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 11Spring 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 11Spring 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 11Spring 2001 x
15 Lecture 17 Applying Brouwer’s FP Theorem Econ 11Spring 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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 Tendall,M
 Perfect Competition

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