix an uppercontour set for consumer 2 (given some given point, and his prefer-ence relation) and choose one of the best points on that set from theperspective of 1’s preferences. The resulting set of points, as we varythe upper contour set for consumer 2, is the set of all Pareto optimalallocations. In the figure it is represented with a thick blue line.

3.1.1. A characterization of Pareto optimal allocations. Let E = (�i, ωi) be an exchange economy in which each preference �i is representedby a utility ui. Consider the maximization problem P (v̄):

(PO-e)

maxx∈RLI+ u1(x1)

s.t. ui(xi) ≥ v̄i ∀i = 2, 3, ..., I,∑Ii=1 xi ≤ ω̄,

where v̄ ∈ RI is a vector of utility values.

Proposition 8. Let each �i be continuous and strictly monotone. Anallocation is Pareto optimal iff there is v̄ ∈ RI for which x solves P (v̄).


3.2. Economies with production. The second model allows for pro-duction. We introduce firms, and consider an economy in which firmsare described by their production possibility sets. Consumers are de-scribed by their preferences and endowments as before, but now theyalso have shares in the profits of the firms. These shares are fixed.

We include production by assuming that there are J firms in the econ-omy, each described by a production possibility set Yj, j = 1, . . . , J .For a production vector yj ∈ Yj, and price vector p ∈ RL+, firms j’sprofits are p · yj. Firms profits need to go to some agent in the econ-omy: this is often called “closing” the model, which roughly meansthat one accounts for all the endogenous quantities in the model. Eachconsumer i will have a share θi,j ≥ 0 in the profits of firm j. We mayhaveθi,j = 0, but all profits go to some agent, so

∑Ii=1 θi,j = 1 for all j.

An private ownership economy is a tuple

E =((Yj)

Jj=1, (�i, ωi, θi)Ii=1

)in which,

• for each j = 1, . . . J , Yj ⊆ RL is a production possibilityset ;• for each i = 1, . . . , I, �i is a preference relation over RL+, andωi is a vector in R

L+;

• for each j = 1, . . . J∑I

i=1 θi,j = 1, and for each i = 1, . . . , Iθi,j ≥ 0.

Let E = ((Yj)Jj=1, (%i, ωi, θi)Ii=1) be a private ownership economy.

• An allocation≤ in E is a pair (x, y), where x = (xi)Ii=1 ∈ RIL+ ,y = (yj)

Jj=1 ∈ RIL, yj ∈ Yj ∀ j = 1, ..., J , and

I∑i=1

xi ≤ ω̄ +J∑j=1

yj.

• An allocation= in E is an allocation≤(x, y), where

I∑i=1

xi = ω̄ +J∑j=1

yj.

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Let Y =∑J

j=1 Yj be the aggregate production set of a private ownership

economy.2 Consider the set:

Y + {ω̄} = {x ∈ RL : ∃ y ∈ Y s.th. x = y + ω̄}.Define PPS = (Y + {w}) ∩ RL+ as the production possibility set ofthe economy. Its boundary is referred to as the economy’s productionpossibility frontier.

Definition 9 (Pareto Optimality). An allocation≤(x, y) in E is Paretooptimal if there is no allocation≤(x

′, y′) such that x′i %i xi for everyi = 1, . . . , I, and = xh �h xh for some h ∈ {1, . . . , I}.

Definition 10 (Walrasian Equilibrium). Let E be a private ownershipeconomy. A Walrasian Equilibrium is a pair (x, y) ∈ RIL+ ×RJL,together with a price vector p ∈ RL+ s.t.

(i) for every i = 1, ..., I, xi ∈ B(p,Mi), and x′i ∈ B(p,Mi) ⇒ xi %x′i, where Mi = p · ωi +

∑Jj=1 θi,jp · yj (consumers optimize by

choosing xi in their budget sets);(ii) for every j = 1, ..., J , yj ∈ Yj, and p · yj ≥ p · y′j ∀ y′j ∈ Yj (firms

optimize profits by choosing yj in Yj);

(iii)∑I

i=1 xi =∑I

i=1 ωi +∑J

j=1 yj (demand equals supply).

There is a simple and useful characterization of Pareto optimal alloca-tions. Let ((Yj)

Jj=1, (%i, ωi, θi)

Ii=1) be a private ownership economy in

which each preference relation �i has a continuous and strictly mono-tone utility representation ui : R

L+ → R.

max(x,y)∈RLI+ ×RLJ

u1(x1)(PO)

subject to ui(xi) ≥ ūi ∀i = 2, 3, ..., I,I∑i=1

xi ≤ ω̄ +J∑j=1

yj,

yj ∈ Yj ∀j = 1, . . . , J.

Proposition 11. Suppose that each preference �i is continuous andstrictly monotone. An assignment (x, y) solves the maximization prob-lem PO if and only if it is Pareto Optimal.

2Define the sum of two sets A,B ⊆ Rn according to Minkowski addition: A+B ={c ∈ Rn : ∃ a ∈ A, b ∈ B s.th. c = a+ b}.


As an illustration, let us analyze the above result for the case of ex-change economies and differentiable utility functions. Let (%i, ωi) bean exchange economy. According to Proposition 11, an assignmentx ∈ RIL+ is Pareto Optimal if and only if it solves the following prob-lem:

maxx∈RLI+

u1(x1)(PO’)

subject to ui(xi) ≥ ūi ∀i = 2, 3, ..., I,I∑i=1

xi ≤ ω̄,

(1)

Assume that ui is a differentiable function, and consider an interiorsolution to the above problem.3

4. Welfare Theorems

4.1. First Welfare Theorem.

Theorem 12 (First Welfare Theorem). Let ((Yj)Jj=1, (%i, ωi, θi)

Ii=1) be

a private ownership economy in which the preference relation of everyconsumer is locally nonsatiated. If (x, y) and p constitute a WalrasianEquilibrium, then (x, y) is Pareto Optimal.

Proof of First Welfare Theorem. Let ((x, y), p) be a Walrasian equilib-rium. Let (x′, y′) ∈ RIL+ × RJL with y′ = (y′j) and y′j ∈ Yj for allj. Suppose that x′i %i xi for every i = 1, . . . , I, and there existsh ∈ {1, . . . , I} such that x′h �h xh. We shall prove that (x′, y′) cannotbe an allocation≤.

First, let us prove that

(2) x′i %i xi =⇒ p · x′i ≥ Wi = p · ωi +J∑j=1

θijp · yj.

Proceed by contradiction: Assume that x′i %i xi, but p · x′i < Wi.Then, there exists ε > 0 such that ||z − x′i|| < ε ⇒ p · z < Wi. Bylocal non-satiation, ∃ z such that z �i x′i and ||z− x′i|| < ε. Therefore,

3This may be achieved by assuming that each individual has strictly convexutility

14 ECHENIQUE

p · z < Wi and z �i xi. This contradicts that consumer i is optimizingwhen choosing xi in B(p, p · ωh +

∑j θi,jp · yj).

Second, note that x′h �h xh implies p·x′h > Wh, again because consumerh is optimizing when choosing xh in B(p, p · ωh +

∑j θh,jp · yj).

Third, note that, since firms are maximizing profits in equilibrium andy′j ∈ Yj for every firm, then p · yj ≥ p · y′j for every j = 1, . . . , J .

Thus, if we put together these inequalities and sum over consumers weobtain that Therefore,

I∑i=1

p · x′i >I∑i=1

(p · ωi +J∑j=1

θi,jp · yj)

= p · ω̄ +I∑i=1

J∑j=1

θi,jp · yj

= p · ω̄ +J∑j=1

(p · yj)I∑i=1

Iθi,j

= p · ω̄ +J∑j=1

(p · yj)

≥ p · ω̄ +J∑j=1

(p · y′j)

= p ·

(ω̄ +

J∑j=1

y′j

)

The first (strict) inequality follows from p·x′i ≥ Wi = p·ωi+∑J

j=1 θijp·yjand p · x′h > Wh. Then we use that

∑j θi,j = 1. The second (weak)

inequality follows from p · yj ≥ p · y′j.

Thus

p ·I∑i=1

x′i > p ·

(ω̄ +

J∑j=1

y′j

),

and therefore (x′, y′) could not be an allocation≤. �

4.2. Second Welfare Theorem. Let E = ((Yj)Jj=1, (%i, ωi, θi)Ii=1) bea private ownership economy (POE).


xx

ω

Figure 3. The second welfare theorem in the Edge-worth box

Definition 13. A Walrasian equilibrium with transfers is a tu-ple (x, y, p, T ), where (x, y) ∈ RIL+ × RJL, p ∈ RL+ (a price vector)and T = (Ti)

Ii=1 ∈ RI (a vector of net transfers), s.t.

(i) for every i = 1, . . . , I, xi ∈ B(p,Mi), and

x′i ∈ B(p,Mi)⇒ xi % x′i,

where Mi = p · ωi +∑J

j=1 θi,jp · yj + Ti (consumers optimize bychoosing xi in their budget sets);

(ii) for every j = 1, . . . , J , yj ∈ Yj, and p · yj ≥ p · y′j ∀ y′j ∈ Yj(firms optimize profits by choosing yj in Yj);

(iii)∑I

i=1 xi =∑I

i=1 ωi +∑J

j=1 yj (demand equals supply).

(iv)∑I

i=1 Ti = 0 (net transfers are “budget balanced”).

Theorem 14 (Second Welfare Theorem). Let

E =((�i, ωi, θi)Ii=1, (Yj)Jj=1

)be a P.O.E. in which each Yj is closed and convex, and each prefer-ence �i is strongly monotone, convex, and continuous. If (x∗, y∗) is aPareto optimal allocation≤in which

∑Ii=1 x

∗i � 0, then there is a price

vector p∗ ∈ RL+ and transfers T = (Ti)Ii=1 such that (x∗, y∗, p∗, T ) is aWalrasian equilibrium with transfers.

Digression: The separating hyperplane theorem. Let p ∈ Rn and α ∈R. The hyperplane defined by (p, α) is:

H(p, α) = {x ∈ Rn : p · x = α}.

