Determine the steady state value of the per capita capital

Question

This is a version of the Solow growth model. Country A is a closed economy with no government expenditure. At time t country A has an aggregate capital stock K_t-1 and a total population Nt. Country A's population grows at a constant rate n and its production function
is Y_t = K_t-1^1-αN_t^α. Denote by d the depreciation rate of capital and s the savings rate.

Assume that 0 < s < 1 and 0 < d < 1. There is no technological progress in this economy.

1. Let define y_t ≡ Y_t/N_t and k_t-1 ≡ K_t-1/N_t in which y stands for output per capita and k denotes capital per capita. Derive the formula of output per capita.

2. Assume the law of motion for the stock of capital is K_t = (1 -d)K_t-1 +I_t where It denotes aggregate investment at time t. Rewrite this equation in terms of the per capita capital stock.

3. Determine the steady state value of the per capita capital, k*, as a function of α, d, n and s.

4. Determine the golden rule quantity of capital per capita in country A, kgr, as a function of α, d, n and s.

5. Assume that country A experiences no population growth and that 5 percent of capital depreciates each year. Assume that country A saves 10 percent of output each year. Determine numerically the steady state level of k*, y* and the golden rule quantity of kgr, ygr, and cgr if labor's share is 0.5.

6. Sketch a diagram to depict your answer from part (e). Make sure that you label everything clearly.

In the following questions, assume that as a result of the war, much of the capital in country A (oil extracting equipment, vehicles, structures etc.) was destroyed. Answer the questions below and use diagrams to support your answer if needed.

7. What will be the effect of this event on per capita capital in country A in the next five years and in the long run?

8. What will be the effect of this event on per capita income in country A in the next five years and in the long run?

9. What will be the effect of this event on the growth rate of per capita income in country A in the next five years and in the long run?

10. Will recovery in country A occur faster if investment by foreigners is permitted?