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Lecture Notes FOUNDATIONS OF ECONOMICS LECTURE NOTES BY OMER TAMUZ California Institute of Technology 2021 Contents 1 Preferences 4 2 Prices and consumer choice 6 3 Production 7 4 Private ownership economies and Walrasian equilibria 8 5 Walrasian equilibria with transfers 10 6 Excess demand 13 7 Compactifying the economy 15 8 Kakutani’s Theorem and Debreu’s Theorem 16 9 Existence of equilibria 17 10 Approximate equilibria 18 11 Scitovsky contours 21 12 The core 22 13 The core via approximate equilibria 25 14 Partial equilibrium: consumers 28 15 Partial equilibrium: production 30 16 Partial equilibrium conditions 31 17 Uncertainty 33 18 Pari-mutuel gambling 34 19 Radner equilibria 39 20 Asset pricing 42 2 Disclaimer This a not a textbook. These are lecture notes. 3 1 Preferences Let X be a metric space. We say that X is connected if it is not the disjoint union of two closed sets. A closed-contour preference on X is a reflexive, transitive and complete binary relation ¹ such that for each x 2 X the upper contour set {x0 2 X : x ¹ x0} and lower contour set {x0 2 X : x0 ¹ x} are closed subsets of X. We say that ¹ is a closed preference if it is a closed subset of X £ X; that is if {(x, x0): x ¹ x0} is a closed subset of X £ X. We say that a subset Q ⊆ X is dense if for each x 2 X there is a sequence (qn)n in Q with limn qn Æ x. Equivalently, X is equal to the closure of Q. Recall that X is separable if it admits a countable dense. Theorem 1.1 (Debreu). For every closed-contour preference ¹ on a separable, connected met- ric space X there is a continuous function u: X ! R such that x ¹ x0 iff u(x) · u(x0). Corollary 1.2. Let X be a connected, separable metric space, and let ¹ be a preference on X. Then the following are equivalent: 1. ¹ is closed-contour. 2. ¹ is closed. Proof. Suppose ¹ is closed-contour. Then by Theorem 1.1 there is a continuous utility func- ! R ¹ · 0 2 · 0 tion u: X that represented . Suppose xn xn for n {1,2,...}. Then u(xn) u(xn). If the two sequences converge to x and x0 respectively, then u(x) · u(x0), and thus {(x, x0): x ¹ x0} is closed. Conversely, assume ¹ is closed. Then {x0 : x0 ¹ x} Æ {(w,w0): w ¹ w0} \ X £ {x} is the intersection of two closed sets, and is therefore closed. We will henceforth assume that all preferences are closed (equivalently, contour-closed). We say that a preference ¹ on a metric space X is locally non-satiated (LNS) if for every x 2 X and " È 0 there is a x0 such that d(x, x0) Ç " and x0  x. We say that ¹ is convex if {x0 : x0 º x} is convex for all x 2 X. We say that it is strictly convex if ¸w Å (1 ¡ ¸)w0  x whenever w º x and w0 º x. Claim 1.3. If ¹ is convex then {x0 : x0  x} is convex for all x 2 X. Suppose X is convex. We will say that ¹ is convex* if 0 0 x  x implies ¸x Å (1 ¡ ¸)x  x for all¸ 2 (0,1). (1.1) Claim 1.4. If X is convex and ¹ is convex* then ¹ is convex. Proof. Let ¹ be represented by u: X ! R. We claim that (1.1) implies that u is quasiconcave, i.e., 0 0 u(¸x Å (1 ¡ ¸)x ) ¸ min{u(x), u(x )}. (1.2) 4 Suppose not, so that the above inequality is violated for some x, x0 and ¸. Denote v(®) Æ u(®x Å (1 ¡ ®)x0), and assume without loss of generality that v(0) · v(1). By assumption we have that v(¸) Ç v(0). It follows from the continuity of v that there is some ¯ 2 (0,¸) such that v(¸) Ç v(¯) Ç v(0) · v(1). But since ¸ 2 (¯,1), it follows from (1.1) that v(¸) È v(¯), and we have reached a contradiction. To complete the proof we claim that the quasiconcavity of u implies (in fact, is equivalent to) the convexity of ¹. Indeed, suppose that x0 º x and x00 º x, and assume without loss of generality that x0 º x00. By the quasiconcavity condition (1.2) xˆ Æ ¸x0 Å (1 ¡ ¸)x00 satisfies xˆ º x00, and so xˆ º x (the proof of the other direction is identical). Claim 1.5. Suppose X is convex, ¹ is convex*, and for every x 2 X there exists x0 2 X with x0  x. Then ¹ is LNS. 2 0  n Æ n¡1 Å 1 0 n  Proof. For each x X, let x x. The sequence x n x n x satisfies x x and intersects every ball around x. 5 2 Prices and consumer choice We consider a setting with L commodities {1,...,L}.A consumption space X is a subset of RL. Let ¹ be a preference on X. A price vector p is an element of RL. Given p and wealth w 2 R, the choice set X ¤(p,w) is given by ¤ 0 0 X (p,w) Æ {x 2 X : p ¢ x · w implies x ¹ x}. (2.1) Equivalently, X ¤(p,w) Æ {x 2 X : x0  x implies p ¢ x0 È w}. Lemma 2.1. Suppose ¹ is LNS, and x¤ 2 X ¤(p,w). Then x º x¤ implies p ¢ x¤ ¸ w. Proof. Suppose towards a contradiction that x º x¤ and p¢ x Ç w. Then there is some " small enough so that p ¢ x0 Ç w for all x0 such that kx ¡ x0k". Since ¹ is LNS, there is some such x0 such that x0  x. Hence x0  x¤, and so it is impossible that x¤ 2 X ¤(p,w). Claim 2.2. Suppose X is convex. Fix a price vector p, w 2 R and x 2 X such that x0  x implies p ¢ x0 ¸ w, and such that there is some xˆ 2 X with p ¢ xˆ Ç w. Then x 2 X ¤(p,w). Proof. Let x0  x. We need to show that p¢x0 È w. By assumption p¢x0 ¸ w, and so it remains to show that p ¢ x0 6Æ w. Assume p ¢ x0 Æ w. Since Á is closed and X is convex, for all ¸ small enough it holds that x00 Æ (1¡®)xÅ®xˆ  x. But for ¸ small enough it also holds that p¢x00 Ç w, in contradiction to the claim hypothesis. 6 3 Production In a setting with L commodities, a production set Y is a subset of RL. Given a price vector p, the optimal production set is ¤ 0 0 Y (p) Æ {y 2 Y : p ¢ y · p ¢ y for all y 2 Y }. (3.1) The indirect utility is given by vY (p) Æ sup p ¢ y. (3.2) y2Y ¤ This is also known as the support function of Y . Note that vY (p) Æ p ¢ y for all y 2 Y (p). For ¸ ¸ 0, we denote ¸Y Æ {¸y : y 2 Y }. The Minkowski sum of two non-empty subsets L Y1,Y2 ⊆ R is given by Y1 Å Y2 Æ {y1 Å y2 : y1 2 Y1, y2 2 Y2}. It is easy to see that 1. v¸Y Æ ¸vY . Å Æ 2. vY1 vY2 vY1ÅY2 . The Minkowski sum of two closed sets is in general not closed. However, if Y1 is open, Å Sthen Y1 Y2 is open for every non-empty Y2, since it is equal to the union of open sets Å y22Y2 Y1 {y2}. We denote Minkowski subtraction by Y1 ¡ Y2 Æ {y1 ¡ y2 : y1 2 Y1, y2 2 Y2}. Claim 3.1. Denote Y Æ Y1 Å ¢¢¢ Å YL. The following are equivalent: P ¤ 1. j yj 2 Y (p). 2 ¤ Æ 2. yj Yj (p) for j 1,..., J. Fix some price vector p, and let y 2 Y ¤(p). How does y change when we change p? That is, if y0 2 Y ¤(p0), what can we say about the relation between y and y0, given p and p0? Claim 3.2. Let y 2 Y ¤(p) and y0 2 Y ¤(p0). Denote ¢y Æ y0 ¡ y and ¢p Æ p0 ¡ p. Then ±p ¢ ¢y ¸ 0. Proof. Since p ¢ y0 · p ¢ y we have that p ¢ ¢y · 0 Likewise, p0 ¢ y0 ¸ p0 ¢ y, and so 0 p ¢ ¢y ¸ 0. Subtracting the first from the second yields the desired result. 7 4 Private ownership economies and Walrasian equilib- ria A private ownership economy consists of the following elements: 1. L commodities. L 2. I consumers, each with a consumption set X i ½ R , a preference ¹i on X i, and an L endowment ei 2 R . 3. J firms, each with a production set Yj. P 4. Each consumer i holds a stake θi j in firm j. We assume i θi j Æ 1 for all j. P P P An allocation is a pair ((xi)i,(yj) j) such that xi 2 X i, yj 2 Yj and i xi Æ i ei Å j yj. The set of all allocations is denoted by A. A Walrasian equilibrium consists of a price vector p together with consumption vectors (xi)i and production vectors (yj) j such that P 2 ¤ Æ ¢ Å θ ¢ 1. For all i, xi X i (p,wi), where wi p ei j i j p yj.
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