16 ECHENIQUE

A hyperplane defines two half-spaces :

H+(p, α) = {x ∈ Rn : p · x ≥ α} and H−(p, α) = {x ∈ Rn : p · x ≤ α}.

Let X, Y ⊆ Rn. We say that the hyperplane H(p, α) properly sepa-rates X and Y if X ⊆ H+(p, α) and Y ⊆ H−(p, α), or Y ⊆ H+(p, α)and X ⊆ H−(p, α), while p · x 6= p · y, for some x ∈ X and y ∈ Y .

Lemma 15 (Separating-hyperplane Theorem). Let X and Y be dis-joint and nonempty convex sets in Rn. Then there exists a hyperplanethat properly separates X and Y .

Proof of Second Welfare Theorem. Let (x∗, y∗) be a Pareto optimal allocation≤inthe hypotheses of the theorem. The first observation is that condition(iii) of the definition of Walrasian equilibrium with transfers (WET)holds, which is equivalent to (x∗, y∗) being an allocation=. Indeed, if we

were to have∑I

i=1 x∗i < ω̄+

∑Jj=1 y

∗j . Then we could modify x

′ by giv-

ing some consumer the difference ω̄+∑J

j=1 y∗j −∑I

i=1 x∗i and (using the

strict monotonicity of preferences), contradict the Pareto optimality of(x′, y′).

We need to prove that there exist a price vector p∗ ∈ RL+ and a transfer-ence schedule T ∈ RL such that conditions (i) to (iv) of the definitionare satisfied. Let

Pi = {zi ∈ RL+ : zi �i x∗i }for i = 1, . . . , I.

Observe that Pi is a convex set as a direct consequence of the convexityof �i.4

Define P =∑I

i=1 Pi, and note that, being the sum of convex sets, P isalso convex. The set P contains all the “aggregate bundles” that canbe disaggregated into individual consumption bundles for which everyindividual is strictly better off than under the individual consumptionbundle x∗i .

Let Y =∑J

j=1 Yj be the aggregate production set, and note that it isconvex, as every firm’s production set is convex. Hence,

PPS = ({ω̄}+ Y ) ∩RL+4Let z, z′ ∈ Pi. Say wlog that z �i z′. Then for any λ ∈ (0, 1) the convexity of

�i means that λz + (1 − λ)z′ �i z′ �i x∗i . Hence, by transitivity of preferences,λz + (1− λ)z′ ∈ Pi.


is also a convex set. Since (x∗, y∗) is Pareto Optimal,

P ∩ PPS = ∅.Otherwise we could produce an allocation≤in which all consumers arebetter off than in x∗ by finding zi ∈ Pi with

∑i zi ∈ {ω̄}+ Y .

Now, by the Separating-hyperplane Theorem (Lemma 15), there existsp∗ ∈ RL such that(3) p∗ · z ≥ p∗ · q for all z ∈ P , and q ∈ ({ω̄}+ Y ).We shall see that p∗ is going to be the price vector in our Walrasianequilibrium with transfers. Observe that the definition of proper sepa-ration implies that p∗ 6= 0. We could, however, in principle have p∗l < 0.We’ll need to do some work to rule out negative prices.

Firstly, we shall prove the following:

(4) x′i �i x∗i =⇒ p∗ · x′i ≥ p∗ · x∗i , for every i.Let x′i �i x∗i . Note that we must have x′i > 0, as�i is strictly monotone.By continuity of �i, ∃ δ ∈ (0, 1) such that (1 − δ)x′i �i x∗i . By strictmonotonicity and the fact that x′i > 0,

x∗h +

(δ

I − 1

)x′i �h x∗h, for every h 6= i.

Therefore, (1−δ)x′i ∈ Pi, and x∗h+(

δI−1

)x′i ∈ Ph for every h 6= i. Then,

x′i +∑h6=i

x∗h = (1− δ)x′i +∑h6=i

(x∗h +

δx′iI − 1

)∈ P.

Since (x∗, y∗) is an allocation=,

I∑h=1

x∗h ∈ ({ω̄}+ Y ).

By (3), then, we obtain that

p∗ ·

(x′i +

∑h6=i

x∗h

)≥ p∗ ·

(x∗i +

∑h6=i

x∗h

).

Simplifying, p∗ · x′i ≥ p∗ · x∗i .

Secondly, let us prove that p∗ > 0. Fix l ∈ {1, . . . , L}. Let el ∈ RLbe the vector that equals 0 in all its entries except for the lth entry,in which it equals 1 (that is, els = 0 ∀ s 6= l and ell = 1). By strictmonotonicity, for any i ∈ 1, . . . , I, x∗i + el �i x∗i . Then, p∗ · (x∗i + el) ≥

18 ECHENIQUE

p∗ · x∗i applying (4). The latter yields p∗ · el = pl ≥ 0. As l is arbitrary,and p∗ 6= 0, we conclude that p∗ > 0.

Knowing that p∗ > 0 we are in a position to strengthen property (4).We shall prove that

(5) x′i �i x∗i =⇒ p∗ · x′i > p∗ · x∗i , for every i.

Note that∑

i x∗i � 0 and p∗ > 0 imply that p∗ ·

∑i x∗i > 0, and

so there exists at least one consumer i with p∗x∗i > 0. For such aconsumer, we show that (5) must hold. To that end, suppose (towardsa contradiction) that x′i �i x∗i but p∗ · x′i ≤ p∗ · x∗i . Given that we haveestablished (4), it must be the case that p∗ · x′i = p∗ · x∗i . Then, bycontinuity of �i, ∃ δ ∈ (0, 1) such that (1− δ)x′i �i x∗i . Then

p∗ · (1− δ)x′i = (1− δ)p∗ · x′i < p∗ · x′i = p∗ · x∗i ,

observe that the strict inequality is due to p∗ · x′i > 0. Note that thelatter contradicts (4), yielding our desired result (5) for the specificconsumer i that we know has p∗ · x∗i > 0.

Now, reasoning as in the proof that p > 0, we see that, in fact, p� 0:Since (5) holds for consumer i, x∗i + el �i x∗i implies that pl > 0. Sop� 0.

Finally, to show that (5) holds for all consumers, note that p � 0implies that p∗ · x∗i > 0 for all i with xi > 0. Then the proof of (5)applies to all consumers with xi > 0. For a consumer i with xi = 0 weknow that if x′i �i xi then x′i > 0 so that p · x′i > 0 = p · xi.

At this point we turn to the definition of transfers. For any i, definethe transfer to i by

(6) Ti = p∗ · x∗i − p∗ · ωi −

J∑j=1

θijp∗ · y∗j .

Now we have all the elements to establish condition (i) of the definitionof Walrasian equilibrium with transfers. Let

Mi = p · ωi +J∑j=1

θijp · yj + Ti,

and note that for every consumer i: p∗ · x∗i = Mi. Thus x∗i ∈ B(p,Mi).By (5), condition (i) is satisfied.


Using the latter result, it is straightforward to verify condition (iv).Note the following:

I∑i=1

Ti =I∑i=1

(p∗ · x∗i − p∗ · ωi −

J∑j=1

θijp∗ · y∗j

)

= p∗ ·

(I∑i=1

x∗i − ω̄ −I∑i=1

J∑j=1

θijy∗j

)

= p∗ ·

(I∑i=1

x∗i − ω̄ −J∑j=1

y∗j

)= 0.

We now proceed to condition (ii). First we shall prove that

(7) p∗ ·I∑i=1

x∗i ≥ p∗ · q ∀ q ∈ {ω̄}+ Y .

Property (7) says that the aggregate bundle∑

i x∗i “maximizes value”

among all the elements of {ω̄}+ Y .

Let q ∈ {ω̄}+ Y . For each n, let

zni = x∗I + (1/n, . . . , 1/n).

By monotonicity of �i, zni ∈ Pi for all i. So zn =∑I

i=1 zni ∈ P . By (3),

(8) p∗ · zn ≥ p∗ · q for every n ∈ N.

Note that zn →∑

i x∗i . Since (8) holds for each n, in the limit as

n→∞ we obtain that p∗ ·∑

i x∗i ≥ q. 5

Now, let us prove condition (ii) of the definition of WET. We mustshow that for every firm j, p∗ · y∗j ≥ p∗ · y′j ∀ yj ∈ Yj. Let y′j ∈ Yj, andnote that by (7):

p∗ ·I∑i=1

x∗i ≥ p∗ · ω̄ + p∗ · y′j +∑k 6=j

p∗ · y∗k.

5The proof of statement 7 relies on showing that aggregate consumption∑

i x∗i

lies on the boundary of the set P . In fact, it lies at the intersection of the boundaryof P and the production possibility set of the economy, PPS = ({ω̄} + Y ) ∩ RL+.Statement 7 says that aggregate consumption maximizes “value” in the PPS.

20 ECHENIQUE

As condition (iii) states∑I

i=1 x∗i = ω̄ +

∑Jj=1 y

∗j , we have:

p∗ · ω̄ +J∑j=1

p∗ · y∗j ≥ p∗ · ω̄ + p∗ · y′j +∑k 6=j

p∗ · y∗k =⇒ p∗ · y∗j ≥ p∗ · y′j.

As we have proved the existence of a price vector p∗ ∈ RL+ and a trans-fer schedule T that satisfy conditions (i) through (iv) of a WalrasianEquilibrium with transfers. �

Terminology: “Aggregation”. Economy-wide variables are called “ag-gregate.” If consumers get the bundles in (x1, . . . , xI), then

∑ni=1 xi is

an aggregate consumption bundle. Similarly, when we add up (excess)demand functions we’ll talk about aggregate (excess) demand.

5. Scitovsky Contours and cost-benefit analysis

Let (�i)Ii=1 be a collection of preferences. The Scitovsky contour atx = (xi)

Ii=1 is

S(x1, . . . , xI) = {I∑i=1

x̃i : x̃i �i xi∀i ∈ {1, . . . , I}}

Recall the definition of upper contour set: let Ui(xi) = {yi ∈ RL+ : yi �ixi}. Then

S(x1, . . . , xI) =I∑i=1

Ui(xi).

