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Unformatted text preview: Solutions to Problem Set 7 EC720.01  Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Thursday, October 29 Consider an economy populated by a large number of identical consumers, each of whom takes s as given, and chooses sequences { c t } ∞ t =0 and { s t } ∞ t =1 to maximize the utility function ∞ X t =0 β t u ( c t ) subject to the budget constraint d t s t c t p t ≥ s t +1 s t for all t = 0 , 1 , 2 ,... . 1. The KuhnTucker Formulation With the Lagrangian for this problem written as L = ∞ X t =0 β t u ( c t ) + ∞ X t =0 π t +1 s t + d t s t c t p t s t +1 the firstorder condition for c t is β t u ( c t ) π t +1 p t = 0 and the firstorder condition for s t is π t +1 + π t +1 d t p t π t = 0 . The first of these two conditions must hold for all t = 0 , 1 , 2 ,... and the second must hold for all t = 1 , 2 , 3 ,... . Together with the binding constraint d t s t c t p t = s t +1 s t for all t = 0 , 1 , 2 ,... , these conditions form a system of three equations in the three unknowns c t , s t , and π t +1 . 1 2. The Maximum Principle The Hamiltonian for the consumer’s problem can be defined as H ( s t ,π t +1 ; t ) = max c t β t u ( c...
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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
 IRELAND
 Economics

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