204Lecture132009 - Economics 204 Lecture 13Wednesday,...

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Unformatted text preview: Economics 204 Lecture 13Wednesday, August 12, 2009 Revised 8/12/09, revisions indicated by ** and Sticky Notes Section 5.5 (Cont.) Transversality Theorem The Transversality Theorem is a particularly convenient formula- tion of Sards Theorem for our purposes: Theorem 1 (2.5, Transversality Theorem) Let X R n and R p be open F : X R m C r with r 1 + max { , n m } Suppose that F ( x, ) = 0 DF ( x, ) has rank m Then there is a set such that \ has Lebesgue measure zero such that , F ( x, ) = 0 D x F ( x, ) has rank m If m = n and , there is a local implicit function x ( ) characterized by F ( x ( ) , ) = 0 where x is a C r function of . the correspondence { x : F ( x, ) = 0 } is lower hemicontinuous at . 1 Interpretation of Tranversality Theorem : a set of parameters (agents endowments and preferences, or players payoff functions). 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, ) = 0 is equilibrium condition, given parameter . Rank DF ( x, ) = m says that, by adjusting either the vari- ables or parameters, it is possible to move F in any direc- tion. **While we only need to know we can do this at equi- libria, i.e. at ( x, ) such that F ( x, ) = 0, in typical appli- cations the parameters allow enough freedom to show that Rank D F ( x, ) = m for all ( x, ). When m = n , Rank D x F ( x, ) = m says det D x F ( x, ) 6 = 0, which says the economy is regular and is the hypothesis of the Implicit Function Theorem; this tells us that the equilibrium correspondence is lower hemicontinuous. Economic correspon- dences like { x : F ( x, ) = 0 } are generally upper hemi- continuous, so regularity in fact tells us the correspondence is continuous. You will see in 201B that regularity, plus a prop- erty of demand functions, tell us that the equilibrium prices are given by a finite number of implicit functions of the parameters (endowments). Parameters of any given economy are fixed. However, we want to study the set of parameters for which the resulting economy is well-behaved....
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204Lecture132009 - Economics 204 Lecture 13Wednesday,...

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