Consider a vector (x1, . . . , xI). Think of an exchange economy withthese I, agent i having preference �i, and endowments being such thatω̄ =

∑Ii=1 xi. Then we can consider the possibility that (x1, . . . , xI) is

a Pareto optimal allocation=of the aggregate bundle∑

i xi.

The set S(x1, . . . , xI) is the set of aggregate bundles that can be dis-aggregated in a way that makes all consumers weakly better than theyare at (x1, . . . , xI). If (x1, . . . , xI) is a Pareto optimal allocation=of theaggregate bundle

∑i xi, then the Scitovsky contour S(x1, . . . , xI) must

be disjoint from the set {z ∈ RL+ : z <∑

i xi}.

Scitovsky contours can be used to evaluate policy decisions. TheKaldor criterion says that a policy change is desirable if every-one can be made better off after the policy change than before. Sci-tovsky contours embody the Kaldor criterion. In particular, suppose


that the group of agents {1, . . . , I} are considering a collective changefrom the aggregate bundle x̄ =

∑i xi to x̂ ∈ RL+. If x̄ is allocated

as in (x1, . . . , xI) and x̂ ∈ S(x1, . . . , xI) then the change is desirablebecause the resulting aggregate bundle can be disaggregated in a waythat makes no consumer worse off than in x̄. Moreover, if x̂ is in theinterior of the contour, the consumers can be made strictly better off.

When each preference �i is convex, continuous and strictly mono-tonic, we can essentially use the the second welfare theorem to de-scribe Scitovsky contours and to operationalize the Kaldor criterion viacost-benefit analysis . The idea is that if (xi) is a Pareto optimalallocation=of

∑i xi, then we obtain prices p

∗ such that p∗ ·z ≥ p∗ ·∑

i xifor all z ∈ S(x1, . . . , xI).6 See Figure 4.

This means that prices p∗ support S(x1, . . . , xI) at∑

i xi: p·z ≥ p·∑

i xifor all z ∈ S(x1, . . . , xI). By observing market prices p∗ we obtaininformation about the shape of S(x1, . . . , xI). In other words, marketprices convey enough information about agents’ preferences to be usefulpolicy tools.

When the group of agents {1, . . . , I} are considering a collective changefrom the aggregate bundle x̄ =

∑i xi to x̂ ∈ RL+. If p · x̂ < p · x̄

then we know that x̂ /∈ S(x1, . . . , xI), and there is therefore no wayto distribute x̂ in a way that will make all the agents better off thanthey are currently in (x1, . . . , xI). Put differently, even without knowingagents’ preferences, if we “price” the difference x̂ − x̄ using prevailingmarket prices p, and the value is negative, we know that the change isfrom x̄ to x̂ is undesirable.

On the other hand if the change is positive then it has some chance ofbelonging to the Scitovsky contour at x̄. Moreover, if preferences aresmooth then the supporting hyperplane defined by p will be a goodlocal description of S(x1, . . . , xI) around x̄. So when the change x̂− x̄is relatively small, we can be relatively confident that x̂ ∈ S(x1, . . . , xI)when p · (x̂− x̄) > 0.

This reasoning is known as cost-benefit analysis. We use prevailingprices to evaluate the change from x̄ to x̂, and decide based on thevalue of the two aggregate bundles at the prevailing prices.

One problem with the Kaldor criterion is that it is possible that x̂ ∈S(x1, . . . , xI) and that there is (x

′i) ∈ RIL+ with x̂ =

∑i x′i, and x̄ ∈

6The reason is that x′i ∈ Ui(xi) implies that p∗ ·x′i ≥ p∗ ·xi. So the prices supporteach of the individual upper contour sets at the consumption bundle xi.

22 ECHENIQUE

x

p∗

p∗ x1,2

x1,1

x2,1

x2,2

S(x1, x2)

Figure 4. Scitovsky contour at x = x1 + x2.

S(x′1, . . . , x′I). Draw such a case for yourself. Think about what it

means.

6. Excess demand functions

6.1. Notation. Let ∆ = {p ∈ RL+ :∑L

l=1 pl = 1} denote the simplex.in RL. The interior of the simplex is denoted by ∆o = {p ∈ ∆ : p� 0},and the boundary by ∂∆ = ∆ \∆o.

For ε > 0, we also use the notation ∆ε = {p ∈ ∆ : pl ≥ ε, l = 1, . . . , L},and

∂∆ε = {p ∈ ∆ε : pl = ε, for some l = 1, . . . , L}.

6.2. Aggregate excess demand in an exchange economy. Sup-pose that (%i, ωi)Ii=1 is an exchange economy in which each %i is con-tinuous, strictly convex, and strictly monotonic. Then we can definea demand function p 7→ x∗i (p, p · ωi), and an excess demand functionp 7→ zi(p) = x∗i (p, p · ωi)− ωi for all consumer i. The aggregate excessdemand function is then

z(p) =I∑i=1

zi(p) =I∑i=1

x∗i (p, p · ωi)− ω̄


This function z : RL++ → RL satisfies a number of basic properties thatwill be essential in our study of equilibrium.

Proposition 16. If (%i, ωi)Ii=1 is an exchange economy in which ω̄ =∑Ii=1 ωi � 0, and each %i is continuous, strictly convex, and strictly

monotonic; then, the aggregate excess demand function satisfies:

(i) (continuity) z is continuous;(ii) (homogeneity) z is homogeneous degree zero;(iii) (Walras Law) ∀ p ∈ RL++, p · z(p) = 0;(iv) (bounded below) ∃ M > 0 s.th. ∀ l, p ∈ RL++, zl(p) > −M ;(v) (boundary condition) If {pn} is a sequence in RL++, and p̄ =

limn→∞ pn, where p̄ ∈ RL0 \ RL++ and p̄ 6= 0, then there is l ∈

{1, . . . , L} such that {zl(pn)} is unbounded.

Proof. Let us proceed in order:

(i) If %i is strictly monotonic, then it is locally non-satitated. Bythe maximum theorem, every demand function x∗i (p, p · ωi) iscontinuous. As z(p) is a linear combination of continuous func-tions, it is continuous.

(ii) The budget set of every consumer i is unchanged if prices aremultiplied by a constant α > 0, i.e, B(p, p ·ωi) = B(αp, αp ·ωi)for every α ∈ R++ with p ∈ RL+. Then, the maximization prob-lem of consumer i is unchanged if the prices are multiplied byα > 0. This shows that the demand function of every consumeris homogeneous of degree zero: x∗i (p, p ·ωi) = x∗i (αp, αp ·ωi) forevery α > 0. Then, by definition, z(αp) = z(p) for every α > 0.

(iii) %i strictly monotonic implies p · x∗i (p, p ·ωi) = p ·ωi for every i,i.e., every consumer’s expenditure level is equal to her income.(Note that otherwise, the consumer would be able to increaseher utility by consuming more from any good). Summing over

24 ECHENIQUE

all consumers, we obtain:

I∑i=1

p · x∗i (p, p · ωi) =I∑i=1

p · ωi

=⇒ p ·I∑i=1

x∗i (p, p · ωi) = p ·I∑i=1

ωi

=⇒ p ·

(I∑i=1

x∗i (p, p · ωi)− ω̄

)= 0

=⇒ p · z(p) = 0

(iv) Note that X = RL+ implies x∗li(p, p · ωi) ≥ 0 for every consumer

i and good l. Hence, zl(p) ≥ − ω̄l for every good l. Let M ∈ Rbe such that M > maxl∈{1,...,L}{ω̄l}, and note that zl(p) > −M∀ l, p ∈ RL++.

(v) Towards contradiction, assume zl(pn) is bounded above for all

l ∈ {1, . . . , L}. Let p̄ ∈ RL+ \RL++ be s.th. p̄ 6= 0. Then, thereare m, k ∈ {1, . . . , L} s.th. p̄k = 0 and p̄m > 0. As the aggregateendowment is strictly positive, ω̄ >> 0, then there exists aconsumer j such that ωmj > 0 and thus p̄ ·ωj > 0. Furthermore,as %j is strictly monotonic, j strictly prefers bundles that havemore of good k, all else equal. Note that this implies that thedemand function x∗j(p, p · ωj) is not well defined at p = p̄ sincej has positive income and good k, which she likes, is free. Tosee this formally, assume by contradiction ∃ x∗j ∈ RL+ such that

x∗j ∈ B(p̄, p̄ · ωj) and x∗j %j xj ∀xj ∈ B(p̄, p̄ · ωj).

Let x̂j = x∗i + εek with ε > 0, where ek ∈ RL is the vector of

zeroes, except for its k-th entry which equals one. As p̄k = 0,then p̄ ·x∗j = p̄ · x̂j ⇒ x̂j ∈ B(p̄, p̄ ·ωj). However, by strict mono-tonicity, x̂j �j x∗j , and we obtain the desired contradiction.

Let {pn} be a sequence in RL++ s.th. pn → p̄. Let zli(pn) bethe sequence of the excess demand of consumer i for good l. Wecan write the aggregate excess demand for any good l as:

zl(pn) = zlj(p

n) +∑i 6=j

zli(pn).

Note that the excess demand of every consumer for good l isbounded below by Mli > ωli. As zl(p

n) is bounded above byhypothesis for every good l, then it must be that zli(p

n) is alsobounded above for every consumer i, j included. Hence, as


zlj(pn) is bounded above and below, it has a convergent subse-

quence, say zlj(pnk). Let z∗lj be the limit of this subsequence.

7

Let z∗j = (z∗lj)

Ll=1 and x

∗j = z

∗j + ωj. As p

n >> 0 for every n,then by strict monotonicity, for every n, pn · x∗j(pn, pn · ωj) =pn · ωj , i.e., consumer j spends all her income. Therefore,pn · zj(pn) = 0 for every n. Then,

limn→∞

pnk ·zlj(pnk) = 0 =⇒ p̄·z∗lj = 0 =⇒ p̄·x∗j = p̄·ωj =⇒ x∗j ∈ B(p̄, p̄·ωj).

