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Unformatted text preview: Solutions to Problem Set 11 EC720.01  Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Tuesday, December 8 Human Capital Accumulation and Economic Growth In this version of the UzawaLucas model, the representative consumer or social planner chooses sequences { c t } t =0 , { u t } t =0 , { h t } t =1 , and { k t } t =1 , to maximize the utility function X t =0 t ln( c t ) , subject to the constraints u t h t h t +1 (1) and k t [(1 u t ) h t ] 1 c t + k t +1 (2) for all t = 0 , 1 , 2 ,... , taking the initial stocks h and k as given. 1. The Bellman equation for the problem can be written as v ( k t ,h t ; t ) = max c t ,u t ln( c t ) + v ( k t [(1 u t ) h t ] 1  c t ,u t h t ; t + 1) or, using the guess that the value function takes the timeinvariant form v ( k t ,h t ; t ) = v ( k t ,h t ) = E + F ln( k t ) + G ln( h t ) where E , F , and G are constants to be determined, E + F ln( k t ) + G ln( h t ) = max c t ,u t ln( c t ) + E + F ln( k t [(1 u t ) h t ] 1  c t ) + G ln( u t h t ) = max c t ,u t ln( c t ) + E + F ln( k t [(1 u t ) h t ] 1  c t ) + G ln( ) + G ln( u t ) + G ln( h t ) ....
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 Fall '09
 IRELAND
 Economics

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