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# homework1 - ˜ k t 1 =(1-˜ δ ˜ k t ˜ σf(1 ˜ k t where...

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ECO 475 Homework 1 Jay H. Hong Due: Review Session, Monday Sep 13, 2010 1 Golden Rule Consider the Solow model with a production function, y t = F ( k t ) and a law of motion for the capital stock, k t +1 = (1 - δ ) k t + σy t , where δ (0 , 1) is the depreciation rate and σ (0 , 1) is the savings rate. F satisfies all appropriate assumptions (which you have to make). Show that The steady state consumption c * can be written as c * = F ( k * ) - δk * , c * is first increasing and then decreasing in σ , (what kinds of assumptions do we need to get this property?) The savings rate σ that maximizes the steady state consumption c * satisfies F 0 ( k * ) = δ . (Note that k * is increasing in σ ) 2 Labor-augmenting Technical Progress and Balanced Growth Consider the Solow growth model where the production function is given by y t = f ( z t h t , k t ) for all t where z t = γ t with γ > 1. Show that (a) : You can define a modified capital accumulation equation
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Unformatted text preview: ˜ k t +1 = (1-˜ δ ) ˜ k t + ˜ σf (1 , ˜ k t ) where ˜ k j = k j γ j . What are ˜ δ, ˜ σ ? You need one additional assumption on f (). What is it? (b) : In a balanced growth path (where ˜ k * = ˜ g ( ˜ k * )) what is the growth rate of k t and y t ? 3 Consumption-Leisure Choice Consider a representative agent whose utility function is given by U ( c,l ) where c is her consumption and l is her leisure. She wants to maximize her utility by solving max c,l U ( c,l ) subject to y = zf (1-l,k ) where k > 0 is given. State minimal assumptions on f and U which guarantee the existance and uniqueness of optimal solution ( c * ,l * ). 4 Finite Horizon Optimal Growth Problem 2.2 (a)(b)(c) from SLP p. 12 1...
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