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201Lecture122006 - Economics 201BSecond Half Lecture 12...

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Economics 201B–Second Half Lecture 12 Quick Romp Through 17.E,F,H 17.E Theorem 1 (Sonnenschein-Mantel-Debreu) Let K be a compact subset of Δ 0 . Given f : K R L satisfying continuity Walras’ Law with Equality ( p · f ( p ) = 0 ) there is an exchange economy with L consumers whose ex- cess demand function, restricted to K , equals f . Proof: Elementary, but far from transparent. Individual pref- erences may be made arbitrarily nice. Corollary 2 There are no comparative statics results for Walrasian Equilibrium in the Arrow-Debreu model; more assumptions are needed. 17.F, Uniqueness : There are no results known under believeable assumptions on individual preferences. 17.H, Tatonnement Stability : d ˆ p dt = ˆ z p ) on R L 1 ++ dp dt = E ( p ) on Δ 0 2 = p R L ++ : p 2 = 1 We would like to know that the solutions converge to the equi- librium price. Scarf gave an example of a non-pathological 1
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exchange economy in which the solutions all circle around the unique Walrasian equilibrium price. There are no known sta- bility results based on reasonable assumptions on individual preferences. Index = +1 is necessary but not sufficient for stability. Modern Approach to Uniqueness and Stability : Assumptions on the Distribution of Agents’ Characteristics. Law of Demand : ( p q ) · ( z ( p ) z ( q )) 0 with strict inequality if p = q The Law of Demand implies uniqueness of equilibrium and Tatonnement stabilty. Hildenbrand: If, for each preference, the density of the income distribu- tion among people holding that preference is decreasing, then the Law of Demand holds. Idea: If demand for a good is a decreasing function of income at some income level, it must first have been an increasing function at lower income levels. Decreasing density of income distribution implies that overall, the increasing part cancels out the decreasing part.
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