Lecture Notes General Equilibrium Theory: Ss205

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LECTURE NOTES GENERAL EQUILIBRIUM THEORY: SS205

FEDERICO ECHENIQUE CALTECH

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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 On Competitve 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 4 ECHENIQUE

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 Chris Shannon’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 some more advanced material. Usually in the last week of the quarter. I’ve never covered all the material here in a single quarter. The current write-up started from Alejandro Robinson’s notes from the class I taught in the Winter of 2015. I have added material, and rewritten the notes each time I have taught the class since. They are a work in progress, so please let me know of any problems you ﬁnd. GENERAL EQUILIBRIUM THEORY 5

1. Preliminary definitions

1.1. Binary relations. Let X be a set. A binary relation on X is a subset of X × X. If B is a binary relation on X, and x, y ∈ X we write x B y to denote that (x, y) ∈ B. A binary relation B is complete if x B y or y B x (or both) for any x, y ∈ X. It is transitive if for any x, 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. In economics we use the term preference relation (or rational preference relation) for weak orders. A preference relation is denoted by . Associated to any preference relation are two more binary relation. The ﬁrst captures indiﬀerence: x ∼ y is x y and y x. The second captures strict preference: x y if x y and it is not the case that x ∼ y.

1.2. Preferences in Euclidean space. A subset A ⊆ Rn is convex if 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 ≤ y and 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 the following deﬁnitions: is

• locally nonsatiated if for any x ∈ X and ε > 0 there is y ∈ X such that kx − yk < ε and y x. • weakly monotone if x ≤ y implies y x and x y implies y x • strongly monotone if x < y implies y x.

Deﬁne the following sets: Let U(x) = {y ∈ X : y x} be the upper contour set of at x and let L(x) = {y ∈ X : x y} be the lower contour 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)y0 x 6 ECHENIQUE

for any y, y0 ∈ U(x) and λ ∈ (0, 1), for any x ∈ X; • and continuous if U(x) and L(x) are closed sets (in X) for all x ∈ X.

The following theorem is due to Debreu. Theorem 1. If a preference relation is continuous on X ⊆ Rn then there 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 consumption space, and is a preference relation on X. As we shall see, we shall require a bit more information when we place the consumer in an economy. The set X represents all the possible consumption bundles that the consumer can choose. The preference is harder to interpret, but the standard view in economics is that is simply a description of the consumer’s choice behavior: speciﬁes what the consumer chooses from each pair of alternative bundles in X.

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

L The consumer chooses from a budget set. Let p ∈ R+ and W ≥ 0. The set L B(p, W ) = {y ∈ R+ : 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 demand correspondence. The theory of the consumer predicts that demand (choices made from budget sets) are optimal choices according to an underlying (rational) preference relation. GENERAL EQUILIBRIUM THEORY 7

The following results are simple observations that you should prove on your 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 a function 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 by φ : 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 is upper hemicontinuous (uhc) if it has closed graph and the image of compact sets are bounded. Note: this is a practical way of understanding uhc. It is how we use it in this class. But you can ﬁnd a general deﬁnition, for example, in Aliprantis 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 p and W . Moreover, the demand correspondence (p, W ) 7→ x∗(p, W ) is upper 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 hemicontinuous. Proposition 5. If is locally nonsatiated, continuous and strictly convex then x∗(p, W ) is a singleton for all p and W . Moreover the demand correspondence (p, W ) 7→ x∗(p, W ) is continuous as a function.

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

1You should be familiar with the supporting hyperplane theorem, but you will in any case prove it as part of your homework. 8 ECHENIQUE

U(x) x

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

A supporting hyperplane at U(x) gives us prices p and income W for which x is a demanded bundle. The reason is that any bundle that is strictly preferred to x must be in the interior of U(x), and therefore not aﬀordable to a consumer with such income, and facing such prices. Observe that we can use the monotonicity of preferences to ensure that the prices obtained are positive. The observation in Figure 1 can be called the “second welfare theorem for an individual consumer,” for reasons that will be clear in a few lectures.

3. Economies

3.1. Exchange economies. Our ﬁrst model of an economy is meant to capture the motivations behind pure economic exchange. I desire oranges but own only apples. You own oranges but no apples. By exchanging goods, we satisfy our needs for the fruits that we lack: a “coincidence of wants” (or “mutual needs”). Prices capture the rate of exchange between goods; how many oranges should I receive for each apple that I give you. The model primitive is a description of a collection of consumers. Each consumer is described by a preference relation and a consumption space, as in the theory of the consumer. We take consumption space L to be R+. An important feature of the theory is to account for who owns and sells the goods that the consumers buy. We also need to account for where consumers’ incomes come from. Each consumer will GENERAL EQUILIBRIUM THEORY 9 be endowed with non-negative quantities of each good. These are the goods that are sold in market, at the prevailing market prices, and consumers’ incomes come from selling the goods that they are endowed with at market prices. Thus, a consumer will be described (ﬁxing con- L sumption space to be R+, by a preference and an endowment vector L in R+. 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 )), I denoted by (i, ωi)i=1, in which each i is a preference relation over L L R+ and each ωi is a vector in R+. I Let E = (%i, ωi)i=1 be an exchange economy. • the aggregate endowment in E is I X ω¯ = ωi; i=1

• An allocation≤ in E is a collection of consumption bundles I L (xi)i=1, where xi ∈ R+, i = 1,...,I, such that I X xi ≤ ω¯ i=1 I PI • an allocation= in E is an allocation≤(xi)i=1 such that i=1 xi = ω¯.

I Deﬁnition 6 (Pareto Optimality). An allocation≤(xi)i=1 is Pareto I optimal in E if there is no allocation≤(yi)i=1 such that yi %i xi for every i = 1,...,I, and yh h xh for some h ∈ {1,...,I}.

Pareto optimality is a basic notion of economic efficiency. In a Pareto optimal allocation, one cannot make some agent better off without making some other agent worse off. Definition 7 (Walrasian Equilibrium). A Walrasian equilibrium I IL L in E is a pair (x, p) such that x = (xi)i=1 ∈ R+ , and p ∈ R+ (a price vector), s.t.:

0 (i) for every i = 1,...,I, xi ∈ B(p, p · ωi), and xi ∈ B(p, p · ωi) ⇒ 0 xi %i xi (all consumers optimize when choosing xi at prices p); PI PI (ii) i=1 xi = i=1 ωi (demand equals supply). 10 ECHENIQUE

B(p, p · ω2)

ω¯2 B(p, p · ω1) ω

ω¯1

Figure 2. The Edgeworth box

When there are two goods (L = 2), an exchange economy with two consumers can be conveniently represented using an Edgeworth box. Consider the following figure. The two agents’ preferences are represented by indifference curves in the figure. The allocation corresponding to the red dot is not Pareto optimal: any point in the orange set describes an allocation that is better for both consumers. Note that each point in the box describes an allocation=, The Pareto optimal allocations are those for which, if we fix an upper contour set for consumer 2 (given some given point, and his preference relation) and choose one of the best points on that set from the perspective of 1’s preferences. The resulting set of points, as we vary the upper contour set for consumer 2, is the set of all Pareto optimal allocations. 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 represented by a utility ui. Consider the maximization problem P (¯v):

max LI u1(x1) x∈R+ (PO-e) s.t. ui(xi) ≥ v¯i ∀i = 2, 3, ..., I, PI i=1 xi ≤ ω,¯ wherev ¯ ∈ RI is a vector of utility values.

Proposition 8. Let each i be continuous and strictly monotone. An allocation is Pareto optimal iﬀ there is v¯ ∈ RI for which x solves P (¯v). GENERAL EQUILIBRIUM THEORY 11

3.2. Economies with production. The second model allows for production. We introduce firms, and consider an economy in which firms are described by their production possibility sets. Consumers are described by their preferences and endowments as before, but now they also have shares in the profits of the firms. These shares are fixed. We include production by assuming that there are J firms in the economy, each described by a production possibility set Yj, j = 1,...,J. L For a production vector yj ∈ Yj, and price vector p ∈ R+, firms j’s profits are p · yj. Firms profits need to go to some agent in the economy: this is often called “closing” the model, which roughly means that one accounts for all the endogenous quantities in the model. Each consumer i will have a share θi,j ≥ 0 in the profits of firm j. We may PI haveθi,j = 0, but all profits go to some agent, so i=1 θi,j = 1 for all j. An private ownership economy is a tuple

J I E = (Yj)j=1, (i, ωi, θi)i=1 in which,

L • for each j = 1,...J, Yj ⊆ R is a production possibility set; L • for each i = 1,...,I, i is a preference relation over R+, and L ωi is a vector in R+; PI • for each j = 1,...J i=1 θi,j = 1, and for each i = 1,...,I θi,j ≥ 0.

J I Let E = ((Yj)j=1, (%i, ωi, θi)i=1) be a private ownership economy.

I IL • An allocation≤ in E is a pair (x, y), where x = (xi)i=1 ∈ R+ , J IL y = (yj)j=1 ∈ R , yj ∈ Yj ∀ j = 1, ..., J, and

I J X X xi ≤ ω¯ + yj. i=1 j=1

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

I J X X xi =ω ¯ + yj. i=1 j=1 12 ECHENIQUE

PJ Let Y = j=1 Yj be the aggregate production set of a private ownership economy.2 Consider the set: L Y + {ω¯} = {x ∈ R : ∃ y ∈ Y s.th. x = y +ω ¯}. L Deﬁne PPS = (Y + {w}) ∩ R+ as the production possibility set of the economy. Its boundary is referred to as the economy’s production possibility frontier.

Deﬁnition 9 (Pareto Optimality). An allocation≤(x, y) in E is Pareto 0 0 0 optimal if there is no allocation≤(x , y ) such that xi %i xi for every i = 1,...,I, and = xh h xh for some h ∈ {1,...,I}. Deﬁnition 10 (Walrasian Equilibrium). Let E be a private ownership IL JL economy. A Walrasian Equilibrium is a pair (x, y) ∈ R+ × R , L together with a price vector p ∈ R+ s.t.

0 (i) for every i = 1, ..., I, xi ∈ B(p, Mi), and xi ∈ B(p, Mi) ⇒ xi % 0 PJ xi, where Mi = p · ωi + j=1 θi,jp · yj (consumers optimize by choosing xi in their budget sets); 0 0 (ii) for every j = 1, ..., J, yj ∈ Yj, and p · yj ≥ p · yj ∀ yj ∈ Yj (ﬁrms optimize proﬁts by choosing yj in Yj); PI PI PJ (iii) i=1 xi = i=1 ωi + j=1 yj (demand equals supply).

There is a simple and useful characterization of Pareto optimal alloca- J I tions. Let ((Yj)j=1, (%i, ωi, θi)i=1) be a private ownership economy in which each preference relation i has a continuous and strictly mono- L tone utility representation ui : R+ → R.

max u1(x1)(PO) LI LJ (x,y)∈R+ ×R

subject to ui(xi) ≥ u¯i ∀i = 2, 3, ..., I, I J X X xi ≤ ω¯ + yj, i=1 j=1

yj ∈ Yj ∀j = 1, . . . , J.

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

2 Deﬁne 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}. GENERAL EQUILIBRIUM THEORY 13

As an illustration, let us analyze the above result for the case of exchange economies and diﬀerentiable utility functions. Let (%i, ωi) be an exchange economy. According to Proposition 11, an assignment IL x ∈ R+ is Pareto Optimal if and only if it solves the following problem:

max u1(x1)(PO’) LI x∈R+

subject to ui(xi) ≥ u¯i ∀i = 2, 3, ..., I, I X xi ≤ ω,¯ i=1 (1)

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

4. Welfare Theorems

4.1. First Welfare Theorem.

J I Theorem 12 (First Welfare Theorem). Let ((Yj)j=1, (%i, ωi, θi)i=1) be a private ownership economy in which the preference relation of every consumer is locally nonsatiated. If (x, y) and p constitute a Walrasian Equilibrium, then (x, y) is Pareto Optimal.

Proof of First Welfare Theorem. Let ((x, y), p) be a Walrasian equilib- 0 0 IL JL 0 0 0 rium. Let (x , y ) ∈ R+ × R with y = (yj) and yj ∈ Yj for all 0 j. Suppose that xi %i xi for every i = 1,...,I, and there exists 0 0 0 h ∈ {1,...,I} such that xh h xh. We shall prove that (x , y ) cannot be an allocation≤. First, let us prove that

J 0 0 X (2) xi %i xi =⇒ p · xi ≥ Wi = p · ωi + θijp · yj. j=1 0 0 Proceed by contradiction: Assume that xi %i xi, but p · xi < Wi. 0 Then, there exists ε > 0 such that ||z − xi|| < ε ⇒ p · z < Wi. By 0 0 local non-satiation, ∃ z such that z i xi and ||z − xi|| < ε. Therefore,

3This may be achieved by assuming that each individual has strictly convex utility 14 ECHENIQUE p · z < Wi and z i xi. This contradicts that consumer i is optimizing P when choosing xi in B(p, p · ωh + j θi,jp · yj). 0 0 Second, note that x h xh implies p·x > Wh, again because consumer h h P h is optimizing when choosing xh in B(p, p · ωh + j θh,jp · yj). Third, note that, since firms are maximizing profits in equilibrium and 0 0 yj ∈ Yj for every firm, then p · yj ≥ p · yj for every j = 1,...,J. Thus, if we put together these inequalities and sum over consumers we obtain that Therefore,

I I J X 0 X X p · xi > (p · ωi + θi,jp · yj) i=1 i=1 j=1 I J X X = p · ω¯ + θi,jp · yj i=1 j=1 J I X X = p · ω¯ + (p · yj) Iθi,j j=1 i=1 J X = p · ω¯ + (p · yj) j=1 J X 0 ≥ p · ω¯ + (p · yj) j=1 J ! X 0 = p · ω¯ + yj j=1

0 PJ The ﬁrst (strict) inequality follows from p·xi ≥ Wi = p·ωi+ j=1 θijp·yj 0 P and p · xh > Wh. Then we use that j θi,j = 1. The second (weak) 0 inequality follows from p · yj ≥ p · yj. Thus I J ! X 0 X 0 p · xi > p · ω¯ + yj , i=1 j=1 0 0 and therefore (x , y ) could not be an allocation≤.

J I 4.2. Second Welfare Theorem. Let E = ((Yj)j=1, (%i, ωi, θi)i=1) be a private ownership economy (POE). GENERAL EQUILIBRIUM THEORY 15

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

Deﬁnition 13. A Walrasian equilibrium with transfers is a tu- IL JL L ple (x, y, p, T ), where (x, y) ∈ R+ × R , p ∈ R+ (a price vector) I I and T = (Ti)i=1 ∈ R (a vector of net transfers), s.t.

(i) for every i = 1,...,I, xi ∈ B(p, Mi), and 0 0 xi ∈ B(p, Mi) ⇒ xi % xi, PJ where Mi = p · ωi + j=1 θi,jp · yj + Ti (consumers optimize by choosing xi in their budget sets); 0 0 (ii) for every j = 1,...,J, yj ∈ Yj, and p · yj ≥ p · yj ∀ yj ∈ Yj (ﬁrms optimize proﬁts by choosing yj in Yj); PI PI PJ (iii) i=1 xi = i=1 ωi + j=1 yj (demand equals supply). PI (iv) i=1 Ti = 0 (net transfers are “budget balanced”).

Theorem 14 (Second Welfare Theorem). Let I J E = (i, ωi, θi)i=1, (Yj)j=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 a PI ∗ Pareto optimal allocation≤in which i=1 xi 0, then there is a price ∗ L I ∗ ∗ ∗ vector p ∈ R+ and transfers T = (Ti)i=1 such that (x , y , p ,T ) is a Walrasian 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: n n H+(p, α) = {x ∈ R : p · x ≥ α} and H−(p, α) = {x ∈ R : p · x ≤ α}.

Let X,Y ⊆ Rn. We say that the hyperplane H(p, α) properly separates 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 disjoint and nonempty convex sets in Rn. Then there exists a hyperplane that properly separates X and Y .

∗ ∗ Proof of Second Welfare Theorem. Let (x , y ) be a Pareto optimal allocation≤in the 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 PI ∗ PJ ∗ 0 were to have i=1 xi < ω¯ + j=1 yj . Then we could modify x by giv- PJ ∗ PI ∗ ing some consumer the differenceω ¯ + j=1 yj − i=1 xi and (using the strict monotonicity of preferences), contradict the Pareto optimality of (x0, y0).

∗ L We need to prove that there exist a price vector p ∈ R+ and a transfer- ence schedule T ∈ RL such that conditions (i) to (iv) of the deﬁnition are satisﬁed. Let L ∗ Pi = {zi ∈ R+ : zi i xi } for i = 1,...,I.

Observe that Pi is a convex set as a direct consequence of the convexity 4 of i. PI Define P = i=1 Pi, and note that, being the sum of convex sets, P is also convex. The set P contains all the “aggregate bundles” that can be disaggregated into individual consumption bundles for which every individual is strictly better off than under the individual consumption ∗ bundle xi . PJ Let Y = j=1 Yj be the aggregate production set, and note that it is convex, as every firm’s production set is convex. Hence, L PPS = ({ω¯} + Y ) ∩ R+

4 0 0 Let z, z ∈ Pi. Say wlog that z i z . Then for any λ ∈ (0, 1) the convexity of 0 0 ∗ i means that λz + (1 − λ)z i z i xi . Hence, by transitivity of preferences, 0 λz + (1 − λ)z ∈ Pi. GENERAL EQUILIBRIUM THEORY 17 is also a convex set. Since (x∗, y∗) is Pareto Optimal, P ∩ PPS = ∅.

Otherwise we could produce an allocation≤in which all consumers are ∗ P better off than in x by finding zi ∈ Pi with i zi ∈ {ω¯} + Y . Now, by the Separating-hyperplane Theorem (Lemma 15), there exists p∗ ∈ RL such that (3) p∗ · z ≥ p∗ · q for all z ∈ P , and q ∈ ({ω¯} + Y ). We shall see that p∗ is going to be the price vector in our Walrasian equilibrium with transfers. Observe that the definition of proper sepa- ∗ ∗ ration implies that p 6= 0. We could, however, in principle have pl < 0. We’ll need to do some work to rule out negative prices. Firstly, we shall prove the following: 0 ∗ ∗ 0 ∗ ∗ (4) xi i xi =⇒ p · xi ≥ p · xi , for every i. 0 ∗ 0 Let xi i xi . Note that we must have xi > 0, as i is strictly monotone. 0 ∗ By continuity of i, ∃ δ ∈ (0, 1) such that (1 − δ)xi i xi . By strict 0 monotonicity and the fact that xi > 0, δ x∗ + x0 x∗ , for every h 6= i. h I − 1 i h h 0 ∗ δ 0 Therefore, (1−δ)xi ∈ Pi, and xh + I−1 xi ∈ Ph for every h 6= i. Then, X X δx0 x0 + x∗ = (1 − δ)x0 + x∗ + i ∈ P. i h i h I − 1 h6=i h6=i

∗ ∗ Since (x , y ) is an allocation=, I X ∗ xh ∈ ({ω¯} + Y ). h=1 By (3), then, we obtain that ! ! ∗ 0 X ∗ ∗ ∗ X ∗ p · xi + xh ≥ p · xi + xh . h6=i h6=i ∗ 0 ∗ ∗ Simplifying, p · xi ≥ p · xi . ∗ L Secondly, let us prove that p > 0. Fix l ∈ {1,...,L}. Let el ∈ R be 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 strict ∗ ∗ ∗ ∗ monotonicity, for any i ∈ 1,...,I, xi + el i xi . Then, p · (xi + el) ≥ 18 ECHENIQUE

∗ ∗ ∗ p · xi 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

0 ∗ ∗ 0 ∗ ∗ (5) xi i xi =⇒ p · xi > p · xi , for every i. P ∗ ∗ ∗ P ∗ Note that i xi 0 and p > 0 imply that p · i xi > 0, and ∗ ∗ so there exists at least one consumer i with p xi > 0. For such a consumer, we show that (5) must hold. To that end, suppose (towards 0 ∗ ∗ 0 ∗ ∗ a contradiction) that xi i xi but p · xi ≤ p · xi . Given that we have ∗ 0 ∗ ∗ established (4), it must be the case that p · xi = p · xi . Then, by 0 ∗ continuity of i, ∃ δ ∈ (0, 1) such that (1 − δ)xi i xi . Then ∗ 0 ∗ 0 ∗ 0 ∗ ∗ p · (1 − δ)xi = (1 − δ)p · xi < p · xi = p · xi ,

∗ 0 observe that the strict inequality is due to p · xi > 0. Note that the latter contradicts (4), yielding our desired result (5) for the specific ∗ ∗ consumer i that we know has p · xi > 0. Now, reasoning as in the proof that p > 0, we see that, in fact, p 0: ∗ ∗ Since (5) holds for consumer i, xi + el i xi implies that pl > 0. So p 0. Finally, to show that (5) holds for all consumers, note that p 0 ∗ ∗ implies that p · xi > 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 we 0 0 0 know that if xi i xi then xi > 0 so that p · xi > 0 = p · xi. At this point we turn to the definition of transfers. For any i, define the transfer to i by

J ∗ ∗ ∗ X ∗ ∗ (6) Ti = p · xi − p · ωi − θijp · yj . j=1

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

J X Mi = p · ωi + θijp · yj + Ti, j=1

∗ ∗ ∗ and note that for every consumer i: p · xi = Mi. Thus xi ∈ B(p, Mi). By (5), condition (i) is satisﬁed. GENERAL EQUILIBRIUM THEORY 19

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

I I J ! X X ∗ ∗ ∗ X ∗ ∗ Ti = p · xi − p · ωi − θijp · yj i=1 i=1 j=1 I I J ! ∗ X ∗ X X ∗ = p · xi − ω¯ − θijyj i=1 i=1 j=1 I J ! ∗ X ∗ X ∗ = p · xi − ω¯ − yj i=1 j=1 = 0.

