Consider an economy in which output is determined by a simple neoclassical production function, Y =...

 

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Consider an economy in which output is determined by a simple neoclassical production function, Y = K (AL), where A is a constant measure of labor efficiency. Capital depreciates at the rate 8 > 0. Population grows at the rate n>0. Consider a representative household that maximizes utility over an infinite horizon, given by: u[c(t)]=- where c(t) is per capita consumption and y<0. The discount rate of the representative household is p, and the initial level of the asset stock of the household is a(0) = ao. _[c(t)] Y a. What are the control and state variables of the above system? Write down the representative household's maximization problem. b. Write down the Hamiltonian for the household's problem. Derive the optimality conditions that will determine the optimal time path for this economy. Interpret them intuitively. c. Write down the core dynamics of the economy (You will need to consider the representative firm's maximization problem as well.) Derive the steady state values of k, c, and y. d. Derive the Golden Rule Level of k. Show that the steady state value of k is less than the KGR. e. Draw a phase diagram in the c-k space to illustrate the dynamics of the system. Mark the arrows of motion. f. Using local stability analysis, show that the system is saddle path stable. g. Derive the approximate convergence rate of the system. Compute its value using the following parameter values: A = 1, a = 1/3, n=0.02, p=0.025, 8-0.05, and y = -1.5. Also, compute the real interest rate, r. Do the values of the convergence rate and r look plausible to you? one- h. Discuss the dynamic adjustment of the above equilibrium system to an unanticipated on time fall in y, graphically. Comment on your results regarding the movements of c and k i. In the short run (i.e., at the time of the shock and during the transition to the new steady state) ii. In the long run (i.e., compare the old steady state with the new one) i. Discuss the dynamic adjustment of the above equilibrium system to an unanticipated one- time increase in the measure of productivity, A, graphically. Comment on your results regarding the movements of c and k i. In the short run Consider an economy in which output is determined by a simple neoclassical production function, Y = K (AL), where A is a constant measure of labor efficiency. Capital depreciates at the rate 8 > 0. Population grows at the rate n>0. Consider a representative household that maximizes utility over an infinite horizon, given by: u[c(t)]=- where c(t) is per capita consumption and y<0. The discount rate of the representative household is p, and the initial level of the asset stock of the household is a(0) = ao. _[c(t)] Y a. What are the control and state variables of the above system? Write down the representative household's maximization problem. b. Write down the Hamiltonian for the household's problem. Derive the optimality conditions that will determine the optimal time path for this economy. Interpret them intuitively. c. Write down the core dynamics of the economy (You will need to consider the representative firm's maximization problem as well.) Derive the steady state values of k, c, and y. d. Derive the Golden Rule Level of k. Show that the steady state value of k is less than the KGR. e. Draw a phase diagram in the c-k space to illustrate the dynamics of the system. Mark the arrows of motion. f. Using local stability analysis, show that the system is saddle path stable. g. Derive the approximate convergence rate of the system. Compute its value using the following parameter values: A = 1, a = 1/3, n=0.02, p=0.025, 8-0.05, and y = -1.5. Also, compute the real interest rate, r. Do the values of the convergence rate and r look plausible to you? one- h. Discuss the dynamic adjustment of the above equilibrium system to an unanticipated on time fall in y, graphically. Comment on your results regarding the movements of c and k i. In the short run (i.e., at the time of the shock and during the transition to the new steady state) ii. In the long run (i.e., compare the old steady state with the new one) i. Discuss the dynamic adjustment of the above equilibrium system to an unanticipated one- time increase in the measure of productivity, A, graphically. Comment on your results regarding the movements of c and k i. In the short run

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a The control and state variables of the above system are the households consumption per capita ct the capital stock Kt and the labor efficiency A The representative households maximization problem is ... View the full answer