401ps8-10 - ECON 401 Autumn 2010 PROBLEM SET VIII (for...

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ECON 401 Hartman Autumn 2010 PROBLEM SET VIII (for Monday, December 6) 1. An economy consists of identical individuals with infinite horizons. The population does not grow. Let yt () and kt () denote output per person and capital per person, respectively, at time t , and assume that the production function is yk 2 . Capital accumulation is given by / dk dt k i k   where i is the rate of investment per person and is the depreciation rate. Let ct () denote consumption per person at time t , and assume that there is also a government which consumes output at the exogenously given and constant rate of g per person. Since output can be allocated to private consumption, to investment, or to government consumption, we must have y ci g  which can be combined with the production function and the capital accumulation equation to give () 2 k c g k   . Consider the problem of maximizing 0 bt a ecd t where b 0 is a parameter reflecting impatience and 01 a . The maximization is subject to the capital accumulation equation and to the predetermined value of k at time 0, k 0 . a. What are the first order necessary conditions for this optimization problem. (You may use the calculus of variations or the Hamiltonian.) What is the transversality condition? Draw a phase diagram for this problem with k on the horizontal axis and c on the vertical axis. Show the optimal trajectory on your phase diagram. b. Find the steady-state values of k and c , in terms of the exogenously given parameters. c. What effect, if any, does an increase in the parameter g have on the steady-state values of k and c ?
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This note was uploaded on 04/04/2011 for the course ECON 401 taught by Professor Staff during the Spring '08 term at University of Washington.

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401ps8-10 - ECON 401 Autumn 2010 PROBLEM SET VIII (for...

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