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

I ∗ X ∗ ∗ (7) p · xi ≥ p · q ∀ q ∈ {ω¯} + Y . i=1 P ∗ Property (7) says that the aggregate bundle i xi “maximizes value” among all the elements of {ω¯} + Y . Let q ∈ {ω¯} + Y . For each n, let

n ∗ zi = xI + (1/n, . . . , 1/n). n n PI n By monotonicity of i, zi ∈ Pi for all i. So z = i=1 zi ∈ P . By (3), (8) p∗ · zn ≥ p∗ · q for every n ∈ N.

n P ∗ Note that z → i xi . Since (8) holds for each n, in the limit as ∗ P ∗ 5 n → ∞ we obtain that p · i xi ≥ q. Now, let us prove condition (ii) of the deﬁnition of WET. We must ∗ ∗ ∗ 0 0 show that for every ﬁrm j, p · yj ≥ p · yj ∀ yj ∈ Yj. Let yj ∈ Yj, and note that by (7):

I ∗ X ∗ ∗ ∗ 0 X ∗ ∗ p · xi ≥ p · ω¯ + p · yj + p · yk. i=1 k6=j

5 P ∗ The proof of statement 7 relies on showing that aggregate consumption i xi lies on the boundary of the set P . In fact, it lies at the intersection of the boundary L of P and the production possibility set of the economy, PPS = ({ω¯} + Y ) ∩ R+. Statement 7 says that aggregate consumption maximizes “value” in the PPS. 20 ECHENIQUE

PI ∗ PJ ∗ As condition (iii) states i=1 xi =ω ¯ + j=1 yj , we have: J ∗ X ∗ ∗ ∗ ∗ 0 X ∗ ∗ ∗ ∗ ∗ 0 p · ω¯ + p · yj ≥ p · ω¯ + p · yj + p · yk =⇒ p · yj ≥ p · yj. j=1 k6=j ∗ L As we have proved the existence of a price vector p ∈ R+ and a transfer schedule T that satisfy conditions (i) through (iv) of a Walrasian Equilibrium with transfers.

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

I Let (i)i=1 be a collection of preferences. The Scitovsky contour at I x = (xi)i=1 is I X S(x1, . . . , xI ) = { x˜i :x ˜i i xi∀i ∈ {1,...,I}} i=1

L Recall the deﬁnition of upper contour set: let Ui(xi) = {yi ∈ R+ : yi i xi}. Then I X S(x1, . . . , xI ) = Ui(xi). i=1

Consider a vector (x1, . . . , xI ). Think of an exchange economy with these I, agent i having preference i, and endowments being such that PI ω¯ = xi. Then we can consider the possibility that (x1, . . . , xI ) is i=1 P a Pareto optimal allocation=of the aggregate bundle i xi.

The set S(x1, . . . , xI ) is the set of aggregate bundles that can be disaggregated in a way that makes all consumers weakly better than they are at (x1, . . . , xI ). If (x1, . . . , xI ) is a Pareto optimal allocation=of the P aggregate bundle i xi, then the Scitovsky contour S(x1, . . . , xI ) must L P be disjoint from the set {z ∈ R+ : z < i xi}. Scitovsky contours can be used to evaluate policy decisions. The Kaldor 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 GENERAL EQUILIBRIUM THEORY 21 that the group of agents {1,...,I} are considering a collective change P L from the aggregate bundlex ¯ = i xi tox ˆ ∈ R+. Ifx ¯ is allocated as in (x1, . . . , xI ) andx ˆ ∈ S(x1, . . . , xI ) then the change is desirable because the resulting aggregate bundle can be disaggregated in a way that makes no consumer worse off than inx ¯. Moreover, ifx ˆ is in the interior of the contour, the consumers can be made strictly better off.

When each preference i is convex, continuous and strictly monotonic, we can essentially use the the second welfare theorem to de- scribe Scitovsky contours and to operationalize the Kaldor criterion via cost-benefit analysis. The idea is that if (xi) is a Pareto optimal P ∗ ∗ ∗ P allocation=of i xi, then we obtain prices p such that p ·z ≥ p · i xi 6 for all z ∈ S(x1, . . . , xI ). See Figure 4. ∗ P P This means that prices p support S(x1, . . . , xI ) at i xi: p·z ≥ p· i xi ∗ for all z ∈ S(x1, . . . , xI ). By observing market prices p we obtain information about the shape of S(x1, . . . , xI ). In other words, market prices convey enough information about agents’ preferences to be useful policy tools. When the group of agents {1,...,I} are considering a collective change P L from the aggregate bundlex ¯ = i xi tox ˆ ∈ R+. If p · xˆ < p · x¯ then we know thatx ˆ ∈ / S(x1, . . . , xI ), and there is therefore no way to distributex ˆ in a way that will make all the agents better off than they are currently in (x1, . . . , xI ). Put differently, even without knowing agents’ preferences, if we “price” the differencex ˆ − x¯ using prevailing market prices p, and the value is negative, we know that the change is fromx ¯ tox ˆ is undesirable. On the other hand if the change is positive then it has some chance of belonging to the Scitovsky contour atx ¯. Moreover, if preferences are smooth then the supporting hyperplane defined by p will be a good local description of S(x1, . . . , xI ) aroundx ¯. So when the changex ˆ − x¯ is relatively small, we can be relatively confident thatx ˆ ∈ S(x1, . . . , xI ) when p · (ˆx − x¯) > 0. This reasoning is known as cost-benefit analysis. We use prevailing prices to evaluate the change fromx ¯ tox ˆ, and decide based on the value of the two aggregate bundles at the prevailing prices. One problem with the Kaldor criterion is that it is possible thatx ˆ ∈ 0 IL P 0 S(x1, . . . , xI ) and that there is (xi) ∈ R+ withx ˆ = i xi, andx ¯ ∈

6 0 ∗ 0 ∗ The reason is that xi ∈ Ui(xi) implies that p ·xi ≥ p ·xi. So the prices support each of the individual upper contour sets at the consumption bundle xi. 22 ECHENIQUE

p∗

S(x1, x2) x

p∗ x1,2 x2,2

x2,1

x1,1

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

0 0 S(x1, . . . , xI ). Draw such a case for yourself. Think about what it means.

6. Excess demand functions

L PL 6.1. Notation. Let ∆ = {p ∈ R+ : 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.

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

6.2. Aggregate excess demand in an exchange economy. Sup- I pose that (%i, ωi)i=1 is an exchange economy in which each %i is continuous, strictly convex, and strictly monotonic. Then we can deﬁne ∗ a demand function p 7→ xi (p, p · ωi), and an excess demand function ∗ p 7→ zi(p) = xi (p, p · ωi) − ωi for all consumer i. The aggregate excess demand function is then I I X X ∗ z(p) = zi(p) = xi (p, p · ωi) − ω¯ i=1 i=1 GENERAL EQUILIBRIUM THEORY 23

L L This function z : R++ → R satisﬁes a number of basic properties that will be essential in our study of equilibrium.

I Proposition 16. If (%i, ωi)i=1 is an exchange economy in which ω¯ = PI i=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; L (iii) (Walras Law) ∀ p ∈ R++, p · z(p) = 0; L (iv) (bounded below) ∃ M > 0 s.th. ∀ l, p ∈ R++, zl(p) > −M; n L (v) (boundary condition) If {p } is a sequence in R++, and p¯ = n L L limn→∞ p , where p¯ ∈ R0 \ R++ and p¯ 6= 0, then there is l ∈ n {1,...,L} such that {zl(p )} is unbounded.

Proof. Let us proceed in order:

(i) If %i is strictly monotonic, then it is locally non-satitated. By ∗ the maximum theorem, every demand function xi (p, p · ωi) is continuous. As z(p) is a linear combination of continuous functions, it is continuous. (ii) The budget set of every consumer i is unchanged if prices are multiplied by a constant α > 0, i.e, B(p, p · ωi) = B(αp, αp · ωi) L for every α ∈ R++ with p ∈ R+. Then, the maximization problem of consumer i is unchanged if the prices are multiplied by α > 0. This shows that the demand function of every consumer ∗ ∗ is homogeneous of degree zero: xi (p, p · ωi) = xi (αp, αp · ωi) for every α > 0. Then, by deﬁnition, z(αp) = z(p) for every α > 0. ∗ (iii) %i strictly monotonic implies p · xi (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 increase her utility by consuming more from any good). Summing over 24 ECHENIQUE

all consumers, we obtain: I I X ∗ X p · xi (p, p · ωi) = p · ωi i=1 i=1 I I X ∗ X =⇒ p · xi (p, p · ωi) = p · ωi i=1 i=1 I ! X ∗ =⇒ p · xi (p, p · ωi) − ω¯ = 0 i=1 =⇒ p · z(p) = 0

L ∗ (iv) Note that X = R+ implies xli(p, p · ωi) ≥ 0 for every consumer i and good l. Hence, zl(p) ≥ − ω¯l for every good l. Let M ∈ R be such that M > maxl∈{1,...,L}{ω¯l}, and note that zl(p) > −M L ∀ l, p ∈ R++. n (v) Towards contradiction, assume zl(p ) is bounded above for all L L l ∈ {1,...,L}. Letp ¯ ∈ R+ \ R++ be s.th.p ¯ 6= 0. Then, there are m, k ∈ {1,...,L} s.th.p ¯k = 0 andp ¯m > 0. As the aggregate endowment is strictly positive,ω ¯ >> 0, then there exists a consumer j such that ωmj > 0 and thusp ¯·ωj > 0. Furthermore, as %j is strictly monotonic, j strictly prefers bundles that have more of good k, all else equal. Note that this implies that the ∗ demand function xj (p, p · ωj) is not well deﬁned at p =p ¯ since j has positive income and good k, which she likes, is free. To ∗ L see this formally, assume by contradiction ∃ xj ∈ R+ such that ∗ ∗ xj ∈ B(¯p, p¯ · ωj) and xj %j xj ∀ xj ∈ B(¯p, p¯ · ωj). ∗ L Letx ˆj = xi + εek with ε > 0, where ek ∈ R is the vector of zeroes, except for its k-th entry which equals one. Asp ¯k = 0, ∗ thenp ¯·xj =p ¯·xˆj ⇒ xˆj ∈ B(¯p, p¯·ωj). However, by strict mono- ∗ tonicity,x ˆj j xj , and we obtain the desired contradiction. n L n n Let {p } be a sequence in R++ s.th. p → p¯. Let zli(p ) be the sequence of the excess demand of consumer i for good l. We can write the aggregate excess demand for any good l as:

n n X n zl(p ) = zlj(p ) + zli(p ). i6=j Note that the excess demand of every consumer for good l is n bounded below by Mli > ωli. As zl(p ) is bounded above by n hypothesis for every good l, then it must be that zli(p ) is also bounded above for every consumer i, j included. Hence, as GENERAL EQUILIBRIUM THEORY 25

n zlj(p ) is bounded above and below, it has a convergent subse- nk ∗ 7 quence, say zlj(p ). Let zlj be the limit of this subsequence. ∗ ∗ L ∗ ∗ n Let zj = (zlj)l=1 and xj = zj + ωj. As p >> 0 for every n, n ∗ n n then by strict monotonicity, for every n, p · xj (p , p · ωj) = n p · ωj , i.e., consumer j spends all her income. Therefore, n n p · zj(p ) = 0 for every n. Then,

nk nk ∗ ∗ ∗ lim p ·zlj(p ) = 0 =⇒ p¯·zlj = 0 =⇒ p¯·xj =p ¯·ωj =⇒ xj ∈ B(¯p, p¯·ωj). n→∞ ∗ To conclude the prove, we will show xj %j xj for every xj ∈ B(¯p, p¯·ωj) to obtain the desired contradiction. Let xj ∈ B(¯p, p¯· n n p ·ωj n n ωj). Let λ = for every n, and note λ → 1 and λ ≥ 0 p¯·ωj n for every n. Sincep ¯ · xj ≤ p¯ · ωj, multiplying both sides by λ n n n n n yieldsp ¯·λ xi ≤ p ·ωj for every n, i.e., λ xj ∈ B(p , p ·ωj). By n ∗ n n n deﬁnition, zj(p ) = xj (p , p · ωj) − ωj . Hence, zj(p ) + ωj %j n λ xj. By continuity of %j, we obtain: n ∗ ∗ n lim zj(p ) + ωj = zj + ωj = xj j xj = lim λ xj. n→∞ % n→∞

P 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 excess demand from an economy, and showed that it satisfies the five basic properties. Now we shall analyze excess demand function as a primitive in 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 excess demand. For example we can incorporate production, and show that a private ownership economy, under some assumptions, has an excess demand function that satisfies the five properties. And there are other models that can also be captured by the demand function as a reduced form. We use property (ii) to restrict the domain in z in various ways. Some times we “normalize” prices, and express all prices in terms of one good. For example in terms of good one: pl/p1, l = 1,...,L. Another PL o normalization is pl/ k=1 pk, so prices are in ∆ :

7 ∗ It is tempting to claim zlj = zlj(¯p), but this may not be true since zlj(p) may no be continuous at p =p ¯. 26 ECHENIQUE

z1(p1, 1) −M

Figure 5. Aggregate excess demand for good 1 with L = 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- O n quence in ∆ andp ¯ = limn→∞ p , with p ∈ ∂∆, then there is l n such that the sequence {zl(p )} is unbounded.

We refer to aggregate excess demand functions satisfying the 5 properties with the understanding that we use homogeneity to go back and L forth between the two domains. Either the domain is R++ and we impose all 5 properties, or the domain is ∆o, and we can obtain a function L on R++ by imposing homogeneity. Speciﬁcally, if we deﬁne an excess demand function z : ∆o → R, then the domain can be extended to all L 1 of R++ by z(p) = z( P p). l pl

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. This allows 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 market for good 1. Whenever z1(p1, 1) = 0 we know that z2(p1, 1) = 0. We present an existence result for excess demand functions satisfying the ﬁve properties. The proof is taken from Geanakoplos (2003). GENERAL EQUILIBRIUM THEORY 27

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

We use the following result, which subsumes the role of the boundary condition.

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

p ∈ ∂∆ε ⇒ p¯ · z(p) > 0.

The proof of Lemma 19 is part of your homework. Now we turn to the proof 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. If f : X → X is continuous, then there is x∗ ∈ X with x∗ = f(x∗).

Proof. Let ε andp ¯ be as in the statement of the lemma. Deﬁne the functions m and φ as follows:

m(˜p, p) =p ˜ · z(p) − kp˜ − pk2

φ(p) = argmaxp˜∈∆ε m(˜p, p), forp, ˜ p ∈ ∆ε. You should prove that φ is a function (it takes singleton values), and that is satisfies the hypotheses of Brower’s fixed point theorem (use the Maximum Theorem). So φ : ∆ε → ∆ε is continuous. Brower’s fixed point theorem implies that there is p∗ ∈ ∆ε with p∗ = φ(p∗). We shall prove that 0 = z(p∗). Notice first that if p∗ is in the interior of ∆ε then the first order condition for the problem in the definition of φ means that

∗ ∂m(˜p, p ) ∗ ∗ ∗ 0 = | ∗ = (z(p ) − 2(˜p − p )) | ∗ = z(p ) ∂p˜ p˜=p p˜=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. Then p¯ · z(p∗) > 0 by the lemma. Let p(λ) = λp¯ + (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∗) − kp(λ) − p∗k2 = λp¯ · z(p∗) + (1 − λ)p∗ · z(p∗) − λ2kp¯ − p∗k2 = λ p¯ · z(p∗) − λkp¯ − p∗k2 , as Walras’ Law implies (1 − λ)p∗ · z(p∗) = 0, and kp(λ) − p∗k2 = λ2kp¯ − p∗k2. Sincep ¯ · z(p∗) > 0 there is λ > 0 small enough that p¯ · z(p∗) − λkp¯ − p∗k2 > 0. ∗ ∗ ∗ Then m(p(λ), p ) > m(p , p ), which is absurd.

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

7.1. The Negishi approach. We give a proof of existence that is based on adjusting welfare weights instead of adjusting prices: this is called the Negishi approach (after Negishi (1960)); you can think of it as a way of using the second welfare theorem to prove existence of Walrasian equilibria. After all, it is easy to see that there must exist Pareto optimal allocations, and the second welfare theorem says that they can be obtained as Walrasian equilibria with transfers. The challenge is to show that they can be obtained as Walrasian equilibria without transfers, meaning transfers that equal zero.

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

Theorem 21. If i is continuous, strictly monotonic and convex, for PI all i = 1,...,I, and i=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 → 2X is a correspondence such that, for all x ∈ X, Π(x) is a nonempty and convex 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∗ ∈ X with x∗ ∈ Π(x∗).8

8The notion of upper hemicontinuous correspondence is more general, but here it reduces to the correspondence having a closed graph. GENERAL EQUILIBRIUM THEORY 29

LI L I A triple (x, p, T ), where x ∈ R+ , p ∈ R+ and T ∈ R is a Walrasian equilibrium with transfers (WET) if

0 0 (1) p · xi = p · ωi + Ti and x xi implies that p · x > p · ωi + Ti, P i i (2) x is an allocation=( xi =ω ¯, or supply = demand), and P i (3) i Ti = 0.

For the rest of this proof, we refer to allocation=as allocations. We shall also need the following version of the second welfare theorem, stated here without proof. Lemma 23 (Second welfare theorem). If x is a Pareto optimal allocation, 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 then x = 0. Let I U = {v ∈ R : 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 x0 be Pareto optimal allocations with 0 vi = ui(xi) = ui(xi)i = 1,...,I . If (x, p, T ) is a Walrasian equilibrium with transfers, then so is (x0, p, T ).

Proof. It is obvious that (x0, p, T ) satisﬁes (2) and (3) in the deﬁnition of Walrasian equilibrium with transfers.

0 0 To prove that (x , p, T ) satisfies (1), note first that if zi i xi then 0 zi i xi as ui(xi) = ui(xi). Therefore, p · zi > p · ωi + Ti. 0 To finish the proof, we need to show that p · xi = p · ωi + Ti. But 0 0 0 xi xi (as ui(xi) = ui(xi)) implies that p · xi ≥ p · ωi + Ti because 0 p · xi < p · ωi + Ti would imply (by strict monotonicity) the existence 0 of a zi i xi with p · zi < p · ωi + Ti. Now, p · xi ≥ p · ωi + Ti for all P 0 P 0 I = 1,...,I, i xi =ω ¯, and i Ti = 0 implies that p · xi = p · ωi + Ti for all i. Lemma 25. There exists M > 0 such that if (x, p, T ) is a Walrasian equilibrium 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 > 0 such 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 exists a 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 has a closed graph.

Proof. First we show that Π(v) is a convex set, for v ∈ U. So let T,T 0 ∈ 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, p0,T 0) are both WET. Let µ ∈ (0, 1),p ˆ = µp + (1 − µ)p0 and Tˆ = µT + (1 − µ)T 0. We shall prove that (x, p,ˆ Tˆ) is a WET. Properties (2) and (3) of WET are immediately satisfies, as (x, p, T ) and (x, p0,T 0) are both WET. To 0 0 0 prove Properties (1) note that p · xi = p · ωi + Ti and p · xi = p · ωi + Ti ˆ implyp ˆ·xi =p ˆ·ωi +Ti. Moreover, zi xi implies that p·zi > p·ωi +Ti, 0 0 0 ˆ and p · zi > p · ωi + Ti . Thusp ˆ· zi > pˆ· ωi + Ti. 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 convergent 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 with vn = 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 that n n n n p · xi = p · ωi + Ti implies that p · xi = p · ωi + Ti. Next, suppose that zi xi. Then the continuity of i implies that for n large enough n n n n zi xi . Then for n large enough, p · zi > p · ωi + Ti . Thus p · zi ≥ p · ωi + Ti. Before we continue, we show that p 0. We know that x is an P P allocation, so i xi =ω ¯, andω ¯ 0. So, p ∈ ∆ means that p · i xi > GENERAL EQUILIBRIUM THEORY 31

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

p · (δzi) = δp · xi < p · xi = p · ωi + Ti, a contradiction of the property we have already established that zi i xi ⇒ p · zi ≥ p · ωi + Ti. For any l, then xi + el i xi (by strict monotonicity) 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 would imply that p·zi > 0 = p·xi as zi > 0 when zi i 0 and p 0. If xi 6= 0 then p · xi > 0, again because p 0. So suppose that p · xi > 0. Then we can just repeat the argument we made for the special consumer above, for which we knew that p · xi > 0. This implies that (1) holds for all consumers.

I PI I Let ∆I = {λ ∈ R+ : i=1 λi = 1} be the simplex in R . Now observe that for each v ∈ U there is a unique value of α > 0 such that αv ∈ ∆I . I In words, the ray deﬁned by v ≥ 0 in R intersects ∆I once and only once. The function that maps v ∈ U into ∆I is one-to-one. It is also possible 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 projection of ∆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 and a closed graph.

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

Deﬁne

η(T ) = argmin{λ · T : λ ∈ ∆I } for T ∈ [−M,M]. The correspondence T 7→ η(T ) has convex values and 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 and has a closed graph. Kakutani’s ﬁxed point theorem implies that there ∗ ∗ ∗ ∗ ∗ ∗ exists (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 that T ∗ = 0, and thereby prove that (x∗, p∗) is a Walrasian equilibrium.

∗ Suppose then, towards a contradiction, that Th > 0 for some i. Then ∗ P ∗ there is j with Tj < 0 as i Ti = 0. But then the deﬁnition of η, and ∗ ∗ λ ∈ η(T ) implies that λh = 0. By deﬁnition of h, then, vh = 0. This means that uh(xh) = 0 which is only possible if xh = 0 (as ui(0) = 0 and 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). o L Let z : ∆ : R+ → R+ be an excess demand function. We say that z satisﬁes 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 economy. Let x∗(p, p · ω) be the demand function of this consumer. Then p∗ ∈ ∆o is an equilibrium price iff x∗(p∗, p∗ · ω) = ω (supply equals demand when there is a single consumer). Let p 6= p∗, and suppose that x∗(p, p · ω) 6= x∗(p∗, p∗ · ω). By WARP, it must 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 price vector. The reason is that the bundle consisting of the endowment is affordable 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 the Strong Weak Axiom, then there is a unique competitive equilibrium.

Proof. By the equilibrium existence theorem, a competitive equilibrium must exist. To see why it must be unique, suppose that there are two diﬀerent competitive equiilbriums, and use the Strong Weak Axiom to reach a contradiction. GENERAL EQUILIBRIUM THEORY 33

We motivated the Strong Weak Axiom in a one consumer economy using the Weak Axiom of Revealed Preference. With more than one consumer, we would need that aggregate demand satisfies WARP. When there are multiple heterogeneous consumers, however, it is very hard to guarantee that an aggregate demand will satisfy WARP. Our next condition may seem to have a lot of economic content, but it turns out to be stronger than SWA. L L Definition 31 (Gross Substitutes). Let z : R++ → R be an excess demand function. We say that z satisfies the gross substitutes prop- 0 L 0 0 erty if, ∀ p, p ∈ R++, pk < pk for some k, and pl = pl ∀ l 6= k imply 0 zl(p) < zl(p ) ∀ l 6= k.

