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probset5 - Problem Set 5 EC720.01 Math for Economists...

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Problem Set 5 EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Thursday, October 15 This problem set asks you to solve a series of dynamic problems that characterize the optimal lending and borrowing behavior of individual consumers, the determination of the equilib- rium interest rate, and the link between equilibrium and optimal resource allocations in an economy in which, for simplicity, it is assumed that all agents live for only two periods. Each consumer is “young” during period t = 0 and “old” during period t = 1 . Consumers are of two types, called “lenders” and “borrowers” for reasons that will (hopefully) become clear when their individual optimization problems are described next. 1. Optimal Lending Consider first the behavior of a “lender” who receives an endowment consisting of one unit of the economy’s single consumption good during period t = 0 when he or she is young. Let c L 0 denote the consumption of this agent during period t = 0 when young and let s L denote the saving of this agent during period t = 0 when young. Assume that this lender earns interest on his or her savings at the rate r between periods t = 0 and t = 1. This lender receives no endowment during period t = 1, and hence must finance consumption c L 1 when old exclusively from his or her savings. Assume that this consumer has an additively time-separable utility function with single-period utility that is logarithmic in form and that utility at when old is discounted relative to utility when young using the discount factor β , which satisfies 0 < β < 1. Now the lender’s dynamic optimization problem can be stated formally as: max c L 0 ,c L 1 ,s L ln( c L 0 ) + β ln( c L 1 ) subject to 1 c L 0 + s L and (1 + r ) s L c L 1 . Note that nonnegativity conditions are not imposed on any of the choice variables, since the form of the utility function implies that the consumer will always want to consume at least
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