To conclude the prove, we will show x∗j %j xj for every xj ∈B(p̄, p̄ ·ωj) to obtain the desired contradiction. Let xj ∈ B(p̄, p̄ ·ωj). Let λ

n =pn·ωjp̄·ωj for every n, and note λ

n → 1 and λn ≥ 0for every n. Since p̄ · xj ≤ p̄ · ωj, multiplying both sides by λnyields p̄ ·λnxi ≤ pn ·ωj for every n, i.e., λnxj ∈ B(pn, pn ·ωj). Bydefinition, zj(p

n) = x∗j(pn, pn · ωj)− ωj . Hence, zj(pn) + ωj %j

λnxj. By continuity of %j, we obtain:

limn→∞

zj(pn) + ωj = z

∗j + ωj = x

∗j %j xj = lim

n→∞λnxj.

�

Observation 17. Note that properties (i)-(iii) do not rely on∑

i ωi � 0,nor on strict monotonicity. Local non-satiation suffices for Walras’ Law.

6.3. Aggregate excess demand. So we derived the aggregate excessdemand from an economy, and showed that it satisfies the five basicproperties. Now we shall analyze excess demand function as a primitivein its own right, “forgetting” the economy that it came from.

This approach allows us to obtain results for many different models,as long as their equilibria are characterized by the zeroes of an excessdemand. For example we can incorporate production, and show thata private ownership economy, under some assumptions, has an excessdemand function that satisfies the five properties. And there are othermodels that can also be captured by the demand function as a reducedform.

We use property (ii) to restrict the domain in z in various ways. Sometimes we “normalize” prices, and express all prices in terms of onegood. For example in terms of good one: pl/p1, l = 1, . . . , L. Another

normalization is pl/∑L

k=1 pk, so prices are in ∆o:

7It is tempting to claim z∗lj = zlj(p̄), but this may not be true since zlj(p) may

no be continuous at p = p̄.

26 ECHENIQUE

p1

−M

z1(p1, 1)

Figure 5. Aggregate excess demand for good 1 withL = 2/

Consider a function z : ∆o → R that satisfies the properties:

(1) z is continuous;(3) z satisfies Walras’ Law , meaning that p · z(p) = 0.(4) z is bounded below , meaning that here is M > 0 such that

zl(p) ≥ −M for all l = 1, . . . , L and p.(5) z satisfies the following boundary condition: If {pn} is a se-

quence in ∆O and p̄ = limn→∞ pn, with p ∈ ∂∆, then there is l

such that the sequence {zl(pn)} is unbounded.

We refer to aggregate excess demand functions satisfying the 5 prop-erties with the understanding that we use homogeneity to go back andforth between the two domains. Either the domain is RL++ and we im-pose all 5 properties, or the domain is ∆o, and we can obtain a functionon RL++ by imposing homogeneity. Specifically, if we define an excessdemand function z : ∆o → R, then the domain can be extended to allof RL++ by z(p) = z(

1∑l plp).

7. Existence of competitive equilibria

Consider the case when L = 2. Then Walras Law implies that p1z1(p)+p2z2(p) = 0. We can normalize the price of good 2 to be 1. Thisallows us to graph excess demand as a function of p1 alone. Moreover,p1z1(p1, 1) + p2z2(p1, 1) = 0 implies that we can focus on the marketfor good 1. Whenever z1(p1, 1) = 0 we know that z2(p1, 1) = 0.

We present an existence result for excess demand functions satisfyingthe five properties. The proof is taken from Geanakoplos (2003).


Theorem 18. Let z : ∆o → R satisfy properties (1)-(5). Then thereis p∗ ∈ ∆O with 0 = z(p∗).

We use the following result, which subsumes the role of the boundarycondition.

Lemma 19. Let z : ∆o → R satisfy properties (1)-(5). Then there isε > 0 and p̄ ∈ ∆ε s.t.

p ∈ ∂∆ε ⇒ p̄ · z(p) > 0.

The proof of Lemma 19 is part of your homework. Now we turn to theproof of the theorem. The proof relies on the following famous result,Brouwer’s fixed-point theorem:

Theorem 20. Let X ⊆ Rn be (nonempty) compact and convex. Iff : X → X is continuous, then there is x∗ ∈ X with x∗ = f(x∗).

Proof. Let ε and p̄ be as in the statement of the lemma.

Define the functions m and φ as follows:

m(p̃, p) = p̃ · z(p)− ‖p̃− p‖2

φ(p) = argmaxp̃∈∆εm(p̃, p),

for p̃, p ∈ ∆ε.

You should prove that φ is a function (it takes singleton values), andthat is satisfies the hypotheses of Brower’s fixed point theorem (use theMaximum Theorem). So φ : ∆ε → ∆ε is continuous. Brower’s fixedpoint theorem implies that there is p∗ ∈ ∆ε with p∗ = φ(p∗). We shallprove that 0 = z(p∗).

Notice first that if p∗ is in the interior of ∆ε then the first order condi-tion for the problem in the definition of φ means that

0 =∂m(p̃, p∗)

∂p̃|p̃=p∗ = (z(p∗)− 2(p̃− p∗)) |p̃=p∗ = z(p∗)

and we are done. To prove the theorem, we need then to show that p∗

must be interior.

Suppose then, towards a contradiction, that p∗ is not interior. Thenp̄ · z(p∗) > 0 by the lemma. Let p(λ) = λp̄ + (1 − λ)p∗. Note that

28 ECHENIQUE

m(p∗, p∗) = p∗ · z(p∗) = 0, by Walras’ Law. Now:m(p(λ), p∗)−m(p∗, p∗) = p(λ) · z(p∗)− ‖p(λ)− p∗‖2

= λp̄ · z(p∗) + (1− λ)p∗ · z(p∗)− λ2‖p̄− p∗‖2

= λ(p̄ · z(p∗)− λ‖p̄− p∗‖2

),

as Walras’ Law implies

(1− λ)p∗ · z(p∗) = 0,and ‖p(λ)− p∗‖2 = λ2‖p̄− p∗‖2.

Since p̄ · z(p∗) > 0 there is λ > 0 small enough thatp̄ · z(p∗)− λ‖p̄− p∗‖2 > 0.

Then m(p(λ), p∗) > m(p∗, p∗), which is absurd. �

Missing: examples of non-existence. Discontinuous offer curves becauseof lack of convexity. One example with lexicographic preferences.

7.1. The Negishi approach. We give a proof of existence that isbased on adjusting welfare weights instead of adjusting prices: thisis called the Negishi approach (after Negishi (1960)); you can thinkof it as a way of using the second welfare theorem to prove existenceof Walrasian equilibria. After all, it is easy to see that there mustexist Pareto optimal allocations, and the second welfare theorem saysthat they can be obtained as Walrasian equilibria with transfers. Thechallenge is to show that they can be obtained as Walrasian equilibriawithout transfers, meaning transfers that equal zero.

Let E = (�i, ωi)Ii=1 be an exchange economy.Theorem 21. If �i is continuous, strictly monotonic and convex, forall i = 1, . . . , I, and

∑Ii=1 ωi � 0, then E has a Walrasian equilibrium.

Our proof of Theorem 21 relies on Kautani’s fixed point theorem:

Theorem 22. Let X ⊆ Rn be compact and convex. If Π : X → 2Xis a correspondence such that, for all x ∈ X, Π(x) is a nonempty andconvex set, and such that the graph

{(x, y) ∈ X ×X : y ∈ Π(x)}is closed (we say that Π is upper hemicontinuous), then there is x∗ ∈ Xwith x∗ ∈ Π(x∗).8

8The notion of upper hemicontinuous correspondence is more general, but hereit reduces to the correspondence having a closed graph.


A triple (x, p, T ), where x ∈ RLI+ , p ∈ RL+ and T ∈ RI is a Walrasianequilibrium with transfers (WET) if

(1) p · xi = p · ωi + Ti and x′i � xi implies that p · x′i > p · ωi + Ti,(2) x is an allocation=(

∑i xi = ω̄, or supply = demand), and

(3)∑

i Ti = 0.

For the rest of this proof, we refer to allocation=as allocations. We shallalso need the following version of the second welfare theorem, statedhere without proof.

Lemma 23 (Second welfare theorem). If x is a Pareto optimal alloca-tion, then there is p ∈ ∆ and T ∈ RI such that (x, p, T ) is a WET.

Let ui be a utility function representing �i. Suppose wlog that u(0) =0. Observe then that strict monotonicity implies that if u(x) = 0 thenx = 0.

Let

U = {v ∈ RI : vi = ui(xi), i = 1, . . . , n for some Pareto optimal allocation x}

be the utility possibility frontier in E .

Lemma 24. Let v ∈ U , and let x and x′ be Pareto optimal allocationswith

vi = ui(xi) = ui(x′i)i = 1, . . . , I

. If (x, p, T ) is a Walrasian equilibrium with transfers, then so is(x′, p, T ).

Proof. It is obvious that (x′, p, T ) satisfies (2) and (3) in the definitionof Walrasian equilibrium with transfers.

To prove that (x′, p, T ) satisfies (1), note first that if zi �i x′i thenzi �i xi as ui(xi) = ui(x′i). Therefore, p · zi > p · ωi + Ti.

To finish the proof, we need to show that p · x′i = p · ωi + Ti. Butx′i � xi (as ui(xi) = ui(x′i)) implies that p · x′i ≥ p · ωi + Ti becausep · x′i < p · ωi + Ti would imply (by strict monotonicity) the existenceof a zi �i xi with p · zi < p · ωi + Ti. Now, p · x′i ≥ p · ωi + Ti for allI = 1, . . . , I,

∑i x′i = ω̄, and

∑i Ti = 0 implies that p · x′i = p · ωi + Ti

for all i. �

Lemma 25. There exists M > 0 such that if (x, p, T ) is a Walrasianequilibrium with transfers, then −M < Ti < M for all i = 1, . . . , I.

30 ECHENIQUE

Proof. Observe that for all WET (x, p, T ), and all i = 1, . . . , I,

p · (xi − ωi) = Ti.The set of allocations, say A, and ∆ is compact, so there exists M > 0such that

−M < min{p·(xi−ωi) : p ∈ ∆, x ∈ A, i = 1, . . . , I} ≤ max{p·(xi−ωi) : p ∈ ∆, x ∈ A, i = 1, . . . , I} < M�

For v ∈ U , let Π(v) ⊆ [−M,M ] be the set of T for which there existsa Pareto optimal x allocation with v = u(x) and (x, p, T ) is a WET.