Like the SWA, the meaning of GS is simple in the case of a single consumer. Consider an economy with one consumer and two goods: coffee and tea. The two goods are substitutes, meaning that if the price of coffee increases, then the demand for tea must increase. Since there is a single consumer, aggregate demand will have the same property. Now, if there are many heterogeneous consumers, then the price increase will have consequences the incomes of the consumers who own coffe, or for whose who own shares in the firms that produce coffee. It is 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 and many goods, the GS property is not very plausible. Proposition 32. If z satisfies the gross substitutes property, then it satisfies 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 property is a suﬃcient condition for uniqueness of competitive equilibrium. We show this fact directly in the next result: L L Theorem 33. If the excess demand function z : R++ → R satisﬁes conditions (i) to (v) and the gross substitutes property, then there is a unique competitive equilibrium in ∆o.

Proof. By Theorem18, properties (i) -(v) from Proposition 16 assure the existence of p∗ ∈ ∆o such that z(p∗) = 0. Let p ∈ ∆o be such that p 6= p∗. Choose λ > 0 such that λp ≥ p∗ and ∃ h for which ∗ ∗ λph = ph. (This is achieved by letting h = argmaxl{pl /pl} and setting ∗ ∗ ∗ λ = ph/ph > 0.) Therefore, λp ≥ p implies that for some k, λpk > pk, + ∗ i.e., the set L = {k ∈ L : λpk > pk}= 6 ∅. Increase the price of each 34 ECHENIQUE good in L+ one by one to get from price vector p∗ to λp. By applying gross substitutes, it must be that the excess demand function of good + ∗ h∈ / L increases with each price increase, so that zh(λp) > zh(p ). As ∗ zh(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- L libria in R++. That is why Theorem 33 qualiﬁes the statment to talk about uniqueness in ∆o.

9. Representative Consumer

I L Consider a collection of preferences (%i)i=1 over R+ such that each i ∗ gives rise to a continuous demand function xi . We want to know when there is a preference relation , giving rise to a demand function x∗ with R PI ∗ P the property that: x (p, p · ω¯) = i=1 xi (p, p · ωi), whereω ¯ = i ωi. Let IL ~ω = (ω1, ω2, . . . , ωI ) ∈ R+ . Note that ~ω represents the distribution of the aggregate endowment in the economy. Under this notation,ω ¯ = ι·~ω, where ι is a vector of ones. We write the aggregate demand function in terms of the distribution of income as: I X ∗ z(p, ~ω) = xi (p, p · ωi) − ι · ~ω. i=1 Deﬁnition 34 (Representative Consumer). A collection of preferences I (%i)i=1 admits a representative consumer if there is a preference R L R relation % on R+ and an associated continuous demand function x : L L R+ × R+ → R+ such that I R X ∗ IL L x (p, p · ω¯) = xi (p, p · ωi) ∀ ~ω ∈ R+ , p ∈ R+. i=1 If a collection of preferences admits a representative consumer, note that its associated demand function satisﬁes R R IL z (p, ω¯) = x (p, p · ω¯) − ω¯ = z(p, ~ω) ∀ ~ω ∈ R+ . In this case, we say that z admits a representative consumer. I Observation 35. Let (%i, ωi)i=1 be an exchange economy. If the collec- I tion of preferences (%i)i=1 admits a representative consumer, then z(p, ~ω0) = z(p, ~ω) ∀, ~ω, ~ω0 s.th.ω ¯ = ι · ~ω = ι · ~ω0. GENERAL EQUILIBRIUM THEORY 35

The observation suggests that a representative consumer will only exist under very stringent conditions. We shall see that homothetic preferences is central to the question of when an economy accepts a representative consumer. Recall that a consumer has a homothetic preference relation if x %i y implies L αx %i αy for all x, y ∈ R+ and α ≥ 0. Theorem 36 (Antonelli’s Theorem). The aggregate excess demand function z admits a representative consumer if and only if there is L a homothetic preference relation % on R+ such that each individual ∗ demand xi 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. If you consider demand when income becomes very small, close to 0, then the income expansion paths have to pass through zero. This means that income expansion curves must behave like they do for homothetic preferences.

9.0.1. Digression: The Cauchy equation. Consider the following equation f(x1 + x2) = f(x1) + f(x2) for all x1, x2 ∈ R. This is an equation in which the unknown is the (real) function f: a so-called functional equation. This particular equation is called Cauchy’s equation. We want to know if Cauchy’s equation has a solution, and what the solution is. It obviously has solutions, for example f(x) = x. The following 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 ∈ Z being 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.

L Proof. (⇐) Let % be a preference on R+ such that it generates every demand function xi and it is homothetic. Let xR be the demand generated by this preference relation. Note that xR(p, m) = mxR(p, 1) for all m as xR is linear homogeneous in income. Then,

I I X ∗ X R xi (p, p · ωi) = x (p, p · ωi) i=1 i=1 I R X = x (p, 1) p · ωi i=1 = xR(p, 1)(p · ω¯) = xR(p, p · ω¯)

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

R z(p, ~ωi) = x (p, p · ω) − ω X ∗ ∗ = xj (p, p · 0) + xi (p, p · ω) − ω j6=i ∗ = xi (p, p · ω) − ω R ∗ =⇒ x (p, p · ω) = xi (p, p · ω) ∀ i ∈ I.

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

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,

R ∗ ∗ X ∗ x (p, m) = x1(p, m1) + x2(p, m2) + xj (p, p · 0) j≥3 R R = x (p, m1) + x (p, m2). GENERAL EQUILIBRIUM THEORY 37

In particular, for a positive integer n, xR(p, nm) = xR(p, (n − 1)m) + R R R k x (p, m). This means that x (p, nm) = nx (p, m). Let q = l ∈ Q+, with k, l ∈ N. Then m xR(p, qm) = kxR p, l lm =⇒ lxR(p, qm) = kxR p, l k =⇒ xR(p, qm) = xR (p, m) l =⇒ 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 to the Cauchy Equation. Let f : R → R be continuous and satisfy that 2 f(x1 + x2) = f(x1) + f(x2) for any (x1, x2) ∈ R . Then ∀ x ∈ R, f(x) = cx for some constant c. In our case, let f(m) = xR(p, m), for a ﬁxed p let c = xR(p, 1). Therefore, xR(p, m) = mxR(p, 1).

I 9.1. Samuelsonian Aggregation. Let (%)i=1 be a collection of pref- L erences where each %i is represented by a utility function ui : R+ → R. Let W : RI → R be a strictly monotone increasing function. We refer to W as a social welfare function. Consider the following problem:

max W (u1(x1), ..., uI (xI ))(P1) IL (x1,...,xI )∈R+ I X subject to p · xi ≤ m. i=1

The problem above implies choosing an aggregate consumption bundle PI IL z = i=1 xi ∈ R for the economy. The objective is to maximize the social welfare in the economy, as given by W . The constraint reﬂects an aggregate budget constraint. 38 ECHENIQUE

Consider the alternative problem: (P2) ( I ) X max max W (u1(x1), . . . , uI (xI )) s.t. xi = z z∈ IL (x ,...,x )∈ IL R+ 1 I R+ i=1 subject to p · z ≤ m.

Therefore, deﬁne the value function of (P1) as

I IL X U(z) = sup{W (u1(x1), . . . , uI (xI )) : (x1, . . . , xI ) ∈ R+ and xi = z}, i=1 and rewrite (P2):

(P2’) max U(z) subject to p · z ≤ m. IL z∈R+

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

∗ ∗ max W (u1(x (p, m1)), ..., uI (x (p, mI )))(P4) I 1 I {mi}i=1 I X subject to mi ≤ m. i=1

Interpret these problems. Problem 1 maximizes social welfare function by choosing an individual bundle for each consumer subject to the aggregate budget constraint. This induces an optimal aggregate bundle z ∈ RL. The second problem is a maximization in two steps. First, given an arbitrary aggregate bundle of consumption z ∈ RL, the social planner maximizes the social welfare function, i.e., decides on the optimal allocation of the aggregate bundle among the consumers. Then, given the income restriction of the economy, the social planner decides the optimal aggregate consumption bundle. Problem P2’ makes explicit that we can write the second problem as that of a representative consumer whose utility function is the value function resulting from the ﬁrst of the two nested problems above. Therefore, the utility function of our representative consumer represents the optimal division of an aggregate consumption bundle among the consumers. Another way of addressing this problem is in terms of income. GENERAL EQUILIBRIUM THEORY 39

I 9.2. Eisenberg’s Theorem. Let (%i)i=1 be a collection of preferences L where each %i is represented by ui : R+ → R. Identify an economy IL IL with a vector of endowments ~ω ∈ R+ (so that for each ~ω ∈ R+ , I (%i, ωi)i=1. An economy has a ﬁxed structure of endowments if I I P there is α = (αi)i=1 ∈ R+ with i αi = 1 and ωi = αiω¯. Theorem 39 (Eisenberg’s Theorem). Consider an economy with a ﬁxed structure of endowments, given by α. Let each %i be represented by a continuous and homogenous degree one utility fuction uii. Then the aggregate demand of the economy is generated by a representative L consumer, whose utility function U : R+ → R is given by:

( I I ) Y αi X (9) U(x) = max ui(xi) s.t. x = xi . (x ,...,x )∈ IL 1 I R+ i=1 i=1

10. Determinacy

I L Consider a collection of preferences (i)i=1 over R+. For each given vector of endowments:

IL ~ω = (ω1, . . . , ωI ) ∈ R+ ,

I we have an exchange economy (i, ωi)i=1. So for ﬁxed preferences I (i)i=1 we can identify a set of possible (exchange) economies with a set of vectors of initial endowments. Let E be an open subset of RIL. Assume that all vectors of endowments P + ~ω ∈ E satisfy i ωi 0, and the consumers’ preferences i are strictly convex, strictly monotone, and continuous. Then each i gives rise to ∗ a demand function xi and aggregate excess demand is

I I ∗ X ∗ X z (p; ~ω) = xi (p, p · ωi) − ωi. i=1 i=1 The function p 7→ z∗(p; ~ω) satisfies properties (1)-(5). Note that we now make 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 to have an excess demand function such as the one depicted below. 40 ECHENIQUE

10.1. Digression: Implicit Function Theorem. REMIND OF THE INVERSE 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 q a parameter of the model. Suppose that the solution to our model is characterized implicitly by f(v, q) = 0.

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

∂f ∂f ∂h ∂f · + = 0 =⇒ h0(q) = − ∂q . ∂v ∂q ∂q ∂f ∂v Note that in order to perform the previous excercise we need to assume ∂f ∂v 6= 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 does not tell us anything the behavior of v. A function f : A ⊆ Rn → R, for which A is open, is said to be of class r if it has 0 ≤ r ≤ ∞ continuous derivatives. We write Cr to denote the property of being of class r. Let g : Rn × Rk → Rm be a C1 function. The Jacobian matrix of g is h dg dg dg dg i Dg(x, y) = Dxg(x, y) Dyg(x, y) = ··· ··· ∂x1 ∂xn ∂y1 ∂yk  ∂g1 ··· ∂g1 ∂g1 ··· ∂g1  ∂x1 ∂xn ∂y1 ∂yk  ......  =  ......  ∂gm ··· ∂gm ∂gm ··· ∂gm ∂x1 ∂xn ∂y1 ∂yk m×(n+k) Theorem 40 (Implicit Function Theorem). Let A ⊆ Rn, and B ⊆ Rm be open sets. Let f : A × B → Rn be Cr with 1 ≤ r ≤ ∞. Let (¯v, q¯) ∈ A × B be such that f(¯v, q¯) = 0. Then, if

Dvf(v, q) (v,q)=(¯v,q¯) is a non-singular matrix, there are open sets A0 ⊆ A, B0 ⊆ B, and a Cr function h : B0 → A0 such that h(¯q) =v ¯, and f(v, q) = 0 for (v, q) ∈ A0 × B0 if and only if v = h(q). Moreover, −1 Dqh(¯q) = − Dvf(v, q) · Dqf(¯v, q¯). (v,q)=(¯v,q¯) GENERAL EQUILIBRIUM THEORY 41

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

10.2. Regular and Critical Economies. We develop methods for analyzing parametrized models. To this end, we focus on exchange economies and interpret endowments as parameters. It is, however, possible to use the same ideas for diﬀerent parametrizations. For example, we could use preferences as the parameters of the model, or ﬁrms’ technologies in a POE.

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

∗ Given an economy ~ω ∈ E, the resulting demand for agent i is xi (p, p · ωi). We assume that demand functions, as functions of prices and in- ∗ L+1 1 come (p, m) 7→ xi (p, m), and with domain R++ are C . This assumption relies on a notion of smooth preferences that we are not going to get into. To make the dependence of the model on its parameters (endowments) explicit, we write the aggregate excess demand as

I I X ∗ X z(p, ~ω) = xi (p, p · ωi) − ωi. i=1 i=1

Before we used homogeneity to normalize prices so that they are in ∆o, but now we will use a diﬀerent normalization. In particular, homogeneity of degree zero implies that p1 pL−1 pL z(p1, . . . , pL) = z ,..., , , pL pL pL

L−1 which means restricting to prices in R++ with the price of the Lth good being ﬁxed at 1. Moreover, by Walras’ law we can restrict attention L−1 L−1 to all market except for one. To this end letz ˆ : R++ × E → R be deﬁned by:

(10)z ˆl(p1, p2, . . . , pL−1; ~ω) = zl(p1, p2, . . . , pL−1, 1; ~ω) for l = 1,...,L − 1. In particular,

zˆ(p1, . . . , pL−1) = 0 iﬀ z(p1, . . . , pL−1, 1) = 0. 42 ECHENIQUE

−M

Figure 6. Aggregate excess demandz ˆ1(p1, 1, ~ω).

Given an economy ~ω, let P(~ω) denote its competitive equilibria. So L−1 P(~ω) = p ∈ R++ :z ˆ(p, ~ω) = 0 . Deﬁnition 41 (Regular Equilibrium). An equilibrium price p∗ ∈ P(~ω) is a regular equilibrium of ~ω if the matrix

Dpzˆ(p; ~ω) p=p∗ is non-singular.

An economy ~ω is said to be a regular economy if all the equilibrium prices 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 case 2I zˆ : R++ ×R → R. Therefore, the condition of regularity is equivalent to the derivative of the excess demand function being diﬀerent from zero at every equilibrium price. Consider the aggregate excess demands in Figure 6. The economy with the green excess demand is regular, while that with the red excess demand is critical. If there are more than two goods, consider an inﬁnitesimal change in prices dp = (dp1, dp2, . . . , dpL−1). The directional derivative of the L functionz ˆ at the price vector p ∈ R++ in the direction of dp is given by Dzˆ(p) · dp.9 If Dzˆ(p) has full rank at p, then Dzˆ(p) · dp 6= 0 for every dp 6= 0. This implies that the normalized excess demand function

9Let f : Rn → R be a differentiable function with gradient vector ∇f(x). The directional 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) = [∇f 1(x),..., ∇f m(x)]0u = [∇f 1(x) · u, . . . , ∇f m(x) · u]0 = [Df 1(x; u), . . . , Df m(x; u)]. GENERAL EQUILIBRIUM THEORY 43 zˆ changes for infinitesimal changes in prices, i.e., we stop being at equilibrium. This is the intuition for our next concept and result. Definition 42 (Local Uniqueness). An equilibrium p∗ ∈ P(~ω) is lo- L−1 cally unique if there exists ε > 0 such that, ∀ p ∈ R++ , kp − p∗k < ε =⇒ p∈ / P(~ω). Proposition 43. Let zˆ be C1 and p∗ ∈ P(~ω) be a regular equilibrium. ∗ Then, p is locally unique. Furthermore, there are neighborhoods B1 of ∗ L−1 ~ω in E, and B2 of p in R++ , and a function h : B1 → B2 such that

zˆ(h(~ω), ~ω) = 0 ∀ ~ω ∈ B1, and −1 ∗ D~ωh(~ω) = − Dpzˆ(p, ~ω) · D~ωzˆ(p , ~ω). p=p∗

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) is a closed set. Moreover P(~ω) is compact because it is a closed subset of a compact set. For each price p ∈ P(~ω), let Np be an open ball with center p and for which p is the only equilibrium price in Np. Such balls exist 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; a contradiction. Definition 45 (Index). The index of p ∈ P(~ω) is defined as: L−1 index(p) = (−1) · sign Dpzˆ(p, ~ω) ,

where Dpzˆ(p, ~ω) is the determinant of the matrix Dpzˆ(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 X index(p) = 1. p∈P(~ω)

The index theorem can be used to establish uniqueness: if you can show that any competitive equilibrium in an economy has index one, then there can only be one equilibrium. Finally, the index theorem implies 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 the graph below the index of each equilibrium in the economy, and verify that they sum up to one.

10.3. Digression: Measure Zero Sets and Transversality. Deﬁnition 48 (Rectangle and Volume). A rectangle in Rn is a set of the form n R = [a1, b1]×[a2, b2]×· · ·×[an, bn] = {x ∈ R : ai ≤ xi ≤ bi, i = 1, . . . , n}, where [ai, bi] = {x ∈ R : ai ≤ x ≤ bi} is an interval in R. The volume of a rectangle R = [a1, b1] × [a2, b2] × · · · × [an, bn] is n Y vol(R) = (bi − ai). i=1 Deﬁnition 49 (Measure Zero). A set A ⊆ Rn is measure zero if ∀ ε > 0 there exist a collection of rectangles R1,R2,... such that ∞ ∞ [ X A ⊆ Ri, and vol(Ri) < ε. i=1 i=1

Consider the following two examples of measure zero sets in R2: a ﬁnite set and a straight line. Under the same reasoning, note that a line segment is not measure zero in R (because it is an interval), but it is in Rn ∀ n > 2. Similarly, a plane is not measure zero in R2, but it is in Rm ∀ m > 2. Observation 50. No open set has measure zero. GENERAL EQUILIBRIUM THEORY 45

Consider a parametrized familiy of equations f : RN+K → RN . Deﬁne a system of equations as f(v, q) = 0, where v ∈ RN is a vector of unknowns, 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) at each (v, q) with f(v, q) = 0, then there is a measure zero set C ⊆ RK K such that, ∀ q ∈ R \ C, the N × N matrix Dvf(v, q) is non-singular whenever 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 that when we are at a critical solution, a change in parameters can “perturb away” the critical point.

10.4. Genericity of regular economies. Return to the assumptions L−1 L−1 1 we made earlier. Suppose thatz ˆ : R++ ×E → R is C and satisﬁes properties (i)-(v).

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

Proof. The result follows from the Transversality Theorem. The matrix Dzˆ(p; ~ω) is (L − 1) × (L − 1 + IL). Write this matrix as Dzˆ(p; ~ω) = Dpzˆ(p; ~ω) Dω1 zˆ(p; ~ω) ··· DωI zˆ(p; ~ω) , where Dωi zˆ(p; ~ω) is the (L−1)×L Jacobian matrix with respect to the endowment vector of consumer i. We shall prove that any one of the matrices Dωi zˆ(p; ~ω) has full rank. If we show that any Dωi zˆ(p; ~ω) has full rank, then we have that Dzˆ(p; ~ω) has L − 1 linearly independent columns and thus it is also full rank.

So lets calculate Dω1 zˆ(p; ~ω), and show that it has full rank. The matrix

Dω1 zˆ(p; ~ω) may be written as Dω1 zˆ(p; ~ω) = Dω11 zˆ(p; ~ω) Dω21 zˆ(p; ~ω) ··· DωL1 zˆ(p; ~ω) h i = dzˆ(p;~ω) dzˆ(p;~ω) ··· dzˆ(p;~ω) ∂ω11 ∂ω21 ∂ωL1 Recall that I X ∗ zˆl(p; ~ω) = xi (p, p · ωi) − ω¯l ∀ l = 1,...,L1. i=1 46 ECHENIQUE

Then, compute for every l = 1,...,L − 1,  ∗ ∂xh1(p,p·ω1) ph − 1 if h = l ∂zˆ (p; ~ω)  ∂m1 l = ∂ω ∗ h1  ∂xh1(p,p·ω1)  ph if h 6= l ∂m1

For notational purposes, we let m1 denote the income of consumer 1.

Therefore we may write the complete matrix Dω1 zˆ(p; ~ω) as

∗ ∗ ∗ ∗  ∂x11 ∂x11 ∂x11 ∂x11  ∂m p1 − 1 ∂m p2 ··· ∂m pL−1 ∂m 1 ∗ ∗ 1 ∗1 ∗1 ∂x21 ∂x21 ∂x21 ∂x21  p1 p2 − 1 ··· pL−1   ∂m1 ∂m1 ∂m1 ∂m1  Dω1 zˆ(p; ~ω) =  . . . . .   ......   ∗ ∗ ∗ ∗  ∂xL−1,1 ∂xL−1,1 ∂xL−1,1 ∂xL−1,1 p1 p2 ··· pL−1 − 1 ∂m1 ∂m1 ∂m1 ∂m1 (L−1)×L See that the L-th column of the previous matrix takes into account the fact that we have normalized pL = 1. Take the L-th column and multiply it by pl to subtract it from the l-th column. This operation does not change the rank of the matrix. Repeating the operation, we obtain:  ∂x∗  −1 0 ··· 0 11 ∂m1 ∂x∗  0 −1 ··· 0 21   ∂m1   . . . . .   ......   ∂x∗  0 0 · · · −1 L−1,1 ∂m1 (L−1)×L

The ﬁrst L−1 columns of the previous matrix are linearly independent, implying that Dω1 zˆ(p; ~ω) has full rank. Therefore, the Jacobian matrix Dzˆ(p; ~ω) has also full rank. Transversality implies that there exists a IL IL measure zero set C ∈ R such that, if ~ω ∈ R \ C, then Dpzˆ(p; ~ω) has full rank for every p ∈ E(~ω). In other words, every economy IL ~ω ∈ R \ 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 k k K k L k L (x , p )k=1 where x ∈ R+ and p ∈ R++ for every k = 1,...,K.

k k K L Deﬁnition 53. A data set (x , p )k=1 is rationalizable by u : R+ → R L if, ∀ y ∈ R++, pk · y ≤ pk · xk =⇒ u(y) ≤ u(xk) ∀ k = 1,...,K. GENERAL EQUILIBRIUM THEORY 47

k k K Deﬁnition 54 (Revealed Preference). Given a data set (x , p )k=1, R L deﬁne a binary relation % on R+ as: R k k k k x % y ⇐⇒ ∃ k s.th. x = x and p · x ≥ p · y, and x R y ⇐⇒ ∃ k s.th. x = xk and pk · xk > pk · y.

