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Econ 403: Advanced Topics in Macroeconomics Answer key to Problem Set #3 : Real Business Cycle Model and A Simple OLG Monetary Model Q1: Consider the following simple RBC model: Preferences are given by E 0 X t =0 β t [ bc 1 - η t + (1 - b ) l 1 - η t ] 1 1 - η 0 < β < 1 (0.1) where l = 1 - n is leisure, technology is given by y t = e z t k α t n 1 - α t (0.2) and the resource constraint: c t + k t +1 - (1 - δ ) k t = y t (0.3) a) Set up the social planner’s problem for the economy and derive the ﬁrst order conditions. The social planner’s problem is max c t ,n t ,k t +1 E 0 X t =0 β t [ bc 1 - η t + (1 - b )(1 - n t ) 1 - η ] l 1 1 - η t (0.4) S.t. c t + k t +1 - (1 - δ ) k t = y t = e z t k α t n 1 - α t (0.5) So the Lagrangian is L = E 0 X t =0 β t n [ bc 1 - η t + (1 - b )(1 - n t ) 1 - η ] 1 1 - η + λ t [ e z t k α t n 1 - α t - c t - k t +1 + (1 - δ ) k t ] o (0.6) So the ﬁrst order conditions imply: ∂L ∂c t = 0 1 1 - η [ bc 1 - η t + (1 - b )(1 - n t ) 1 - η ] 1 1 - η - 1 (1 - η ) bc - η t - λ t = 0 (0.7) 1

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∂L ∂n t = 0 1 1 - η [ bc 1 - η t +(1 - b )(1 - n t ) 1 - η ] 1 1 - η - 1 (1 - η )(1 - b )(1 - n t ) - η ( - 1)+ λ t (1 - α ) e z t k α t n - α t = 0 (0.8) ∂L
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