Spring 2009 Problem Set 1 Solutions

Spring 2009 Problem Set 1 Solutions - Department of...

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Department of Economics University of Maryland Economics 325 Intermediate Macroeconomic Analysis Problem Set 1 Suggested Solutions Professor Sanjay Chugh Spring 2009 Instructions : Written (typed is strongly preferred, but not required) solutions must be submitted no later than 11:00am on the date listed above (either in class or in the Economics Department Main Office, Tydings Hall 3105). Your solutions, which likely require some combination of mathematical derivations, economic reasoning, graphical analysis, and pure logic, should be thoroughly presented and not leave the reader (i.e., your TAs and I) guessing about what you actually meant. You must submit your own independently-written solutions. You are permitted (in fact, encouraged) to work in groups to think through issues and ideas, but your “writing up” of solutions should be done independently of anyone else. Under no circumstances will multiple verbatim identical solutions be considered acceptable. There are three problems in total, each with multiple subparts.
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2 Problem 1: Optimal Choice in the Consumption-Savings Model During a Credit Crunch: A Numerical Analysis. Consider a two-period economy (with no government and hence neither government spending nor taxation), 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 real income in period 1 is y 1 = 10 and the consumer’s real income in period 2 is y 2 = 22. Suppose that the real interest rate between period 1 and period 2 is ten percent (i.e., r = 0.10), and also suppose the consumer begins period 1 with real net wealth (inclusive of interest) of (1+ r ) a 0 = 2. Set up the lifetime Lagrangian formulation of the consumer’s problem, and use it to answer part a, b, and c. Show all steps in your logic/arguments. Solution: The Lagrangian in real terms, using the given functional form for utility, is 22 1 2 1 0 1 ln ln (1 ) 11 y c c c y r a c rr % & $ $ $ $ $ ( ( ) * $ $ + , , where the term in square brackets (when set equal to zero) is simply the LBC in real terms. The first-order conditions of this problem are (recognizing that the FOC with respect to ! is simply the LBC in real terms): 1 2 1 0 1 0 1 c cr (# ( # $ Combining these two equations to eliminate the multiplier as usual gives the consumption-savings optimality condition for this particular utility function: 2 1 1 c r c # $ . In what follows, you must use the consumption-savings optimality condition along with the LBC (which together constitute two equations in the two unknowns c 1 and c 2 ) to proceed. a. I s it possible to numerically compute the consumer’s optimal choice of consumption in period 1? If so, compute it; if not, explain why not.
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This note was uploaded on 09/07/2011 for the course ECON 325 taught by Professor Chugh during the Spring '08 term at Maryland.

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Spring 2009 Problem Set 1 Solutions - Department of...

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