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answer8 - Solutions to Problem Set 8 EC720.01 Math for...

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Unformatted text preview: Solutions to Problem Set 8 EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Thursday, November 5 1. Life Cycle Saving The consumer chooses sequences { c t } T t =0 and { k t } T +1 t =1 to maximize T X t =0 β t c 1- σ t- 1 1- σ subject to the constraints k = 0 given , w t + r t k t- c t ≥ k t +1- k t for all t = 0 , 1 ,...,T , and k T +1 ≥ k * > . a. The Hamiltonian for the consumer’s problem can be defined as H ( k t ,π t +1 ; t ) = max c t β t c 1- σ t- 1 1- σ + π t +1 ( w t + r t k t- c t ) . b. According to the maximum principle, the solution to the consumer’s dynamic optimiza- tion problem is characterized by the first-order condition β t c- σ t- π t +1 = 0 , the pair of difference equations π t +1- π t =- H k ( k t ,π t +1 ; t ) =- π t +1 r t and k t +1- k t = H π ( k t ,π t +1 ; t ) = w t + r t k t- c t , the initial condition k = 0 , and the terminal or transversality condition π T +1 ( k T +1- k * ) = 0...
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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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answer8 - Solutions to Problem Set 8 EC720.01 Math for...

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