Spring 2009 Problem Set 2 Solutions

Spring 2009 Problem Set 2 Solutions - Department of...

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Department of Economics University of Maryland Economics 325 Intermediate Macroeconomic Analysis Problem Set 2 Suggested Solutions Professor Sanjay Chugh Spring 2009 Due: Monday, May 11, 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 must 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: Intertemporal Consumption Leisure Framework – A Numerical Analysis (20 points). Consider the two-period intertemporal consumption-leisure economy. Suppose the representative consumer’s lifetime utility function is given by 1 1 2 2 1 1 2 2 ( , , , ) ln( ) ln( ) v c l c l c l c l ! " " " , where, as usual, c denotes consumption and l denotes hours of leisure (hence 168 – l is an individual’s hours of labor during a week). Assume that the representative consumer begins period 1 with zero assets. The period-1, period-2, and lifetime budget constraints in this model, expressed in real terms, are thus given, respectively, by 1 1 1 1 1 2 2 1 2 2 2 (1 ) (168 ) ) ) (168 ) c a t w l c a r a t w l " ! # $ $ # " ! " $ " # $ $ # 2 2 2 2 1 1 1 1 ) (168 ) ) (168 ) 11 c t w l c t w l rr # # " ! # # " "" The tax rates in the two periods are 12 0.5 tt ! ! , and the real wages in the two periods are 1 20 w ! and 2 22 w ! . Note that you are NOT given a numerical value for the real interest rate (you will solve for this in part b below). a. (10 points) Is it possible to solve numerically for the representative consumer’s optimal choices of consumption in each of the two periods? If so, do so, showing any important steps in your logic/computation. If not, briefly describe the economic and/or mathematical issue(s) that prevents doing so. ( Note : If you can solve without setting up and solving a Lagrangian, you may do so.) Solution: Construct the three optimality conditions in the model. To do this, you needed to compute the four marginal utility functions, which are: 1 1 1 c v c ! , 2 2 1 c v c ! , 1 1 l v ! , and 2 1 l v ! . With these, the three optimality conditions are 1 1 2 2 1 2 1 22 2 1 2 1 ) 1/ 1 ) 1 l c l c c c v tw vc v v c r ! ! # ! ! # ! ! " The first two of these expressions are the optimal consumption-leisure margins within period 1 and period 2, respectively, while the third expression is the optimal consumption-savings margin linking period 1 and period 2. Note that leisure 1 l and 2 l do not appear in any of these optimality conditions (this observation is critical for part c below).
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Spring 2009 Problem Set 2 Solutions - Department of...

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