R We refer to the binary relation % as the revealed preference relation. Definition 55 (Weak Axiom of Revealed Preference). A data set satisifes the Weak Axiom of Revealed Preference (WARP) if there is k R l l R k no k and l such that x % x and x x . Definition 56. A data set satisfies the Generalized Axiom of Revealed Preference (GARP) if there is no sequence xk1 , xk2 , ..., xkn such that k R k R R k k R k x 1 % x 2 % ··· % x n , and x n x 1 . Observation 57. If L ≤ 2, WARP and GARP are equivalent. If L ≥ 3, then GARP =⇒ WARP, but not conversely.

k k K Theorem 58 (Afriat’s Theorem). Consider a data set (x , p )k=1. The following statements are equivalent:

(1) the data set is rationalizable by a locally nosatitated utility function; (2) the data set satisﬁes GARP; (3) there are numbers U k, λk > 0 for k = 1,...,K, such that U k ≤ U l + λlpl · (xk − xl); (4) the data set is rationalizable by a strictly monotonic and concave utility function.

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

11.2. Sonnenschein-Mantel-Debreu Theorem: Anything goes. In this section we consider a set of important results regarding what can be obtained as an equilibrium for a well-behaved economy. The answer is that, in a sense, anything can happen. The Sonnenschein-Mantel- Debreu Theorem says that, oﬀ the boundaries of the simplex, any function that satisﬁes the basic properties of an excess demand function can be 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 > 0 I there exists an exchange economy (%i, ωi)i=1 with I = L and each %i strictly monotonic, continuous and strictly convex such that the aggregate 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 into individual aggregate excess demand functions that satisfy WARP. For i each individual consumer i, we let ω = ei, the ith unit vector (recall that I = L). So agent i is endowed with one unit of good i. Then let 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 regression of ωi = ei on the line spanned by p. Then gi(p) will satisfy Walras law, as it was chosen to live in the orthogonal complement to p. It is also continuous, and satisﬁes WARP because it is the outcome of an optimization problem. Moreover, if hi(p) > 0 is a scalar, then the function p·hi(p)gi(p) is also continuous and satisﬁes WARP and Walras Law. Theorem 60 (Mantel). Let f : ∆o → RL be C2 and satisfy Walras’ I Law. Then, ∀ ε > 0, there exists an exchange economy (%i, ωi)i=1 with each %i homothetic, monotonic and strictly convex such that its aggregate excess demand function coincides with f on ∆ε. Corollary 61. For any compact set K ⊆ ∆o, there exists an exchange economy with I = L and each %i strictly monotonic, continuous, and strictly convex such that K is the set of equilibrium prices of the economy.

11.3. Brown and Matzkin: Testable Restrictions On Com- petitve Equilibrium. If we imagine that we can observe more than just prices, so that we can see how equilibrium varies with endowments (or, observe outcomes “on the equilibrium manifold”), then we can avoid the negative conclusion in the SMT theorem. These ideas are due 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, the I equilibrium manifold for a ﬁxes set of preferences (i)i=1 is the set

10See Shafer and Sonnenschein (1982) or Chambers and Echenique (2016) for a sketch of the main ideas behind the proof. GENERAL EQUILIBRIUM THEORY 49 of price and endowment combinations for which markets clear in the resulting exchange economy. Suppose we have the following two observations for an economy with two agents and two goods: (p1, ω1) and (p2, ω2), i.e., we have two observations of the equilibrium manifold. GRAPH OF EQUILIBRIUM MANIFOLD Each endowment vector ω1 and ω2 induces an Edgeworth Box economy. For illustrative purposes, suppose that each Edgewroth Box is given by: GRAPH OF TWO SEPARATE EDGEWROTH BOXES Superimpose both Edgeworth Boxes by aligning the origin of consumer 1’s consumption set, and draw the equilibrium price vectors as follows: GRAPH OF TWO EDGEWROTH BOXES SUPERIMPOSED As the economy is in equilibrium under the two observations, then it must be that both individuals are maximizing their respective utilities. However, note that this two choices violate WARP for consumer 1. There are no preferences %1 and %2 for which there exist alloca- 0 0 1 tions (x1, x2) and (x1, x2) such that ((x1, x2), p ) is a Walrasian Equi- 1 1 0 0 2 librium of the economy ((%1, ω1), (%2, ω2)) and ((x1, x2), p ) is a Wal- 2 2 rasian Equilibrium of the economy ((%1, ω1), (%2, ω2)). Therefore, this pair of observations is not consistent with the notion of competitive equilibrium.

12. The Core

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

I IL • An allocation≤ in E is a vector x = (xi)i=1 ∈ R+ , such that PI PI i=1 xi ≤ i=1 ωi =ω. ¯ I IL • An allocation= in E is a vector x = (xi)i=1 ∈ R+ , such that PI PI i=1 xi = i=1 ωi =ω. ¯ • A nonempty subset S ⊆ {1,...,I} of agents is called a coalition. • Let S be a coalition. A vector (yi)i∈S is an S-allocation≤ if P P i∈S yi ≤ i∈S ωi. • Let S be a coalition. A vector (yi)i∈S is an S-allocation= if P P i∈S yi = i∈S ωi. 50 ECHENIQUE

Deﬁnition 62. A coalition S blocks the allocation≤x in E if there exists an S-allocation≤(yi)i∈S such that yi i xi ∀ i ∈ S.

• An allocation≤is weakly Pareto optimal if it not blocked by the coalition I of all consumers, • individually rational if no coalition consisting of a single consumer blocks it, • and a core allocation if there is no coalition that blocks it.

Let C(E) be the set of core allocation≤of E. We refer to C(E) as the core of the economy E. Let P (E) be the set of Pareto Optimal allocation≤of the economy E, and let W (E) be the set of Walrasian Equilibrium allocation≤.

IL Note that C(E), W (E), and P (E) are subsets of R+ . Deﬁnition 63. A coalition S weakly blocks the allocation x if there exists an S-allocation≤(yi)i∈S such that yi i xi ∀ i ∈ S, and yj j xj for some j ∈ S. Observation 64. If each preference relation is continuous and strictly monotonic, then a coalition blocks an allocation if and only if it weakly blocks it.

Proof. (⇒) If a coalition blocks an allocation, then it weakly blocks it by the deﬁnition of strict preference. (⇐) Let x be an allocation and S a coalition that weakly blocks it. Then, there exists an S-allocation (yi)i∈S such that yi %i xi ∀ i ∈ S, and yj j xj for some j ∈ S. By continuity of %j, there exists δ ∈ (0, 1) such δyj that (1 − δ)yj j xj. By strict monotonicity of each %i, |S|−1 + yi i yi δyj ∀ i ∈ S \{j}. Let zj = (1 − δ)yj and zi = |S|−1 + yi ∀ i ∈ S \{j}. z = (zi)i∈S is an S-allocation since

X X δyj X X z = (1 − δ)y + + y = y = ω . i j |S| − 1 i i i i∈S i∈S\{j} i∈S i∈S

By transitivity, zi i xi for every i ∈ S, so S blocks the allocation x.

For the remainder of our treatment of the core, we will work with preferences that are continuous and strictly monotonic. So there will be no diﬀerence between weak blocks and blocks. We will also refer GENERAL EQUILIBRIUM THEORY 51 to allocations, and not distinguish between allocation≤and allocation=. (The only relevant allocations will be allocation=.)

12.1. Pareto Optimality, The Core and Walrasian Equiilbria. Observation 65. If each preference relation is continuous and strictly monotonic, then all core allocations are Pareto Optimal, i.e., C(E) ⊆ P (E).

• An allocation x of E is individually rational if xi %i ωi ∀ i = 1,...,I. Observation 66. If x ∈ C(E), then x is individually rational. Example 67. Let I = 2. Given continuous and strictly increasing preferences, the core is the intersection of the set of Pareto optimal allocations and the set of individually rational allocations. We have already shown that every core allocation is both Pareto Optimal and individually rational.

The following is a sort of strong version of the ﬁrst welfare theorem, except that the notion of blocking we’re using now precludes the need for local non-satiation. Theorem 68. Every Walrasian Equilibrium allocation is a core allocation, i.e., W (E) ⊆ C(E).

∗ ∗ IL L Proof. Let (x , p ) ∈ R+ × R++ be a Walrasian Equilibrium. Towards contradiction, suppose x∗ ∈/ C(E). Then, there exists a coalition S and P P an S-allocation (ys)s∈S such that s∈S ys = s∈S ωs, and, ∀s ∈ S, ∗ ∗ ys s xs. Since x is an equilibrium allocation, the latter implies ∗ ∗ ∗ ∗ ys ∈/ B(p , p · ωs) for every s ∈ S, i.e., p · ys > p · ωs. Summing over s and pricing both resulting bundles, we obtain a contradiction: ! ! ∗ X ∗ X p · ys > p · ωs . s∈S s∈S Thus, we conclude W (E) ⊆ C(E).

12.2. Debreu-Scarf Core Convergence Theorem. We next dis- cuss the validity of the competitive hypothesis: the idea that agents act as price-takers. We start from bargaining outcomes, in which agents will typically posses market power. The bargaining outcome is modeled through the game theoretic notion of the core, which says that no 52 ECHENIQUE gains from multilateral exchange are left unexploited. Note that we do not model the detailed institutional framework for trade (the extensive- form game, if we were to adopt a non-cooperative paradigm). Instead we are agnostic about how trade takes place, and impose that whatever the outcome is, it has to exhaust all the gains from trade. So it has to be in the core. The second, crucial, idea is to ensure that no agent is special. We replicate an economy, so that there will be many copies of the same agent. Each agent will have a large number of identical “twins.” This will reduce the bargaining power of each individual agent. In the limit all the core outcomes must be Walrasian allocations.

I Deﬁnition 69 (Replica Economy). Let E = (i, ωi)i=1 be an exchange economy. and N ≥ 1 an integer. The N-th replica of E is the exchange economy N E = (%i,n, ωi,n)i=1,...,I,n=1,...,N in which agents are indexed by (i, n) with i = 1,...,I, n = 1,...,N, and satisfy that:

∀ n = 1,...,N, %i=%i,n and ωi = ωi,n.

Note that the replica E N has IN agents. Deﬁnition 70 (Equal Treatment Property). An allocation

x = (xi,n)i=1,...,I,n=1,...,N N of E has the equal treatment property if xi,n = xi,m for every n, m = 1,...,N and i = 1,...,I.

Lemma 71. Let %i be strictly monotonic, continuous and strictly convex for every i = 1,...,I. Every allocation in C(E N ) has the equal treatment property.

Proof. The result is trivial for N = 1. Let N ≥ 2, and consider the contrapositive statement. Let x be an allocation of E N that does not have the equal treatment property. For each i = 1,...,I, let n(i) be the agent (i, n(i)) such that

xi,n %i xi,n(i) ∀ n = 1,...,N. Hence, n(i) is the (weakly) worst oﬀ agent among all the replicas of i. Let S = {n(1), n(2), ..., n(I)} be a coalition of all such agents. We shall prove that coalition S weakly blocks the allocation x. This is suﬃcient for x∈ / C(E N ) by continuity and strict monotonicity. GENERAL EQUILIBRIUM THEORY 53

Deﬁne the bundle yi,n(i) by:

N 1 X y = x ∀ i. i,n(i) N i,n n=1

As xi,n %i xi,n(i) ∀ n = 1, ..., N, then yi,n(i) %i xi,n(i) ∀ i by (strict) convexity. As x does not satisfy the equal treatment property, there is at least one i ∈ {1,...,I} and one m ∈ {1,...,N} such that xi,m 6= xi,n(i). Therefore, for such i, by strict convexity we have

11 yi,n(i) i xi,n(i).

I Finally, let us prove that (yi,n(i))i=1 is an S-allocation.

I I N X X 1 X y = x i,n(i) N i,n i=1 i=1 n=1 I N 1 X X = x N i,n i=1 n=1 I N 1 X X = ω since x is an allocation of E N N i,n i=1 n=1 I 1 X = Nω since ω = ω for every n and i N i i,n i i=1 I I X X = ωi = ωi,n(i). i=1 i=1

I Therefore, we have proved that (yi,n(i))i=1 is an S-allocation that (weakly) N blocks x, so x∈ / C(E ).

11 If N = 2 the result is straightforward. Let N ≥ 3, and note that xi,m %i xi,n(i) 1 1 and xi,m 6= xi,n(i)) imply 2 xi,m+ 2 xi,n(i) i xi,n(i) by strict convexity. Furthermore, as xi,n %i xi,n(i) for every n = 1, ..., N, then by convexity: 1 X x . N − 2 %i i,n(i) n/∈{m,n(i)} Therefore, see that 2/N ∈ (0, 1) and note that by strict convexity we obtain: 2 1 1 2 1 X x + x + 1 − = y x . N 2 i,m 2 i,n(i) N N − 2 i,n(i) i i,n(i) n/∈{m,n(i)} 54 ECHENIQUE

By the previous lemma, we can represent the core allocations of E N as IL N IL vectors in R+ . So we shall write C(E ) as a subset of R+ . The same is true of Walrasian allocations.

I Observation 72. Let E = (i, ωi)i=1 be an exchange economy in which preferences i are continuous, strictly monotonic and strictly convex.

(i) The core of a replica economy decreases with the number of replicas: ∀ N,C(E N ) ⊇ C(E N+1) ⊇ C(E N+2) ⊇ · · · (ii) The equilibrium allocations of E N may be represented as allocations in E, i.e., the elements in W (E N ) can be represented in IL R+ . (iii) W (E N ) = W (E) for every N.

(iv) An equilibrium allocation of E is in the core of every replica economy E N : ∞ \ W (E) ⊆ C(E N ). N=1

Statement (iii) needs some explanation. We know that W (E N ) are in the core, and therefore satisfy the equal treatment property. So we can represent W (E N ) as a subset of RIL for all N. Now, in a Walrasian equilibrium (x, p) of E N , any two agents of the same type face the same budget, and have the same preferences. So a consumption bundle is optimal for each one of them iﬀ it is optimal for the original prototype in E. By the equal treatment property, the supply = demand property holds for E N iﬀ it holds in E. The following result is due to Debreu and Scarf (1963).12

Theorem 73 (Debreu-Scarf Core Convergence Theorem). Let E = (%i I , ωi)i=1 be an exchange economy. Suppose that, for all i = 1,...,I, i is continuous, strictly monotonic and strictly convex, and that ωi > 0. Suppose also that ω¯ 0. Then x∗ ∈ C(E N ) ∀ N ≥ 1 implies that x∗ ∈ W (E). In other words, ∞ \ W (E) = C(E N ). N=1

12The proof here follows Kim Border’s notes on Debreu-Scarf. GENERAL EQUILIBRIUM THEORY 55

0 0 ∗ {xi : xi i xi }

−ωi Pi

Figure 7. Proof of Debreu-Scarf Theorem

∗ ∞ N Proof. Let x ∈ ∩N=1C(E ). Deﬁne L ∗ Pi = {zi ∈ R : zi + ωi i xi }, I 13 and let P be the convex hull of ∪i=1Pi. ∗ Note that individual rationality of x rules out that 0 ∈ ∪Pi. By imposing the core property we get much more than individual rationality, and will be able to rule out that 0 ∈ P . For this it is important that we work with arbitrary replicas of E as we need to represent convex combinations with rational coeﬃcients as coalitions in some (arbitrarily large) economy. We shall prove that 0 ∈/ P . Suppose then towards a contradiction that 0 ∈ P . P is an open set, so there is a neighborhood V of 0 with V ⊆ P . Then there is a neighborhood V 0 in V contained in L L 0 R−− = {−z : z ∈ R++}. See Figure 8. Now, since V ⊆ P there is a collection

z1, . . . , zI

13Note the diﬀerence with the approach in the second welfare theorem. We work I PI I with ∪i=1Pi instead of i=1 Pi, and ∪i=1Pi may not be a convex set. 56 ECHENIQUE

V 0

Pj zj

Figure 8. Proof of Debreu-Scarf Theorem with zi ∈ Pi for all i, and I X 0 αizi ∈ V . i=1 The function I X (˜α1,..., α˜I ) 7→ α˜izi i=1 0 0 PI 0 0 is continuous, so there is (α1, . . . , αI ) with i=1 αizi ∈ V and for which 0 αi ∈ Q+ for all i = 1,...,I. 0 Let βi ∈ N and N be such that αi = βi/N. Consider a coalition S N in E with βi copies of agent type i. Deﬁne the vector (yh)h∈S with yh = zi + ωi for the type of agent i that h is a member of. This means ∗ that yh h xh, as zi ∈ Pi. In addition: I X X yh = βi(zi + ωi) h∈S i=1 I I X βi X = N z + β ω N i i i i=1 i=1 X ωh, h∈S as I X βi z 0. N i i=1 GENERAL EQUILIBRIUM THEORY 57

Agents’ preferences are monotonic, so the coalition S blocks x∗ in E N , contradicting that x∗ ∈ C(E N ).14 The rest of the proof is similar to the proof of the second welfare theorem, but simpler since we do not need to go from individual bundles to aggregate bundles. The set P is convex and nonempty, and 0 ∈/ P . By the separating hyperplane theorem, there is p∗ ∈ RL, non-zero, such that p∗ · z ≥ 0 ∗ for all z ∈ P . Then xi i xi implies that xi − ωi ∈ Pi, which means ∗ ∗ that p · xi ≥ p · ωi.

k ∗ k ∗ Let xi = xi + (1/k, . . . , 1/k) for k ≥ 1. Then xi i xi , which means ∗ k ∗ ∗ ∗ ∗ ∗ k that p · xi ≥ p · ωi for all k. Then p · xi ≥ p · ωi as xi = limk→∞ xi . ∗ ∗ ∗ PI ∗ ∗ ∗ ∗ PI ∗ Then p ·xi ≥ p ·ωi for all i, and i=1(p ·xi −p ·ωi) = p · i=1(xi − ∗ ∗ ∗ ∗ ωi) = 0 (as x is an allocation), imply that p · xi = p · ωi for all i. ∗ ∗ ∗ ∗ ∗ So we have established that if xi i xi then p · xi ≥ p · xi = p · ωi ∗ ∗ Now we can show that p > 0. Strict monotonicity implies that xi + ∗ ∗ ∗ ∗ ∗ ∗ ∗ el i x . Then, p · (xi + el) ≥ p · xi gives pl ≥ 0. Since p 6= 0 we have p∗ > 0. We haveω ¯ 0. So p∗ > 0 implies that p∗ ·ω¯ > 0. Then there is j with ∗ ∗ ∗ ∗ ∗ p · ωj > 0. For such j, if we had some xj j xj and p · xj = p · xj we would obtain a contradiction by the same reasoning as in the proof of the second welfare theorem: namely that there would be δ > 0 with ∗ ∗ ∗ ∗ ∗ ∗ (1 − δ)xj j xj but p · (1 − δ)xj = p · (1 − δ)xj < p · xj . Now, using what we know about this particular consumer j, we can ∗ ∗ ∗ ∗ ∗ show that p 0. We have that xj + el j x implies p · (xj + el) > ∗ ∗ p · xj . So pl > 0. ∗ Finally, let i be any consumer. Then p 0 and ωi > 0 implies that ∗ ∗ ∗ p · ωi = p · xi > 0. This implies that we can repeat the argument ∗ ∗ we made above for consumer j, and show that xi i xi then p · xi > ∗ ∗ p · xi .

14 P We obtain a vector (yh)h∈S that sums up to less than h∈S ωh but this implies that S blocs since we can increase yh a little bit so as to satisfy the equality with the sum of endowments. 58 ECHENIQUE

13. Partial equilibrium

Economists often want to focus on the market for one good in isolation. The ability to do so depends on making certain assumptions, which we proceed to spell out. Consider an exchange economy in which each preference is quasilinear. I Speciﬁcally, let E = (i, ωi)i=1 be an exchange economy in which each i is represented by a utility function

L X ui(x1, . . . , xL) = vi(x1) + xl. l=2

Al consumers regard goods l = 2,...,L as perfect substitutes. As a consequence, in equilibrium, these goods should have the same price (meaning that consumers should exchange one unit of good l for one unit of good l0, for all l, l0 6= 1). We can normalize the price of all those PL goods to be 1, and regard good l=2 xl as a composite good. Denote this composite good by m, and x1 by x. Eﬀectively, then E becomes a two-good economy in which consumers have utility ui(x, m) = vi(x) + m. Let p be the price of good x. Consider the maximization problem for consumer i:

max(x,m) vi(x) + m s.t. px + m ≤ W.

Allow negative consumption of m (a common, but not innocent assumption). Then the problem becomes to maximize

vi(x) + (W − px).

∗ Note that the solution x (p) is the solution to maximizing vi(x) − px and does not depend on W . In contrast the demand for “money” is m∗(p, W ) = W − px∗(p). These facts about x∗ and m∗ are very important. All “wealth eﬀects,” meaning all changes in income W , translate one for one into changes in the demand for m. The demand for x does not depend on W .

2 Assume: vi is strictly increasing, C and concave. GENERAL EQUILIBRIUM THEORY 59

v0(x) x

Figure 9. Quasi-linear demand

The FOC for the consumer’s problem is then ( ≤ 0 ifx = 0 v0(x) − p = 0 ifx > 0 We should emphasize that it is this simple because we are allowing for m < 0.

