204Lecture132007 - Economics 204 Lecture 13–Wednesday,...

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Unformatted text preview: Economics 204 Lecture 13–Wednesday, August 15, 2007 Revised 8/15/07, revisions indicated by ** Section 5.5 (Cont.) Transversality Theorem The Transversality Theorem is a particularly convenient formula- tion 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 { , 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 ) = 0 ⇒ 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 ) , ω ) = 0 x ∗ is a C r function of ω , and the correspondence ω → x ∗ ( ω ) is lower hemicontinuous. Interpretation of Tranversality Theorem • Ω: a set of parameters (agents’ endowments and preferences, or players’ payoff functions). 1 • X : a set of variables (price vectors, or strategies). • R m is the range of F (excess demand, or best-response strate- gies). • F ( x, ω ) = 0 is equilibrium condition, given parameter ω . • Rank DF ( x, ω ) = m says that, by adjusting either the vari- ables 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 Function Theorem. This will tell us that the equilib- rium prices are given by a finite number of implicit functions of the parameters (endowments), and the equilibrium correspon- dence is thus lower hemicontinuous. • Parameters of any given economy are fixed. 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 direc- tion, then for almost all parameter values, all equilibria are regular, and hence there are finitely many equilibria, the equilibria are implicitly defined C r functions of the parameters, and the equilibrium correspondence is lower hemicontinuous.” • If n < m , Rank D x F ( x, ω ) ≤ min { m, n } = n < m...
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This note was uploaded on 03/30/2009 for the course ECON 0204 taught by Professor Staff during the Summer '08 term at Berkeley.

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204Lecture132007 - Economics 204 Lecture 13–Wednesday,...

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