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Unformatted text preview: University of California, Davis Date: September 6, 2007 Department of Economics Time: 5 hours Macroeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE Directions: Answer all questions; the questions are equally weighted 1. Imagine a representative consumer with time separable, logarithmic utility, V = (1 & & ) E t X 1 j =0 & j ln C t + j ; (1) where < & < 1 . Consumption is an element of an exogenous random vector X t which evolves according to X t +1 = ¡ + AX t + " t +1 ; (2) where " t ¡ iid N (0 ; &) : Suppose that A has a single unit eigenvalue, with all other eigenvalues less than 1 in magnitude. For convenience, assume that ln C t is the &rst element of the vector X t : Assume ¡su¢ cient discounting.£ (This is intentionally vague; the context should become clear as you develop your answer.) Derive the consumer£s value function. (Hint: There are at least two ways to solve this, a recursive and a nonrecursive approach. The recursive approach is a lot easier. The nonrecursive approach turns into an algebraic quagmire.) 2. Consider a two-period overlapping generations model in which people work when young and retire when old. Consumers maximize ln( c 1 t ) + (1 + ¢ ) & 1 ln( c 2 t +1 ) ; subject to the ¤ow budget constraints...
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