NonconvexHandout - Economics 201B Nonconvex Preferences and...

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Unformatted text preview: Economics 201B Nonconvex Preferences and Approximate Equilibria 1 The Shapley-Folkman Theorem The Shapley-Folkman Theorem is an elementary result in linear algebra, but it is apparently unknown outside the mathematical economics literature. It is closely related to Caratheodorys Theorem, a linear algebra result which is well known to mathematicians. The Shapley-Folkman Theorem was first published in Starr [3], an important early paper on existence of approximate equilibria with nonconvex preferences. Theorem 1.1 (Caratheodory) Suppose x con A , where A R L . Then there are points a 1 , . . . , a L +1 A such that x con { a 1 , . . . , a L +1 } . Theorem 1.2 (Shapley-Folkman) Suppose x con ( A 1 + + A I ) , where A i R L . Then we may write x = a 1 + + a I , where a i con A i for all i and a i A i for all but L values of i . We derive both Caratheodorys Theorem and the Shapley-Folkman Theorem from the following lemma: Lemma 1.3 Suppose x con ( A 1 + + A I ) where A i R L . Then we may write x = I X i =1 m i X j =0 ij a ij (1) with I i =1 m i L ; a ij A i and ij > for each i, j ; and m i j =0 ij = 1 for each i . Proof: 1 1. Suppose x con ( A 1 + + A I ). Then we may write x = m X j =0 j I X i =1 a ij = I X i =1 m X j =0 j a ij (2) with j > 0, m j =0 j = 1. Letting ij = j and m i = m for each i , we have an expression for x in the form of equation 1....
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This note was uploaded on 08/01/2008 for the course ECON 201B taught by Professor Anderson during the Spring '06 term at University of California, Berkeley.

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NonconvexHandout - Economics 201B Nonconvex Preferences and...

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