Problem_Set_10

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

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Unformatted text preview: 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 . 5 1 + x . 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 . 5 1 + x . 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 . 5 1 + x . 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...
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This note was uploaded on 10/03/2009 for the course ECON 6170 taught by Professor Mitra during the Fall '08 term at Cornell.

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

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