204Lecture132007

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 03/30/2009 for the course ECON 0204 taught by Professor Staff during the Summer '08 term at Berkeley.

### Page1 / 15

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online