PS_10_Solution_Key

PS_10_Solution_Key - TA Yankun Wang Economics 617 Problem...

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TA: Yankun Wang Economics 617 Problem Set 10 Solution Key 1. [Using the Kuhn-Tucker Necessity Theorem] Let a, c be arbitrary positive parameters, satisfying a > c. De fi ne the following functions: F ( x ) = a x for x [0 , a ] G ( x ) = R x 0 F ( z ) dz for x [0 , a ] Consider the following constrained maximization problem: Maximize xF ( x ) cx + y subject to y G ( x ) xF ( x ) x a ( x, y ) 0 ( P ) (a) Use Weierstrass theorem to show that problem ( P ) has a solution. Solution: G ( x ) = Z x 0 F ( z ) dz = ax x 2 2 , and the problem becomes: Maximize x 2 + ( a c ) x + y subject to y x 2 2 x a ( x, y ) 0 It’s easy to verify that the constraint set is nonempty, closed and bounded; the objective function is continuous as a polynomial. Thus Weierstrass the- orem tells us problem ( P ) has a solution. (b) Show that if x, ˆ y ) is any solution to problem ( P ) , then ˆ x > 0 , ˆ y > 0 . Proof: Suppose ˆ x = 0 , by constraint y x 2 2 , we must have ˆ y = 0 . But consider ( x , y ) = ( a c 2 , ( a c ) 2 8 ) . If we let f ( x, y ) = x 2 + ( a c ) x + y, f ( x , y ) > f (0 , 0) = 0 , contradicting the optimality of x, ˆ y ) . Now suppose ˆ x > 0 , ˆ y = 0 . Then let ( x , y ) = ³ ˆ x, ˆ x 2 2 ´ . Again, f ( x , y ) > f x, ˆ y ) , and contraction. 1
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(c) Using (b), apply the Kuhn-Tucker necessity theorem to obtain the solution to problem ( P ) . [Remember to verify the Arrow-Hurwicz-Uzawa Constraint Quali fi cation]. Solution: It is tempting to let G 1 ( x, y ) = x 2 2 y, G 2 ( x, y ) = a x. However, if we check the bordered Hessian of G 1 ( x, y ) = x 2 2 y, we will fi nd that it is not quasi-concave on R 2 ++ . Or notice that h 1 ( x ) = x 2 2 is convex, and h 2 ( y ) = y is convex, and thus G 1 is convex, and thus quasi-convex. Thus in order to use the necessity theorem, we have to do some transformation. Just by noticing
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