0 ∗ ∗ ∗ Now we have that vi(xi (p)) = p when xi (p) > 0, so that xi is the 0 inverse of the marginal utility vi. The indirect utility of consumer i is Z ∞ ∗ ∗ ∗ ∗ ∗ ψi(p, W ) = vi(xi (p))+m (p, W ) = (vi(xi (p)) − pxi (p))+W = xi (s)ds+W, p R ∞ ∗ when the (improper) integral p xi (s)ds is well deﬁned. See Figure 10 The term Z ∞ ∗ ∗ ∗ CSi(p) = vi(xi (p)) − pxi (p) = xi (s)ds p is usually called consumer surplus. It is a measure of the well being of a consumer, in the obvious sense that it’s (up to the value of W ) ∗ her indirect utility. It has a simple meaning, for magnitude x < xi (p), the consumer was willing to pay an additional v0(x) in moiney for a bit more of the good, but only paid p. So the cosumer gained v0(x) − p, If we “add” up all these gains, stemming from the excess willingness ∗ R xi (p) 0 to pay over what was actually paid, we obtain 0 [vi(x) − p]dx = ∗ ∗ vi(xi (p)) − pxi (p). Moreover, ∗ Z xi (p) Z +∞ 0 ∗ 0 ∗ vi(x) = s iﬀ xi (s) = x, hence [vi(x) − p]dx = xi (s)ds 0 p 60 ECHENIQUE p

CSi(ˆp)

pˆ

∗ xi (p) ∗ x xi (ˆp)

Figure 10. Consumer surplus

Note that the formula expressing consumer surplus as a function of demand allows us to empirically measure consumer surplus, if we can empirically estimate consumer demand (and we often can).

∗ PI ∗ 13.1. Aggregate demand and welfare. Let X (p) = i=1 xi (p), which deﬁnes a utility v (by integrating its inverse). So there exists obviously a representative consumer. PI PI Note also that the sum i=1 ψi(p, Wi) is, up to the value of i=1 Wi, equal to

I I Z ∞ Z ∞ I Z ∞ X ∗ ∗ X ∗ X ∗ ∗ (vi(xi (p))−pxi (p)) = xi (s)ds = xi (s)ds = X (s)ds. i=1 i=1 p p i=1 p We can often estimate aggregate demand, which gives us a measure of collective welfare. In fact, consider the problem of ﬁnding a Pareto optimal allocation

max v1(x1) + m1  vi(xi) + mi ≥ u¯i∀i = 2,...,I P P s.t. i xi ≤ i ωi,x P P  i mi ≤ i ωi,m

P P This problem is equivalent to maximizing vi(xi) subject to xi ≤ P i i i ωi,x. GENERAL EQUILIBRIUM THEORY 61

The Lagrangian is ! X X X X X L((xi); λ) = vi(xi)+λ ωi,x − xi = [vi(xi)−λxi]+λ ωi,x i i i i i

The ﬁrst-order conditions are the same as for individual maximization when λ = p. This is a version of the welfare theorems. So we can ∗ set the solution to be xi (λ). Then the value of the lagrangian at the PI maximum is equal to i=1 CSi(λ). So in a sense what we’re tying to do is to maximize the sum of consumer surpluses.

13.2. Production. Suppose that instead x is produced from “money” using a cost function c : R+ → R+. This means that there is a production possibility set

Y = {(x, −c(x)) : x ∈ R+}.

Pareto eﬃcient allocation can be obtained as before by solving:

max v1(x1) + m1 v (x ) + m ≥ u¯ ∀i = 2,...,I  i i i i PI x ≤ PI ω + xf s.t. i=1 i i=1 i,x PI m ≤ PI ω − mf  i=1 i i=1 i,m (xf , −mf ) ∈ Y

PI Suppose that i=1 ωi,x = 0. Then at a solution we must have that P f f f f P i xi = x . Then (x , −m ) ∈ Y means that m = c( i xi). Finally, P P f i mi = i ωi,m − m implies that eﬃciency is characterized by

I I X X max vi(xi) − c( xi) (x1,...,x )≥0 I i=1 i=1

Assume: c is C1, with c0 > 0, and c is strictly convex.

0 0 The FOCs for an interior solution demands that v (xi) = c (x), where P i x = i xi. This solution can be decentralized by setting the price of 0 0 0 ∗ x to be p = c (x). Then vi(xi) = p = c (x), xi = xi (p), and a ﬁrm maximizes proﬁts by choosing x at prices p. 62 ECHENIQUE

13.3. Public goods. A public good is a good that all consumers must consume in the same quantity. If the public good consumed by i is x, then all consumers consume the quantity x of the public good. Each consumer j obtains then utility vj(x). Eﬃciency demands that n X max vi(x) − c(x). x≥0 i=1 The FOCs for an interior solution now mean that n X 0 0 vi(x) = c (x) i=1

Pn 0 0 The condition i=1 vi(x) = c (x), equating marginal cost to the sum of marginal utilities is often called Samuelson’s condition for eﬃcient provision of public goods.15 See Figure 11. Let x∗ be the solution to this maximization problem, so x∗ is the (unique) eﬃcient level of the public good.

p c0

0 0 v1 + v2 0 v2 0 v1 x

Figure 11. Samuelson’s condition for public good provision

Compare to private provision of public good. Suppose that the price of the public good is p. Then each agent i is choosing a quantity xi to buy, at price p. Then i solves the problem X max vi(xi + xj) − pxi. xi≥0 j6=i

15Paul Samuelson introduced this way of analyzing public goods in Samuelson (1954). GENERAL EQUILIBRIUM THEORY 63

The FOC is then 0 X vi( xj) − p = 0 if xi > 0 j 0 X vi( xj) − p ≤ 0 if xi = 0. j

0 P Proﬁt maximization requires that p = c ( j xj). 0 0 0 Suppose, to simplify the exposition, that v1 < v2 < ··· < vI . 0 0 Then since p does not depend on i we must have vI (ˆx) = p and vi(ˆx) − p < 0 for i = 1,...,I −1. Hence xi = 0 for i = 1,...,I −1, and xI =x ˆ. 0 P 0 0 0 0 Note that since vi > 0 we have that i vi(ˆx) > vI (ˆx) = c (ˆx). Since c P 0 is monotone increasing and i vi is strictly decreasing, xˆ < x∗. In words, there is underprovision of the public good.

13.4. Lindahl equilibrium. Imagine that we could fool consumers into believing that the quantity consumed of the public good only depends on their individual purchases. Sort of like selling the Hollywood sign to a very gullible tourist. They would then pay an (individualized!) price pi for each unit of the public good, and solve the maximization problem max vi(xi) − pixi xi≥0

0 ∗ The FOC for an interior solution is vi(xi) = pi. Now let x be the 0 ∗ eﬃcient level of the public good and set pi = vi(x ). It is then optimal for each consumer i to purchase the same level x∗ of the public good. The ﬁrm solves the problem of X max( pi)q − c(q). q≥0 i

P P 0 ∗ ∗ Since i pi = i vi(x ) and x satisfies the Samuelson condition of P 0 ∗ optimality, we have that i pi = c (x ). The firm optimizes by choosing the level x∗ of public good when it sells it to each consumer i at price pi. This outcome is called a Lindahl equilibrium. The idea is to illustrate the role of the inability to exclude agents from consuming the public 64 ECHENIQUE good as the source of inefficiency in the private provision of public goods.

14. Two-sided matching model with transferable utility

We turn to a model of a two-sided market due to Shapley and Shubik (1971): B is a set of buyers and S is a set of sellers. The sets B and S are finite, nonempty, and disjoint. Each buyer seeks to buy one and only one unit of an indivisible good. Each seller has one unit to sell, but sellers are different from each other, and offer potentially different goods.

If i ∈ B buys from j ∈ S, they generate a surplus αi,j. Here αi,j is given; part of the primitive of the model. You can think of buyer i getting some value from consuming j’s good, and of j having some cost of providing that good to i. Then αi,j would be the diﬀerence between value and cost.

When i buys from j, she obtains some utility ui and j gets some profit vj, so that ui + vi = αi,j. For example, imagine that the cost for j is zero. Then i’s utility is ui = αi,j − vj, where vj is the price (and therefore the profit) that j gets from i. In this case, i buys from j if αi,j − vj ≥ αi,h − vh for all h. Put differently,

ui + vh ≥ αi,h, for all i and h.

A matching is a matrix (xi,j)i∈B,j∈S such that xi,j ≥ 0, and for all (i, j) ∈ B × V , X xi,h ≤ 1 h∈V X xh,j ≤ 1. h∈B

If xi,j ∈ {0, 1} we can interpret xi,j = 1 as i buying from j. The above inequalities mean that each buyer buys from at most one seller, and each seller sell to at most one buyer.

An assignment is a pair of vectors ((ui)i∈B, (vj)j∈S) such that ui ≥ 0, P vj ≥ 0 and such that there exists a matching xi,j such that ui + P P i∈B j∈S vj = i∈B,j∈S αi,jxi,j. An assignment (u, v) is in the core if

ui + vj ≥ αi,j. GENERAL EQUILIBRIUM THEORY 65

The core terminology arises from a notion of blocks. A pair (i, j) can block an assignment if ui + vj < αi,j by trading among themselves and sharing the surplus αi,j. In this model, only pairs can “generate value.” So the only relevant coalitions are pairs, and the core is the set of assignments that no pair can block. At the same time, the core requirement ensures that any agent is optimizing in the sense that

ui = max{αi,h − vh : h ∈ S} vj = max{αh,j − uh : h ∈ B}

We can now analyze the model by means of linear programming duality. There will be a duality between the problem of efficiently (meaning surplus-maximizing), matching buyers and sellers, and the problem of finding a core assignment. Consider the problem of efficiently matching buyers and sellers. This is the problem of choosing a matching between buyers and sellers to maximize total surplus. P max(x ) xi,jαi,j  i,j i∈B,j∈S xi,j ≥ 0  P s.t. (∀i ∈ B) j∈S xi,j ≤ 1  P (∀j ∈ S) i∈B xi,j ≤ 1.

This is a linear program. We set up the corresponding Lagrange multiplier, and use the minmax theorem. Let ui be the Lagrange multiplier P associated to the constraint that j∈S xi,j ≤ 1. Let vj be the Lagrange P multiplier associated to i∈B xi,j ≤ 1. X X X X X L(x;(u, v)) = xi,jαi,j + ui(1 − xi,j) + vj(1 − xi,j). i∈B,j∈S i∈B j∈S j∈S i∈B

Then we have that max min L(x;(u, v)) = min max L(x;(u, v)). (xi,j ) ((ui),(vj )) ((ui),(vj )) (xi,j )

Note that X X X X X L(x;(u, v)) = xi,jαi,j + ui(1 − xi,j) + vj(1 − xi,j) i∈B,j∈S i∈B j∈S j∈S i∈B X X X = xi,j(αi,j − ui − vj) + ui + vj. i∈B,j∈S i∈B j∈S 66 ECHENIQUE

Hence, X X X max min L(x;(u, v)) = min max ui+ vj+ xi,j(αi,j−ui−vj). (xi,j ) ((ui),(vj )) ((ui),(vj )) (xi,j ) i∈B j∈S i∈B,j∈S

But then xi,j are Lagrange multipliers in the problem of minimizing P P i∈B ui + j∈S vj subject to (αi,j − ui − vj) ≤ 0. This gives us the dual to the problem we started from: P P min ui + vj  i∈B j∈S u + v ≥ α  i j i,j s.t. ui ≥ 0  vj ≥ 0

Observation 74. If xi,j solves the surplus maximization problem, and (u, v) solves the dual, then X X X αi,jxi,j = ui + vj, i,j i j and (u, v) is a core assignment. Note that this holds for any pair of solutions to the two problems. We can pair up any optimal matching with any core assignment. Observation 75. There is a solution to the surplus maximization problem in which xi,j ∈ {0, 1} and where, if xi,j = 1, then ui + vj = αi,j.

Proof. The extreme points of the set of matchings consist of matrices with 0−1 values. The primal problem is a linear programming problem, so a solution always exists that is a extreme point. As for the second statement, complementary slackness in the dual problem says that X 0 = xi,j (αi,j − ui − vj) . i∈B,j∈S | {z } ≤0

So xi,j > 0 implies that ui + vj = αi,j.

Suppose that αi,j > 0 for all i, j, and that |B| = |S|. Then the matching constraints will hold with equality at a solution, and all agents will be matched to someone. Proposition 76. Let (u, v) and (u0, v0) be assignments in the core. Let 0 0 u¯i = max{ui, ui} and vj = min{vj, vj}. Then (¯u, v) is an assignment in the core. GENERAL EQUILIBRIUM THEORY 67

Proof. Note that for any i ∈ B and j ∈ S,u ¯i + vj ≥ αi,j (for say that vj = vj thenu ¯i + vj ≥ ui + vj ≥ αi,j), andu ¯i, vj ≥ 0.

Choose one optimal matching xi,j that is an extreme point. Then 0 0 xi,j = 1 means that ui + vj = ui + vj. Thenu ¯i = ui iﬀ vj = vj. So u¯i + vj = ui + vj. Thus, X X X X X X ui+ vj = xi,j(ui+vj) = xi,j(¯ui+vj) = u¯i+ vj. i∈B j∈S i∈B,j∈S i∈B,j∈S i∈B j∈S

The last equality uses the fact that if xi,j = 0 for all j ∈ S then 0 0 ui = ui = 0, and analogously for vj, vj. Thus (¯u, v) satisﬁes the constraints of the dual program, and has the same value for the objective function.

0 An analogous result is true if we take the maximum of vj and vj and 0 the minimum of ui and ui. This is called the lattice structure of the core assignments. It means that buyers share some interests with other buyers, and sellers with other sellers. There are common interests for agents on the same side of the market, and opposing interest for agents on opposite side of the market. As a consequence we have

∗ ∗ Corollary 77. There exists core assignments (u , v∗) and (u∗, v ) such that for any core assignment (u, v),

∗ ui ≥ui ≥ u∗i ∗ vj ≥vj ≥ v∗i

∗ ∗ Think of (u , v∗) and (u∗, v ) as core assignments with minimal, respec- tively maximal, prices.

14.1. Pseudomarktes.

n Pn n 14.1.1. Notation: The simplex {x ∈ R+ : j=1 xj = 1} in R is de- n n n Pn noted by ∆ ⊆ R , while the set {x ∈ R+ : j=1 xj ≤ 1} is denoted n n by ∆− ⊆ R . When n is understood, we simply use the notation ∆ and ∆−.

n A function u : ∆− → R is 68 ECHENIQUE

• concave if, for any x, z ∈ ∆−, and λ ∈ (0, 1), λu(z) + (1 − λ)u(x) ≤ u(λz + (1 − λ)x); • quasi-concave if, for each x ∈ ∆−, the set {z ∈ ∆− : u(z) ≥ u(x)} is convex. • semi-strictly quasi-concave if, for any x, z ∈ ∆−, u(z) < u(x) and λ ∈ (0, 1) imply that u(z) < u(λz + (1 − λ)x). • expected utility if it is linear. In this case we identify u with a vector u ∈ Rn and denote u(x) as u · x. • C1 if it can be extended to a continuously diﬀerentiable function deﬁned on an open set that contains ∆−.

i A discrete allocation problem is a tuple (N,O, (u )i∈I , where N = i |O| {1, . . . , n} is a finite set of agents, O is a set of objects, and u : ∆− → R is a utility function for agent i. There is an implicit normalization in ui. We suppose that an agent P gets the outside option ∅ with probability 1 − o xo, and u takes this into account. Let L = |O|. Allocations and Pareto optimality. An allocation in Γ is a vector x ∈ LN i N i L R+ , which we write as x = (x )i=1, with x ∈ ∆−, such that X i xs = qs i∈I i for all i ∈ I and all s ∈ S. When xs ∈ {0, 1} for all i and all s, x is a deterministic allocation. The notion of efficiency comes in three flavors: An allocation x is weak Pareto optimal (wPO) if there is no allocation y such that ui(yi) > ui(xi) for all i; ε-weak Pareto optimal (ε-PO), for ε > 0, if there is no allocation y such that ui(yi) > ui(xi) + ε for all i; and Pareto optimal (PO) if there is no allocation y such that ui(yi) ≥ ui(xi) for all i and uj(yj) > uj(xj) for some j. Equilibrium. A Hylland-Zeckhauser equilibrium is a pair (x, p) N L such that x ∈ ∆− , and p = (po)o∈O ∈ R+ is a price vector such that

PN i (1) i=1 x = (1,..., 1); and (2) xi maximizes i’s utility within his budget: i i i i i i x ∈ argmax{u (z ): z ∈ ∆− and p · z ≤ α + (1 − α)p · ω };

The following important result is due to Hylland and Zeckhauser (1979). GENERAL EQUILIBRIUM THEORY 69

Theorem 78. Suppose that each ui is continuous, monotonic and quasi-concave. Then there exists a Hylland-Zeckhauser equilibrium (x, p) in which x is Pareto eﬃcient and envy-free.

Note: in this model the ﬁrst welfare theorem fails. It is possible to construct examples of HZ equilibria that are Pareto dominated. The theorem says, however, that there exists one that is Pareto eﬃcient. Let i B (p) = {x ∈ ∆− : p · x ≤ 1} di(p) = argmax{ui(x): x ∈ Bi(p)} di(p) = argmin{p · x : x ∈ di(p)} N X zi(p) = di(p) − {(1,..., 1)} and z(p) = zi(p). i=1

Letp ¯ > N. Lemma 79. di is upper hemi-continuous on [0, p¯]L

We omit the proof of Lemma 79. Consider the correspondence φ : [0, p¯]L → [0, p¯]L deﬁned by

φl(p) = {min{max{0, ζl + pl}, p¯} : ζ ∈ z(p)}. Lemma 80. φ is upper hemi-continuous, convex- and compact- valued.

The proof of Lemma 80 essentially follows from Lemma 79. By Kakutani’s ﬁxed point theorem there is p∗ ∈ [0, p¯]L with p∗ ∈ φ(p∗). We shall prove that p∗ is an equilibrium price. Note that there exists ζ ∈ z(p∗) such that ∗ ∗ (11) pl = min{max{0, ζl + pl }, p¯}. Lemma 81. p∗ · ζ ≥ 0.

∗ ∗ Proof. If p · ζ < 0 then there is some good l with pl > 0 and ζl < 0. ∗ By Equation 11, then, we cannot have pl =p ¯ because ζl < 0 and then ∗ ∗ ∗ ζl + pl < p¯. So pl = max{0, pl + ζl}, which is not possible as ζl < 0 ∗ and pl > 0. ∗ Lemma 82. pl < p¯ for all l ∈ [L] 70 ECHENIQUE

∗ Proof. Suppose towards a contradiction that there is l for which pl =p ¯. ∗ ∗ Then pl > 0, so Equation 11 means thatp ¯ ≤ ζl + pl = ζl +p ¯. Let P i i i ∗ ζ = i x − 1, with x ∈ d (p ). Note that p∗ · xi ≤ 1 for each i, so, adding over i, we obtain that ∗ P ∗ ∗ ∗ ∗ p · i xi ≤ N. Subtract p · 1 and we obtain that p · ζ ≤ N − p · 1. ∗ ∗ ∗ Now, p · 1 ≥ pl =p ¯. But we assumed thatp ¯ > N. So p · ζ < 0, contradicting Lemma 81. Lemma 83. ζ = 0

Proof. By Lemma 82 and Equation (11), ∗ ∗ (12) pl = max{0, ζl + pl } for all l ∈ [L].

Equation 12 implies two things. First, that ζl > 0 is not possible for ∗ any l. Hence ζ ≤ 0. Second, that if ζl < 0 then pl = 0.

Suppose then, towards a contradiction, that that ζl < 0 for some good ∗ l, and correspondingly that pl = 0. Now, ζl < 0 and ζ ≤ 0 means that X X X i X X i 0 > ζl = ( xl − 1) = ( xl − 1). l l i i l P i So there is some agent i for which l xl < 1. Agent i can then increase his consumption of good l without violating the constraint that con- ∗ sumption lie in ∆−. Given that pl = 0, the increase in consumption of good l would also not violate the budget constraint. So there exist a bundle in Bi(p) with strictly more of good l, and the same amount of every other good, than xi. This contradicts the strict monotonicity of i i i ∗ u , and the fact that x ∈ d (p ).

I learned this proof from Antonio Miralles (HZ’s proof seems to be in the original unpublished working paper, but I do not have a copy).

15. General equilibrium under uncertainty

Uncertainty is modeled through a set S (ﬁnite) of states of the world. If the uncertainty is over the weather in the future, S could be S = {rain, shine}. Agents trade and consume contingent goods; deﬁned as functions from S into Rn. The interpretation is that there are n physical goods, and that a contingent good x is a contract that delivers x(s) ∈ Rn upon the realization of s ∈ S. GENERAL EQUILIBRIUM THEORY 71

|S|n n|S| Note that each contingent good is a vector in R+ . A vector x ∈ R+ speciﬁes a level of consumption of each physical good conditional on a state of the world. For example if S = {s1, s2, s3} and n = 2 then the vector (2, 3, 1, 2, 2, 2) implies consumption of (2, 3) if the ﬁrst state of the world, s1, occurs, but (1, 2) if the second state of the world, s2, occurs. Let L = n|S|. An exchange economy now has the meaning that trade takes place in contingent goods.

15.1. Two-state, two-agent economy. Consider the following example: I = 2, n = 1 and S = {s1, s2}. Suppose that each agent i has preferences represented by a utility

Ui(x1, x2) = πui(x1) + (1 − π)ui(x2).

These are expected utility preferences. Suppose that ui is strictly in- 1 creasing, C , and strictly concave. Agents’ endowments are (ω1, ω2) withω ¯1 =ω ¯2. This assumption means that there is no aggregate risk (or no systemic risk). All risk is idiosyncratic. We can represent the Pareto optimal allocations in the Edgeworth box. Pareto optimality is characterized by 0 0 πu1(x1) πu2(¯ω1 − x1) 0 = 0 , (1 − π)u1(x2) (1 − π)u2(¯ω2 − x2) for interior allocations. This means that x1 = x2 for, suppose that that were not the case. Suppose wlog that x1 > x2; so we have as a consequence ofω ¯1 =ω ¯2 thatω ¯1 − x1 < ω¯2 − x2. Then the strict concavity of u1 and u2 would mean that 0 0 πu1(x1) π πu2(¯ω1 − x1) 0 < < 0 . (1 − π)u1(x2) (1 − π) (1 − π)u2(¯ω2 − x2)

That xi,1 = xi,2 means that agents will get full insurance. They are both risk averse and have the same beliefs about the states of the world. Agents have incentives to trade with each other until they achieve full insurance. There is no aggregate risk, so full insurance is feasible. At a competitive equilibrium, prices will satisfy p π 1 = , p2 (1 − π) 72 ECHENIQUE so prices will reﬂect the agents’ shared beliefs about the state of the world. Note that, while these properties are very appealing, they rest of very particular assumptions on the economy: no aggregate risk, risk aver- sion, and expected utility preferences with a common belief over the state of the world.

