Econ617Fall2005SecondExam - p = ( p 1 , p 2 , p 3 ) = (1 ,...

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T.Mitra, Fall 2005 Economics 617 Second Exam (October 26, 2005) 1. This exam has three questions. You have 1 hour and 15 minutes to write the exam. 2. This is a closed-book exam. 3. If you f nd any question ambiguous, explain your confusion and make whatever assumptions you think are necessary to answer the question. Clearly state any additional assumption you make. GOOD LUCK! 1
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1. [Quadratic Forms: (i) = 15 points; (ii) = 15 points] An n × n matrix (not necessarily symmetric ) is called positive quasi- de f nite if the symmetric n × n matrix B, de f ned by B =( A + A 0 ) is positive de f nite. [This is a de f nition]. Suppose A is an n × n matrix (not necessarily symmetric), which is positive quasi-de f nite. (i) Show that for every h R n , with h 6 =0 , we must have h 0 Ah > 0 . (ii) Show that the determinant of A is non-zero. 2. [Extension of Weierstrass Theorem: (i) = 15 points; (ii) = 25 points] (i) Let f : R 3 + R + be de f ned by: f ( x 1 ,x 2 ,x 3 )=m in { x 1 ,x 2 ,x 3 } Show that f is continuous on R 3 + . (ii) Let f be de f
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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).

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Econ617Fall2005SecondExam - p = ( p 1 , p 2 , p 3 ) = (1 ,...

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