201Lecture72006

201Lecture72006 - Economics 201BSecond Half Lecture 7...

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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...
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201Lecture72006 - Economics 201BSecond Half Lecture 7...

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