Lemma 26. The correspondence v 7→ Π(v) is convex valued and hasa closed graph.

Proof. First we show that Π(v) is a convex set, for v ∈ U . So letT, T ′ ∈ Π(v). By Lemma 24 there is a Pareto optimal x and p, p ∈ ∆such that v = u(x) and (x, p, T ) and (x, p′, T ′) are both WET.

Let µ ∈ (0, 1), p̂ = µp + (1 − µ)p′ and T̂ = µT + (1 − µ)T ′. Weshall prove that (x, p̂, T̂ ) is a WET. Properties (2) and (3) of WETare immediately satisfies, as (x, p, T ) and (x, p′, T ′) are both WET. Toprove Properties (1) note that p ·xi = p ·ωi +Ti and p′ ·xi = p′ ·ωi +T ′iimply p̂ ·xi = p̂ ·ωi+ T̂i. Moreover, zi � xi implies that p ·zi > p ·ωi+Ti,and p′ · zi > p′ · ωi + T ′i . Thus p̂ · zi > p̂ · ωi + T̂i. This shows that Π(v)is a convex set.

Now we prove that v 7→ Π(v) has a closed graph. Let {vn} be a con-vergent sequence in U and {T n} be a convergent sequence in [−M,M ],with T n ∈ Π(vn) for all n. Let xn be a Pareto optimal allocation withvn = u(xn) for all n. Choose pn ∈ ∆ such that (xn, pn, T n) is a WET.By the compactness of the set of allocations, and the compactness of∆, after considering a subsequence, we can suppose that (xn, pn, T n)convergence to a triple (x, p, T ).

To finish the proof, we have to prove that (x, p, T ) is a WET. Again,properties (2) and (3) are immediate. To prove (1) note first thatpn · xni = pn · ωi + T ni implies that p · xi = p · ωi + Ti. Next, supposethat zi � xi. Then the continuity of �i implies that for n large enoughzi � xni . Then for n large enough, pn · zi > pn · ωi + T ni . Thus p · zi ≥p · ωi + Ti.

Before we continue, we show that p � 0. We know that x is anallocation, so

∑i xi = ω̄, and ω̄ � 0. So, p ∈ ∆ means that p ·

∑i xi >


0. Therefore, there is some consumer i for which p · xi > 0. We showthat (1) holds for this consumer. For consumer i, we can rule out thatzi �i xi and p ·zi = p ·ωi+Ti = p ·xi because there would then exist, bycontinuity of �i, δ ∈ (0, 1) with δzi �i xi. But then (and this argumentshould be familiar by now),

p · (δzi) = δp · xi < p · xi = p · ωi + Ti,

a contradiction of the property we have already established that zi �ixi ⇒ p · zi ≥ p · ωi + Ti. For any l, then xi + el �i xi (by strictmonotonicity) and therefore pl > 0. Thus p� 0.

Now consider an arbitrary consumer i and suppose that zi �i xi. Con-sider first the possibility that p · xi = 0. If xi = 0 then zi �i xi wouldimply that p ·zi > 0 = p ·xi as zi > 0 when zi �i 0 and p� 0. If xi 6= 0then p · xi > 0, again because p� 0. So suppose that p · xi > 0. Thenwe can just repeat the argument we made for the special consumerabove, for which we knew that p · xi > 0. This implies that (1) holdsfor all consumers. �

Let ∆I = {λ ∈ RI+ :∑I

i=1 λi = 1} be the simplex in RI . Now observethat for each v ∈ U there is a unique value of α > 0 such that αv ∈ ∆I .In words, the ray defined by v ≥ 0 in RI intersects ∆I once and onlyonce. The function that maps v ∈ U into ∆I is one-to-one. It is alsopossible to show that it is onto, and that it and its inverse is continuous.Let h be its inverse. The function h is called the radial projectionof ∆I onto U .

Lemma 27. The function h : ∆I → U is a continuous bijection.

Lemma 28. The correspondence λ 7→ Π(h(λ)) has convex values anda closed graph.

Proof. This follows from Lemma 26. If λn is a sequence, T n ∈ Φ(λn)for each n, and (λ, T ) = limn→∞(λ

n, T n) �

Define

η(T ) = argmin{λ · T : λ ∈ ∆I}for T ∈ [−M,M ]. The correspondence T 7→ η(T ) has convex valuesand a closed graph (an application of the maximum theorem).

Consider the correspondence

F : ∆I × [−M,M ]→ ∆I × [−M,M ]

32 ECHENIQUE

defined by F (λ, T ) = η(T ) × Π(h(λ). Then F is convex valued andhas a closed graph. Kakutani’s fixed point theorem implies that thereexists (λ∗, T ∗) ∈ ∆I × [−M,M ] with (λ∗, T ∗) ∈ F ((λ∗, T ∗)).

Let (x∗, p∗, T ∗) be a WET with h(λ∗) = u(x∗). We shall show thatT ∗ = 0, and thereby prove that (x∗, p∗) is a Walrasian equilibrium.

Suppose then, towards a contradiction, that T ∗h > 0 for some i. Thenthere is j with T ∗j < 0 as

∑i T∗i = 0. But then the definition of η, and

λ∗ ∈ η(T ∗) implies that λh = 0. By definition of h, then, vh = 0. Thismeans that uh(xh) = 0 which is only possible if xh = 0 (as ui(0) = 0and ui is strictly monotone). Then

0 = p∗ · xh = p∗ · ωh + T h.Hence T h ≤ 0, contradicting that T h > 0.

8. Uniqueness

Definition 29 (Strong Weak Axiom for Excess Demand functions).Let z : ∆o : RL+ → R+ be an excess demand function. We say that zsatisfies the Strong Weak Axiom if:

[z(p∗) = 0, p 6= p∗] =⇒ p∗ · z(p) > 0.

The strong weak axiom is motivated by a one-consumer exchange econ-omy. Let x∗(p, p · ω) be the demand function of this consumer. Thenp∗ ∈ ∆o is an equilibrium price iff x∗(p∗, p∗ · ω) = ω (supply equalsdemand when there is a single consumer).

Let p 6= p∗, and suppose that x∗(p, p ·ω) 6= x∗(p∗, p∗ ·ω). By WARP, itmust be the case that p∗ · x∗(p, p · ω) > p∗ · x∗(p∗, p∗ · ω) = p∗ · ω, i.e.,the new optimal bundle was unaffordable under the equilibrium pricevector. The reason is that the bundle consisting of the endowment isaffordable under any price vector, and x∗(p∗, p∗ · ω) = ω. Therefore,

p∗ · (x∗(p, p · ω)− ω) = p∗ · z(p) > 0.

Theorem 30. If z : ∆o → RL satisfies conditions (i) to (v) and theStrong Weak Axiom, then there is a unique competitive equilibrium.

Proof. By the equilibrium existence theorem, a competitive equilibriummust exist. To see why it must be unique, suppose that there are twodifferent competitive equiilbriums, and use the Strong Weak Axiom toreach a contradiction. �


We motivated the Strong Weak Axiom in a one consumer economy us-ing the Weak Axiom of Revealed Preference. With more than one con-sumer, we would need that aggregate demand satisfies WARP. Whenthere are multiple heterogeneous consumers, however, it is very hardto guarantee that an aggregate demand will satisfy WARP. Our nextcondition may seem to have a lot of economic content, but it turns outto be stronger than SWA.

Definition 31 (Gross Substitutes). Let z : RL++ → RL be an excessdemand function. We say that z satisfies the gross substitutes prop-erty if, ∀ p, p′ ∈ RL++, pk < p′k for some k, and pl = p′l ∀ l 6= k implyzl(p) < zl(p

′) ∀ l 6= k.

Like the SWA, the meaning of GS is simple in the case of a single con-sumer. Consider an economy with one consumer and two goods: coffeeand tea. The two goods are substitutes, meaning that if the price ofcoffee increases, then the demand for tea must increase. Since thereis a single consumer, aggregate demand will have the same property.Now, if there are many heterogeneous consumers, then the price in-crease will have consequences the incomes of the consumers who owncoffe, or for whose who own shares in the firms that produce coffee. Itis hard to know what the final impact of these consequences will be.It could result in lower demand for tea. In all, with many agents andmany goods, the GS property is not very plausible.

Proposition 32. If z satisfies the gross substitutes property, then itsatisfies the Strong Weak Axiom.

A proof of Proposition 32 can be found in Arrow and Hahn (1971).

By Theorem 30, Proposition 32 implies that the gross substitutes prop-erty is a sufficient condition for uniqueness of competitive equilibrium.We show this fact directly in the next result:

Theorem 33. If the excess demand function z : RL++ → RL satisfiesconditions (i) to (v) and the gross substitutes property, then there is aunique competitive equilibrium in ∆o.

Proof. By Theorem18, properties (i) -(v) from Proposition 16 assurethe existence of p∗ ∈ ∆o such that z(p∗) = 0. Let p ∈ ∆o be suchthat p 6= p∗. Choose λ > 0 such that λp ≥ p∗ and ∃ h for whichλph = p

∗h. (This is achieved by letting h = argmaxl{p∗l /pl} and setting

λ = p∗h/ph > 0.) Therefore, λp ≥ p∗ implies that for some k, λpk > p∗k,i.e., the set L+ = {k ∈ L : λpk > p∗k} 6= ∅. Increase the price of each

34 ECHENIQUE

good in L+ one by one to get from price vector p∗ to λp. By applyinggross substitutes, it must be that the excess demand function of goodh /∈ L+ increases with each price increase, so that zh(λp) > zh(p∗). Aszh(p

∗) = 0, and zh(λp) = λzh(p) by homogeneity of z, then zh(p) > 0.Therefore, p is not an equilibrium. �

By homogeneity of degree zero, there is of course always multiple equi-libria in RL++. That is why Theorem 33 qualifies the statment to talkabout uniqueness in ∆o.

9. Representative Consumer

Consider a collection of preferences (%i)Ii=1 over RL+ such that each �i

gives rise to a continuous demand function x∗i . We want to know whenthere is a preference relation�, giving rise to a demand function x∗ withthe property that: xR(p, p · ω̄) =

∑Ii=1 x

∗i (p, p · ωi), where ω̄ =

∑i ωi.

