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Problem 7.1: Country A and B has the production function: Y = F(K, L) = KV2 . 1/2 A. Does this production function have constant returns to scale? B. What is the per-worker production function, Y/L=f(K/L) 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. Find the steady state level of capital per worker, the steady-state level 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 gross investment minus depreciation, calculate capital stock per worker, income per worker, and consumption per worker over time. How many years will it be before consumption per worker in Country B is higher than the level of consumption per worker in country A. Problem 7.2: In the discussion of German and Japanese postwar growth, the text describes what happens when part of the capital stock is destroyed in a war. By contrast, suppose that a war does not affect the capital stock, but that casualties reduce the labor force. A. What is the immediate impact on total output and on output per person? B. Assuming that the saving rate is unchanged and that the economy was in a steady state before the war, what happens subsequently to output per worker in the postwar economy? Is the growth rate of output per worker after the war smaller or greater than normal? Problem 7.3: Consider an economy described by the production function: Y = F(K, L) = K03 . [0.7 A. What is the per-worker production function? B. Assuming no population growth or technological progress, find the steady-state capital stock per worker, output per worker, and consumption per worker as a function of the saving rate and the depreciation rate. C. Assume that the depreciation rate is 10 percent per year. Make a table showing steady- state capital per worker, output per worker, and consumption per worker for saving rates of 0 percent, 10 percent, 20 percent, and 30 percent and so on. What saving rate maximizes output per worker? What saving rate maximizes consumption per worker? D. Use calculus to find the marginal product of capital. Add to your table the marginal product of capital net of depreciation for each of the saving rates. Problem 7.4. "Devoting a larger share of national output to investment would help restore rapid productivity growth and rising living standards". Do you agree with this claim? Explain. Problem 7.5: (somewhat hard) One view of the consumption function is that workers have high propensities to consume and capitalists have low propensities to consume. To explore the implications of this view, suppose that an economy consumes all wage income and saves all capital income. Show that if the factors of production earn their marginal product, this economy reaches the Golden rule level of capital. (Hint: Begin with the identity that saving equals investment. Then use the steady-state condition that investment is just enough to keep up with depreciation and population growth, and the fact that saving equals capital income in this economy.) Problem 7.6: Many demographers predict that the US will have zero population growth in the 21" century, in contrast to an average population growth of about 1 percent per year in the 20" century. Use the Solow model to forecast the effect of this slowdown in population growth on the growth of total output and the growth of output per person. Consider the effects both in the steady state and in the transition between steady states. Problem 7.9: Empirically, as an indicator of the saving rate is investment/GDP. In a closed economy, saving equals investment. And in an open economy saving may be close to investment. No question here. Problem 7.10: Assume the Solow growth model with population growth and with no technological progress: Derive steady-state expressions for capital per worker, output per worker, the real wage and the real return to capital. Y = F(K, L) = A. K" . ['- As functions of the exogenous variables: What happens to the real wage and the real return to capital if the saving rate increases?3. THE CLASSICAL MODEL FOR THE CLOSED ECONOMY= THE KEYNESIAN MODEL FOR A CLOSED ECONOMY IN THE LONG RUN Problem 3.7: The government raises taxes by 100 billions USD. If the marginal propensity to consume is 0.6, what happens to public, private, and national saving and to investment? Do they rise or fall? By what amounts? Problem 3.8: Suppose that an increase in consumer confidence raises consumers' expectations about their future income and thus increases the amount they want to consume today. This might be interpreted as an upward shift in the consumption function. How does this shift affect investment and the interest rate. Problem 3.9 Consider an economy described by the following equations: Y=C+I+G, Y=5000, G=1000, T=1000, C=250+0.75(Y-T), 1=1000-50r. A. In this economy, compute private saving, public saving, and national saving. B. Find the equilibrium interest rate. C. Now suppose that G rises to 1250. Compute private saving, public saving, and national saving. D. Find the new equilibrium interest rate. E. Now suppose T decreases to 750 (and G=1000). Compute private, public and national saving, and find the new equilibrium interest rate. Problem 3.10: Suppose that the government increases taxes and government purchases by equal amounts. What happens to the interest rate and investment in response to this balanced budget change? Does your answer depend on the marginal propensity to consume? Problem 3.11: When the government subsidizes investment, such as with an investment tax credit, the subsidy often applies to only some types of investment. Suppose that there are two types of investment in the economy: business investment and residential investment. And suppose that the government institutes an investment tax credit only for business investment. A. How does this policy affect the demand curve for business investment? The demand curve for residential investment. B. Draw the economy's supply and demand for loanable funds. How does this policy affect the supply and demand for loanable funds? What happens to the equilibrium interest rate? C. Compare the old and the new equilibrium. How does this policy affect the total quantity of investment? The quantity of business investment? The quantity of residential investment? Problem 3.12: If consumption depended on the interest rate, how would that affect the conclusions reached in this chapter about the effects of fiscal policy