325_MidtermExamSolutions

# 325_MidtermExamSolutions - Department of Economics...

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Department of Economics University of Maryland Economics 325 Intermediate Macroeconomic Analysis Midterm Exam – Suggested Solutions Professor Sanjay Chugh Fall 2011 NAME: Each problem’s total number of points is shown below. Your solutions should consist of some appropriate combination of mathematical analysis, graphical analysis, logical analysis, and economic interpretation, but in no case do solutions need to be exceptionally long. Your solutions should get straight to the point – solutions with irrelevant discussions and derivations will be penalized. You are to answer all questions in the spaces provided. You may use one page (double-sided) of notes. You may not use a calculator. TOTAL PART 1 / 50 TOTAL PART 2 / 50 TOTAL / 100

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1 Problem 1: Two-Period Economy (25 points). Consider a two-period economy (with no government and hence no taxes), in which the representative consumer has no control over his income. The lifetime utility function of the representative consumer is ±² 12 1 2 ,l n ucc c c ³ , where ln stands for the natural logarithm (that is not a typo – it is only 1 c that is inside a ln(.) function, c 2 is not inside a ln(.) function). Suppose the following numerical values: the nominal interest rate is 0.02 i , the nominal price of period-1 consumption is 1 100 P , the nominal price of period-2 consumption is 2 102 P , and the consumer begins period 1 with zero net assets. a. (3 points) Is it possible to numerically compute the real interest rate ( r ) between period one and period two? If so, compute it; if not, explain why not. Solution: The inflation rate is easily computed as 2 2 1 101 11 0 . 0 1 100 P P S ´ ´ . Then, using the exact Fisher equation, 2 . 0 1 . 0 1 i r ³ ³ ³ , so that 0 r . b. (14 points) Set up a sequential Lagrangian formulation of the consumer’s problem, and compute first-order conditions in order to answer the following: i) is it possible to numerically compute the consumer’s optimal choice of consumption in period 1? If so, compute it; if not, explain why not. ii) is it possible to numerically compute the consumer’s optimal choice of consumption in period 2? If so, compute it; if not, explain why not. Solution: The sequential Lagrangian for this problem (here cast in real terms, but you could have case it in nominal terms as well) is 11 1 1 2 2 1 2 (, ) [ ] [ ( 1 ) ] uc c y c a y ra c O ³´ ´ ³ ³ ³ ´ , where 1 and 2 are the multipliers on the period-1 and period-2 budget constraints. The first- order condition with respect to 1 c is 112 1 0 ´ , with respect to 2 c 212 2 0 ´ , and with respect to 1 a is (1 ) 0 r ´³ . The third FOC allows us to conclude ) r ³ . Substituting this into the FOC on 1 c gives 2 ( r ³ . Next, the FOC on 2 c allows us to obtain 22 1 2 . Substituting this into the previous expression gives us ( r ³ , or 1 r ³ , which of course is the usual consumption- savings optimality condition. Using the given functional form, the consumption-savings optimality condition for this problem can be expressed as 1 1/ 1 1 c r ³ , which immediately allows us to conclude 1 1 c r ³ , which completes part i. However, 2 c cannot be
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## This document was uploaded on 11/01/2011 for the course ECON 325 at Maryland.

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325_MidtermExamSolutions - Department of Economics...

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