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Unformatted text preview: University of California, Davis Date: September 2, 2004 Department of Economics Time: 4 hours Microeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE PLEASE ANSWER FOUR QUESTIONS (OUT OF FIVE) Question 1. In this question we consider utility functions of the additively separable form ) ( ) ,..., ( 1 1 j L j j L x u x x u = = , (1.1) where L j x u j j ,..., 1 , ) ( ' = > . Throughout this question, we assume that the UMAX[ p , w ] problems (where the price vector p and the wealth level w are strictly positive) have unique, interior solutions, and that all functions are (twice continuously) differentiable. Except for part 1(a), we consider a single consumer. 1(a) Suppose that all consumers have preferences representable by the same utility function, of form (1.1). Can we be sure that a positive representative consumer exists for an unrestricted domain of wealth vectors? If YES, justify your claim. If NO, provide a counterexample. 1(b) Write the L +1 first order equalities of the UMAX[ p , w ] problem, denoting its solution by )) , ( ~ ),..., , ( ~ ( ) , ( ~ 1 w p x w p x w p x L = and its (positive) Lagrange multiplier by ) , ( w p . 1(c) By differentiating the just obtained first-order equalities with respect to the parameters, obtain ( L +1) 2 equalities involving the partial derivatives of the Walrasian demand function. 1(d) Show that ) , ( ) , ( w p w w p v = , where ) , ( w p v is the indirect utility function, and interpret in words....
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