ps2ans08 - Econ 387L: Macro II Spring 2008, University of...

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Econ 387L: Macro II Spring 2008, University of Texas Instructor: Dean Corbae Problem Set #2- Due 1/29/08 Consider a stochastic growth model with the following speci f cation. Preferences: U ( C t ,H t t )= ³ C t + λ t h (1 H t ) 1 η 1 1 η 1 ψ 1 1 ψ where λ t =(1 γ ) 1 λ t 1 ε t 1 =1 . (1) and log ε t is i.i.d. N (0 ε ) . Agents discount the future at rate β< 1 . Technology: Y t =(1+ g ) t (1 θ ) K θ t H 1 θ t (2) where K 0 is given, 1 H t 0 . Information: Households must choose H t before knowing the shock to preferences λ t but choose K t +1 after its realization. 1) Derive the stochastic Euler equation for the savings choice. Answer: U K ( C t t t βE t n U K ( C t +1 t +1 t +1 ) h (1 + g ) ( t +1)(1 θ ) θK θ 1 t +1 H 1 θ t +1 +(1 δ ) io (3) where U K ( C t t t h C t + λ t ³ (1 H t ) 1 η 1 1 η ´i ψ 2) Derive the equation describing the labor/leisure choice. Answer: E t n U K ( C t t t )(1 θ ) K θ t (1 + g ) t (1 θ ) H θ t + U H ( C t t t ) o =0 (4) where U H ( C t t t h C t + λ t ³ (1 H t ) 1 η 1 1 η ´i ψ [ λ t (1 H t ) η ] 3) Along a possible balanced growth path, assume that ε t is constant and equal to its expected value of 1. Under what conditions on parameters might a balanced growth path where K t ,C t ,Y t all grow at rate κ while H t remains constant exist? Is the set empty?
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ps2ans08 - Econ 387L: Macro II Spring 2008, University of...

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