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Lecture Notes General Equilibrium Theory: Ss205 LECTURE NOTES GENERAL EQUILIBRIUM THEORY: SS205 FEDERICO ECHENIQUE CALTECH 1 2 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 2 X we write x B y to denote that (x; y) 2 B. A binary relation B is complete if x B y or y B x (or both) for any x; y 2 X. It is transitive if for any x; y; z 2 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 pref- erence 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 2 A for any x; y 2 A and any λ 2 (0; 1). We use the following notational conventions: For vectors x; y 2 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 2 X and " > 0 there is y 2 X such that kx − yk < " 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) = fy 2 X : y xg be the upper contour set of at x and let L(x) = fy 2 X : x yg be the lower contour set of at x. The preference relation is • convex if U(x) is a convex set for all x 2 X; • strictly convex if λy + (1 − λ)y0 x 6 ECHENIQUE for any y; y0 2 U(x) and λ 2 (0; 1), for any x 2 X; • and continuous if U(x) and L(x) are closed sets (in X) for all x 2 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 consump- tion 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 2 R+ and W ≥ 0. The set L B(p; W ) = fy 2 R+ : p · y ≤ W g 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 ) = fx 2 B(p; W ): x y for all y 2 B(p; W )g: 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 2 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 f(x; y) 2 A × B : y 2 φ(x)g 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 2 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 hemi- continuous. 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 prefer- ences . 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.
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