204Final082107 - Economics 204Final ExamAugust 21, 2007,...

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Unformatted text preview: Economics 204Final ExamAugust 21, 2007, 9am-12pm Each of the four questions is worth 25% of the total Please use three separate bluebooks, one for each of the three Parts Part I 1. State and prove the Supremum Property. 2. Prove that if a finite set X has exactly n elements, then 2 X , the set of all subsets of X , has exactly 2 n elements. Hint : use induction. Part II 3. Consider the function f ( x, y ) = x 3 y 3 + 2 x 2 + 2 xy + 8 y 2 9 x 15 y 2 (a) Compute the first order conditions for a local maximum or minimum of f . Show that the first order conditions are satisfied at the point ( x , y ) = (1 , 1). (b) Compute D 2 f ( x , y ) and give the quadratic Taylor polynomial for f at the point ( x , y ). (c) Find the eigenvalues of D 2 f ( x , y ) and determine whether f has a local max, a local min, or a saddle at ( x , y ). (d) Does f have a global max, a global min, or neither, at ( x , y )?...
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This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at University of California, Berkeley.

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