15.2. Pari-mutuel betting. Agents’ probabilistic assessments are aggregated into a single “odds ratio.” We can think of the pari-mutuel market as a system for aggregating information in the form of different priors. Suppose that there are I agents betting on L horses. Each agent i has beliefs over which horse will win the race in the form of a probability distribution πi ∈ ∆L. So πi,l is the probability that i assigns to the lth horse winning the race. Suppose that for each l there is i with πi,l > 0, a monotonicity condition. Agents are going to bet on horses with the objective of maximizing their expected payoff. Let βi,l be how much i bets on l (expressed in monetary units), for i = 1,...,I and l = 1,...,L. In the pari-mutuel system, if horse l wins, then the total amount of bets in the race get allocated to the agents who bid on l, and it is allocated proportional to their bets. So agent i gets a payoff that equals

I L ! βi,l X X β . PI h,k h=1 βh,l h=1 k=1 PI PL The total amount of bets are h=1 k=1 βh,l, and these are distributed proportionally among the agents who bet on horse l (if no-one bets on l this magnitude is not deﬁned). The parimutuel odds on horse l are:

" I L # 1 X X β − 1 to 1. PI h,k h=1 βh,l h=1 k=1 So, for example, if the odds on horse 3 are 5-to-1, it means that a bet of $1 dollar on horse 3 pays $6 (meaning a net gain of $5 on the $1).16

16Of course, in an actual horse race, the track’s proﬁt is ﬁrst subtracted from P P the total proceeds h k βh,k. GENERAL EQUILIBRIUM THEORY 73

We assume that agents have a budget bi > 0 for their bets, so they P are constrained by βi,l = bi (wlog we assume equality). Moreover, l P assume (as a normalization) that i bi = 1. This means that if horse PI l wins, then i gets a payoff of βi,l/ h=1 βh,l. Or that the parimutuel PI −1 odds on horse l are [( h=1 βh,l) − 1] to 1. We can translate the problem into a more familiar model if we express the agents’ decisions as the purchase of an asset, instead of as a monetary bet. Each agent i chooses a number xi,l ∈ R+ of tickets to bet on horse l. So that if horse l wins then they obtain a monetary payoff in L the amount of xi,l. The expected payoff from a vector of tickets x ∈ R+ P 17 is l πi,lxi,l.

Now, βi,l is the amount of money that i bets on l; so if pl is the price for betting on l, meaning the price of one unit of xi,l, then βi,l = plxi,l. So, now if l wins then i gets a payoﬀ equal to

βi,l = xi,l. pl PI This payoﬀ must equal βi,l/ h=1 βh,l, so we have I X pl = βh,l h=1

So we have two representations of the problem. In the ﬁrst, each agent L P P i chooses βi ∈ R to maximize πi,lβi,l/pl subject to βi,l = bi. + P l l And we know that l pl = 1. In the second representation of the L P problem we have each agent i choosing xi ∈ R to maximize πi,lxi,l P + l subject to plxi,l = bi. In this case we require that “in equilibrium” P l xi,l = 1. These are market clearing constraints for each l. They i P make sense because of horse l wins then there is 1 = i bi monetary units to be distributed among the agents.

A pari-mutuel equilibrium is a pair (β, p), where β = (βi,l) is such P 0 that βi maximizes agent i’s expected payoﬀ ( l πi,lβi,l/pl; taking p P 0 as given) subject to her budget constraint ( l βi,l = bi), and pl = PI h=1 βh,l. P P P P P Given that pl = i βi,l, we have l pl = l i βi,l = i bi = 1, so we can think of p as a vector of “market probabilities.” Moreover, recall

17The vector of tickets should not be confused with a portfolio of L Arrow- Debreu securities because the payoﬀ upon horse l winning depends on how much other agents bet on horse l. 74 ECHENIQUE

P 1−pl that the track odds are 1/ βi,l − 1 = to 1. Therefore, prices p i pl correspond one-to-one to track odds. In the second representation of the problem: Agent i’s problem is to P P maximize l πi,lxi,l subject to l plxi,l = bi. The solution is given by πi,l πi,l xi,l > 0 ⇒ θi = , where θi = max{ : 1 ≤ l ≤ L}. pl pl

Let x = (xi,l) denote a vector of ticket choices. Consider the function I L ! X X φ(x) = bi log πi,lxi,l i=1 l=1

∗ Suppose that x solves the problem of maximizing φ subject to xi,l ≥ 0 P ∗ PL ∗ and i xi,l ≤ 1. Note that x exists, and that l=1 πi,lxi,l > 0 for all i; hence

∂φ(x)) biπi,l = ∂x PL ∗ i,l x=x∗ k=1 πi,kxi,k is well deﬁned. The following result is due to Eisenberg and Gale (1959). Theorem 84. A pari-mutuel equilibrium (β, p) exists in which

∂φ(x)) (13) pl = max{ : 1 ≤ i ≤ I} ∂xi,l x=x∗ ∗ (14) βi,l = plxi,l. Moreover, equilibrium price p is unique.18

The idea in the proof is to consider the problem of allocating the vector 1 = (1,..., 1) ∈ RL among the agents i ∈ {1,...,I} so as to maximize I bi the social welfare function W (U1,...,UI ) = Π U , where Ui(xi) = P i=1 i l πi,lxi,l. The prices p support the resulting solution (xi) (and are therefore obtained as dual variables from the problem of allocating the vector 1).

Proof. Consider the choice of x = (xi) ≥ 0 in the program to maximize P φ subject to the constraint that i xi,l ≤ 1. The relevant Lagrangian

18The equilibrium βs are not generally unique. GENERAL EQUILIBRIUM THEORY 75 is

L I X X L(x; λ) = φ(x1, . . . , xI ) + λl(1 − xi,l)(15) l=1 i=1 X X X X (16) = [bi log( πi,lxi,l) − λlxi,l] + λl i l l l

Note that x∗ satisﬁes the (KKT) ﬁrst-order condition:

( ∗ ∂φ(x) biπi,l = λl if xi,l > 0 = P ∗ ∗ ∂xi,l x=x∗ l πi,lxi,l ≤ λl if xi,l = 0

So we have pl = λl when p is deﬁned as in the statement of the theorem.

∗ We shall prove that xi is utility maximizing, which will imply that ∗ (βi,l), with βi,l = plxi,l, is utility maximizing. Observe that

∗ X ∗ X ∗ X xi,lbiπi,l plxi,l = plxi,l = (P ∗ ) = bi, ∗ ∗ k πi,kxi,k l l:xi,l>0 l:xi,l>0 as x∗ satisﬁes the KKT conditions.

∗ Note that x maximizes L(x, (λl)), given λl. That’s just the standard saddle-point property of Lagrangians. Given the expression on Equa- ∗ P P tion 16, this means that xi maximizes bi log( l πi,lxi,l) − i λlxi,l. 0 PL Recall that λl = pl. So if yi is in i s budget set, we have l=1 λlyi,l ≤ PL ∗ P P ∗ l=1 λlxi,l. So we must have that l πi,lyi,l ≤ l πi,lxi,l. ∗ P ∗ Now, βi,l = λlxi,l and 1 = i xi,l implies that

X ∗ X λl = λlxi,l = βi,l. i i

We now turn to uniqueness. Suppose that (β, p) and (β0, p0) are two pari-mutuel equilibria. We shall prove that p = p0.

0 0 For each i, let µi = max{πi,l/pl : 1 ≤ l ≤ L} and µi = max{πi,l/pl : 1 ≤ l ≤ L}. Observe that βi,lµipl = βi,lπi,l because pl canceles out when βi,l > 0 (and both sides of the equation are zero otherwise). Since, 0 0 0 0 0 0 0 πi,l ≤ plµl, we obtain that βi,lµipl ≤ βi,lµipl. Similarly, βi,kµipk ≤ 0 βi,kµipk. 76 ECHENIQUE

Given that the elements of p and p0 are strictly positive, these inequalities imply that 0 0 0 0 pk 0 0 0 pl βi,lβi,kµi ≤ βi,lβi,kµi ≤ βi,lβi,kµi . pk pl 0 Cancel out µi, which is strictly positive, and summing up: 0 0 0 0 X 0 pk X 0 pk X 0 pl X pl bi βi,k = βi,lβi,k ≤ βi,lβi,k = bi βi,l pk pk pl pl k l,k l,k l

0 0 P 0 pk P pl Again, bi > 0 so β ≤ βi,l . Adding over i gives k i,k pk l pl 0 X 0 pk X 0 pk ≤ pl = 1. pk k l But then, using the Cauchy-Schwartz inequality we have that: 0 0 2 0 2 X 0 2 X √ pk 2 X X pk X (pk) 1 = ( pk) = ( pk √ ) ≤ pk √ = ≤ 1. pk pk pk k k k k k

This means that √ p0 P √k 2 ( k pk p ) k = 1, 0 2 P P pk pk √ k k pk √ 0 √ so the correlation between the vectors pk and pk/ pk is = 1. Thus √ 0 √ there is a scalar θ such that pk = θpk/ pk. Which implies that p and 0 0 p are collinear. Since they are probabilities, we must have p = p .

In a pari-mutuel market, then, equilibrium “market probabilities” are given by biπi,l pl = max{P ∗ : 1 ≤ i ≤ I}. k πi,kxi,k So that the market probability that horse l wins the race is, in a sense, the largest weighted subjective probability that some agent assigns to l winning. Example 85. Suppose that all agents agree on the probability that each horse wins, so that πi,l = πl for all i. Then in any parimutuel equilibrium (β, p) an agent i will bet only on horse for which πl/pl is maximal. This means that πl/pl = πk/pk for all l, k (otherwise we P would have βi,l = 0 for all i for some l, which by pl = i βi,l would give pl = 0, and this is not possible). Hence πl = pl, as both the (πl) and (pl) vectors add up to one. GENERAL EQUILIBRIUM THEORY 77

Example 86. Suppose that I = L = 2 and π1 = (1/2, 1/2) = (b1, b2). We claim that p = (1/2, 1/2) regardless of π2. First, if π2 = π1 we obtain the desired conclusion by the argument in Example 85. So suppose that this is not the case and wlog that π2,1 < π1,1. If p1 > p2 then both agents will only bet on horse 2 so we must have p1 < p2. This P means that agent 1 bets on 1 and agent 2 bets on 2. Now, pl = i βi,l so p1 = b1 and p2 = b2. Hence p1 = p2, a contradiction.

15.3. Arrow-Debreu and Radner equilibria. Let (i, ωi) be an exchange economy, with L = n|S| goods. We assume that each i is represented by a utility function Ui. We refer to the Walrasian equilibria of such an economy as Arrow-Debreu equilibria (and to contingent commodities as Arrow-Debreu commodities, but more about this later). Now, in general we have that events unfold over time. In an Arrow- Debreu model one would trade contingent commodities, contingent on all possible events and dates at which the events might occur. It turns out that a simple insight shows that such complexity is not really needed. Suppose that there are two time periods: t = 0 and t = 1. We can re-interpret the exchange economy model as trade in contingent goods taking place at time t = 0, and consumption taking place in t = 1. Suppose instead that at time t = 0 agents buy contingent quantities of one good only: say good 1. Then agents solve the following problem Pi: maxz ∈RS ,x ∈RL Ui(xi) (i i q · z ≤ 0 s.t. i ps · xs ≤ ps · ωi,s + p1,szi,s ∀s ∈ S

Each zi is a portfolio of state-contingent contracts for delivery of good S 1. The vector q ∈ R+ is a vector of prices at time t = 0 for the state- contingent quantities of good 1. The prices ps are called spot prices, and are valid for each of the spot markets that may open at time t = 1.

I I Deﬁnition 87. An allocation (xi)i=1 with portfolio choices (zi)i=1, spot prices (ps)s∈S, and contingent-good prices q constitute a Radner equilibrium if

(1) For each i = 1,...,I,(xi, zi) solve consumer i’s problem Pi, given (ps) and q. 78 ECHENIQUE

(2) For each s ∈ S:

I I X X xi,s = ωi,s, i=1 i=1 and PI (3) i=1 zi = 0.

Proposition 88. Suppose that each Ui is strictly monotonic.

(1) Let (x∗, p∗) be an Arrow-Debreu equilibrium. Then there is ∗ I ∗ ∗ ∗ ∗ (zi )i=1 and q such that (x , p , z , q ) is a Radner equilibrium. ∗ ∗ ∗ ∗ ∗ (2) Let (x , p , z , q ) be a Radner equilibrium, then there is µs > 0, ∗ ∗ ∗ s ∈ S, such that (x , (µsps)s∈S) is an Arrow-Debreu equilibrium.

16. Asset Pricing

We now turn to a diﬀerent two-period model. We assume consumption at both times t = 0 and t = 1. We also assume that there is a single physical good. So time is indexed by t = 0, 1. There is uncertainty at time t = 0, captured through a state space, and uncertainty is resolved at time t = 1.

• Let S = {s1, s2, . . . , sm} be a set of states. • A column vector c ∈ R1+m is called a cash ﬂow. 0 m • A column vector a = (a1, a2, . . . , am) ∈ R is called an asset, where ak is the payment of asset a (in terms of the good) in period 1 under state sk for k = 1, . . . , m. • Let {a1, a2, . . . , aJ } be a collection of J assets. Collect them all in a matrix A with j-th column equal to aj. That is: a1 a2 ··· aJ A = m×J

Examples of assets include: a risk-free asset arf = (1, 1,..., 1) ∈ Rm, AD or the Arrow-Debreu security ak = ek = (0,..., 0, 1, 0,..., 0) which delivers a unit of the good if and only if the realized state is sk.

19 Another example is an option. Suppose that sj = j, for example the state could be the value of a stock market index, and consider an

19We’re describing an option to buy, a so-called call option. An option to sell is termed a put option. GENERAL EQUILIBRIUM THEORY 79 option to buy the index at a ﬁxed “strike” price p. We can write this as an asset a = (0,..., 0, j − p, (j + 1) − p, . . . m − p), where j − p ≥ 0 and (j − 1) − p) < 0. An option to buy at price p will only be exercised when the price exceeds p, and will give a payoﬀ s − p.20

j j Let qj ∈ R+ be the price of asset a . Purchasing one unit of asset a generates the following cash ﬂow: j j j 0 1+m (−qj, a1, a2, . . . , am) ∈ R . j The cash ﬂow generated by selling one unit of asset a at price qj is: j j j 0 1+m (qj, −a1, −a2,..., −am) ∈ R .

J • Let q = (q1, . . . , qJ ) ∈ R+ be the vector of asset prices, where j qj is the price of asset a . • Deﬁne W as the following matrix: −q W = A (1+m)×J

Note that the j-th column of W , Wj, is the cash ﬂow generated by purchasing one unit of asset aj. That is:

−qj j  a1  Wj =  .   .  j am (1+m)×1 • A column vector z ∈ RJ is called a portfolio. A portfolio z generates a cash ﬂow W z. In period zero, the porfolio z pays PJ − j=1 zjqj, and in period 1 if state k ∈ {1, . . . , m} is realized, PJ j it pays j=1 zjak. That is, (17)  PJ  −q1 · · · −qj · · · −qJ  z1  − j=1 zjqj 1 j J PJ j −q a ··· a ··· a z2  zja1  W z = z =  1 1 1    =  j=1  A  ......   .   .   . . . . .   .   .  1 j J PJ j am ··· am ··· am zJ z a j=1 j m (1+m)×1

20More jargon: when s > p we say that the option is “in the money;” otherwise it is “out of the money.” 80 ECHENIQUE

We denote by hW i the set of cash flows that can be achieved through a portfolio of the assets a1, . . . , aJ at prices q. hW i = {W z : z ∈ RJ } Observation 89. hW i is a linear subspace of R1+m. It is the linear subspace generated by the columns of W . Definition 90. A market is a pair (A, q). Definition 91. An arbitrage opportunity is a cash flow c ∈ R1+m such that c > 0. We say that a market (A, q) is free of arbitrage opportunities if there is no arbitrage opportunity in hW i.

So a market is free of arbitrage opportunities iff there is no portfolio z ∈ RJ such that W z > 0. This means that there is no portfolio that generates a strictly positive payoff in some state or time, without also generating some negative payoff.

1+m • Let ω ∈ R+ be an endowment vector. • The budget set associated to market (A, q) and endowment ω is given by: B(ω, A, q) = {x ∈ R1+m : ∃ z ∈ RJ s.th. x ≤ ω + W z}. Theorem 92 (Fundamental Theorem of arbitrage pricing). Let (A, q) be a market. The following statements are equivalent:

(i) for any continuous and strictly monotonic utility function u : 1+m R+ → R the following problem has a solution (18) max u(x) subject to x ∈ B(ω, A, q); x (ii) the market (A, q) is free of arbitrage opportunities; 1+m (iii) ∃ π ∈ R++ such that πW = 0; 1+m (iv) B(ω, A, q) is compact and there is π ∈ R++ such that 1+m B(ω, A, q) ⊆ {x ∈ R+ : π · x ≤ π · ω}.

Proof. (i) ⇒ (ii). If c ∈ R1+m is an arbitrage opportunity in market (A, q), then there exists a portfolio z∗ s.th. c = W z∗ > 0. For any x = ω + W zˆ ∈ B(ω, A, q), x < x0 = x + c = ω + W (ˆz + z∗) ∈ B(ω, A, q) So u(x) < u(x0) as u is strictly monotonic. Then (18) has no solution. GENERAL EQUILIBRIUM THEORY 81

(ii) ⇒ (iii). Suppose that (A, q) admits no arbitrage opportunity. Then 1+m hW i ∩ R+ = {0}. 1+m 1+m Let ∆ ⊆ R+ be the simplex in R+ , so ( ) 1+m X ∆ = p ∈ R+ : pi = 1 . i

Then, hW i ∩ ∆ = ∅. hW i is a closed and convex set, and ∆ is compact and convex. Then, ∃ π ∈ R1+m such that (19) π · c < π · p ∀ c ∈ hW i and p ∈ ∆ by the Strict Separating-hyperplane Theorem.21 Note that 0 ∈ hW i 1+m and el (the l-th unit vector) in R is in ∆, so π · 0 < π · el = πl ∀ l. Hence, π 0. To show that πW = 0, suppose towards a contradiction that πW 6= 0. Then we can choose z ∈ RJ such that πW z > 0. Then for all α we have X απW z = πW (αz) ≤ {π · p : p ∈ ∆}. This is a contradiction because we can choose α > 0 to make απW z arbitrarily large while p 7→ π · p is a bounded function over p ∈ ∆. (iii) ⇒ (iv). Let x ∈ B(ω, A, q). Then x ≤ ω + W z ⇒ πx ≤ πω, as πW z = 0. Therefore, B(ω, A, q) ⊆ {x ∈ R1+m : πx ≤ πω}. The right-hand side set of the above equation is a standard “Arrow- Debreu budget set,” so it is compact. B(ω, A, q) is a closed subset of a compact set, so it is compact. (iv) ⇒ (i). If the budget set B(ω, A, q) is compact, then problem (18) has a solution because u is continuous.

21Compare this version of the Separating-hyperplane Theorem with the one given in Lemma 15. Note that it is necessary that at least one of the sets is compact. Consider X = {(x, y) ∈ R2 : x ≥ 0, y ≥ 1/x} and Z = {(x, y) ∈ R2 : y ≤ 0}. 82 ECHENIQUE

16.1. Digression: Farkas Lemma. In the proof of Theorem 92 we basically proved the following useful result. Lemma 93. Let W be a (n × m) matrix. Then one and only one of the following statements is true.

(1) There is z ∈ Rm such that W z > 0. n (2) There is π ∈ R++ such that πW = 0.

Proof. In the proof of the theorem, we proved that (1) being false implies that (2) must be true. All that is left is to observe that (1) and (2) cannot hold at the same time because it would lead to 0 = πW z > 0, which is absurd.

16.2. State prices. The vector π in the theorem may be intriguing. It has the interpretation as a price vector: in fact as a vector of Arrow- Debreu prices.

−q1 · · · −qj · · · −qJ  1 j J −q  a1 ··· a1 ··· a1  πW = π = π  . . . . .  = 0 A  ......  (1+m)×J 1 j J am ··· am ··· am

Therefore, for every j = 1,...,J,

−qj  j m a1 X (π , π , . . . , π )   = π (−q ) + π aj = 0. 0 1 m  .  0 j i i  .  i=1 J am We may write m X πi (20) q = aj ∀ j = 1, . . . , J. j π i i=1 0

Hence, we may write the price of asset j, qj, at time t = 0 as a weighted j j sum of it future payments under the different states a1, . . . , am. The j πi weight on the payoff a in state si, is . i π0 Normalize the vector π by defining π π π π¯ = 1, 1 , 2 ,..., m . π0 π0 π0 πW = 0 implies that π W =πW ¯ = 0. π0 GENERAL EQUILIBRIUM THEORY 83

Therefore, rewrite (20) as:

m X j (21) qj = π¯iai ∀ j = 1, . . . , J. i=1 m From the expression above it becomes clear thatπ ¯ ∈ ∆ ⊆ R++ is a price vector. The price of a unit of good (of money, say) at time t = 0 is 1. The price of a unit of good at time t = 1 conditional on state si occurring isπ ¯i. We refer toπ ¯ as a normalized state price vector, and call the expression for qj given in (21) the present value price of asset aj. The state price vectorπ ¯ allows us to price assets. For example, let yi ∈ R for every state i = 1, . . . , m, and deﬁne the column vector 0 m y˜ = (y1, y2, . . . , ym) ∈ R , so thaty ˜ is an asset. We can useπ ¯ to Pm calculate the price of assety ˜ as: i=1 π¯iyi. A risk-free asset is arf = (1,..., 1) ∈ Rm. In a market with no arbitrage, then, the price of the risk free asset is given by: m rf X (22) q = π¯i. i=1

Deﬁne the rate of return of the risk free asset as 1 1 (23) Rrf = = . rf Pm q i=1 π¯i The rate Rrf is known as the risk-free rate.

