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Econ617Fall2005SecondExam

Econ617Fall2005SecondExam - p = p 1 p 2 p 3 =(1 1 0...

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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 fi 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 fi nite if the symmetric n × n matrix B, de fi ned by B = ( A + A 0 ) is positive de fi nite. [This is a de fi nition]. Suppose A is an n × n matrix (not necessarily symmetric), which is positive quasi-de fi 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 fi ned by: f ( x 1 , x 2 , x 3 ) = min { x 1 , x 2 , x 3 } Show that f is continuous on R 3 + . (ii) Let f be de fi ned as in (i) above, and let
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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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Econ617Fall2005SecondExam - p = p 1 p 2 p 3 =(1 1 0...

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