Let~ω = (ω1, ω2, . . . , ωI) ∈ RIL+ .

Note that ~ω represents the distribution of the aggregate endowment inthe economy. Under this notation, ω̄ = ι ·~ω, where ι is a vector of ones.We write the aggregate demand function in terms of the distributionof income as:

z(p, ~ω) =I∑i=1

x∗i (p, p · ωi)− ι · ~ω.

Definition 34 (Representative Consumer). A collection of preferences(%i)Ii=1 admits a representative consumer if there is a preferencerelation %R on RL+ and an associated continuous demand function x

R :RL+ ×R+ → RL+ such that

xR(p, p · ω̄) =I∑i=1

x∗i (p, p · ωi) ∀ ~ω ∈ RIL+ , p ∈ RL+.

If a collection of preferences admits a representative consumer, notethat its associated demand function satisfies

zR(p, ω̄) = xR(p, p · ω̄)− ω̄ = z(p, ~ω) ∀ ~ω ∈ RIL+ .In this case, we say that z admits a representative consumer.

Observation 35. Let (%i, ωi)Ii=1 be an exchange economy. If the collec-tion of preferences (%i)Ii=1 admits a representative consumer, then

z(p, ~ω′) = z(p, ~ω) ∀, ~ω, ~ω′ s.th. ω̄ = ι · ~ω = ι · ~ω′.


The observation suggests that a representative consumer will only existunder very stringent conditions.

We shall see that homothetic preferences is central to the questionof when an economy accepts a representative consumer. Recall thata consumer has a homothetic preference relation if x %i y impliesαx %i αy for all x, y ∈ RL+ and α ≥ 0.

Theorem 36 (Antonelli’s Theorem). The aggregate excess demandfunction z admits a representative consumer if and only if there isa homothetic preference relation % on RL+ such that each individualdemand x∗i is generated by %.

You may have seen a result on aggregate demand and Gorman forms,and you may be wondering what the relation is to Antonelli’s theorem.The Gorman form characterizes demand with linear expansion paths(Engel curves), but the usual result on the Gorman form is local. Ifyou consider demand when income becomes very small, close to 0, thenthe income expansion paths have to pass through zero. This meansthat income expansion curves must behave like they do for homotheticpreferences.

9.0.1. Digression: The Cauchy equation. Consider the following equa-tion f(x1 + x2) = f(x1) + f(x2) for all x1, x2 ∈ R. This is an equationin which the unknown is the (real) function f : a so-called functionalequation . This particular equation is called Cauchy’s equation .We want to know if Cauchy’s equation has a solution, and what thesolution is. It obviously has solutions, for example f(x) = x. Thefollowing result characterizes all continuous solutions.

Proposition 37. Let f : R→ R be continuous and satisfy that f(x1 +x2) = f(x1)+f(x2) for all x1, x2 ∈ R. Then there is c ∈ R (a constant)such that f(x) = cx. Note that c = f(1).

Proof. First, let n > 1 be a positive integer and r ∈ R. Note that

f(nr) = f(r+(n−1)r) = f(r)+f((n−1)r) = f(r)+f(r)+f((n−2)r) = . . . = nf(r).

In second place, let q = n/m ∈ Q be a rational number, with n,m ∈ Zbeing positive integers. Then f(n/m) = nf(1/m), which means that

mf(n/m) = mnf(1/m) = nf(m/m) = nf(1).

Hence f(q) = qf(1). The same holds true when q ≤ 0.

36 ECHENIQUE

Now, since f(q) = qf(1) for all rational numbers q, and f is continuous,f(x) = xf(1) for all real numbers x. �

9.0.2. Proof of Antonelli’s theorem.

Proof. (⇐) Let % be a preference on RL+ such that it generates everydemand function xi and it is homothetic. Let xR be the demand gen-erated by this preference relation. Note that xR(p,m) = mxR(p, 1) forall m as xR is linear homogeneous in income. Then,

I∑i=1

x∗i (p, p · ωi) =I∑i=1

xR(p, p · ωi)

= xR(p, 1)I∑i=1

p · ωi

= xR(p, 1)(p · ω̄)= xR(p, p · ω̄)

(⇒) Let p ∈ RL++ and ω ∈ RL+. Let ~ωi = (0, . . . , 0, ω, 0, . . . , 0), with ωin the i-th position. Then

z(p, ~ωi) = xR(p, p · ω)− ω

=∑j 6=i

x∗j(p, p · 0) + x∗i (p, p · ω)− ω

= x∗i (p, p · ω)− ω=⇒ xR(p, p · ω) = x∗i (p, p · ω) ∀ i ∈ I.

As the consumer i, price vector p and endowment ω are arbitrary, xR =x∗i ∀i. Let%R generate xR, so that%R also generates x∗i . We shall provethat xR is linear homogeneous in income, i.e., xR(p, λ ·m) = λxR(p,m)∀ λ > 0, (p,m) ∈ RL++ × RL+. This is a neccesary and sufficient forhomotheticity.

First, let m = m1 +m2, and choose for i = 1, 2, p · ωi = mi and ωj = 0∀j 6= i. Since we have a representative consumer,

xR(p,m) = x∗1(p,m1) + x∗2(p,m2) +

∑j≥3

x∗j(p, p · 0)

= xR(p,m1) + xR(p,m2).


In particular, for a positive integer n, xR(p, nm) = xR(p, (n − 1)m) +xR(p,m). This means that xR(p, nm) = nxR(p,m). Let q = k

l∈ Q+,

with k, l ∈ N. Then

xR(p, qm) = kxR(p,m

l

)=⇒ lxR(p, qm) = kxR

(p,lm

l

)=⇒ xR(p, qm) =

(k

l

)xR (p,m)

=⇒ xR(p, qm) = qxR (p,m)

Finally, since xR is continuous by hypothesis:

xR(p, λm) = λxR(p,m) ∀λ > 0.

�

Observation 38. The exercise in the proof is known as the solution tothe Cauchy Equation. Let f : R → R be continuous and satisfy thatf(x1 + x2) = f(x1) + f(x2) for any (x1, x2) ∈ R2. Then ∀ x ∈ R,f(x) = cx for some constant c. In our case, let f(m) = xR(p,m), for afixed p let c = xR(p, 1). Therefore, xR(p,m) = mxR(p, 1).

9.1. Samuelsonian Aggregation. Let (%)Ii=1 be a collection of pref-erences where each %i is represented by a utility function ui : RL+ → R.

Let W : RI → R be a strictly monotone increasing function. We referto W as a social welfare function .

Consider the following problem:

max(x1,...,xI)∈RIL+

W (u1(x1), ..., uI(xI))(P1)

subject to p ·I∑i=1

xi ≤ m.

The problem above implies choosing an aggregate consumption bundlez =

∑Ii=1 xi ∈ RIL for the economy. The objective is to maximize the

social welfare in the economy, as given by W . The constraint reflectsan aggregate budget constraint.

38 ECHENIQUE

Consider the alternative problem:

maxz∈RIL+

{max

(x1,...,xI)∈RIL+W (u1(x1), . . . , uI(xI)) s.t.

I∑i=1

xi = z

}(P2)subject to p · z ≤ m.

Therefore, define the value function of (P1) as

U(z) = sup{W (u1(x1), . . . , uI(xI)) : (x1, . . . , xI) ∈ RIL+ andI∑i=1

xi = z},

and rewrite (P2):

maxz∈RIL+

U(z) subject to p · z ≤ m.(P2’)

Let x∗i (p,mi) be the demand function generated by %i, where mi is theincome of consumer i. Consider the following problem:

max{mi}Ii=1

W (u1(x∗1(p,m1)), ..., uI(x

∗I(p,mI)))(P4)

subject toI∑i=1

mi ≤ m.

Interpret these problems. Problem 1 maximizes social welfare functionby choosing an individual bundle for each consumer subject to the ag-gregate budget constraint. This induces an optimal aggregate bundlez ∈ RL. The second problem is a maximization in two steps. First,given an arbitrary aggregate bundle of consumption z ∈ RL, the socialplanner maximizes the social welfare function, i.e., decides on the op-timal allocation of the aggregate bundle among the consumers. Then,given the income restriction of the economy, the social planner decidesthe optimal aggregate consumption bundle. Problem P2’ makes ex-plicit that we can write the second problem as that of a representativeconsumer whose utility function is the value function resulting from thefirst of the two nested problems above. Therefore, the utility functionof our representative consumer represents the optimal division of anaggregate consumption bundle among the consumers. Another way ofaddressing this problem is in terms of income.


9.2. Eisenberg’s Theorem. Let (%i)Ii=1 be a collection of preferenceswhere each %i is represented by ui : RL+ → R. Identify an economywith a vector of endowments ~ω ∈ RIL+ (so that for each ~ω ∈ RIL+ ,(%i, ωi)Ii=1. An economy has a fixed structure of endowments ifthere is α = (αi)

Ii=1 ∈ RI+ with

∑i αi = 1 and ωi = αiω̄.

Theorem 39 (Eisenberg’s Theorem). Consider an economy with afixed structure of endowments, given by α. Let each %i be representedby a continuous and homogenous degree one utility fuction uii. Thenthe aggregate demand of the economy is generated by a representativeconsumer, whose utility function U : RL+ → R is given by:

(9) U(x) = max(x1,...,xI)∈RIL+

{I∏i=1

(ui(xi)

)αis.t. x =

I∑i=1

xi

}.

10. Determinacy

Consider a collection of preferences (�i)Ii=1 over RL+. For each givenvector of endowments:

~ω = (ω1, . . . , ωI) ∈ RIL+ ,

we have an exchange economy (�i, ωi)Ii=1. So for fixed preferences(�i)Ii=1 we can identify a set of possible (exchange) economies with aset of vectors of initial endowments.

Let E be an open subset of RIL+ . Assume that all vectors of endowments~ω ∈ E satisfy

∑i ωi � 0, and the consumers’ preferences �i are strictly

convex, strictly monotone, and continuous. Then each �i gives rise toa demand function x∗i and aggregate excess demand is

z∗(p; ~ω) =I∑i=1

x∗i (p, p · ωi)−I∑i=1

ωi.

