325_PracticePS4_Soln - Department of Economics University...

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Department of Economics University of Maryland Economics 325 Intermediate Macroeconomic Analysis Practice Problem Set 4 Suggested Solutions Professor Sanjay Chugh Spring 2011 1. Optimal Choice in the Consumption-Savings Model with Credit Constraints: A Numerical Analysis. Consider our usual two-period consumption-savings model. Let preferences of the representative consumer be described by the utility function 12 1 2 (, ) , uc c c c E ± where 1 c denotes consumption in period one and 2 c denotes consumption in period two. The parameter is known as the subjective discount factor and measures the consumer's degree of impatience in the sense that the smaller is , the higher the weight the consumer assigns to present consumption relative to future consumption. Assume that 1/1.1. For this particular utility specification, the marginal utility functions are given by 112 1 1 2 ucc c and 212 2 2 c . The representative household has initial real financial wealth (including interest) of 0 (1 ) 1 ra ± The household earns 1 5 y units of goods in period one and 2 10 y units in period two. The real interest rate paid on assets held from period one to period two equals 10% (i.e., 0.1 r ). a. Calculate the equilibrium levels of consumption in periods one and two. ( Hint: Set up the Lagrangian and solve.) b. Suppose now that lenders to this consumer impose credit constraints on the consumer. Specifically, they impose the tightest possible credit constraint – the consumer is not allowed to be in debt at the end of period one, which implies that the consumer’s real wealth at the end of period one must be nonnegative ( 1 0 a t ). What is the consumer’s choice of period-one and period-two consumption under this credit constraint? Briefly explain, either logically or graphically or both. c. Does the credit constraint described in part b enhance or diminish welfare (i.e., does it increase or decrease lifetime utility)? Specifically, find the level of utility under the credit constraint and compare it to the level of utility obtained under no credit constraint. Suppose now that the consumer experiences a temporary increase in real income in period one to 1 9 y , with real income in period two unchanged. d. Calculate the effect of this positive surprise in income on 1 c and 2 c , supposing that there is no credit constraint on the consumer.
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2 e. Finally, suppose that the credit constraint described in part b is back in place. Will it be binding? That is, will it affect the consumer’s choices? Solution: a. The consumer’s problem is to maximize lifetime utility (given by 12 (, ) uc c subject to the LBC. The Lagrangian for this problem is thus 22 0 1 1 (, ,) (, ) ( 1 ) 11 y c Lc c ra y c rr OO §· ± ± ± ± ² ² ¨¸ ± ± ©¹ , where we must include the nonzero initial real wealth (inclusive of interest) 0 (1 ) ± . The first-order conditions with respect to 1 c and 2 c are 112 212 0 0 1 ucc r O ² ² ± C o m b i n i n g t h e s e , w e g e t t h e u s u a l c o n s u m p t i o n - s a v i n g s o p t i m a l i t y c o n d i t i o n , (, ) ( 1 ) (, ) rucc ± (ie, the MRS equals the slope of the LBC). Using the
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325_PracticePS4_Soln - Department of Economics University...

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