2004ps - Homework 10 Spring 2004, Econ 702 Problem 1...

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Homework 10 Spring 2004, Econ 702 Problem 1 Suppose we have an economy with CRTS technology and log utility. Le tthelaborproduc t iv i tygrowa tra te γ. Show that normalizing the variables by growth rate of labor productivity to make this economy stationary, do not e f 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 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 f 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, f nal good, intermediate goods, R&D with f 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 e et e t 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 et t t ) = k α st h μu σ t + (1 - μ )( λk ρ et + (1 - λ ) s ρ t ) σ ρ i 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 et t t ) (6)
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2004ps - Homework 10 Spring 2004, Econ 702 Problem 1...

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