Y=0.5√ √N, where Y is output. K is capital, and N is the number of workers The steady-state level of capital per worker in terms of the saving rate, s, and the depreciation rate, 6, is KIN=(Property format your expression using the tools in the palette. Hover over tools to see keyboard shortcuts. E.g., a superscript can be created with the character) The steady-state level of output per worker in terms of the saving rate, s, and the depreciation rate, 6, is VIN- (Property format your expression using the tools in the palette) The equation for steady-state consumption per worker in terms of the saving rate, s, and the depreciation rate, 6, is CIN- (Property format your expression using the tools in the palette)
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- A CES production function with physical and human capital Consider the CES production function in terms of physical capital, K, and human capital, H: where 0 a. Set up the Hamiltonian and find the first-order conditions. b. What is the optimal relation between K and H? Substitute this relation into the given production function to get a relation between Y and K. What does this “reduced-form” production function look like? c. What is the steady-state value of the ratio of physical to human capital, (K/H)∗? d. Describe the behavior of the economy over time if the initial condition is such that K(0)/H(0)? e. Suppose that the inequality restrictions IK ≥ 0 and IH ≥ 0 apply. How do these constraints affect the dynamics if the economy begins with K(0)/H(0)∗?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%, 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…
- Economics, physical capital represents the uildings or machines used by a business to produce product. The marginal product of physical capital presents the rate of change of output product with spect to physical capital (informally, if you increase e size of your factory a little, how much more Foduct can you create?). articular model tells us that the output product Y is given, a function of capital K, by Y = AKªL'-a ere A is a constant, L is units of labor (assumed to be stant), and a is a constant between 0 and 1. Determine marginal product of physical capital predicted by this del. ned with CamScannerIn Wonderland production per worker (y) depends on capital per worker() such the y=10vk. Every year 15% of the capital stock depreciates, while workers in Wonderland save 10% of their income. Every year the population grows ratas te of 3% (c) The country of Neverland is identical to Wonderland in terms of output per worker, the savings rate, the depreciation rate and population growth. They differ in one respect: Wonderland has capital per worker of 10, whereas Neverland has capital per worker of 20. Which country experiences a higher growth rate of output per worker and how will their growth rates evolve over time?1. Consider an economy where the production function is Y = K0.5 (LE)0.5 The depreciation rate is = 0.04, the savings rate is s = 0.2, the popula- tion growth rate is n = 0.03 and technology growth rate is g = 0.03. (a) What is the 'per effective worker' production function? (b) Find the steady state levels of capital per effective worker (k*), in- come per effective worker (y*), investment per effective worker (¿*) and consumption per effective worker (c"). (c) Find the golden rule levels of capital per effective worker (kg), income per effective worker (y), investment per effective worker (it) and consumption per effective worker (c2). Also find sg, that is the level of the savings rate that would lead the economy to the golden rule steady state. (d) Suppose the government pursues policies that change the savings rate from s = 0.2 to sg. What is the immediate effect on income per effective worker and consumption per effective worker? What is the long run effect on income per…
- Table 1 for the Production Function. 1. Capital per hour of work (k/l) 10 20 30 40 50 Show Transcribed Text 2 C Output per hour of work (Y/L) 14 22 28 32 34 Plot the above production function information. What is the equilibrium if capital per hour Hour of work (K/L) =40. Thank You!!2. An economy has a production function: Y 3KtLt. (1) The economy has a saving rate of 24 percent, a depreciation rate of 3 percent, 1 for all period (no population growth). and Lt Yt Lt (a) What is the per-worker production function, yt f(kt)? Define yt Ки Lt and kt (b) Find the equation for the evolution of capital per worker in terms of kt and kt+1 (c) Find the long-run growth rate of output per worker. Now the economy has the following production function: Y 3Kt (2) but savings rate, depreciation rate, and population remain the same4. If the production function is given by Y=K2L'3 and the saving rate saving rate (s) equals the depreciation rate (8). Find the numerical values of the steady state capital and output?
- Question 1 Consider the following economy with production function: Y = AK"L-a where Y is total output; K is capital; L is labour force; A is the level of technology (exogenous); s is the saving rate; n is the population growth rate; 0 < a < 1. Assume the rate of capital depreciation (5) to be zero. Hence, capital accumulation equation is: K = sY (a) Does this production function exhibitconstantreturns to scale to capital and labour? Explain. Derive production function per worker. (b) Derive an equation showing how the growth rate of capital per worker depends on s, n, and A. Explain its growth implications and use a graph to support your answer. (c) Find out the expression for the steady state capital per worker andoutputper worker. How do they depend on s, n, and A? Explain. (d) Instead of assuming saving rate (s) to be a constant, suppose we assume that "s" depends on y,y =(equivalently on k, k = . In particular, s = e is constant. Find the expression for the growth rate of capital…1. Over a period of 10,000 years, the population of Brobdingnag is estimated to have increased from 4 million to 170 million. Assuming that the level of income per worker was constant, there are only two factors of production, Land (X) and Labour (L), and the exponent on land in the production function is one-third, what was the annual growth rate of productivity, A? a) 0.17% b) 0.42% c) 0.03% d) 0.01%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].