Econ617Fall2006PS10 - T. Mitra Fall, 2006 Economics 617...

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T. Mitra Fall, 2006 Economics 617 Problem Set 10 [For practice only; do not hand in solutions] 1. [Using the Kuhn-Tucker Necessity Theorem] Let a, c be arbitrary positive parameters, satisfying a>c . De±ne the following functions: F ( x )= a xf o r x [0 ,a ] G ( x ± x 0 F ( z ) dz for x [0 ] 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. (b) Show that if x, ˆ y ) is any solution to problem ( P ) , then ˆ x> 0 , ˆ y> 0 . (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±cation]. (d) Denoting this solution by x, ¯ y ) , show that: ( i ) F x c ( ii y = G x ) c ¯ x 2. [Increasing Transformations of Quasi-Concave and Concave Functions] Denote R n + by X, and let u be a function from X to R . Let g be a function from u ( X ) to R . De±ne the composite function, v, from X to R by: v ( x g ( u ( x
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Econ617Fall2006PS10 - T. Mitra Fall, 2006 Economics 617...

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