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Unformatted text preview: Solutions to Problem Set 6 EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Thursday, October 22 1. The Permanent Income Hypothesis The consumer chooses c , c 1 , and s to maximize the utility function ln( c ) + ln( c 1 ) subject to the constraints w c + s and w 1 + (1 + r ) s c 1 . a. With the Lagrangian for the consumers problem defined as L ( c ,c 1 ,s, , 1 ) = ln( c ) + ln( c 1 ) + ( w- c- s ) + 1 [ w 1 + (1 + r ) s- c 1 ] , the first-order conditions can be written as 1 c *- * = 0 , c * 1- * 1 = 0 , and- * + * 1 (1 + r ) = 0 and the constraints, which will bind at the optimum, can be written as w- c *- s * = 0 and w 1 + (1 + r ) s *- c * 1 = 0 . Combine the budget constraints to obtain w + w 1 1 + r = c * + c * 1 1 + r , which says that the present value of the consumers income will equal the present value of his or her consumption over the two periods. Now combine the first-order conditions to obtain 1 c * = (1 + r ) c * 1 1 or c * 1 = (1 + r ) c * . Substitute this last expression into the present value budget constraint to find the desired solution c * = 1 1 + w + w 1 1 + r , then substitute this solution into the previous expression to obtain c * 1 = (1 + r ) 1 + w + w 1 1 + r ....
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This note was uploaded on 02/19/2010 for the course ECON 720 taught by Professor Ireland during the Fall '09 term at BC.
- Fall '09