714s10_fin_sol - Noah Williams Department of Economics...

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Economics 714 Department of Economics Macroeconomic Theory University of Wisconsin Spring 2010 F INAL A NSWER K EY by Tim Lee 1. Let’s begin by defining a CE. D EFINITION 1 A competitive equilibrium is an allocation { c t } t = 0 and sequence of prices { p t } t = 0 such that (i) Given prices, allocation solves consumers’ problem: max c t t = 0 β t u ( c i t ) s.t. t = 0 p t c i t t = 0 p t e i t = t = 0 p 2 t + i - 1 ( 1 + g ) 2 t + i - 1 for i = 1, 2 , (ii) Markets clear, i.e. c 1 t + c 2 t = e 1 t + e 2 t = ( 1 + g ) t . (a) Attaching the Lagrange multiplier λ i on each consumers’ budget constraint, the consumers’ f.o.c.’s are β t u 0 ( c i t ) = λ i p t c i t = f ( λ i p t / β t ) , (1) where f ( · ) is the inverse function of marginal utility. Normalizing p 0 = 1, the two budget constraints and market clearing conditions form the following system t = 0 p t f ( λ 1 p t / β t ) = k = 0 p 2 k ( 1 + g ) 2 k t
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This note was uploaded on 06/19/2011 for the course ECON 714 taught by Professor Staff during the Spring '08 term at University of Wisconsin.

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714s10_fin_sol - Noah Williams Department of Economics...

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