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NonconvexHandout

# NonconvexHandout - Economics 201B Nonconvex Preferences and...

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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 Caratheodory’s 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 Caratheodory’s 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 i =1 m i j =0 λ ij a ij (1) with I i =1 m i L ; a ij A i and λ ij > 0 for each i, j ; and m i j =0 λ ij = 1 for each i . Proof: 1

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1. Suppose x con ( A 1 + · · · + A I ). Then we may write x = m j =0 λ j I i =1 a ij = I i =1 m j =0 λ j a ij (2) with λ j > 0, m j =0 λ j = 1. Letting λ ij = λ j and m i = m for each i , we have an expression for x in the form of equation 1. 2. Suppose we have any expression for x in the form of equation 1 with I i =1 m i > L . Then the set { a ij a i 0 : 1 i I, 1 j m i } (3) contains I i =1 m
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NonconvexHandout - Economics 201B Nonconvex Preferences and...

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