Problem_Set_10

# Problem_Set_10 - T Mitra Fall 2008 Economics 6170 Problem...

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T. Mitra, Fall, 2008 Economics 6170 Problem Set 10 [For practice only; do not hand in solutions] 1. [Using Kuhn-Tucker Sufficiency Theory by Contracting the Domain] Let p be an arbitrary positive real number. Consider the following con- strained optimization problem: Maximize x 0 . 5 1 + x 0 . 5 2 subject to px 1 + x 2 px 3 + x 4 ( x 3 ) 2 + ( x 4 ) 2 1 ( x 1 , x 2 , x 3 , x 4 ) R 4 + ( R ) (a) To solve problem ( R ) , first solve problem ( S ) given below: Maximize x 0 . 5 1 + x 0 . 5 2 subject to px 1 + x 2 px 3 + x 4 ( x 3 ) 2 + ( x 4 ) 2 1 ( x 1 , x 2 , x 3 , x 4 ) R 4 ++ ( S ) Define X = R 4 ++ , f ( x ) = x 0 . 5 1 + x 0 . 5 2 , g 1 ( x ) = px 3 + x 4 - px 1 - x 2 , g 2 ( x ) = 1 - [( x 3 ) 2 + ( x 4 ) 2 ] , where x = ( x 1 , x 2 , x 3 , x 4 ) X. Write down and solve the Kuhn-Tucker conditions for problem ( S ) , and denote the solution of the Kuhn-Tucker conditions by (¯ x, ¯ λ ) X × R 2 + . (b) Show that ¯ x solves problem ( S ) , and (¯ x, ¯ λ ) satisfies: f ( x ) + ¯ λg ( x ) f x ) + ¯ λg x ) for all x X (c) Use (b) and the continuity of f , g 1 and g 2 on R 4 + to establish that ¯ x solves ( R ) . 2. [Using Kuhn-Tucker Sufficiency Theory by Expanding the Domain] Let a, b be arbitrary positive numbers, satisfying a > b > 1 .

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