Reference no: EM133655839

Assignment

In this problem, we study a growth model in which technological progress is endogenous. Production and capital accumulation take place as in a standard Solow growth model:

Y_t = K_t^α (A_tL_yt )^1-α

K_t+1 = (1- δ )K_t + γY_t.

A fraction γ_A of the population is involved in the production of new ideas and a fraction 1-γ_A A is involved in the production of output.

Hence,

L_Yt = (1- γ_A)L_t
L_At = γ_AL_t,

where L_t is the population at date t. The population grows at the rate η , so L_t+1 = (1+η)L_t. Finally, we assume that new ideas are produced as follows:

A_t+1 - A_t = 1/μ At^Φ L_At^λ

where 0 < λ ≤ 1 and Φ < 1

a. Derive an expression for the steady state growth rate of A_t, call it ¯g.

b. As in Assignment 3, let γ = 0.2, α = 0.33 and δ = .08. Suppose population grows at a rate of 1.5 percent per year and that per capita output grows at a rate of 1.5 percent per year along a steady state growth path (also as in Assignment 3).

Also, let γ A = .05. What value for g is consistent with these facts? In addition, assume that Φ = 0 and λ = 1 . Did the growth rate of per capita income play any role in determining the value for g? In any event, does the calibrated model imply per capita output growth of 1.5 percent per year as in the data? Explain.

c. Let μ = 10, A₀ = 100 and  L₀ = 1000. Simulate the path of gt for 250 periods. Print a plot of your simulation.

d. Assuming the same parameter values and same L₀, find the value of A₀ so that the economy is in steady state in period 0. Using this value of A₀ instead of the one used in part (c)-that is, assume that we start in steady state-suppose that γ_A was increased to 0.1 in period 5. Create a plot showing what happens to response to this change g_t in response to this change.

e. Now, expand the simulation in part (d) to include other elements of the Solow growth model. Assume, as in part (d), the economy is initially in a steady state. This requires that you find the steady state for the other elements of the model, in particular the stock of capital (or, more precisely, K_t/A_tL_t ). Provide a plot that shows what happens to the log of per capita output for 50 periods following the change in γ_A. In the same plot, show the log of y with no change in γ_A.