PS7_Solution_Key - TA Yankun Wang Economics 617 Problem Set 7 Solution Key 1[Convex Sets(a Define the set n Rn i i1 n 1 This set is called in the

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Economics 617 Problem Set 7 Solution Key 1. [Convex Sets] (a) Define the set: Δ n R n : i 1 n i 1 This set is called in the unit simplex in R n . Show that Δ n is a convex subset of R n . Solution: Let , ′′ Δ n , and 0, 1 . We need to prove:  1 ′′ Δ n . It’s obvious that  1 ′′ R n . i 1 n  i 1 i ′′ i 1 n i 1 i 1 n i ′′ 1 1, where we have used the facts that i 1 n i 1 , i 1 n i ′′ 1. Therefore,  1 ′′ Δ n . By definition, Δ n is convex. (b) Let a R  n . Define the set: T x R n : ax 1 Is T a convex set in R n ? Explain. Remark: for n 1, T x R : x 1/ a , T is convex by definition. For n 2, T is a closed half space, which is also convex. Solution: Let x , x ′′ T , and 0, 1 . It’s easy to verify that x 1 x ′′ R n . a x 1 x ′′ ax 1  ax ′′ 1 1, where the inequality follows because ax 1, ax ′′ 1 and 0, 1 . 2. [Concave Functions] (a) Let A be a convex subset of R m , and f a real-valued function on A . Show that f is concave on A if and only if the set: V  x , A R : f x is a convex subset of R m 1 . Solution: (i) the "only if" part: Suppose f is concave on A . Let x , and x ′′ , ′′ be elements of V , and 0, 1 . We know f x and f x ′′ ′′ . Mutliply the first equation by
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This note was uploaded on 08/29/2009 for the course ECON 617 taught by Professor Staff during the Fall '08 term at Cornell University (Engineering School).

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PS7_Solution_Key - TA Yankun Wang Economics 617 Problem Set 7 Solution Key 1[Convex Sets(a Define the set n Rn i i1 n 1 This set is called in the

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