midterm387s03answer

midterm387s03answer - Econ 387L Macro II Spring 2003...

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Econ 387L: Macro II Spring 2003, University of Texas Instructor: Dean Corbae Midterm Exam Answers Consider the following investment problem faced by a & rm. The & rm&s revenues are given by R ( K t ,Z t ) and there are costs of adjusting its capital holdings C ( K t +1 ,K t ) where K t is the capital stock at the beginning of period t , K t +1 0 is capital chosen at the beginning of period t to come on line in t +1 , Z t is a technology innovation, and K 0 > 0 is given.Gross investment is given by I t = K t +1 (1 δ ) K t and the relative price of investment goods in terms of consumption goods is p t . Firms discount the future at rate β (0 , 1) .The & rm&s problem is max { K t +1 } t =0 E " X t =0 β t { R ( K t t ) C ( K t +1 t ) p t [ K t +1 (1 δ ) K t ] } # . We will assume the following: R ( K t t )= Z t K t , C ( K t +1 t γ 2 & I t K t 2 K t ,and p t = p, t. There are 4 parts to this question. 1. What are the & rst order conditions? Interpret them in terms of the marginal cost of adding capital versus the marginal bene & t. Answer: The f.o.c. is given by p t + ∂C ( K t +1 t ) ∂K t +1 = βE t ∂R ( K t +1 t +1 ) t +1 ( K t +2 t +1 ) t +1 + p t +1 (1 δ ) or γ & I t K t + p = t Z t +1 + γ 2 & I t +1 K t +1 ¶‰& I t +1 K t +1 +2(1 δ ) ± + p (1 δ ) (1) The left hand side is the marginal cost of adding capital (the pecuniary cost plus the adjustment cost) in period t . The right hand side is the marginal bene & tinperiod t from adding the capital; the marginal product of capital less the costs of adjustment plus the sales price.
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midterm387s03answer - Econ 387L Macro II Spring 2003...

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