The function p 7→ z∗(p; ~ω) satisfies properties (1)-(5). Note that we nowmake the dependence of z∗ on ~ω explicit. As a function of p (meaning,for fixed ~ω, z∗ satisfies properties (i)-(v).

As an example, consider an economy with two goods. Normalize p2 = 1,and focus on p1 and z1. In this case, we would expect the economy tohave an excess demand function such as the one depicted below.

40 ECHENIQUE

10.1. Digression: Implicit Function Theorem. REMIND OF THEINVERSE Fn THM using a picture.

Here’s a quick calculation to remind you of what the IFT says.

Let f : R2 → R be a C1 function. Let v be a variable of interest and qa parameter of the model. Suppose that the solution to our model ischaracterized implicitly by f(v, q) = 0.

We want to find a function h defined on a neighborhood of a q0 suchthat f(v0, q0) = 0, v0 = h(q0) and f(h(q), q) = 0 for all q in theneighborhood. Taking derivatives, we find that

∂f

∂v· ∂h∂q

+∂f

∂q= 0 =⇒ h′(q) = −

∂f∂q

∂f∂v

.

Note that in order to perform the previous excercise we need to assume∂f∂v6= 0. This is intuitive: as f(v, q) = 0 describes our solution, if f

does not vary in v, even though f may change as we vary q, this doesnot tell us anything the behavior of v.

A function f : A ⊆ Rn → R, for which A is open, is said to be of classr if it has 0 ≤ r ≤ ∞ continuous derivatives. We write Cr to denotethe property of being of class r.

Let g : Rn ×Rk → Rm be a C1 function. The Jacobian matrix of g is

Dg(x, y) =[Dxg(x, y) Dyg(x, y)

]=[

dg∂x1

· · · dg∂xn

dg∂y1

· · · dg∂yk

]=

∂g1∂x1

· · · ∂g1∂xn

∂g1∂y1

· · · ∂g1∂yk

.... . .

......

. . ....

∂gm∂x1

· · · ∂gm∂xn

∂gm∂y1

· · · ∂gm∂yk

m×(n+k)

Theorem 40 (Implicit Function Theorem). Let A ⊆ Rn, and B ⊆ Rmbe open sets. Let f : A × B → Rn be Cr with 1 ≤ r ≤ ∞. Let(v̄, q̄) ∈ A×B be such that f(v̄, q̄) = 0. Then, if

Dvf(v, q)∣∣∣(v,q)=(v̄,q̄)

is a non-singular matrix, there are open sets A′ ⊆ A, B′ ⊆ B, anda Cr function h : B′ → A′ such that h(q̄) = v̄, and f(v, q) = 0 for(v, q) ∈ A′ ×B′ if and only if v = h(q). Moreover,

Dqh(q̄) = −[Dvf(v, q)

∣∣∣(v,q)=(v̄,q̄)

]−1·Dqf(v̄, q̄).


This version of the IFT can be found in Mas-Colell (1989).

10.2. Regular and Critical Economies. We develop methods foranalyzing parametrized models. To this end, we focus on exchangeeconomies and interpret endowments as parameters. It is, however,possible to use the same ideas for different parametrizations. For ex-ample, we could use preferences as the parameters of the model, orfirms’ technologies in a POE.

Fix a collection of preferences �i on RL+, 1 ≤ i ≤ I. For each collectionof endowments ~ω = (ω1, . . . , ωI) ∈ RIL+ we have an exchange economy(�i, ωi)Ii=1. With fixed preferences, we identify an economy with thevector of endowments ~ω, and let E ⊆ RIL++ be a set of economies.Assume that E is open.

Given an economy ~ω ∈ E, the resulting demand for agent i is x∗i (p, p ·ωi). We assume that demand functions, as functions of prices and in-come (p,m) 7→ x∗i (p,m), and with domain RL+1++ are C1. This assump-tion relies on a notion of smooth preferences that we are not going toget into.

To make the dependence of the model on its parameters (endowments)explicit, we write the aggregate excess demand as

z(p, ~ω) =I∑i=1

x∗i (p, p · ωi)−I∑i=1

ωi.

Before we used homogeneity to normalize prices so that they are in∆o, but now we will use a different normalization. In particular, ho-mogeneity of degree zero implies that

z(p1, . . . , pL) = z

(p1pL, . . . ,

pL−1pL

,pLpL

),

which means restricting to prices in RL−1++ with the price of the Lth goodbeing fixed at 1. Moreover, by Walras’ law we can restrict attentionto all market except for one. To this end let ẑ : RL−1++ × E → RL−1 bedefined by:

(10) ẑl(p1, p2, . . . , pL−1; ~ω) = zl(p1, p2, . . . , pL−1, 1; ~ω)

for l = 1, . . . , L− 1. In particular,

ẑ(p1, . . . , pL−1) = 0 iff z(p1, . . . , pL−1, 1) = 0.

42 ECHENIQUE

p1

−M

Figure 6. Aggregate excess demand ẑ1(p1, 1, ~ω).

Given an economy ~ω, let P(~ω) denote its competitive equilibria. So

P(~ω) ={p ∈ RL−1++ : ẑ(p, ~ω) = 0

}.

Definition 41 (Regular Equilibrium). An equilibrium price p∗ ∈ P(~ω)is a regular equilibrium of ~ω if the matrix

Dpẑ(p; ~ω)∣∣∣p=p∗

is non-singular.

An economy ~ω is said to be a regular economy if all the equilibriumprices in P(~ω) are regular. An economy that is not regular is critical .

Recall our initial example of an economy with two goods. In that caseẑ : R++×R2I → R. Therefore, the condition of regularity is equivalentto the derivative of the excess demand function being different from zeroat every equilibrium price. Consider the aggregate excess demands inFigure 6. The economy with the green excess demand is regular, whilethat with the red excess demand is critical.

If there are more than two goods, consider an infinitesimal changein prices dp = (dp1, dp2, . . . , dpL−1). The directional derivative of thefunction ẑ at the price vector p ∈ RL++ in the direction of dp is given byDẑ(p) · dp.9 If Dẑ(p) has full rank at p, then Dẑ(p) · dp 6= 0 for everydp 6= 0. This implies that the normalized excess demand function

9Let f : Rn → R be a differentiable function with gradient vector ∇f(x). Thedirectional derivative of f in the direction of vector u ∈ Rn is given by Df(x;u) =∇f(x) · u. If f : Rn → Rm, then Df(x;u) = [∇f1(x), . . . ,∇fm(x)]′u = [∇f1(x) ·u, . . . ,∇fm(x) · u]′ = [Df1(x;u), . . . , Dfm(x;u)].


ẑ changes for infinitesimal changes in prices, i.e., we stop being atequilibrium. This is the intuition for our next concept and result.

Definition 42 (Local Uniqueness). An equilibrium p∗ ∈ P(~ω) is lo-cally unique if there exists ε > 0 such that, ∀ p ∈ RL−1++ ,

‖p− p∗‖ < ε =⇒ p /∈ P(~ω).

Proposition 43. Let ẑ be C1 and p∗ ∈ P(~ω) be a regular equilibrium.Then, p∗ is locally unique. Furthermore, there are neighborhoods B1 of~ω in E, and B2 of p

∗ in RL−1++ , and a function h : B1 → B2 such thatẑ(h(~ω), ~ω) = 0 ∀ ~ω ∈ B1,

and

D~ωh(~ω) = −[Dpẑ(p, ~ω)

∣∣∣p=p∗

]−1·D~ωẑ(p∗, ~ω).

Proof. Inmediate from the Implicit Function Thm �

Proposition 44. A regular economy has a finite number of equilibria.

Proof. Consider prices as subsets of the simplex, ∆, a re-normalization.Local uniqueness is preserved under the re-normalization.

Note that P(~ω) = z−1(0; ~ω). Since z is a continuous function, P(~ω) isa closed set. Moreover P(~ω) is compact because it is a closed subset ofa compact set. For each price p ∈ P(~ω), let Np be an open ball withcenter p and for which p is the only equilibrium price in Np. Such ballsexist because equilibrium prices are locally unique. Now

P(~ω) ⊆ ∪p∈P(~ω)Np,an open cover. So P(~ω) has a finite subcover, and therefore is unique.

Another proof is as follows: Suppose towards a contradiction that P(~ω)is infinite. Let pn be a sequence of distinct prices in P(~ω). pn is in ∆so it has a convergent subsequence that we, in an abuse of notation,also denote by pn. If p

∗ = limn→∞ pn then p∗ ∈ P(~ω) by continuity

of aggregate excess demand. But then p∗ is not locally isolated; acontradiction. �

Definition 45 (Index). The index of p ∈ P(~ω) is defined as:

index(p) = (−1)L−1 · sign(∣∣∣Dpẑ(p, ~ω)∣∣∣) ,

where∣∣∣Dpẑ(p, ~ω)∣∣∣ is the determinant of the matrix Dpẑ(p, ~ω).

44 ECHENIQUE

Note that for every regular economy ~ω, index(p) ∈ {−1, 1} ∀ p ∈ P(~ω).We state without proof the following theorem.

Theorem 46 (Index Theorem). If ~ω is a regular economy, then∑p∈P(~ω)

index(p) = 1.

The index theorem can be used to establish uniqueness: if you canshow that any competitive equilibrium in an economy has index one,then there can only be one equilibrium. Finally, the index theoremimplies the following curiosity.

Corollary 47. A regular economy has an odd number of equilibria.

Consider our initial example of an economy with two goods. See in thegraph below the index of each equilibrium in the economy, and verifythat they sum up to one.

10.3. Digression: Measure Zero Sets and Transversality.

Definition 48 (Rectangle and Volume). A rectangle in Rn is a setof the form

R = [a1, b1]×[a2, b2]×· · ·×[an, bn] = {x ∈ Rn : ai ≤ xi ≤ bi, i = 1, . . . , n},where [ai, bi] = {x ∈ R : ai ≤ x ≤ bi} is an interval in R. The volumeof a rectangle R = [a1, b1]× [a2, b2]× · · · × [an, bn] is

vol(R) =n∏i=1

(bi − ai).

