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# 2004ps - Homework 10 Spring 2004 Econ 702 Problem 1 Suppose...

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Homework 10 Spring 2004, Econ 702 Problem 1 Suppose we have an economy with CRTS technology and log utility. Let the labor productivity grow at rate γ. Show that normalizing the variables by growth rate of labor productivity to make this economy stationary, do not e ff ect the optimal decision of the agents. Problem 2 Show that lim σ 1 c t 1 σ 1 1 σ = log c t (1) Problem 3 Suppose we have an economy standart CRRA preferences without leisure and following technology, Y t = A t K α t N 1 α t , A t +1 A t = γ, A 0 , N 0 , K 0 given (2) Solve for the balanced growth rate and verify it is not γ. Problem 4 (Lucas Human Capital Model) We have seen in class the we can have ’human capital’ as a reproducable factor of production. Suppose we have the following CRTS technology and laws of motion for the inputs where both inputs are reproducable in tems of output, Y = AH 1 θ K θ (3) K 0 = (1 δ k ) K + i k H 0 = (1 δ H ) H + i h and standard CRRA preferences without leisure. Set up the problem of the social planner, de fi ne and compute balanced growth path of this economy. Problem 5 (Romer Endogenous Growth) Consider the model of Romer we cov- ered in class with production composed of three sectors, fi nal good, intermediate goods, R&D with fi nal and intermediate good technology and law of motion for variety of new intermediate goods, Y t = L β 1 t Z A t 0 x t ( i ) 1 β di (4) (1 δ ) K t 1 + i t = K t = Z A t 0 ηx t ( i ) di , K 0 given (5) A t +1 = A t + L 2 t ζA t , A 0 given (6) L 1 t + L 2 t = 1 (7) Assume CRRA preferences, formulate and solve the SPP’s problem and using the class notes, show that the decentralized allocation is not optimal by comparing the euler equations. 1

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Problem Set 11 Econ 702, Spring 2004 April 15, 2004 Problem 1 (Romer Endogenous Growth) Define the decentralized problem in Romer’s endogenous growth model that we went over in class. Get a formula for the balanced growth path in equilibrium and compare to the social planner’s problem (Look at Problem 5 in Homework 10 for the social planner’s problem). Problem 2 (Capital-Skill Complementarity) Consider an economy with two sectors: one producing consumption goods and capital structures and the other producing capital equipment. Output of these two sectors is given by: c t + x st = A t G ( k c st , k c et , u c t , s c t ) (1) x et = q t A t G ( k e st , k e et , u e t , s e t ) (2) where c t is consumption, x st is investment in structures and x et is output of the equipment sector. The superscripts of the input variables denote the sector where they are used (c for consumption and e for equipment). k st and k et are the stocks of capital structures and equipment. They evolve according to the following laws of motion, k s,t +1 = (1 - δ s ) k st + x st (3) k e,t +1 = (1 - δ e ) k et + x et (4) The labor inputs are unskilled labor u t and skilled labor s t . The function G is common to both sectors and is given by, G ( k st , k et , u t , s t ) = k α st μu σ t + (1 - μ )( λk ρ et + (1 - λ ) s ρ t ) σ ρ 1 - α σ (5) Total output, in consumption units, is defined by, y t = c t + x st + x et q t = A t G ( k st , k et , u t , s t ) (6)

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