325_Fall2010_FinalExamSolutions

325_Fall2010_FinalExamSolutions - Department of Economics...

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Department of Economics University of Maryland Economics 325 Intermediate Macroeconomic Analysis Final Exam Suggested Solutions Professor Sanjay Chugh Fall 2010 NAME: The Exam has a total of four (4) problems and pages numbered one (1) through twelve (12). 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 two pages (double-sided) of notes. You may not use a calculator. Problem 1 / 25 Problem 2 / 20 Problem 3 / 15 Problem 4 / 40 TOTAL / 100
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1 Problem 1: Consumption and Savings in the Two-Period Economy (25 points). Consider a two-period economy in which the government collects only lump-sum taxes from the representative consumer, and in which the representative consumer has no control over his pre- tax income. The lifetime utility function of the representative consumer is ± ² 1 2 1 2 , ln ln u c c c c ³ , where, as usual, 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 pre-tax income is 26, and the present discounted value of ALL lifetime tax payments is 6. Suppose that the real interest rate between period 1 and period 2 is zero (i.e., r = 0), and 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 2 1 2 2 1 1 , 1 1 1 c y c y t r t r r ³ ´ ³ ´ ³ ³ ³ in which the right hand side is the present value of lifetime disposable (i.e., after-tax) income (the notation is standard from Chapter 7). The Lagrangian is thus 2 2 1 2 1 1 2 1 ( , ) 1 1 1 y c u c c y c t r r t r O ª º ³ ³ ´ ´ ´ ´ « » ³ ³ ¬ ¼ ³ , where O 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 ( , ) 0 ( , ) 0 1 u c c u c c r O O ´ ´ ³ (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 2 1 2 ( , ) 1 ( , ) u c c
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