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 y t ( ) and k t ( ) denote output per person and capital per person, respectively, at time t , and assume that the production function is y k 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 c t ( ) 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 c i g   which can be combined with the production function and the capital accumulation equation to give ( ) 2 k t k c g k   . Consider the problem of maximizing 0 bt a e c dt where b 0 is a parameter reflecting impatience and 0 1 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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