An economy has production function Y = K1/2L1/2, a savings rate of 60 percent. There is no technological progress or labour force growth. The depreciation rate initially 10 percent, increases to 40 percent. There are 400 workers in this economy. When the depreciation rate is 40 percent, output will be equal to dollars.
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- Question 2Assume production function is given by:Y= K(1/2) L(1/2)a. Write the production function in per worker terms (y=f(k))b. Assume that the per worker level of capital in the steady state is 4, the depreciation rate is 5% per year, and population growth is 5% per year. Does this economy have “too much” or “too little” capital? How do you know? [Show your work].(Long Run) The country of Prosperous has: Production function: Y = 2 K1/2 (AN)1/2, where Y=output;K=capital; A=technology; and N=labor Saving rate (s): 20% per year Depreciation rate (d): 10% per year Labor growth rate (n): 2% per year Rate of technological change (g): 8% per year a. Transform the production function into the relation between output per effective worker (Y/AN) and capital per effective worker (K/AN). Does the function exhibit diminishing return?Solow economy where TFP = 1, capital starts off at K0=1, the savings rate is 50%, and the annual depreciation rate of capital is 10%. What is the steady-state capital stock?
- If the depreciation rate increases from 5% to 10%, it will ___ in the long run. decrease the level of output increase the growth rate of output decrease the growth rate of output not change the level of outputSuppose that the production function for an economy is given by Y = K1/4L3/4. The depreciation rate is 4%, the saving rate is 12%, Suppose that now there is labor augmenting technology, the rate of labor-augmenting technological is 2 percent. At what rate does total output, output per worker, and output per effective worker grow?2. An economy has a production function:Yt = 3(squaredKt)(squaredLt). The economy has a saving rate of 24 percent, a depreciation rate of 3 percent,and Lt = 1 for all period (no population growth). There is no technologicalprogress. (a) What is the per-worker production function, yt = f(kt)? Define yt =YtLtand kt =KtLt.(b) Find the equation for the evolution of capital per worker in terms of ktand kt+1. (c) Find the long-run growth rate of output per worker. Now the economy has the following production function:Yt = 3Kt but savings rate, depreciation rate, and population remain the same. (d) What is the per-worker production function, yt = f(kt)? Define yt =Yt/Lt (e) Find the equation for the evolution of capital per worker in terms of ktand kt+1. (f) Find the long-run growth rate of output per worker. (g) Explain why the economy with production function (2) explain persistent growth without the assumption of exogenous technologicalprogress. How does this differ from the economy…
- 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?Fred’s Frisbees is trying to determine how many Frisbee pressing machines to buy for its new factory. The real price of a new pressing machine is 7600 Frisbees. The depreciation rate on these Frisbee presses is equal to 12% per year. The real interest rate is 10%. The expected future marginal product of these fabricating machines is given by the expression 5000 – 4K, measured in Frisbees. (a) What is the user cost of capital? (b) What is the profit-maximizing number of Frisbee presses for Fred’s to purchase for its new factory? (c) Before purchasing the machines Fred’s finds that it will be subject to a new tax of 60 percent on all its revenue. Now what is the profit maximizing number of machines for Fred’s to purchase?Solve for the steady state of K with: An Investment (I) function with a rate of growth of 0.4 times the square root of K and a depreciation (D) function of 0.03 * K. Now graph what this looks like. Hint: you need to set the production function equal to the depreciation function and solve for k.
- 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.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?Suppose that the production function for an economy is given by Y = K1/4L3/4. The depreciation rate is 4%, the saving rate is 12%, Explain how the economy goes from one steady state to another.