4. Consider the following endogenous growth model: a representative household's life-time utility function is given as: The production function is: Le t=0 -pt c(t)1-0 1-0 - 1 -dt. Y(t) = AK(t)ªH₂(t)1-a where Hp(t) denotes human capital used in production. Human capital accumulation equation is:
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- Suppose the per-worker production function is: y = A(1-gA) Where ga is the fraction of all workers that produce technologies. Further, suppose the growth of technology is given by the following equation growth of A = (g//m)(L) Suppose L = 1 and m = 7, and that initially ga = 0.7. If ga fell to 0.8 the level of output per %3D worker would: fall Impossible to say rise stay the sameSuppose the per-worker production function is: y = A(1-ga) Where ga is the fraction of all workers that produce technologies. Further, suppose the growth of technology is given by the following equation growth of A = (ga/m)(L) Suppose L = 1 and m = 7, and that initially ga = 0.7. If g, fell to 0.8 the level of output per worker would: Impossible to say fall stay the same O riseAssume that a country's production function is Y = K1/2L1/2 and there is no population growthor technological change.a. What is the per-worker production function y = f (k)?b. Assume that the country possesses 40,000 units of capital and 10,000 units of labor. What isY? What is labor productivity computed from the per-worker production function? Is thisvalue the same as labor productivity computed from the original production function?c. Assume that 10 percent of capital depreciates each year. What gross saving rate isnecessary to make the given capital–labor ratio the steady-state capital–labor ratio? (Hint:In a steady state with no population growth or technological change, the saving ratemultiplied by per-worker output must equal the depreciation rate multiplied by the capital–labor ratio.)d. If the saving rate equals the steady-state level, what is consumption per worker?
- Assume that a country's production function is Y = K1/2L1/2 and there is no population growthor technological change.a. What is the per-worker production function y = f (k)?b. Assume that the country possesses 40,000 units of capital and 10,000 units of labor. What isY? What is labor productivity computed from the per-worker production function? Is thisvalue the same as labor productivity computed from the original production function?c. Assume that 10 percent of capital depreciates each year. What gross saving rate isnecessary to make the given capital–labor ratio the steady-state capital–labor ratio? (Hint:In a steady state with no population growth or technological change, the saving ratemultiplied by per-worker output must equal the depreciation rate multiplied by the capital–labor ratio.)d. If the saving rate equals the steady-state level, what is consumption per worker? Only D, other option answeredwhere H(0) = a# _ ]s(ß) Consider a simplified version of endogenous growth model discussed in the class. The growth rate is determined by: Region 1: For 0 se s$(1-m), s(B) 9t = H(8), + a' 1- Region 2 and 3: For (1- n) s 0 s 1, 9t = a#s(ß) %3D Note that s(B) Assume the following parameter values: 1+B° aH = 2, a² = 1, B = 1, 7 = 0.25 Q1: Suppose 0 = 0.25. Derive the economic growth rate. Q2: Suppose 0 = 0.5. Derive the economic growth rate. (Q3-Q5 will follow in the next page.)Write out the equation for output growth with capital, labor and total factor productivityas determinants of growth. Suppose that the shares of capital and labor are respectively0 .3 and 0.7. If labor supply grows by 10% what would be the growth rate of outputassuming that there is no change in the other determinants? What would happen to percapita output if labor supply growth is entirely due to population growth?
- Exercise 4: Growth and capital over-accumulation = Suppose two countries, A and B, with the same production function Y KaL¹-a. The value of a is 0.30, the growth rate of population is 2% and the depreciation rate is 5%. a) Show that with price-taking firms the share of labor must be 1 a. b) Compute the stock of capital, output and consumption per unit of labor in the steady state if the savings rates were 25% for country A and 35% for country B. c) Compare both economies to the Golden Rule. d) Explain what would happen to both countries if suddenly their savings rate became the Golden Rule savings rate.Write the production function (human capital (H) is included in TFP) and the logarithmic form of this production function then explain why the GDP growth will be at the end ofa given long period lower than at the beginning of this period, assuming that the average contribution of TFP to GDP average growth during the period is constant.Assume that a country's production function is Y= AKº³Lº7. The ratio of capital to output is 3, the growth rate of output is 3 percent, and the depreciation rate is 4 percent. Capital is paid its marginal product. a. What is the marginal product of capital in this situation? (Hint: The marginal product of capital may be computed using calculus by differentiating the production function and using the capital-output ratio or by using the fact that capital's share equals MPK multiplied by K divided by Y.) b. If the economy is in a steady state, what must be the saving rate? (Hint: The saving rate multiplied by Y must provide for gross growth of (8 + n + g)K, where 8 is the depreciation rate.) c. If the economy decides to achieve the Golden Rule level of capital and actually reaches it, what will be the marginal product of capital? d. What must the saving rate be to achieve the Golden Rule level of capital?
- Assume a Cobb-Douglas production with capital share 1/3; total factor productivity equal to 2, total population in an economy is equal to 1, the steady-state level of capital stock is 8.0. Then the steady-state level of output is about: a. 4.0 b. 2.0 c. 22.6 d. 8.0 e. 45.4Economics, 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 CamScannerQuestion 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…