Bounded the sets dimensions are finite lecture 17

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Unformatted text preview: ssociating points in a set with another set of points. – Closed: The set contains its edge. – Convex: If the set contains two points, it also contains all points on the line connecting the two points. – Bounded: The set’s dimensions are finite. Lecture 17 Econ 11--Spring 2001 Example for a Univariate Mapping 13 – A continuous function is a continuous mapping. – The set X = [0,1] is closed, bounded, and convex. • A fixed point of f(x) is a point x* such that f(x*)= x*. • On a graph of f(x), it’s fixed points are on the 45o line. Lecture 17 Example: Brouwer’s Fixed Point Theorem f (x ) Econ 11--Spring 2001 14 Back to Walras’ Economy • The next step to finding an equilibrium is to “normalize” the n prices so that they add to one. 45o line 1 – For the new price set, divide each price by the sum of all the prices. P′ = Pj ∀j j f(x*) n ∑P i i =1 – Redefining prices in this way will not change demand because of zero degree homogeneity. x* Lecture 17 1 Econ 11--Spring 2001 x 15 Lecture 17 Applying Brouwer’s FP Theorem Econ 11--Spring 2001 An Equilibrium Exists • After normalizing prices, the pr...
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This note was uploaded on 02/11/2012 for the course ECON 51 at Stanford.

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