16.2.1. A risk-neutral probability measure. For any asset j, the expected rate of return on j is the expected value of the random j j j ˜j 22 variable s 7→ as/q , which we denote by (Rs)s∈S, or by R . The expectation of R˜j depends of course on the probability measure used. It turns out that a particular probability measure is useful. Let

π¯i pi = Pm , so that p = (p1, . . . , pm) ∈ ∆ h=1 π¯h is a probability distribution over S. The probability distribution p is termed the risk free probability measure. The name comes from

22 A random variable s 7→ Xs is denoted by X˜. 84 ECHENIQUE the following calculation: m qj X = p aj = E a˜j. qrf h h p h=1

Thus, m 1 X aj Rrf = = p s = E R˜j. qrf i qj p s i=1 So the expected return of asset j, when the expectation is calculated using risk-neutral probabilities, equals the risk-free rate.

16.3. Application: Put-call parity. An option to buy an asset aj at a price K is called a call option. The price K is called the strike price. Since the option is exercised only when proﬁtable, i.e., if the market j j + price is higer than K, the call option on a has payoﬀ (ai − K) in state i = 1, . . . , m.23 Hence, the price of the call option is given by: m c X j + q = π¯i(ai − K) . i=1

An option to sell an asset aj at strike price K is called a put option. Since the option is exercised only when proﬁtable, i.e., if the market j j + price is lower than K, the put option on a has payoﬀ (K − ai ) in state i = 1, . . . , m. Hence, the price of the put option is given by: m p X j + q = π¯i(K − ai ) . i=1

Note that m m c p X j + X j + q − q = π¯i(ai − K) − π¯i(K − ai ) i=1 i=1 m X j = π¯i(ai − K) i=1 m j X = q − π¯iK i=1 = qj − Kqrf .

23For any x ∈ R, x+ = max{x, 0}. GENERAL EQUILIBRIUM THEORY 85

Then, we obtain: (24) qj − Kqrf = qc − qp. Equation (24) is called put-call parity. The left hand side of (24) is the payoﬀ from selling one unit of asset j and borrowing K dollars at the risk-free rate. The right-hand-side is the expense that results from purchasing a call option and selling a put option for the same strike price of K. No-arbitrage demands that these two quantities must be the same because they result in the same state-contingent payoﬀs.

16.4. Market incompleteness. Let (q, A) be a financial market, and define the matrix W as before. Suppose that the market is free of arbitrage. Definition 94. The market (q, A) is complete if dim(hW i) = |S| = m. Otherwise say that the market (q, A) is incomplete.

When a market is complete, agents can use the assets to carry out transfers of the good (of “money,” or “wealth”) across states. Proposition 95. The market (q, A) is complete iﬀ dim(hAi) = m.

Proof. When (q, A) is free of arbitrage, q =π ¯ · A. So q is a linear combination of the rows of A. Therefore, A and W have the same row rank.

The importance of this proposition is that market completeness is a matter of A, which is exogenous. Not prices q. In a more general model with more than one good, market completeness is however dependent on prices. When the market is free of arbitrage, then πW = 0 implies that π ∈ hW i⊥, so that hW i⊥ 6= ∅. Since π 6= 0, dim(hW i⊥) ≥ 1. Observation 96. A market is complete iﬀ there is a unique (up to a scalar multiple) solution to the system of equations πW = 0. The uniqueness is unique up to a scalar multiple: For example, normalizing M π0 = 1, when the market is complete there is a uniqueπ ¯ ∈ R++ with q = πA.

To see why Observation 96 is true: Let hW i⊥ denote the set of vectors orthogonal to the vectors in hW i, called the orthogonal complement of hW i: hW i⊥ = {z ∈ R1+m : z · y = 0 for all y ∈ hW i}. 86 ECHENIQUE

Note that R1+m = hW i + hW i⊥. Given lack of arbitrage, dim(hW i⊥) ≥ 1. The subspace hW i⊥ has dimension 1 iff the dimension of hW i is m. And π is unique (up to a scalar multiple) iff hW i⊥ has dimension 1. Uniqueness of the state prices that result from the “fundamental theorem” is a matter of market completeness. For this reason, Observa- tion 96 is some times called the “second fundamental theorem of asset pricing.” Theorem 97. In the one-good, two-period model with C∞ utilities. If the market is incomplete then there is a set of vectors of endowments of measure zero such that for all endowments outside of this set, every equilibrium allocation is Pareto inefficient.

The theorem is stated informally. Since market completeness in this model is a matter of A, not q, we can make sense of incompleteness in- dependently of prices (which are endogenous, of course, and determined in equilibrium). A natural question to ask is about Pareto efficiency constrained by the transfers across states that are possible given the asset structure A. It turns out that, in the one-good model, equilibrium allocations are constrained Pareto efficient, but this is no longer true when we allow for multiple goods. In fact, generically, equilibrium allocations are constrained Pareto inefficient.

16.5. The Capital Asset Pricing Model (CAPM). The tradi- tional CAPM is : ER˜j = Rrf + β(ER˜m − Rrf ), where Rm is the “market” rate of return. In practice, Rm is taken to be the return of some market index such as the SP500. The CAPM equation is a linear regression: Cov(R˜j, R˜m) β = . V(R˜m)

The CAPM means that the expected return of an asset j is given by the risk-free rate + a risk premium β(ERm − Rrf ) that depends on the “beta” of the asset j. The beta of an asset depends on how closely it is related to the market returns. An asset that varies closely with the GENERAL EQUILIBRIUM THEORY 87 market returns (has high beta) has high systemic risk and therefore commands a larger risk premium. The idea behind the CAPM is that optimally choosing a portfolio will allow an agent to fully diversify all risk that is idiosyncratic to the asset. So there is no risk premium for such “diversifiable” risk. Systemic risk cannot be diversified away, and it is reflected in the expected return of the asset.

16.5.1. A CAPM formula resulting from no arbitrage. The expected returns in the CAPM are calculated according to some given probability distribution over states (not necessarily the risk free measure). Let pˆ ∈ ∆ be a probability measure on {1, . . . , m}. Then n n n X X π¯i X (25) qj = π¯ aj = pˆ aj = θ p aj = E θã˜j. i i pˆ i i i i i pˆ i=1 i=1 i i=1 We use θ˜ to denote the random variable i 7→ π¯i . pî Equation 25 is important. The random variable θ˜ is a stochastic discount factor. It says how a payoff in each state should be “discounted” so as to give the correct, arbitrage free, price qj. We see many instances of such stochastic discount factors in asset pricing, this is perhaps the simplest one. ˜ ˜ ˜ ˜ For any two random variables X and Y , we have that Covpˆ(X, Y ) = ˜ ˜ ˜ ˜ EpˆXY − EpˆXEpˆY . So, j ˜ j ˜ j ˜ j q = Epˆθa˜ = Covpˆ(θ, a˜ ) + EpˆθEpâ˜ . ˜ P rf rf Now, note that Epˆθ = i π¯i = q = 1/R . Then, E a˜j Rrf (26) E R˜j = pˆ = Rrf − Cov (θ,˜ a˜j) = Rrf − Cov (˜ν, R˜j) pˆ qj qj pˆ pˆ (whereν ˜ = Rrf θ˜). Equation (26) is a “CAPM-like” formula. It says that the expected return on an asset j equals the risk-free rate of return plus a term that depends on the correlation of the asset returns with a “market variable”ν ˜. The variableν ˜ is common to all assets, that is why it is a market variable. 88 ECHENIQUE

16.6. Consumption CAPM. Consider a consumer with income I, who can invest in J assets. The (random) rate of return on asset j is R˜j. Asset 1 is a risk-free asset.

Let x0 denote consumption (of “money”) on date 0. So that the agent uses I − x0 to invest in assets for consumption on date 1. Investment in asset j is j zj = (I − xo)η , j J J P j where η = (η )j=1 ∈ R+ with j η = 1. Then the random payoﬀ of a portfolio deﬁned by η is

J " J # X j ˜j 1 X j ˜j 1 x˜1 = (I − xo) η R = (I − xo) R + η (R − R ) . j=1 j=2

The consumer’s problem is n h io 1 PJ j ˜J 1 max u(x0) + δEu (I − xo) R + j=2 η (R − R )  0 ≤ x ≤ I  0 st ηj ≥ 0 PJ j  j=2 η ≤ 1

Suppose that u : R+ → R is smooth, monotonic and concave. Sup- pose also that we can focus on interior solutions. Then the ﬁrst order conditions that characterize a solution are:

0 0 ˜ u (x0) = δEu (˜x1)R

˜ 1 P j ˜j 1 where R = R + j η (R − R ), and

0 ˜j 1 δEu (˜x1)(I − x0)(R − R ) = 0, j = 2, . . . , J.

0 0 ˜ The first equality, u (x0) = δEu (˜x1)R is a so-called Euler equation, describing the intertemporal trade-off between consumption and saving. The second collection of equations serve to give a CAPM formula for the returns of each of the J − 1 risky assets. Specifically, since δ > 0 and (I − x0) > 0 we have that

0 ˜j 1 Eu (˜x1)(R − R ) = 0, j = 2, . . . , J. GENERAL EQUILIBRIUM THEORY 89

Recall the properties of the covariance: 0 ˜j 1 0 = Eu (˜x1)(R − R ) 0 ˜j 1 0 ˜j 1 = Cov(u (˜x1), R − R ) + Eu (˜x1)E(R − R ) 0 ˜j 0 ˜j 1 = Cov(u (˜x1), R ) + Eu (˜x1)E(R − R )

Then ˜j 1 1 0 ˜j ER = R − 0 Cov(u (˜x1), R ). Eu (˜x1)

If asset j is positively correlated with consumption then it is negatively correlated with marginal utility (concavity of u). So it commands a positive “risk premium” and ER˜j > R1.

16.7. Lucas Tree Model. We turn to a one-agent exchange economy with many goods. Specifically, consumption occurs over time, and it is uncertain. Endowments are also stochastic, and arrive over time. The problem is to characterize prices that support the autarky equilibrium where the agent consumes the endowment. There is a single good, “fruit,” in each period, and a single asset: a “tree.” The tree pays off “dividends,” a random production of fruit in every period. Time is infinite, and ranges from t = 0, 1,... In period t, the production of fruit is realized, and a spot market opens up in fruit. The consumer can therefore sell and purchase fruit in the spot market. She can also buy trees. We shall normalize the price of fruit in each spot market to be 1, and determine the price qt of trees in period t. To sum up, in period t the consumer has an income derived from her holdings of trees, and how much fruit these have produced. If each tree has a dividend dt and she holds st trees, then her income in period t is wt = st(qt + dt). This income can be used to purchase fruit (at a price of 1) for consumption, ct, and trees. If she buys st+1 trees then she spends at = st+1qt on trees. So her budget constraint for period t is ct + at ≤ wt. The rate of return on trees is

qt+1 + dt+1 Rt+1 = , qt composed of a capital gain component qt+1/qt, and a dividend payoﬀ. Note that Rt+1at = st+1(qt+1 + dt+1) = wt+1 90 ECHENIQUE

The consumer seeks to maximize the expected discounted sum of per- period utility. The consumer has a utilty function u : R+ → R over fruit. Suppose that u is C1, strictly increasing and concave. The consumer has a discount factor δ ∈ (0, 1).

P∞ t MaxE δ u(ct) ( t=0 c + a ≤ w s.t t t t wt+1 = Rt+1at

Assume that {Rt} follows a Markov process. The Bellman equation is v(w, R) = sup{u(c) + δE[v(Ra,˜ R˜)|R]: c + a = w}.

Consider the problem on the right-hand-side of the Bellman equation max u(w − a) + δE[v(Ra,˜ R˜)|R] s.t. 0 ≤ a ≤ w Suppose that the solution is interior and that the value function v is 0 ∂v(w,R) diﬀerentiable. Use v1 to denote ∂w . Then the ﬁrst-order condition for a maximum in the right-hand side of the Bellman equation is 0 0 ˜ ˜ ˜ u (w − a) = δE[v1(Ra, R)R|R]. Moreover, the the envelope theorem gives us that

0 0 v1(w, R) = u (w − a).

Thus, along an optimal solution we must have that u0(c) = δE[u0(˜c)R˜|R].

Speciﬁcally, if {ct} is an optimal path we must have that

0 0 (27) u (ct) = δEt[u (ct+1)Rt+1], where Et means expectation conditional of Rt. Equation (27) is an Euler equation. We can write the Euler equation as 0 u (ct+1) 1 = δEt 0 Rt+1, u (ct) GENERAL EQUILIBRIUM THEORY 91 and the Euler equation implies that 0 u (ct+1) qt = Et δ 0 (qt+1 + dt+1). u (ct)

| {zt+1 } Mt t+1 Here Mt is the stochastic discount factor for consumption in period t + 1 at time t. Then

t+2 t+2 Etqt+1 = EtEt+1Mt+1 (qt+2 + dt+2) = EtMt+1 (qt+2 + dt+2) by the “law of iterated expectations.” Note also that if we deﬁne t+τ τ 0 0 t+1 t+2 t+2 Mt = δ u (ct+τ )/u (ct) then Mt Mt+1 = Mt . Hence we obtain that

t+1 t+2 t+1 qt = EtMt Mt+1 (qt+2 + dt+2) + EtMt dt+1) t+2 t+1 = EtMt (qt+2 + dt+2) + EtMt dt+1) t+2 t+2 t+1 = EtMt qt+2 + EtMt dt+2 + EtMt dt+1).

Continuing in this fashion we obtain that

T X t+τ t+T qt = Et Mt dt+τ + EtMt qt+T . τ=1 t+T If we assume that limT →∞ EtMt qt+T = 0, which means ruling out t+τ “bubbles” where agents do not expect at time t that the process {Mt qt+τ } converges to zero, then we obtain that

T ∞ X t+τ X t+τ qt = lim Et Mt dt+τ = Et Mt dt+τ , T →∞ τ=1 τ=1 assuming that we can pass the limit inside the expectation. Thus the price of the tree in period t is the discounted expected sum of future dividends. The discount factor is a subjective stochastic discount factor; it depends on marginal utility and pure time discounting. Consider now an equilibrium of this economy. There is a single tree and a single consumer. So equilibrium (market clearing) in the period t spot market requires that ct = dt and st = 1. Hence we have that

∞ 0 X u (dt+τ ) q = E δτ d . t t u0(d ) t+τ τ=1 t 92 ECHENIQUE

16.8. Risk-neutral consumer and the martingale property. The 0 0 consumer is risk-neutral when u is linear, so that u (ct+1)/u (ct) = 1. Then we have

(28) qt = Etδ(qt+1 + dt+1).

So adjusting for dividends and time-discounting, the price of trees follows a martingale process. This is the eﬃcient markets hypothesis. As we shall see next, the martingale property holds in the absence of risk neutrality, using the risk-free equivalent measure to calculate expectations (or the equivalent martingale measure).

16.9. Finite state space. We can simplify the calculations a bit, and gain some intuition from considering the model with a ﬁnite set of possi- 1 L ble dividends. Suppose that dt takes values in the set D = {d , . . . , d }. j When dt = d then dt+1 is drawn from D according to the probability l j distribution Pj = (Pj,1,...,Pj,L). So Pr(dt+1 = d |d + t = d ) = Pi,l. In this case we can write the price of trees as a function of the value of dt, and we obtain that

L X u0(dl) q(dj) = P δ (q(dl) + dl). j,l u0(dj) l=1

¯ Deﬁne a new probability Pj by P δu0(dl)/u0(dj) P u0(dl) P¯ = j,l = j,l j,l P 0 k 0 j P 0 k k Pj,kδu (d )/u (d ) k Pj,ku (d ) ¯ and denote expectation with respect to this probability by Ej. Let P 0 k 0 j −1 Rj = ( k Pj,kδu (d )/u (d )) : think of Rj as the rate of return on a risk-less asset (one that pays one unit of fruit in the next period, regardless of the state). Then we have j −1 ¯ ˜l ˜l q(d ) = Rj Ej(q(d ) + d ). Note that this is an equation like (28), expressing that the dividend- adjusted price follows a martingale-type property. ¯ Now the probabilities Pj,l deﬁne a probability measure over sequences h l of dividends. For example the probability that dt+2 = d and dt+1 = d GENERAL EQUILIBRIUM THEORY 93

j ¯ ¯ given that dt = d is Pj,lPl,h. And the marginal probability that dt+2 = h j d given that dt = d is 0 l 0 h X X Pj,lu (d ) Pl,hu (d ) P¯2 = P¯ P¯ = j,h j,l l,h P 0 k P 0 k Pj,ku (d ) Pl,ku (d ) l l k k

∗ Let Pj be the resulting probability measure on sequences of dividends j dt+1, dt+2,..., given that dt = d ; and denote its expectation operator ∗ by Ej .

j Then if dt = d we have that the “cum dividend” price of the tree is j j ¯ −1 ∗ −1 −1 qˆt = q(dt) + dt = d + q(d ) = EjRj qˆt+1 = Et Rt Rt+1qˆt+2.

Hence, for any τ > 1,

∗ qˆt+τ qˆt − dt = Et τ−1 . Πt0=tRt0 So that the price of the tree is equal to the expected discounted future price at t + τ, when the expectation is calculated according to the equivalent martingale measure. In other words, using the equivalent martingale probability measure, discounted prices follow a martingale process.

16.10. Logarithmic utility. Macroeconomists like to consider the case when u(c) = ln c so that 0 u (dt+τ ) dt 0 dt+τ = dt+τ = dt. u (dt) dt+τ Hence, ∞ X δ q = d E δτ = d t t t 1 − δ t τ=1

So the dividend-price ratio dt/qt = (1 − δ)/δ is constant.

16.11. CRRA utility. Another common assumption is that u is of the CRRA form: c1−σ − 1 u(c) = . 1 − σ 0 0 σ In this case, u (ct+τ )/u (ct) = (ct/ct+τ ) . Therefore, ∞ X dt (29) q = E δτ ( )σd . t t d t+τ τ=1 t+τ 94 ECHENIQUE

¯ 1−σ Suppose that {dt} is iid and that d = Etdt+τ . Then, ∞ qt X δ = E δτ d1−σ = d.¯ dσ t t+τ 1 − δ t τ=1

1−σ Suppose that dt is iid log-normal with mean = 1. Then Etdt+τ = X 1−σ (1−σ)X 2 2 E(e ) = Ee , where X ∼ N(µ, ν ) with µ = −ν /2 so that dt has mean = 1. Hence,

1−σ 2 2 Etdt+τ = exp((1 − σ)µ + (1/2)(1 − σ) ν ) = exp(−(1 − σ)ν2/2 + (1/2)(1 − σ)2ν2) = exp(−σ(1 − σ)ν2/2). Thus qt δ σ(σ−1)ν2/2 σ = e , dt 1 − δ and therefore

2 log qt = σ log dt + σ(σ − 1)ν /2 + log(δ/(1 − δ)) and we have 2 V log qt = σ V log dt. This means that (log) prices will be more volatile than (log) dividends if and only if σ > 1. Now suppose instead that dividends follow a random walk. Speciﬁcally suppose that the log dt+1 = log dt + εt, where {εt} are iid normally 2 distributed with mean zero and variance ν . Then dt+1/dt is iid log- ¯ normal with mean 1. Hence d = Edt+1/dt. From the Euler equation we have that

dt σ qt = δEt( ) (dt+1 + qt+1), dt+1 So the price-dividend ratio obeys

qt dt σ−1 dt σ−1 qt+1 (30) = δEt( ) + δEt( ) . dt dt+1 dt+1 dt+1 Now, as before, we have that d d E( t )σ−1 = E( t+1 )1−σ = δ exp(−σ(1 − σ)ν2/2) dt+1 dt GENERAL EQUILIBRIUM THEORY 95

We conjecture that there is an equilibrium in which the price-dividend ratio is a constant γ. Then Equation (30) implies that

σ(σ−1)ν2/2 2 δe γ = δeσ(σ−1)ν /2(1 + γ) ⇒ γ = . 1 + δeσ(σ−1)ν2/2

So we get a similar relation between the log of price and dividend:

2 σ(σ−1)ν2/2 log qt = log dt + σ(σ − 1)ν /2 + log δ − log(1 + δe )

16.11.1. Government expenditure and taxes. Suppose that goverment expenditure is a fraction of dividends, so that gt = εtdt, and {εt} follows a Markov process with εt ∈ (0, 1). Government expenditure is ﬁnanced with a lump-sum tax Tt levied on the cosumer, so that the consumer’s L budget constraint becomes ct + at ≤ wt − Tt. Let qt be the resulting price on trees. Now market clearing demands that ct + gt = dt, and thus ct = (1 − εt)dt. Then the Euler equation determines that

0 ∞ 0 u ((1 − εt+1)dt+1) X u ((1 − εt+τ )dt+τ ) qL = δE (qL +d ) = E δτ d t t u0((1 − ε )d ) t+1 t+1 t u0((1 − ε )d ) t+τ t t τ=1 t t

Suppose that the consumer has logarithmic utility u(c) = log c. Then we obtain that

∞ ∞ X (1 − εt)dt X 1 qL = E δτ d = (1 − ε )d E δτ t t (1 − ε )d t+τ t t t (1 − ε ) τ=1 t+τ t+τ τ=1 t+τ

Now suppose in contrast that government expenditure is ﬁnanced through A a tax on trees. Let the resulting price on trees be qt Then the con- A sumer’s income becomes wt = st((qt − Tt) + dt) and the budget constraint ct + at ≤ wt. The rate of return on trees, however, must factor in the tax payments for owners of trees. Hence,

A qt+1 − Tt+1 + dt+1 Rt+1 = A . qt 96 ECHENIQUE

We need Tt = gt = εtdt. So the asset pricing formula (with logarithmic utility) becomes

A ct A qt = δEt (qt+1 − Tt+1 + dt+1) ct+1 ct A = δEt (qt+1 + (1 − εt+1)dt+1) ct+1

(1 − εt)dt A = δEt (qt+1 + (1 − εt+1)dt+1) (1 − εt+1)dt+1 ∞ X (1 − εt)dt = E δτ (1 − ε )d t (1 − ε )d t+τ t+τ τ=1 t+τ t+τ δ = (1 − ε )d 1 − δ t t

Note that εt ∈ (0, 1) implies that X 1 X δ δτ E > δτ = . t 1 − ε 1 − δ τ t τ L A Hence, qt > qt . This diﬀerence in price is purely an accounting exercise. In either case, consumption is (1 − εt)dt as government expenditure “crowds out” private consumption. For the lump sum tax, income is directly adjusted by taxes. For the tax on trees, income has to be adjusted by making trees cheaper.

