. Explain the following questions.

Consider the Solow-Swan model of growth. Imagine that the production function is Y = AKaLl'\" 1. Use the production function to compute output per capita, y = Y / L, as a function of capital per person, k = K/ L. 2. Dene the fundamental equation of the Solow-Swan model. Please show all the steps. Furthermore, imagine that the savings, depreciation, and population growth rates take the values 5 = 0.2,. 5 = 0.1 and n = 0.01. You do not know the value of A. 3. Use the fundamental equation of the Solow-Swan model to compute the growth rate of capital per person as a function of k. 4. In the steady~state, the growth rate of capital is zero. Using the parameters assumed above, nd the steady-state level of the capital stock, k". 5. Imagine that this country is in its steady state so its capital stock is k". Imagine that the country receives a gift of one unit of capital from the world bank (so, suddenly, the capital stock is k'+1). Can you say what is going to happen to the growth rate nnmediately after the donation? Why? What will the capital stock be in the long run? Explain. Question 2 Consider the Solow growth model. Recall that upper-case variables denote aggegate variables and lower-case variables denote per worker [or per capita} variabla. Capital is assumed to evolve according to the following equation: K'=I+{1 d]|K, (1} where K' denct aggregate capital in the future period and K is capital in the current period. There is a closed-economy model without a government, so in a competitive equilibrium the income-expenditure identity is Y = U+I. {a} Use the income-expenditure identity and equation [1) to nd an expression for the evolution of per- worker capital. In other words, show that equation [1] can be rewritten as follows: r_ 33115:] {1dlk k _ 1+n + 1+n' (9} [b] Denote the steady-state level of capital per worker as In". Show that in the steady state, equation [2] implies the following equilibrium equation: azk\") = [n + on. (3} For the rest of this question, suppose there are two countries with an aggregate production function Y = zKD-3N-T. Further suppose the following values for the model's parameters: a = 0.25, d = Ill and n = [1.02 in both muntries. (c) Suppose 2,1. = 1 in Country A. Use equation {3) and the given values for the model's parameters and production function to calculate the steady-state level of income per capita, y}, and the steady-state level of capital per worker, in}, in Country A. (d) Suppose 23 = 2 in Country B, but otherwise the countries are equivalent. Because 23 :5 2,4, Country B will have a higher standard of Living {i.e., per capital income} in the initial period. In the long run, will Country A ever converge with Country E in terms of per capita income? Explain your answer. {e} How is the Solow model inconsistent with the growth fact that there is no correlation between the level of output per capita in 1960 and the average growth rate in output per capita since that time?I 1. Country A and country B both have the production function 1' = F(K,L)= 41:71.. (5 Points) Does this production function have constant returns to scale? Explain. (5 Points) What is the per-worker production function, y = fat)? (10 Points) Assume that neither country experiences population growth or technological progress and that 5 percent of capital depreciates each year. Assume further that country A saves 10 percent of output each year and country B saves 20 percent of output each year. Using your answer from part (b) and the steady-state condition that investment equals depreciation, nd the steady-state level of capital per worker for each country. Then nd the steady-state levels of income per worker and consumption per worker. (15 Points) What saving rate maximizes output per worker for each country? What saving rate maximizes consumption per worker for each country? And nd the Golden Rule level of consumption, output, investment per worker for each country. (15 Points) Suppose that both countries start off with a capital stock per worker of 2. What are the levels of income per worker and consumption per worker? Remembering that the change in the capital stock is investment less depreciation, use a calculator or a computer spreadsheet to show how the capital stock per worker will evolve over time in both countries. For each year, calculate income per worker and consumption per worker. How many years will it be before the consumption in country B is higher than the consumption in country A? Country A and country B both have the production function Y=F(K, L)=K3\"L \"r\". (a) Does this production function have constant rettu'us to scale? Explahi. {h} What is the per-worker production function, y = f (k)? (c) Assume that neither country experiences population growth or technological progress and that 3 percent of capital depreciatcs each year. Assume further that country A saves 25 percent of output each year and country B saves 30 percent of output each year. Using your answer from part {b} and the steady-state condition that investment equals depreciation, nd the steady-state love] of capital per worker for each country. Then nd the steady-state levels of income per worker and consumption per worker. {{1} Suppose that both countries start off with a capital stock per worker of 3. What are the levels of income per worker and consumption per worker?I Remembering that the change in the capital stock is investment less depreciation. use a calculator or a computer spreadsheet to show how the capital stock per worker will evolve over time in both countries. For each year, calculate income per worker and consumption per worker. How many years will it be before the consumption in country B is higher than the consumption in country A? (c) Find the golden rule level of capital per worker in each country and the saving rate that supports it respectively