204Lecture132009

# 204Lecture132009 - Economics 204 Lecture 13–Wednesday...

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Unformatted text preview: Economics 204 Lecture 13–Wednesday, 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 Sard’s Theorem for our purposes: Theorem 1 (2.5’, Transversality Theorem) Let ∗ ∗ X ⊆ R n and Ω ⊆ R p be open F : X × Ω → R m ∈ C r with r ≥ 1 + max { , n − m } Suppose that F ( x, ω ) = 0 ⇒ DF ( x, ω ) has rank m Then there is a set Ω ⊆ Ω such that Ω \ Ω has Lebesgue measure zero such that ω ∈ Ω , F ( x, ω ) = 0 ⇒ D x F ( x, ω ) has rank m If m = n and ω ∈ Ω , • there is a local implicit function x ∗ ( ω ) characterized by F ( x ∗ ( ω ) , ω ) = 0 where x ∗ is a C r function of ω . • the correspondence ω → { x : F ( x, ω ) = 0 } is lower hemicontinuous at ω . 1 Interpretation of Tranversality Theorem • Ω: 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, 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 direc- tion. **While we only need to know we can do this at equi- libria, i.e. at ( x, ω ) such that F ( x, ω ) = 0, in typical appli- cations the parameters ω allow enough freedom to show that Rank D ω F ( x, ω ) = m for all ( x, ω ). • 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 tells us that the equilibrium correspondence is lower hemicontinuous. Economic correspon- dences like ω → { x : F ( x, ω ) = 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 13–Wednesday...

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