TABLE 9-2 Accounting for Economic Growth in the United States SOURCES OF GROWTH Output Growth Capital Labor Total Factor Productivity Years ?Y/Y = ??K/K + (1 ? ?)?L/L + ?A/A (average percentage increase per year) 1948–2016 3.4 1.3 1.0 1.1 1948–1973 4.3 1.3 1.0 1.9 1973–2016 3.0 1.2 1.0 0.7 Data from: U.S. Department of Labor. Data are for the non-farm business sector. Parts may not add to total due to rounding.
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6 percent per year, and the capital share of output is 1/3. Therefore, we can use the growth accounting equation to calculate the contributions of capital, labor, and total factor productivity (TFP) to output growth: Output growth = Capital growth + Labor growth + Show more…
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1. Consider the Solow growth model with depreciation of capital and labour-force growth. The fundamental equation of the Solow model is: Δk = s f(k) - (n+∂) k where s = MPS=APS, k = capital per worker, n= growth rate of labour, and ∂ = the rate of depreciation of capital. a) Show the steady state value of k in a diagram. What happens to the steady-state value of k when s increases? b) Other things being the same, what happens to the steady-state value of k as n declines. Show this in a diagram. 2. The following equations are given for the Solow model: Y = K.25 L.75 (dL/dt)/L = .02 the growth rate of the labour force APS=MPS= s = 0.16 the depreciation rate, ∂ = 0.08 Based on the above equations and other related equations, derive the following differential equation of the Solow model and find the steady- state values of y and k. dk/dt +(n + ∂) k = sk^̑̑ where k= K/L = capital per worker. 3. The following equations are given for the Solow model with population growth (n), depreciation ( ∂ ) , and Harrod-neutral technological progress (g). Y = K.20 N.80 N = L.A (dL/dt)/L = .01 the growth rate of the labour force and (dA/dt) /A = g the Harrod-neutral technological progress = .02 APS=MPS= s = 0.20 the depreciation rate, ∂ = 0.10 a )Based on the above equations and other related equations, derive the following differential equation of the Solow model and find the steady- state values of y^ and k^. dk^/dt +(n + ∂ +g) k^ = s k^̑̑ where k^= capital per worker measured in efficiency units Find the steady-state values of k^ and y^. b) i) Show in a diagram sf(k^) and (n+∂ +g)k^ and the steady-state value of k^. ii) Show the effect of a decrease in n on k^.
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In 1928, mathematician Charles Cobb and economist Paul Douglas studied the growth of the American economy between the years 1899 and 1922. The function that they used to model production, P, over this time period was P(L, K) = 1.01L^0.75 * K^0.25. Here, L is the amount of labor input and K is the amount of capital input. a) Calculate the partial derivatives ∂P/∂L (the marginal productivity of labor) and ∂P/∂K (the marginal productivity of capital). b) Based on economic data published by the U.S. government, in the year 1910, L = 147 and K = 208. What are the marginal productivity of labor and the marginal productivity of capital in the same year? c) In the year 1910, which would benefit production more: an increase in labor input or an increase in capital input? Explain your answer. d) Use a linear approximation to approximate the value of P in the year 1911. You may use the fact that in 1911, L = 148 and K = 216.
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