This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Economics 201BSecond Half Lecture 7 Existence of Walrasian Equilibrium Theorem 1 (Brouwers Fixed Point Theorem) Suppose A R L is nonempty, convex, compact, and f : A A is continuous. Then f has a fixed point, i.e. x A f ( x ) = x Proof (in very special case L = 1 ) : A is a closed interval [ a, b ]. Let g ( x ) = f ( x ) x g ( a ) = f ( a ) a g ( b ) = f ( b ) b By the Intermediate Value Theorem, there exists x [ a, b ] such that g ( x ) = 0. Then f ( x ) = g ( x ) + x = x 1 General Case: Much harder, but a wonderful result due to Scarf gives an ecient algorithm to find approximate fixed points: > x  f ( x ) x  < Sketch of Idea of Scarf Algorithm : Suppose A is L 1 dimensional. Let A be the price simplex A = p R L + : L X ` =1 p ` = 1 Triangulate A , i.e. divide A into a set of simplices such that the intersection of any two simplices is either empty or a whole face of both. Label each vertex in the triangulation by L ( x ) = min { ` : f ( x ) ` < x ` } Each simplex in the triangulation has L vertices. A simplex is completely labelled if its vertices carry each of the labels 1 , . . . , L exactly once; it is...
View Full
Document
 Spring '06
 ANDERSON
 Economics

Click to edit the document details