PS9_Solution_Key - TA: Yankun Wang Economics 617 Problem...

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TA: Yankun Wang Economics 617 Problem Set 9 Solution Key 1. [On Slater’s Condition] Let X = R 2 + , and f,g 1 and g 2 be functions from X to R , de f ned by: f ( x 1 ,x 2 )=2 x 1 + x 2 g 1 ( x 1 2 )=1 ( x 1 + x 2 ) g 2 ( x 1 2 )=( x 1 + x 2 ) 1 Consider the following optimization problem: Max f ( x 1 2 ) subject to g j ( x 1 2 ) 0 for j { 1 , 2 } and x X ( P ) (a) Show that ˆ x =(1 , 0) solves problem ( P ) . Solution: In order to have g j ( x 1 2 ) 0 for j =1 , 2 at the same time, we must have ( x 1 + x 2 ) 1=0 . Thus x 1 x 2 ; f ( x 1 2 x 1 + x 2 =2 x 2 . To maximize f ,wewant x 2 to be as small as possible, and the minimum of x 2 is 0 . When x 2 =0 1 x 2 . Remark: The problem can be view at a utility maximization problem with linear utility function. Think about the economic explanation as well as the graphical illustration for this problem. Just notice that neither the graph nor the intuition serves as the formal proof. (b) Note that X is a convex set in R 2 , and that 1 and g 2 are concave functions from X to R . Show that Slater’s Condition is not satis f ed. Solution: If we know what the slater’s condition is, this is kind of obvious: we can never have a point x X such that 1 ( x 1 + x 2 ) > 0 as well as ( x 1 + x 2 ) 1 > 0 , i.e., x 1 + x 2 < 1 and x 1 + x 2 > 1 . Remark: Make sure you know how to use the de f nition to prove that linear functions are both concave and convex. (c) Does there exist ˆ λ R 2 + such that x, ˆ λ ) is a saddle point? Explain. Solution: We want to know if we can f nd ˆ λ R 2 + such that: 2 x 1 + x 2 +( ˆ λ 1 ˆ λ 2 )(1 ( x 1 + x 2 )) 2 2 , for all x X, X = R 2 + . where the de f nition of saddle point has been applied. If we let ˆ λ 1 ˆ λ 2 , the above inequality becomes: 2 x 2 2 , and this is always true for x 2 0 . Thus by de f nition we have f nd that x, ˆ λ ) is a saddle point as long as: ˆ x , 0) , ˆ λ 1 ˆ λ 2 , and ˆ λ R 2 + . Remark: The Lecture Notes has an example (right after Theorem 45 ) in 1
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which Slater’s Condition is violated, and we can’t f nd a saddle point. Try to understand the di f erence between that example and this problem. 2. [Application of the theorem: “GM SP”] Let a (0 , 1) and b (0 , 1) be given parameters. Let A be the set de f ned by: A = { ( x, y, z ) R 3 + : y x, z bx and [ y + a ( z bx )] 1 } Consider the following constrained optimization problem: Maximize y subject to ( x, y, z ) A and z x ( B ) Problem ( B ) has a solution (you do not have to show this); denote an arbitrary solution of problem ( B ) by x, ¯ y, ¯ z ) . Show that there is ¯ p> 0 such that: y pz ¯ px ¯ y for all ( x, y, z ) A Proof: The title of the problem serves as a hint: use Theorem 45 in the Lecture Notes. It is given that x, ¯ ¯ z ) is a constrained global maximum. Let g ( x, y, z )= z x, f ( x, y, z y, X = A, we need to check f rst that A is convex, f,g are concave. All of these can be done by checking the de f nitions of convex sets and concave functions. Is the Slater’s Condition satis f ed? Yes, since (0 , 0 , 1) in an element in A, and g (0 , 0 , 1) = 1 > 0 .
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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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PS9_Solution_Key - TA: Yankun Wang Economics 617 Problem...

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