a. How would the growth rate of ideas behave if a <1? b. Assuming a = 1, solve for the new growth rate of knowledge and growth rate of output per capita. C. Assuming a = 1, sketch the graph of Inyt against time t.
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- 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.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?Suppose that output is given by the neoclassical production function Y (t) = F (K(t), L(t), A(t)) satisfying Assumptions 1 and 2, and that we observe output, capital, and labor at two dates t and t + T . Suppose that we estimate TFP growth between these two dates using the equation Where g(t, t + T ) denotes output growth between dates t and t + T , and other growth terms are defined similarly, while αK(t) and αL(t) denote the factor shares at the beginning date. Let x(t, t + T ) be the true TFP growth between these two dates. Show that there exist functions F such that x(t, t ˆ + T )/x(t, t + T ) can be arbitrarily large or small. Next show the same result when the TFP estimate is constructed using the end-date factor shares: Explain the importance of differences in factor proportions (capital-labor ratio) between the beginning and end dates in these results.
- Suppose the economy is on a balanced growth path in the Romer model, and then, in the year 2030, ̄l (lower case L bar) decreasesimmediately and permanently to the new level ̄l′ (lower case L bar one). (a) Solve for the new growth rate of knowledge and yt. (b) Make a graph of yt over time using a ratio scale.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.Consider the simple innovation model studied in class. In this model, the society allocates a fraction "gamma" of workforce to innovation and the remaining fraction to production. Suppose that, for simplicity, total workforce is normalized to unity: L = 1. If the productivity growth rate is defined by d A t/dt = "theta"* "gamma" * A_t and real GDP per worker by yt = At* (1 - "gamma"), what is the long-run growth rate of y_t?
- 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.Country A and country B both have the production function Y = F(K, L) = K^0,5L^0,5 A. Does this production function have constant returns to scale? Explain. B. What is the per-worker production function, y = f(k)? C. 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, find the steady-state level of capital per worker for each country. Theen find the steady-state levels of income per worker and consumption per worker. D. 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…The output of an economy is characterized by a Cobb-Douglas production function with constant return to scale and an output elasticity with respect to capital equal to 0.3. Also given are the following parameters: 30% saving rate, 5% depreciation rate, 2% population growth rate, and the technology factor is 2. a). Find the capital-labour ratio, and also the output, consumption and investment on a per capita basis in the steady state equilibrium. b). Is a government policy that raises the saving rate to 40% socially desirable? Explain your answer with reference to a comparison of this new steady state equilibrium and the initial steady state equilibrium in a) above. c). If the initial capital-labour ratio is 30, is the economy operating efficiently in the sense that welfare cannot be improved? Explain concisely. What if the initial capital-labour ratio is 15? Again explain concisely.
- Consider an economy in which the labour force grows by 2.7 percent per annum, physical capital grows by 4 percent per annum and human capital grows by 1.8 percent per annum. Suppose 45 percent of national income goes to labour and 40 percent to capital. Use a constant returns to scale production function to answer the following growth accounting questions: (a) If the Solow residual were zero what rate of growth would the economy achieve? (b) The country's actual rate of growth has been 4.5 percent per annum, which is faster than the growth rate generated by the accumulation of capital and labour stocks. Calculate the value of the residual.Suppose that the economy is summarized by the following: Technology (Production Function): Yt = 10 (Kt)0.3 (Lte)0.7 Consumption function: Ct = 0.8Yt Depreciation rate: 8% (i.e. δ= 0.08) Population growth: 2% (i.e. n = 0.02) Technological growth: 4% (i.e. g = 0.04) 1. Assuming that in 2013 the US economy is in the steady state and L2013 = Le2013 = 8, what is the value of ke2014, ye2014, ce2014 , k2014, y2014, and c2014 ?In the 1990s, developed countries agreed to double their aid to Africa by 2015. A report by the United Nations conference on Trade and Development noted that sceptics had raised concerns about how much effect the doubling of aid would have on output and incomes in Africa, if the quantity of other inputs such as human capacity and institutions were to remain fixed. It also pointed to the divisions between the sceptics with some suggesting the return would diminish when aid reached only 4% of GDP, while others thought they would diminish only when it had reached 50%. It should be added that even if the returns do begin to diminish, they could still be very important.In 1887, Cecil Rhodes created the De Beers Consolidated mines Company, which controlled about 90% of the total world supply of rough uncut diamonds with its South African mines. Until 2001, De Beers produced about half of the world’s diamonds in its mines and marketed about 80% of the world’s diamonds. Diamond producing…