17. Large economies and approximate equilibrium

17.1. The Shapley-Folkman theorem. Suppose a set of sheep and goats are out to pasture on a field. If, for every set of four animals, there is a (straight) fence separating the goats from the sheep, then there is a fence that separates all the sheep from all the goats. This bucolic theorem about separating sheep and goats is an example of a “combinatorial” separation property. The next two results are general combinatorial separation results. The first (which we are not going to prove) is called Kirchberger’s theorem. The second is called Radon’s theorem: it is very simple and highlights the main idea behind these results. Theorem 98. Kirchberger’s theorem] Let A and B be finite subsets of Rn. If, for any set C of cardinality at most n + 2, A ∩ C and B ∩ C GENERAL EQUILIBRIUM THEORY 97 can be strictly separated by a hyperplane, then A and B can be strictly separated by a hyperplane. Theorem 99 (Radon’s theorem). Let A ⊆ Rn be a finite set with cardinality at least n + 2. Then A can be partitioned into A1 and A2 such that

cvh(A1) ∩ cvh(A2) 6= ∅

Proof. Suppose wlog that A has cardinality n+2 and let A = {a0, . . . , an+1}. Consider the set {a1 − a0, . . . , an+1 − a0}. Such a set cannot be linearly independent, so there exists λ1, . . . , λn+1, not all zero, for which Pn+1 Pn+1 i=1 λi(ai − a0) = 0. Deﬁne λ0 = − i=1 λi, I = {i ∈ {0, . . . , n + 1} : λi > 0} and J = {i ∈ {0, . . . , n + 1} : λi < 0}. These two sets are Pn+1 nonempty because not all λi are zero, and i=0 λi = 0. Moreoever, Pn+1 P P Pn+1 λi = 0 implies that λi = − λi, and λi(ai −a0) = i=0 P P i∈I i∈J i=1 0 that i∈I λiai = − i∈J λiai.

Now let A1 = {ai : i ∈ I or λi = 0} and A2 = {ai : i ∈ J}. Thus

X λi X −λi X λi P ai = P ai = P ai ∈ cvh(A1)∩cvh(A2). λj − λj λj i∈I j∈I i∈J j∈j i∈J j∈j Theorem 100 (Caratheodory). Let A ⊆ Rn and x ∈ cvh(A). Then there are a1, . . . , an+1 ∈ A such that x ∈ cvh({a1, . . . , an+1}). n Theorem 101 (Shapley-Folkman). Let A1,...,AK ⊆ R and

x ∈ cvh(A1 + ... + AK ).

Then there is ai ∈ cvh(Ai), i = 1,...,K, such that x = a1 + ... + aK and ai ∈ Ai for all but at most n values of i.

The next lemma and its use in proving these theorems is taken from Bob Anderson’s lecture notes.

n Lemma 102. Let A1,...,AK ⊆ R and

x ∈ cvh(A1 + ... + AK ). Then K m X Xi x = λi,jai,j, i=1 j=0

Pmi PK where ai,j ∈ Ai for all i, j, λi,j > 0, j=0 λi,j = 1, and i=1 mi ≤ n. 98 ECHENIQUE P Observe that Lemma 102 says that x = i a¯i, wherea ¯i ∈ cvh(Ai), but it bounds the number of points in the support of the K convex mi combinations (λi,j)j=0.

Proof of Caratheodory’s theorem. Let x ∈ cvh(A). By Lemma 102 with K = 1 there is a0, a1, . . . , am ∈ A and λ0, λ1, . . . , λm with x = Pm j=0 λjaj and m ≤ n.

Proof of the Shapley-Folkman theorem. Let ai,j, λi,j i = 1,...,K, j = 1, . . . , mi be as in the statement of Lemma 102. mi is an integer. So PK mi > 0 means that mi ≥ 1. Thus, i=1 mi ≤ n means that mi = 0 for all but at most n values of i ∈ {1,...,K}. In mi = 0 then λ0 = 1. Thus m X X Xi x = ai,0 + λi,jai,j .

i:mi=0 i:mi≥1 j=0 | {z } ∈cvh(Ai)

The proof of Lemma 102 relies on familiar ideas. A large number of vectors in Rn cannot be linearly independent, so any linear combination of a large number of vector can be simpliﬁed. In the lemma, we talk about convex combinations, so the relevant concept is that of “aﬃne independence:” we translate the vectors ai,j and work with the fact that the collection {ai,j − ai,0 : i = 1, . . . , K, j = 1, . . . , mi} cannot be P linearly independent when i mi > n. P P Proof. Let x ∈ cvh( i Ai). Then there is λj ∈ (0, 1) and xj ∈ i Ai, Pm P with j = 0, . . . , m, such that x = j=0 λjxj and j λj = 1. Moreover, PK for each j there is ai,j ∈ Ai, i = 1,...,K, and i=1 ai,j = xj. Thus, K m X Xi (31) x = λi,jai,j i=1 j=0 where mi = m and λi,j = λj for all i, j. Now, we shall prove that P for each representation of x in the form of (31) with i mi > n there exists another representation of x in the form of (31) with a strictly P smaller value of i mi. P Suppose then that i mi > n. Consider the set of vectors

{ai,j − ai,0 : 1 ≤ i ≤ K and 1 ≤ j ≤ mi}. GENERAL EQUILIBRIUM THEORY 99

n P This subset of R has cardinality i mi > n. Therefore, it is linearly dependent.

Let βi,j, not all zero, be such that K m X Xi 0 = βi,j(ai,j − ai,0). i=1 j=1 Then, for any t, K m K m X Xi X Xi x = λi,jai,j + t βi,j(ai,j − ai,0) i=1 j=0 i=1 j=1 K " m m # X Xi Xi = (λi,j + tβi,j)ai,j + (λi,0 − t βi,j)ai,0 i=1 j=1 j=1

∗ ∗ Pmi Let λi,j(t) = λi,j + tβi,j and λi,0(t) = λi,0 − t j=1 βi,j, and observe Pmi ∗ Pmi Pmi that j=0 λi,j(t) = j=1(λi,j + tβi,j) + λi,0 − t j=1 βi,j = 1, and that ∗ λi,j(0) = λi,j > 0.

Now note that the βi,j are not all zero, so that there must exist at ∗ least one λi,j(t) that is strictly monotone (increasing or decreasing) as Pmi ∗ a linear function of t. Since j=0 λi,j(t) = 1 for all t, there must in ∗ ∗ fact exist (i, j) for which λi,j(t) is strictly monotone decreasing: λi,j(t) ∗ is a linear function of t, so there is t for which λi,j(t) = 0. Finally, ∗ ˆ 0 0 λi,j(0) > 0 for all (i, j) means that there exists t and some (i , j ) for ∗ ˆ ∗ ˆ which λi0,j0 (t) = 0 and λi,j(t) ≥ 0 for all (i, j).

ˆ ˆ i=K,j=m ˆ i ∗ ˆ Given such t, deﬁne now new coeﬃcients (λi,j)i=1,j=0 from λi,j(t) by keeping only the ones that are strictly positive. This gives us PK Pk i=1 mˆ i < i=1 mi.

Finally, I want to state the following approximate version of Caratheodory due to Sid Barman. Remarkably, this approximation does not depend on n.

n Theorem 103. Let x ∈ cvh({x1, . . . , xK }) ⊆ R , ε > 0 and p an integer with 2 ≤ p < ∞. Let γ = max{kxkkp : 1 ≤ k ≤ K}. Then there is a vector x0 that is a convex combination of at most 4pγ2 ε 0 of the vectors x1, . . . , xK such that kx − x kp < ε. 100 ECHENIQUE

17.2. Approximate equilibrium. As a simple consequence of the Shapley-Folkman theorem we obtain a result on the existence of approximate Walrasian equilibria. This says that, when preferences are not necessarily convex, we can obtain an approximate equilibrium where demand equals supply and “most” agents (when I is large relative to L) are optimizing.

I Theorem 104. Let E = (i, ωi)i=1 be an exchange economy with L goods. Suppose that each preference i is continuous and strictly mono- L I IL tone, and that ω¯ 0. Then there exists p ∈ R++ and (xi)i=1 ∈ R+ such that:

(1) p · xi ≤ p · ωi for all i. PI PI (2) i=1 xi = i=1 ωi. (3) The property

0 0 xi i xi ⇒ p · xi > p · ωi holds for all but at most L consumers.

Proof. Under the assumptions we have made, the aggregate excess de- ∗ mand correspondence of each consumer i, zi , is well deﬁned. For each L ∗ L ∗ p ∈ R++, zi (p) ⊆ R is nonempty and compact, and p 7→ zi (p) is ∗ upper hemi-continuous. Moreover, p · ζi = 0 for all ζi ∈ zi (p) (Walras’ ∗ Law). The sets zi (p) are bounded below by −1kω¯k∞. ∗ P ∗ Let z = i zi be the aggregate excess demand correspondence of the economy E. It follows from the standard argument that if pn → p¯ 6= 0 withp ¯l = 0 for some l, then

∗ n ∞ = lim inf{kζik∞ : ζi ∈ zi (p )}. n→∞ Thus, the assumptions needed for existence of a competitive equilibrium are satisﬁed by z∗, with the exception of the convexity of the set z∗(p). The correspondence p 7→ cvhz∗(p) is obviously convex-valued. It turns out that it continues to satisfy the properties mentioned above (I omit the proof, which is not diﬃcult), So, by the standard existence theorem (in its correspondence form) the exists p∗ with 0 ∈ cvh(z∗(p∗)).

∗ ∗ By the Shapley-Folkman theorem, there exists ζi ∈ cvh(zi (p )), i = ∗ ∗ 1,...,I, with ζi ∈ zi (p ) for all but at most L values of i, such that P ∗ 0 = i ζi. Deﬁne xi = ζi + ωi. GENERAL EQUILIBRIUM THEORY 101

We could get a variation on this result by taking the agents who are not optimizing and instead giving them a bundle in their demand cor- respondences. If we can bound the diﬀerence between any point in the demand correspondence of agent i, and in its convex hull, then we can obtain a result on the existence of an approximate equilibrium where all agents are optimizing, but where demand may not be exactly equal to supply. There will instead be a “small” gap between supply and demand. However, to prove such a result we need a lower bound on prices. This is done, for example in a theorem due to (Hildenbrand et al., 1973).24 Instead, I present a theorem due to Anderson et al. (1982). It estab- lishes the existence of an approximate equilibrium under weak assumptions on preferences: not only avoiding convexity, but even monotonicity. The equilibrium is approximate in the sense that excess demands are bounded above. The main idea in this theorem is to work with√ an unusual domain of prices, where prices are bounded below by 1/ I, and then bound the value of excess demand. Since prices cannot be too low, an upper bound on the value of excess demand implies a bound on (physical) excess demand. The focus on the value of excess demand also means that we shall not require monotonicity of preferences, or Walras Law. I Theorem 105. Let E = (i, ωi)i=1 be an exchange economy with L goods. Suppose that each preference i is continuous. There exists L I IL p ∈ R++ and (xi)i=1 ∈ R+ such that

• p · xi ≤ p · ωi; • yi i xi implies that p · yi > p · ωi; and • Aggregate excess demand is bounded above in the following sense: L I 1 X X L + 1 [ (x − ω )]+ ≤ √ max{kω k : 1 ≤ i ≤ I}. I i,l i,l i 1 l=1 i=1 I

Proof. Let

L 1 M = {p ∈ R : √ ≤ pl ≤ 1, l = 1,...,L}. + I ∗ L For each p ∈ M, let zi (p) ⊆ R be consumer i’s excess demand at ∗ p. Note that zi (p) is a nonempty and compact set, but that it may ∗ not be convex. Note that p 7→ zi (p) is upper hemicontinous. Let 24Starr (1969), who introduced many of these ideas and where the ﬁrst statement of the Shapley-Folkman theorem appears. 102 ECHENIQUE

∗ PI ∗ L z = i=1 zi (p). Let K ⊆ R be a compact and convex set such that z∗(M) ⊆ K. Define a correspondence φ : M × K 7→ M × K by φ(p, ζ) = argmax{q · ζ : q ∈ M} × cvh(z∗(q)). The correspondence φ is in the hypotheses of Kakutani’s fixed point theorem. Let (p, ζ) be a fixed point of φ, so (p, ζ) ∈ φ(p, ζ). The idea here is to calculate the value of excess demand at a fixed point. That explains the choice of M. The bound on value will imply a bound on physical quantities.

PI ∗ By the Shapley-Folkman theorem, since ζ ∈ cvh( i=1 zi (p)), there exists (ζ1, . . . , ζI ) and C ⊆ {1,...,I} with |C| = I − L such that

∗ • ζi ∈ zi (p) for all i ∈ C; ∗ • ζi ∈ cvh(z (p)) for all i∈ / C and P i • ζ = i ζi.

0 ∗ 0 0 Choose ζi ∈ zi (p) for i∈ / C and let xi = ωi + ζi. Observe that 0 0 0 ζi,l − ζi,l = xi,l − (ζi,l + ωi,l) ≤ xi,l. Therefore, L L √ X 0 X 0 p · ωi max{ζi,l − ζi,l, 0} ≤ xi,l ≤ ≤ Ikωik1, min{pl : 1 ≤ l ≤ L} l=1 l=1 where the second inequality follows as PL x0 ≤ p·xi and l=1 i,l min{pl:1≤l≤L} p · xi ≤ p · ωi. The third inequality follows from the deﬁnition of M. This implies that, for any q ∈ M, L √ X 0 X X 0 q · (ζi − ζi) ≤ max{ζi,l − ζi,l, 0} ≤ L I max{kωik1 : i∈ / C}. i/∈C i/∈C l=1

Note that p·ζi ≤ 0 for all i. By deﬁnition of φ, and since (p, ζ) ∈ φ(p, ζ), q · ζ ≤ p · ζ ≤ 0 for any q ∈ M. So we have that, for any q ∈ M, I √ X X 0 X X 0 q·( ζi+ ζi) = q · ζi +q· (ζi−ζi) ≤ L I max{kωik1 : 1 ≤ i ≤ I}. i∈C i/∈C i=1 i/∈C | {z } ≤0 GENERAL EQUILIBRIUM THEORY 103 √ Choose in particular q ∈ M such that ql = 1 if ζl > 0 and ql = 1/ I if ζl ≤ 0. Then we have that L X X max{ζl, 0} = ql max{ζl, 0} l=1 l 1 X = q · ζ − √ ζl I l:ζl≤0 √ 1 X ≤ L I max{kωik1 : 1 ≤ i ≤ I} + √ ω¯l I l:ζj ≤0 √ I ≤ L I max{kωik1 : 1 ≤ i ≤ I} + √ max{kωik1 : 1 ≤ i ≤ I} I √ = (L + 1) I max{kωik1 : 1 ≤ i ≤ I}. The ﬁrst inequality follows from the bound on q·ζ we established above, and the fact that if ζl < 0 then −ζl ≤ ω¯l, so X − ζl ≤ kω¯k1.

l:ζl≤0

The second inequality holds because kω¯k1 ≤ I max{kωik1 : 1 ≤ i ≤ I}.

The idea in this proof is that if (p, ζ) is a ﬁxed point of φ, and if P ∗ we could show that ζ = i ζi with ζi ∈ zi (p) for all i, then p · ζ ≤ 0 (compliance with budget constraints) and a lower bound on pl provides an upper bound on ζl.

17.3. Core convergence revisited. The Debreu-Scarf theorem shows that core allocations of large replica economies converge to Walrasian equilibrium allocations. Using the Shapley-Folkman theorem, it is possible to show a version of this result that does not need the machinery of replica economies, and can also dispense with convexity of preferences. The theorem, which is due to Anderson (1978), ensures the existence of a price vector at which agents are on average close to minimizing expenditure.25

I Theorem 106. Let E = (i, ωi)i=1 be an exchange economy with L ∗ goods in which each preference i is monotonic. If x is in the core of L E then here exists a price vector p ∈ R+, p > 0, such that

25See also Dierker (1975). 104 ECHENIQUE

(1) I 1 X 2M |p · (x∗ − ω | ≤ I i i I i=1 (2) I 1 X 2M | inf{p · (y − ω ): y x∗}| ≤ , I i i i i i I i=1 where

M = L max{ωi,l : 1 ≤ i ≤ I, 1 ≤ l ≤ L}

The second statement in the theorem deserves an explanation. It says that if one were to make (on average) agents’ incomes a bit smaller, then there would be no bundle aﬀordable that would be preferred to the agent’s consumption bundle in x∗. For an exact Walrasian equilibrium, ∗ note that we would have | inf{p · (yi − ωi): yi i xi }| = 0. Incidentally, the theorem does not explicitly say that demand equals supply but ∗ P ∗ P note that x is an allocation, so i xi = i ωi.

17.3.1. Proof of Theorem 106. Let θ = max{kωik∞ : i = 1,...,I}, so M = Lθ. Let z = −Lθ1 = −M1.

Let ∗ X Bi = {yi − ωi : yi i xi } ∪ {0} and B = Bi i Note that B is a hybrid of the approaches we used for the second welfare theorem and the Debreu-Scarf core convergence theorem. It is an “aggregate” object in the sense that it adds up the translated upper contour sets, but by including the null vector we can omit agents in the sum and obtain an aggregate for any coalition of agents. Note also that Bi may not be convex even if preferences are convex. Since x∗ is in the core, we have:

L Lemma 107. B ∩ (−R++) = ∅.

We shall ﬁrst prove that L cvh(B) ∩ (z − R++) = ∅. Suppose, towards a contradiction, that there exists b ∈ cvh(B) and z0 0 such that z + z0 = b. By the Shapley-Folkman theorem there is GENERAL EQUILIBRIUM THEORY 105

0 P 0 (bi) and I ⊆ I with b = i bi, bi ∈ Bi for all i ∈ I and bi ∈ cvh(Bi) for all i ∈ I \ I0, and |I \ I0| ≤ L. Then

X 0 X X bi = z + z − bi z − bi. i∈I0 i/∈I0 i/∈I0 ∗ Now, bi ∈ Bi means that either bi = 0 or bi = yi − ωi with yi i xi . Either way, bi ≥ −ωi. So for any bi ∈ cvh(Bi), bi ≥ −ωi. Therefore, using the deﬁnition of θ, we obtain that X 0 − bi ≤ |I \ I | θ1 ≤ Lθ1. i/∈I0 P Thus i∈I0 bi z + Lθ1 = 0 ∗ ∗ 0 ∗ 0 Deﬁne (bi ) by setting bi = bi for all i ∈ I and bi = 0 for all i∈ / I . Thus P ∗ P ∗ P i bi ∈ B, but i bi = i∈I0 bi 0. A contradiction of Lemma 107. L Now, by the separating hyperplane theorem, cvh(B) ∩ (z − R++) = ∅ implies that there is p ∈ RL, p 6= 0, such that sup{p · z˜ :z ˜ z} ≤ inf{p · b : b ∈ cvh(B)} In fact, by standard arguments, p > 0; and, using a normalization, we can take p to satisfy p · 1 = 1.26

∗ By monotonicity, for all k ≥ 1, xi + 1(1/k) − ωi ∈ Bi, so ∗ p · (xi + 1(1/k) − ωi) ≥ inf{p · Bi}. Hence ∗ p · (xi − ωi) ≥ inf{p · Bi}.

Now, Bi ⊆ B (as 0 ∈ Bj for all j 6= i). This implies that ∗ p · (xi − ωi) ≥ inf{p · Bi} ≥ inf{p · B} ≥ sup{p · z˜ :z ˜ z} = p · z = −M, where the last equality uses the normalization imposed on p.

P ∗ Let S = {i ∈ I : p · (xi − ωi) < 0}. Then i p · (xi − ωi) = 0 implies that 1 X 1 X 1 X −2 X |p·(x −ω )| = p·(x −ω )− p·(x −ω ) = p·(x −ω ). I i i I i i I i i I i i i i/∈S i∈S i∈S 26 We have p > 0 because if pl < 0 then we may choosez ˜ z to make p · z˜ arbitrarily large. So p ≥ 0, but we also know that p 6= 0. 106 ECHENIQUE

Lemma 108. X p · (xi − ωi) ≥ −M. i∈S

∗ P ∗ Proof. For all k, xi +1(1/k)−ωi ∈ Bi. So i∈S(xi +1(1/k)−ωi) ∈ B. Then X ∗ p · (xi + 1(1/k) − ωi) ≥ p · z = −M. i∈S The result follows as k is arbitrary.

By Lemma 108, 1 X −2 X 2M |p · (x − ω )| = p · (x − ω ) ≤ I i i I i i I i i∈S

Now lets turn to the issue of expenditure minimization. Let

I 1 X A = | inf{p · (y − ω ): y x∗}| I i i i i i i=1

∗ ∗ Let Li = {yi − ωi : yi i xi }. By monotonicity, p · (xi + 1(1/k) − ωi) ∈ ∗ p · Li for all k. So p · (xi − ωi) ≥ inf p · Li. Note that if i ∈ S then ∗ 0 ≥ p · (xi − ωi) ≥ p · Li. So inf p · Li = inf p · Bi. Hence −1 X 1 X A = inf p · B + | inf p · L | I i I i i∈S i/∈S I −1 X 1 X = inf p · B + {inf p · B + | inf p · L |} , I i I i i i=1 i/∈S 1 P where the second equality results from adding and subtracting I i/∈S inf p· Bi.

For i∈ / S there are two possibilities. The ﬁrst is that inf p · Li > 0. Then inf p · Bi = 0 and thus ∗ inf p · Bi + | inf p · Li| = inf p · Li ≤ p · (xi − ωi)

The second is that inf p · Li ≤ 0, so that inf p · Bi = inf p · Li and therefore ∗ inf p · Bi + | inf p · Li| = inf p · Bi − inf p · Li = 0 ≤ p · (xi − ωi), where the last inequality follows from i∈ / S. GENERAL EQUILIBRIUM THEORY 107

As a consequence we obtain that −1 X 1 X A ≤ inf p · B + p · (x∗ − ω ). I i I i i i∈I i/∈S

P Now, i inf p · Bi = inf p · B ≥ p · z = −M. So that −1 X M inf p · B ≤ . I i I i P ∗ P ∗ On the other hand, i/∈S p · (xi − ωi) = − i∈S p · (xi − ωi) ≤ M, by Lemma 108. Thus A ≤ M/I + M/I.

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