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Unformatted text preview: p = ( p 1 , p 2 , p 3 ) = (1 , 1 , 0) . Consider the following constrained maximization problem: Max f ( x 1 , x 2 , x 3 ) subject to p 1 x 1 + p 2 x 2 + p 3 x 3 1 and ( x 1 , x 2 , x 3 ) ( M ) Use the extension of Weierstrass theorem to show that problem ( M ) has a solution. 3. [Implicit Function Theorem: (i) = 15 points; (ii) = 15 points] Let A be an m n matrix and B be a non-singular m m matrix. Let F : R n R m R m be de f ned by: F ( x, y ) = Ax By for all x R n , y R m (i) Use the implicit function theorem to show that there is an open set X containing n and an open set Y containing m and a unique function g : X Y such that: (a) F ( x, g ( x )) = 0 for all x X (b) g (0 n ) = 0 m . (ii) Using (i), show that g ( x ) = B 1 Ax for all x X. 2...
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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).
- Fall '08