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1 Answers to the Sept 08 macro prelim - Long Questions 1. Suppose that a representative consumer receives an endowment of a non-storable consumption good. The endowment evolves exogenously according to ° ln C t = ° + ± ° ln C t ° 1 + ²" t ; (1) where ° is the difference operator and " t is an iid N (0 ; 1) random variable. The consumer’s preferences are E t P 1 j =0 ³ j ln C t + j : (a) Solve recursively for the value of the endowment process. (I.e., write the Bellman equation and solve for the value function.) (b) What is the marginal value of an increase in the mean growth rate of the endowment? (c) What is the marginal value of a reduction in volatility, as measured by a decline in ² ? (d) Provide intuition about the relative magnitudes of the marginal values in (b) and (c). ANSWER: a. Begin by writing the endowment process as ln C t ° ln C t ° 1 = ° + ± (ln C t ° 1 ° ln C t ° 2 ) + ²" t ; (2) ln C t = ° + (1 + ± ) ln C t ° 1 ° ± ln C t ° 2 + ²" t : This can be expressed in companion form as X t = m + AX t ° 1 + ± " t ; (3) where X t = ° ln C t ln C t ° 1 ± ; m = ° ° 0 ± ; A = ° 1 + ± ° ± 1 0 ± ; ± = ° ² 0 ± : (4) The state vector for this problem is X t ; and the Bellman equation is V ( X t ) = ln C t + ³E t V ( X t +1 ) : (5) Conjecture that the value function is linear in X t ; 1 V ( X t ) = b 0 + b 1 X t ; (6) 1 This is obvious. It follows from the fact that expected utility is linear in expectations of ln C t + j and that expectations of ln C t + j are linear in X t : 1

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where b 0 is a scalar and b 1 is a row vector conformable with X t : Substitute the guess into (5), b 0 + b 1 X t = ln C t + ³E t ( b 0 + b 1 X t +1 ) ; (7) = e 1 X t + ³ [ b 0 + b 1 ( m + AX t )] ; = ³ ( b 0 + b 1 m ) + ( e 1 + ³b 1 A ) X t ; where e 1 = [1 0] : To solve for ( b 0 ; b 1 ) ; equate powers of X t on both sides of this equation. b 0 = ³ ( b 0 + b 1 m ) ; (8) b 1 = e 1 + ³b 1 A: Assuming the inverse exists, 2 the second condition can be solved to °nd b 1 = e 1 ( I ° ³A ) ° 1 : (9) Substitute this solution into the condition for b 0 to °nd b 0 = e 1 ( I ° ³A ) ° 1 m 1 ° ³ : (10) Thus the value function can be expressed as V ( X t ) = e 1 ( I ° ³A ) ° 1 m 1 ° ³ + e 1 ( I ° ³A ) ° 1 X t ; (11) = e 1 ( I ° ³A ) ° 1 ° m 1 ° ³ + X t ± : For later use, de°ne a ij to be the ( i; j ) th element of ( I ° ³A ) ° 1 : It follows that e 1 ( I ° ³A ) ° 1 = ( a 11 a 12 ) : (12) Hence the value function can also be expressed as V ( X t ) = a 11 ° 1 ° ³ + a 11 ln C t + a 12 ln C t ° 1 : (13) b. The average growth rate of consumption is ² g = °= (1 ° ± ) : Hence we can write the value function in terms of mean growth as V ( X t ) = a 11 (1 ° ± ) 1 ° ³ ² g + a 11 ln C t + a 12 ln C t ° 1 : (14) Taking the derivative wrt ² g gives the marginal value of an increase in growth, @V ( X t ) @ ² g = a 11 (1 ° ± ) 1 ° ³ > 0 : (15) 2 The eigenvalues of A are 1 and °: Since consumption growth is weakly autocorrelated in US data, 0 < ° < 1 : Since ± < 1 ; the eigenvalues of ±A are both inside the unit circle. 2
c. Since V ( X t ) does not depend on ²; it follows that a reduction in volatility has a marginal value of @V ( X t ) = 0 : d. Intuition: an increase in mean growth raises expected ln C in every future period.

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