201Lecture112006

201Lecture112006 - Economics 201BSecond Half Lecture 11...

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Economics 201B–Second Half Lecture 11 Tranversality Theorem, and Generic Regularity Theorem 1 (2.5’, Transversality Theorem) Let X × Ω R n + p be open F : X × Ω R m C r with r 1+max { 0 ,n m } If F ( x, ω )=0 DF ( x, ω ) has rank m then for all ω except for a set of Lebesgue measure zero, F ( x, ω D x F ( x, ω ) has rank m In particular, if m = n , there is a local implicit function x ( ω ) characterized by F ( x ( ω ) and x C r . Interpretation of Tranversality Theorem Ω: a set of parameters. In our case, Ω = R LI ++ ,th es e to f strictly positive endowment proFles, p = LI . X : a set of variables. In our case, X = R L 1 ++ e f strictly positive prices normalized by p L =1 . R m is the range of F .Inou rca s e , F ( x, ω )=ˆ z ( x ), when the endowment proFle is ω , m = n = L 1. 1
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F ( x, ω )=0s a y st h a t x is an equilibrium price when the endowment profle is ω . rank DF ( x, ω )= m = L 1 says that, by adjusting either the prices x or the endowments ω , it is possible to move F z in any direction in R L 1 rank D x F ( x, ω m = L 1saysdet D x F ( x, ω ) 6 =0 ,wh ich says the economy is regular and is the hypothesis oF the Im- plicit ±unction Theorem. This will tell us that the equilibrium prices are given by a fnite number oF implicit Functions oF the parameters (endowments). Parameters oF any given economy are fxed. However, we want to study the set oF parameters For which the resulting economy is well-behaved. Theorem says the Following: “IF, whenever ˆ z p ) = 0, it is possible by perturbing the endowments and adjusting the prices to move ˆ z in any direction in R L 1 , then For almost all endowments, the
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201Lecture112006 - Economics 201BSecond Half Lecture 11...

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