201Lecture72006

# 201Lecture72006 - Economics 201B–Second Half Lecture 7...

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Unformatted text preview: Economics 201B–Second Half Lecture 7 Existence of Walrasian Equilibrium Theorem 1 (Brouwer’s 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 eﬃcient algorithm to find approximate fixed points: ∀ ε > ∃ x ∗ ε | f ( x ∗ ε ) − x ∗ ε | < ε 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 201B–Second Half Lecture 7...

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