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midterm_solutions - Noah Williams Department of Economics...

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Noah Williams Economics 714 Department of Economics Macroeconomic Theory University of Wisconsin Spring 2009 Midterm Examination Solutions Instructions: This is a 75 minute exam with two questions worth a total of 100 points. Points are indicated at the start of each question. Allocate your time wisely. In order to get full credit, you must give a clear, concise, and correct answer, including all necessary explanations and calculations. Notes, books, and calculators are not permitted. 1. ( 50 points ) Consider the following version of a Lucas asset pricing model, in which an agent wants to minimize fluctuations in consumption around a mean. For simplicity we normalize the mean to zero. In other words, a representative agent has preferences: - 1 2 E 0 X t =0 β t c 2 t over the single nonstorable consumption good c t (“fruit”). The endowment process x t is also mean zero and follows a Gaussian AR process: x t +1 = ρx t + ε t +1 where 0 < ρ < 1 and ε t +1 is an i.i.d. N (0 , σ 2 ) random variable. (a) ( 15 points ) Define a recursive competitive equilibrium with a market in claims to the endowment process (“trees”), with pricing function p ( x ). Solution. A recursive competitive equilibrium is a value function V ( a, x ) , a pair of policy functions c ( a, x ) and a 0 ( a, x ) , and a pricing function p ( x ) such that ( i ) V ( a, x ) = max c,a 0 { u ( c ) + βE [ V ( a 0 , x 0 ) | x ] } st c + p ( x ) a 0 = ( p ( x ) + x ) a x 0 = ρx + ε 0 ( ii ) c ( a, x ) , a 0 ( a, x ) attain the maximum above, ( iii ) markets clear: a 0 = 1 , c (1 , x ) = x. Here we can substitute in u ( c ) = - 1 2 c 2 and the expectation can be written as: E [ V ( a 0 , x 0 ) | x ] = Z V ( a 0 , ρx + ε 0 ) d Φ( ε 0 ) where Φ is the standard normal distribution function. 1
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(b) ( 15 points ) Characterize the recursive competitive equilibrium and provide an expression for the pricing function p ( x ). Solution. Combining the FOC u 0 ( c ) p ( x ) = βE [ V a ( a 0 , x 0 ) | x ] and the envelope condition V a ( a, x ) = u 0 ( c ) ( p ( x ) + x ) we get the Euler equation u 0 ( c ) p ( x ) = βE u 0 ( c 0 ) ( p ( x 0 ) + x 0 ) | x .
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