midterm387s08answers - Econ 387L: Macro II Spring 2008,...

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Econ 387L: Macro II Spring 2008, University of Texas Instructor: Dean Corbae Midterm Exam I. Consider the following OG model of production. Each period a new generation of individuals are born and live for 2 periods. There is population growth of size n 0 so that the measure of young alive at time N t =(1+ n ) t N 0 with N 0 =1 (i.e. the measure of agents of born in period t =0 is normalized to 1) and the initial old are taken to be of measure N 1 / (1 + n ) . Each agent from a generation born at time t has preferences given by U t =ln( c t t )+ β ln( c t t +1 ) where c g t denotes individual consumption (i.e. per capita) of generation g at time t (so that c t t represents consumption in youth and c t t +1 represents consumption in old age of a generation t individual) and β (0 , 1) such that β (1 + n ) < 1 . Each individual can only work at most a unit amount of time in youth. There is a constant returns to scale production function where aggregate output Y t is given by Y t = K α t L 1 α t where K t is aggregate capital, L t is aggregate labor input, and α (0 , 1) . There is complete depreciation of capital (i.e. δ =1) so that the law of motion for aggregate capital is given by K t +1 = I t with K 0 given. In what follows let per capita and aggregate variables be related as x t = X t /N t so that k t = K t t . 1. Suppose that a social planner chooses { c t t ,c t 1 t ,k t +1 } t =0 to maximize the following welfare function 1 1+ n ln( c 1 0 X t =0 β t N t U t . a. (2.5 points) What is the per capita feasibility constraint facing the planner in any given period t ? b. (5 points) What are the first order conditions for this problem? c. (2.5 points) Manipulate the first order conditions to arrive at a condition like an Euler equation. d. (5 points) What do the first order conditions imply about consumption of the young and the old in any given period (i.e. c t t and c t 1 t )? e. (2.5 points) Assume that the sequence { k t +1 } t =0 implied by the first order conditions converges. What is the steady state capital stock in the planner’s solution? What is the implied steady state interest rate? f. (15 points) Linearize the necessary conditions around the steady state. What conditions need to be satisfied for the system to be locally stable? If you are running out of time, at least discuss how you would do this step. g. (5 points) Is there balanced growth of aggregate output? At what rate? h. (2.5 points) How do these results relate to what you would find in a steady state of the RBC model? 1
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Answer: A1a. In aggregate terms the feasibility constraint is: K α t N 1 α t = K t +1 + N t c t t + N t 1 c t 1 t or in per capita terms: K α t N 1 α t N t = N t +1 N t · K t +1 N t +1 + c t t + N t 1 N t c t 1 t ⇐⇒ k α t =(1+ n ) k t +1 + c t t + 1 (1 + n ) c t 1 t (1) A1b. The Lagrangian for this problem is £ = 1 1+ n ln( c 1 0 )+ X t =0 β t (1+ n ) t £ ln( c t t )+ β ln( c t t +1 ) ¤ + X t =0 λ t k α t (1 + n ) k t +1 c t t 1 (1 + n ) c t 1 t ¸ .
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This note was uploaded on 08/06/2008 for the course ECON 387 taught by Professor Corbae during the Spring '07 term at University of Texas at Austin.

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midterm387s08answers - Econ 387L: Macro II Spring 2008,...

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