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probset7 - Problem Set 7 EC720.01 Math for Economists...

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Problem Set 7 EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Thursday, October 29 This problem asks you to derive some of the implications of a model of stock prices and con- sumption due originally to Robert Lucas, “Asset Prices in an Exchange Economy,” Econo- metrica , November 1978, pp.1429-1445. Consider an economy populated by a large number of identical consumers, each of whom has preferences described by the additively-time separable utility function X t =0 β t u ( c t ) , (1) where the discount factor satisfies 0 < β < 1 and the single-period utility function u is strictly increasing, strictly concave, and satisfies lim c 0 u 0 ( c ) = , this last assumption allowing us to ignore nonnegativity constraints on consumption in all of the analysis that follows. Each consumer finances his or her consumption by trading equity shares in the economy’s productive assets: let’s call them “fruit trees.” Each share in each tree provides a dividend in the form of d t pieces of “fruit” during each period t = 0 , 1 , 2 , ... , where fruit is the economy’s only consumption good. Let s t denote the number of shares carried by a representative consumer into each period t = 0 , 1 , 2 , ... , and let p t denote the price of each share in each tree during each period t = 0 , 1 , 2 , ... . Then, as sources of funds during each period t = 0 , 1 , 2 , ... , the representative consumer has his or her dividend payments d t s t and the total value p t s t of the shares carried into the period. And as uses of funds during each period t = 0 , 1 , 2 , ... , the consumer has his or her consumption c t and the value p t s t +1 of the shares that he or she will carry into period t + 1. The representative consumer therefore faces the budget constraint ( d t + p t ) s t c t + p t s t +1 for all t = 0 , 1 , 2 , ... . For the analysis that follows, it is useful, though not essential, to divide
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