Economics 164 assignment derive an expression for the

Question

Economics Assignment-

In this assignment, the goal is to study the evolution of per capita output in the Solow growth model with population growth and technological progress. Assume that total output is determined by Y_t = K^α_t(A_tL_t)^1-α, where A_t is labor augmenting technical progress. Here, A_t+1 = (1+g)A_t, and g > 0. Suppose that the savings rate is equal to a fraction s and the capital depreciates at the rate δ, where 0 < δ ≤ 1. Hence, K_t+1 = (1- δ)K_t + sY_t. Finally, assume that the population grows at the rate η > 0 each period.

A. Let kˆ be capital per effective unit of labor. That is, kˆ_t = K_t/A_tL_t. Derive a difference equation expressing kˆ_t+1 as a function of kˆ. Graph this difference equation. What are the steady states of this difference equation? Are they stable or unstable?

B. Derive an expression for the capital-output ratio, K_t/Y_t, as a function of kˆ_t. If the parameters of the model stay constant, what can you say about the capital-output ratio in the long run? Does it grow? Derive the long run value of this ratio.

C. Derive an expression for per capita output, y_t as a function of kˆ_t. Again, what is the value of y_t in the long run? Derive an expression for this. What is the growth rate of y_t according to this expression? Does this depend on the growth rate of population? On the growth rate of technical progress?

D. Let s = 0.2, α = 0.33 and let 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. What value for g does this imply?

In simulations a - d below, each simulation starts with parameter values set as described above. Also assume that A₀ = 1 and L₀ = 1.

a. Suppose that capital per effective unit of labor is equal to its steady state value in period 0. In period 5, there is a permanent increase in the growth rate of technical progress from 1.5 percent per year to 3 percent per year. Simulate and graph the level and growth rate of per capita output for 50 periods (beginning with period 1). Use two separate graphs.

b. Assuming the economy starts at the same point as in part a, suppose that there is a great invention that doubles A_t in period 5. The growth rate after that, however, remains at 1.5 percent per year. Simulate and graph the level and growth rate of per capita output for 50 periods.

c. Now suppose there is a permanent increase in the growth rate of population from 1.5 percent to 3 percent per year in period 5 and no change in growth rate of technical progress. Again, simulate and graph per capita income and its growth rate for 50 periods.

d. Finally, assume that there is a massive wave of immigration that increases the level of the population by 20 percent in period 5 but that the growth rate after that remains at 1.5 percent per year. Again, simulate and graph per capita income and its growth rate for 50 periods.

e. Of these four cases, which led to a permanent change in the growth rate of living standards? Which led to the biggest permanent change in the level of living standards?