Consider an AK model of endogenous growth. If the aggregate production function is given by Y-1.5K and the depreciation rate is 17.5%: a. What is the minimum savings rate such that this economy will experience growth in the long run? b. Discuss the pros and cons of pushing a high saving rate in this economy.
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- Consider an economy’s production function is Y=K^1/3 N^2/3 and that both the saving rate and the depreciation rate are equal to 0.15. A. What is the steady-state level of capital per worker? B. What is the steady-state level of output per worker? Now suppose that the economy has reached its steady-state in period t, and then, in period t+1, the saving rate doubles to 0.30. The depreciation rate remains constant at 0.15. C. Solve for the new steady-state levels of capital per worker and output per worker. D. Calculate the path of capital per worker and output per worker over the first three periods after the change in the saving rate.Consider a two-sector model of growth, with two kinds of investment opportunities—one with a diminishing marginal product and one with a constant marginal product. a. What does the production function for this problem look like? b. Characterize the set of equilibria for this model. Does output in any of the equilibria have nonzero per capita growth? c. What can this model help us explain that strict endogenous and neoclassical growth models cannot?Consider an economy that has access to a production technology Y = AKαL1−α where Y is output, A is the level of technology, K is capital and L is the amount of labor in the economy. Capital evolves according to K˙ = sY (thus, the depreciation rate δ = 0). The x˙ population growth rate is n. (Throughout, gx = x , where x can be any of the variables in the model.) (a) Assume that technology is determined by A = BKφ What sort of endogenous growth model is this? Find K/K in terms of the K, L, and other parameters of the model.
- In an economy characterized by a Cobb-Douglas production function (without technical progress), labour’s share of income is 70% and the depreciation rate is 3% per annum. The economy is in a steady state with GDP growth at 4% per year and with a capital output ratio of 2. Find the saving rate and the marginal product of capital. At time t the saving rate in this economy increases to a new constant level, with the outcome that the economy converges to the Golden Rule steady state. What is the new savings rate, capital output ratio and marginal product of capital?Consider an economy with the following aggregate production function: Y = 3K1/3(AL)2/3 Capital grows through investment but also decays due to wear and tear at a constant rate δ per period. Assume that A is growing at the exogenous rate g, that L is growing at the exogenous rate n, and that households save a constant proportion s of their income. (a ) Find the steady state level of the capital per effective worker (k*), output per effective worker (y*) and consumption per effective worker (c*) - in terms of the parameters of the model. (B) What is the level of k (k**) that maximizes consumption? (C) Given a depreciation rate of 7%, population growth rate of 2%, technological progress of 1% and a saving rate of 30%, calculate the steady state levels of k, y and c. (D) To move to the level of capital that maximizes consumption, how should the saving rate be changed? Explain. (E) Calculate the saving rate needed to reach the golden rule level of capital per effective worker.Consider an economy with a Cobb-Douglas production function with α = 1/3 that begins in steady state with a growth rate of technological progress of g of 2 percent. Consider what happens when g increases to 3 percent. (a) What is the growth rate of output per worker before the change? What happens to this growth rate in the long run? (b) Perform a growth accounting exercise for the economy, decomposing the growth rate in output per capita into components contributed by capital per capita growth and technology growth. What is the contribution of the change in g to output per capita growth according to this formula? (c) In what sense is the growth accounting result in part b producing a misleading picture of this experiment? Explain why this is the case.
- In class we argued that if people could accumulate human as well as physical capital, the production function would look like the “AK” production function. • (a) If the production function is AK and the savings rate is constant at rate “s”, and the rates of depreciation and populati on growth are δ and n respectively, what would the growth rate of the economy be? • (b) What would be the macroeconomic consequences of decreasing the savings rate in this economy? • (c) What would be the consequences of an increase in fertility in this economy? • (d) Would the consequences of decreasing fertility be UNAMBIGUOUSLY GOOD? • (e) Can human capital grow without bounds? Explain why or why not (make sure you discuss the physical nature of human capital). • (f) What is the growth rate of the economy (in the absence of technological progress) if human capital cannot grow without bounds?3 Assume a closed economy, perfectly elastic labor supply, and linear technology. Suppose the incremental capital-output ratio (ICOR) is 3, the depreciation rate is 3%, and the gross savings rate is 10%. Use the Harrod-Domar growth equation to determine the rate of growth. What would the gross savings rate have to be to achieve 5% growth? Assuming a perfectly elastic labour supply, state one criticism of this model from an exogenous growth theory viewpoint and another criticism of this model from an endogenous growth theory viewpoint.7. 1. Consider a neoclassical growth economy described by the following.•Yt = K0.3t ·L0.7t (aggregate production function)•s = 0.35 (saving rate)•δ = 0.10 (depreciation rate)•n = 0.01 (population growth rate)•L1 = 120 (initial population)•K1 = 160 (initial capital stock)•g = 0 (technological growth rate)Compute K, Y , k, y, and c for the first three periods. Please report numerical answersto two decimal points. (a) K1 = ; Y1 = ; k1 = ; y1 = ; c1 =(b) K2 = ; Y2 = ; k2 = ; y2 = ; c2 =(c) K3 = ; Y3 = ; k3 = ; y3 = ; c3 =
- c. You are given the following information about an economy. Y = C + I Y = F(K, L) The aggregate production function for this economy exhibits constant returns to scale and the marginal products of labor and capital are both subject to diminishing returns. s = saving rate (assume this is constant) per year δ= depreciation rate (assume this is a constant) per year y = Y/L k = K/L k* = steady state of capital per worker (K/L) and sf(k) < δk. i. What is sf(k) ii. What is δk? iii. Interpret the meaning of sf(k) < δk?A country produces output using a production function: Y = F(K, L) = AxK0.5 L0.5 where L is labour, K is capital and A is total factor productivity. Prove that this function exhibits constant returns to scale. Assume that A = 3 and its growth rate is zero. The country has a population growth of 1.5% per year (0.015) and capital depreciates at a rate of 10% (0.10) per year. If the country saves 30% (0.3) of national income, find the steady state levels of capital per worker, as well as consumption and income per worker in the Solow growth model corresponding to this country. What would be the rate of growth of GDP per capita in steady state? Calculate the “Golden Rule” level of capital stock in steady state. Is this economy dynamically efficient?Draw a graph with the following elements: A Solow growth (Y) function where Y= square root of K An investment (I) function where I = % * Y A depreciation (D) function where D = % * K