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answer5 - Solutions to Problem Set 5 EC720.01 - Math for...

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Unformatted text preview: Solutions to Problem Set 5 EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Thursday, October 15 1. Optimal Lending The lender’s dynamic optimization problem can be stated formally as: max c L ,c L 1 ,s L ln( c L ) + β ln( c L 1 ) subject to 1 ≥ c L + s L and (1 + r ) s L ≥ c L 1 . a. The Lagrangian for this consumer’s problem can be defined as L ( c L ,c L 1 ,s L ,λ L ,λ L 1 ) = ln( c L ) + β ln( c L 1 ) + λ L (1- c L- s L ) + λ L 1 [(1 + r ) s L- c L 1 ] . b. According to the Kuhn-Tucker theorem, the lender’s optimal choices of c L * , c L * 1 , and s L * , together with the associated values of λ L * and λ L * 1 , must satisfy the first-order conditions L 1 ( c L * ,c L * 1 ,s L * ,λ L * ,λ L * 1 ) = 1 /c L *- λ L * = 0 , L 2 ( c L * ,c L * 1 ,s L * ,λ L * ,λ L * 1 ) = β/c L * 1- λ L * 1 = 0 , and L 3 ( c L * ,c L * 1 ,s L * ,λ L * ,λ L * 1 ) =- λ L * + λ L * 1 (1 + r ) = 0 the constraints L 4 ( c L * ,c L * 1 ,s L * ,λ L * ,λ L * 1 ) = 1- c L *- s L * ≥ and L 5 ( c L * ,c L * 1 ,s L * ,λ L * ,λ L * 1 ) = (1 + r ) s L *- c L * 1 ≥ , the nonnegativity conditions λ L * ≥ and λ L * 1 ≥ , and the complementary slackness conditions λ L * (1- c L *- s L * ) = 0 and λ L * 1 [(1 + r ) s L *- c L * 1 ] = 0 . 1 c. Combine the first-order conditions to obtain c L * = 1 λ L * 1 (1 + r ) and c L * 1 = β λ L * 1 . Combine the constraints to obtain 1 = c L * + c L * 1 1 + r . Next, substitute the first two of these last three conditions into the third to solve for λ L * 1 = 1 + β 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.

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answer5 - Solutions to Problem Set 5 EC720.01 - Math for...

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