Q3. Consider the Ramsey growth model with technological progress and government. Output is determined by a neoclassical production function with labor-augmenting technological progress: Y = F[K, AL] where the level of productivity A grows at the constant rate x>0. Capital DOES NOT depreciate (8=0). Assume that the population grows at rate n>0. Suppose that the economy is initially at its steady state with t =T = g = 0 and 7, =T, >0. Tax revenue is redistributed through lump-sum transfers, v. Consider a representative household that maximizes utility over an infinite horizon, given by: c( 1-0 where oft) is per capita consumption. Discount rate of the representative household is p and the initial level of the asset stock of the household is a(0) = ao. u[c()]=[001 a. 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. Explain how taxation changes the intuition behind these optimality conditions. c. Write down the core dynamics of the economy (You will need to consider representative firm's maximization problem as well.) Derive the steady state values of k, , and. d. Draw a phase diagram in the - k space to illustrate the dynamics of the system. Mark the arrows of motion. e. Discuss the dynamic adjustment of the above equilibrium system to an unanticipated one- time increase in the rate of tax on capital income, Tk, graphically. Comment on your results regarding the movements of and k i. In the short run ii. In the long run f. Show that the steady state saving rate is decreasing in the g. Suppose there are many economies like this one. Households' preferences are the same in each country, but the tax rates on capital income may vary across countries. Do citizens in low-Tn, high-k* , high-saving countries have any incentive to invest in low-saving countries? Why or why not? Q3. Consider the Ramsey growth model with technological progress and government. Output is determined by a neoclassical production function with labor-augmenting technological progress: Y = F[K, AL] where the level of productivity A grows at the constant rate x>0. Capital DOES NOT depreciate (8=0). Assume that the population grows at rate n>0. Suppose that the economy is initially at its steady state with t =T = g = 0 and 7, =T, >0. Tax revenue is redistributed through lump-sum transfers, v. Consider a representative household that maximizes utility over an infinite horizon, given by: c( 1-0 where oft) is per capita consumption. Discount rate of the representative household is p and the initial level of the asset stock of the household is a(0) = ao. u[c()]=[001 a. 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. Explain how taxation changes the intuition behind these optimality conditions. c. Write down the core dynamics of the economy (You will need to consider representative firm's maximization problem as well.) Derive the steady state values of k, , and. d. Draw a phase diagram in the - k space to illustrate the dynamics of the system. Mark the arrows of motion. e. Discuss the dynamic adjustment of the above equilibrium system to an unanticipated one- time increase in the rate of tax on capital income, Tk, graphically. Comment on your results regarding the movements of and k i. In the short run ii. In the long run f. Show that the steady state saving rate is decreasing in the g. Suppose there are many economies like this one. Households' preferences are the same in each country, but the tax rates on capital income may vary across countries. Do citizens in low-Tn, high-k* , high-saving countries have any incentive to invest in low-saving countries? Why or why not
