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answer7 - Solutions to Problem Set 7 EC720.01 Math for...

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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 0 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 Kuhn-Tucker 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 first-order condition for c t is β t u 0 ( c t ) - π t +1 p t = 0 and the first-order 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
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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 t ) + π t +1 d t s t - c t p t .
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