Definition 49 (Measure Zero). A set A ⊆ Rn is measure zero if ∀ε > 0 there exist a collection of rectangles R1, R2, . . . such that

A ⊆∞⋃i=1

Ri, and∞∑i=1

vol(Ri) < ε.

Consider the following two examples of measure zero sets in R2: a finiteset and a straight line.

Under the same reasoning, note that a line segment is not measure zeroin R (because it is an interval), but it is in Rn ∀ n > 2. Similarly, aplane is not measure zero in R2, but it is in Rm ∀ m > 2.

Observation 50. No open set has measure zero.


Consider a parametrized familiy of equations f : RN+K → RN . Definea system of equations as f(v, q) = 0, where v ∈ RN is a vector ofunknowns, and q ∈ RK is a vector of parameters.

Theorem 51 (Transversality Theorem). Let f : RN+K → RN be C1.If the N × (N +K) Jacobian matrix Df(v, q) has full rank (rank N) ateach (v, q) with f(v, q) = 0, then there is a measure zero set C ⊆ RKsuch that, ∀ q ∈ RK \ C, the N ×N matrix Dvf(v, q) is non-singularwhenever f(v, q) = 0.

The rough idea behind the theorem is that when Df(v, q) is full rank,the function f can change locally in any direction. This means thatwhen we are at a critical solution, a change in parameters can “perturbaway” the critical point.

10.4. Genericity of regular economies. Return to the assumptionswe made earlier. Suppose that ẑ : RL−1++ ×E → RL−1 is C1 and satisfiesproperties (i)-(v).

Theorem 52. There is a measure zero set C ⊆ RIL+ such that, if~ω ∈ E \ C, then ~ω is a regular economy.

Proof. The result follows from the Transversality Theorem.

The matrix Dẑ(p; ~ω) is (L− 1)× (L− 1 + IL). Write this matrix as

Dẑ(p; ~ω) =[Dpẑ(p; ~ω) Dω1 ẑ(p; ~ω) · · · DωI ẑ(p; ~ω)

],

where Dωi ẑ(p; ~ω) is the (L−1)×L Jacobian matrix with respect to theendowment vector of consumer i. We shall prove that any one of thematrices Dωi ẑ(p; ~ω) has full rank. If we show that any Dωi ẑ(p; ~ω) hasfull rank, then we have that Dẑ(p; ~ω) has L − 1 linearly independentcolumns and thus it is also full rank.

So lets calculate Dω1 ẑ(p; ~ω), and show that it has full rank. The matrixDω1 ẑ(p; ~ω) may be written as

Dω1 ẑ(p; ~ω) =[Dω11 ẑ(p; ~ω) Dω21 ẑ(p; ~ω) · · · DωL1 ẑ(p; ~ω)

]=[

dẑ(p;~ω)∂ω11

dẑ(p;~ω)∂ω21

· · · dẑ(p;~ω)∂ωL1

]Recall that

ẑl(p; ~ω) =I∑i=1

x∗i (p, p · ωi)− ω̄l ∀ l = 1, . . . , L1.

46 ECHENIQUE

Then, compute for every l = 1, . . . , L− 1,

∂ẑl(p; ~ω)

∂ωh1=

∂x∗h1(p,p·ω1)

∂m1ph − 1 if h = l

∂x∗h1(p,p·ω1)∂m1

ph if h 6= l

For notational purposes, we let m1 denote the income of consumer 1.Therefore we may write the complete matrix Dω1 ẑ(p; ~ω) as

Dω1 ẑ(p; ~ω) =

∂x∗11∂m1

p1 − 1 ∂x∗11

∂m1p2 · · · ∂x

∗11

∂m1pL−1

∂x∗11∂m1

∂x∗21∂m1

p1∂x∗21∂m1

p2 − 1 · · · ∂x∗21

∂m1pL−1

∂x∗21∂m1

......

. . ....

...∂x∗L−1,1∂m1

p1∂x∗L−1,1∂m1

p2 · · ·∂x∗L−1,1∂m1

pL−1 − 1∂x∗L−1,1∂m1

(L−1)×L

See that the L-th column of the previous matrix takes into accountthe fact that we have normalized pL = 1. Take the L-th column andmultiply it by pl to subtract it from the l-th column. This operationdoes not change the rank of the matrix. Repeating the operation, weobtain:

−1 0 · · · 0 ∂x∗11

∂m1

0 −1 · · · 0 ∂x∗21

∂m1...

.... . .

......

0 0 · · · −1 ∂x∗L−1,1∂m1

(L−1)×L

The first L−1 columns of the previous matrix are linearly independent,implying that Dω1 ẑ(p; ~ω) has full rank. Therefore, the Jacobian matrixDẑ(p; ~ω) has also full rank. Transversality implies that there exists ameasure zero set C ∈ RIL such that, if ~ω ∈ RIL \ C, then Dpẑ(p; ~ω)has full rank for every p ∈ E(~ω). In other words, every economy~ω ∈ RIL \ C is regular, where C is a measure zero set. �

11. Observable Consequences of Competitive Equilibrium

11.1. Digression on Afriat’s Theorem. A data set as a collection(xk, pk)Kk=1 where x

k ∈ RL+ and pk ∈ RL++ for every k = 1, . . . , K.

Definition 53. A data set (xk, pk)Kk=1 is rationalizable by u : RL+ → R

if, ∀ y ∈ RL++,

pk · y ≤ pk · xk =⇒ u(y) ≤ u(xk) ∀ k = 1, . . . , K.


Definition 54 (Revealed Preference). Given a data set (xk, pk)Kk=1,define a binary relation %R on RL+ as:

x %R y ⇐⇒ ∃ k s.th. x = xk and pk · xk ≥ pk · y,and x �R y ⇐⇒ ∃ k s.th. x = xk and pk · xk > pk · y.

We refer to the binary relation %R as the revealed preference relation.

Definition 55 (Weak Axiom of Revealed Preference). A data setsatisifes the Weak Axiom of Revealed Preference (WARP) if there isno k and l such that xk %R xl and xl �R xk.

Definition 56. A data set satisfies the Generalized Axiom of RevealedPreference (GARP) if there is no sequence xk1 , xk2 , ..., xkn such that

xk1 %R xk2 %R · · · %R xkn , and xkn �R xk1 .

Observation 57. If L ≤ 2, WARP and GARP are equivalent. If L ≥ 3,then GARP =⇒ WARP, but not conversely.

Theorem 58 (Afriat’s Theorem). Consider a data set (xk, pk)Kk=1. Thefollowing statements are equivalent:

(1) the data set is rationalizable by a locally nosatitated utility func-tion;

(2) the data set satisfies GARP;(3) there are numbers Uk, λk > 0 for k = 1, . . . , K, such that

Uk ≤ U l + λlpl · (xk − xl);

(4) the data set is rationalizable by a strictly monotonic and concaveutility function.

See Chambers and Echenique (2016) for a proof and detailed discussionof Afriat’s theorem.

11.2. Sonnenschein-Mantel-Debreu Theorem: Anything goes.In this section we consider a set of important results regarding what canbe obtained as an equilibrium for a well-behaved economy. The answeris that, in a sense, anything can happen. The Sonnenschein-Mantel-Debreu Theorem says that, off the boundaries of the simplex, any func-tion that satisfies the basic properties of an excess demand function canbe obtained as the excess demand of a well-behaved exchange economy.Note that properties (iv) and (v) are about the boundary behavior.Homogeneity will be subsumed in the use of ∆o as the domain of f .

48 ECHENIQUE

Theorem 59 (Sonnenschein-Mantel-Debreu Theorem). Let f : ∆o →RL be a continuous function that satisfies Walras’ Law. For any ε > 0there exists an exchange economy (%i, ωi)Ii=1 with I = L and each %istrictly monotonic, continuous and strictly convex such that the aggre-gate excess demand function of this economy coincides with f on ∆ε.

The proof of the Sonnenschein-Mantel-Debreu Theorem is complicated,but here are some of the basic ideas.10 The proof “decomposes” f intoindividual aggregate excess demand functions that satisfy WARP. Foreach individual consumer i, we let ωi = ei, the ith unit vector (recallthat I = L). So agent i is endowed with one unit of good i. Thenlet gi(p) be the projection of ei onto the orthogonal complement of p.You can think of gi(p) as the residuals of a least squares regressionof ωi = ei on the line spanned by p. Then gi(p) will satisfy Walraslaw, as it was chosen to live in the orthogonal complement to p. Itis also continuous, and satisfies WARP because it is the outcome ofan optimization problem. Moreover, if hi(p) > 0 is a scalar, then thefunction p·hi(p)gi(p) is also continuous and satisfies WARP and WalrasLaw.

Theorem 60 (Mantel). Let f : ∆o → RL be C2 and satisfy Walras’Law. Then, ∀ ε > 0, there exists an exchange economy (%i, ωi)Ii=1with each %i homothetic, monotonic and strictly convex such that itsaggregate excess demand function coincides with f on ∆ε.

Corollary 61. For any compact set K ⊆ ∆o, there exists an exchangeeconomy with I = L and each %i strictly monotonic, continuous, andstrictly convex such that K is the set of equilibrium prices of the econ-omy.

11.3. Brown and Matzkin: Testable Restrictions On Com-petitve Equilibrium. If we imagine that we can observe more thanjust prices, so that we can see how equilibrium varies with endowments(or, observe outcomes “on the equilibrium manifold”), then we canavoid the negative conclusion in the SMT theorem. These ideas aredue to Brown and Matzkin (1996).

Recall that the equilibrium manifold is defined by {(p, ~ω) : z(p; ~ω) =0} for an economy with exchange demand function z. In words, theequilibrium manifold for a fixes set of preferences (�i)Ii=1 is the set

10See Shafer and Sonnenschein (1982) or Chambers and Echenique (2016) for asketch of the main ideas behind the proof.


of price and endowment combinations for which markets clear in theresulting exchange economy.

Suppose we have the following two

lecture notes general equilibrium theory: ss205fede/lecture_notes/echenique_ge.pdf · the ideas and...

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