Spring 2009 Midterm Exam Solutions

# Spring 2009 Midterm Exam Solutions - Department of...

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Department of Economics University of Maryland Economics 325 Intermediate Macroeconomic Analysis Midterm Exam Professor Sanjay Chugh Spring 2009 Suggested Solutions NAME: TA’S NAME: The Exam has a total of four (4) problems. 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 intuition, 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. Problem 1 / 20 Problem 2 / 30 Problem 3 / 20 Problem 4 / 30 TOTAL / 100

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1 Problem 1: Consumption and Savings in the Two-Period Economy (20 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 !" 1 2 1 2 , ln ln u c c c c #\$ , where ln stands for the natural logarithm. We will work here in nominal terms: suppose the consumer’s present discounted value of ALL lifetime NOMINAL income is 52. For part a of this problem, suppose also the following: 1. The nominal interest rate between period 1 and period 2 is zero (i.e., i = 0, which is roughly what the nominal Federal Funds interest rate is currently). 2. The consumer begins period 1 with zero net assets. 3. Nominal prices of consumption in the two periods are 1 2 P # and 2 2 P # . a. (14 points) Set up the lifetime Lagrangian formulation of the consumer’s problem, 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. iii) is it possible to numerically compute the consumer’s nominal asset position at the end of period 1? If so, compute it; if not, explain why not. Solution: We know that with zero initial assets, the nominal LBC of the consumer is 2 2 2 2 1 1 1 1 , 11 P c P y Pc P y ii \$ # \$ \$ \$ where the notation is standard from class. The Lagrangian is thus 2 2 2 2 1 2 1 1 1 1 ( , ) P y P c u c c P y ri i % & \$ \$ ( ( ) * \$ \$ + , , where of course is the Lagrange multiplier (note there’s only one multiplier since this is the lifetime formulation of the problem not the sequential formulation of the problem). The first- order conditions with respect to 1 c and 2 c (which are the objects of choice) are, as usual: 1 1 2 1 2 2 1 2 ( , ) 0 ( , ) 0 1 u c c P P u c c i ( # ( # \$ (And of course the FOC with respect to the multiplier just gives back the LBC.) Proceeding exactly as we did in class, these FOCs can be combined into 1 1 2 2 1 2 2 ( , ) 1 1 ( , ) 1 u c c i r u c c - \$ # \$ , where the last equality follows by the Fisher Equation. Note that because we know both i and inflation
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Spring 2009 Midterm Exam Solutions - Department of...

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