204Lecture132009web

204Lecture132009web - Economics 204 Lecture 13Wednesday,...

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Economics 204 Lecture 13–Wednesday, August 12, 2009 Section 5.5 (Cont.) Transversality Theorem The Transversality Theorem is a particularly convenient formulation of Sard’s Theorem for our purposes: 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 } Suppose that F ( x, ω )=0 DF ( x, ω ) has rank m Then there is a set Ω 0 Ω such that Ω \ Ω 0 has Lebesgue measure zero such that ω Ω 0 ,F ( x, ω D x F ( x, ω ) has rank m If m = n and ω 0 Ω 0 , there is a local implicit function x ( ω ) characterized by F ( x ( ω ) where x is a C r function of ω . the correspondence ω →{ x : F ( x, ω } is lower hemicontinuous at ω 0 . 1
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Interpretation of Tranversality Theorem Ω: a set of parameters (agents’ endowments and preferences, or players’ payoF functions). X : a set of variables (price vectors, or strategies). R m is the range of F (excess demand, or best-response strategies). F ( x, ω ) = 0 is equilibrium condition, given parameter ω . Rank DF ( x, ω )= m says that, by adjusting either the variables or parameters, it is possible to move F in any direction. When m = n ,Rank D x F ( x, ω m says det D x F ( x, ω ) 6 = 0, which says the economy is regular and is the hypothesis of the Implicit ±unction Theorem; this tells us that the equilibrium correspondence is lower hemicontinuous. Economic correspondences like ω →{ x : F ( x, ω )=0 } are generally upper hemicontinuous, so regularity in fact tells us the correspondence is continuous. You will see in 201B that regularity, plus a property of demand functions, tell us that the equilibrium prices are given by a ²nite number of implicit functions of the parameters (endowments). Parameters of any given economy are ²xed. However, we want to study the set of parameters for which the resulting economy is well-behaved. Theorem says the following: “If, whenever F ( x, ω ) = 0, it is possible by perturbing the parameters and variables to move F in any direction, then for almost all parameter values, all equilibria 2
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are regular, the equilibria are implicitly defned C r Functions oF the parameters, and the equilibrium correspondence is lower hemicontinuous.” You will see in 201B that the reg-
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This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at Berkeley.

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204Lecture132009web - Economics 204 Lecture 13Wednesday,...

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