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Fall 2008 Midterm Exam Solutions

# Fall 2008 Midterm Exam Solutions - 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 2008 NAME: The Exam has a total of five (5) problems and pages numbered one (1) through eleven (11). 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 / 25 Problem 2 / 25 Problem 3 / 11 Problem 4 / 15 Problem 5 / 24 TOTAL / 100

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1 Problem 1: Consumption and Savings in the Two-Period Economy (25 points). Consider a two-period economy (with no government), 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 purely real terms: suppose the consumer’s present discounted value of ALL lifetime REAL income is 26. Also suppose the consumer begins period 1 with zero net assets. a. (17 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 real 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 LBC of the consumer is 22 11 , cy rr \$ # \$ \$ \$ where the notation is standard from class. The Lagrangian is thus 1 2 1 1 ( , ) yc u c c y c % & \$ \$ ( ( ) * \$\$ + , , 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 2 1 2 1 ( , ) 0 ( , ) 0 1 u c c u c c r ( # ( # \$ (And of course the FOC with respect to the multiplier just gives back the LBC.) Also as usual, these FOCs can be combined to give the consumption-savings optimality condition, 1 1 2 1 2 1 2 ( , ) 1 ( , ) u c c r u c c . With the given utility function, the marginal utility functions are 1/ uc # and # , so the consumption-savings optimality condition in this case becomes 2 1 1 /1 c c r # \$ .
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Fall 2008 Midterm Exam Solutions - Department of Economics...

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