Answer the relation mrs lc mrt lc mp n still holds so

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ANSWER: The relation MRS l,C = MRT l,C = MP N still holds. So, the Competitive Equilibrium allocation is the same as Planner’s allocation. 7
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Solow Growth Model (30 points total) Question 6. True or False (15 points) Answer based on the Solow Growth Model seen in class: T (a) With only populational growth, the aggregate variables converge to a stable growth rate equal to n (population growth rate) and steady state output per worker growth is zero F (b) Economies with higher savings rate have higher rates of per worker output growth in steady state T (c) Economies with higher savings rate have higher level of output per worker in steady state F (d) The higher the steady state capital per worker k * , the higher the steady state con- sumption level T (e) The source of permanent growth in output per worker is ever-increasing total factor productivity z Question 7. (15 points) Consider aggregate ouput is produced using a Cobb-Douglas production function Y = zK α N 1 - α where 0 < α < 1. Then, for a given level of savings rate s , depreciation rate d , population growth rate n and productivity z . (a) (5 pts) Write per worker production funciton ANSWER: Y N = zK α N 1 - α N Y N = z K N α N N 1 - α y = zk α (b) (5 pts) Calculate the golden rule steady state capital per worker k * gr ANSWER: Consumption in steady state can be written as: c = (1 - s ) zf ( k * ) = zf ( k * ) - szf ( k * ) = zf ( k * ) - ( n + d ) k * 10
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In the golden rule, consumption in steady state is maximized: max k * zf ( k * ) - ( n + d ) k * and the optimal condition is: zf 0 ( k * gr ) = n + d αz ( k * gr ) α - 1 = n + d k * gr = n + d αz 1 α - 1 = αz n + d 1 1 - α (c) (5 pts) Calculate the golden rule savings rate s gr and consumption c * gr ANSWER: s gr = ( n + d ) k * gr z ( k * gr ) α = αz ( k * gr ) α z ( k * gr ) α = α c * gr = (1 - α ) zf ( k * g r ) = (1 - α ) z αz n + d 1 1 - α 11
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