## PROBLEMS AND APPLICATIONS

1. Country A and country B both have the production function

Y = F(K, L) = K1/2L1/2.

1. Does this production function have constant returns to scale? Explain.

2. What is the per-worker production function, y = f(k)?

3. 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. Then find the steady-state levels of income per worker and consumption per worker.

247

4. 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 to show how the capital stock per worker will evolve over time in both countries. For each year, calculate income per worker and consumption per worker. How many years will it be before the consumption in country B is higher than the consumption in country A?

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 directly affect the capital stock, but that casualties reduce the labour force. Assume that the economy was in a steady state before the war, the savings rate is unchanged, and the rate of population growth after the war returns to normal.

1. What is the immediate impact of the war on total output and on output per person?

2. 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?

3. Consider an economy described by the production function: Y = F(K, L) = K0.3L0.7.

1. What is the per-worker production function?

2. 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.

3. 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, 30 percent, and so on. (You will need a calculator with an exponent key for this.) What saving rate maximizes output per worker? What saving rate maximizes consumption per worker?

4. (Harder) 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. What does your table show?

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.

5. 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.)

6. Many demographers predict that Canada will have zero population growth in the twenty-first century, in contrast to average population growth of about 1.6 percent per year in the last 75 years. 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.

7. In the Solow model, population growth leads to steady-state growth in total output, but not in output per worker. Do you think this would still be true if the production function exhibited increasing or decreasing returns to scale? Explain. (For the definitions of increasing and decreasing returns to scale, see Chapter 3, “Problems and Applications,” Problem 2.)

8. Consider how unemployment would affect the Solow growth model. Suppose that output is produced according to the production function Y = Kα [(1 – u)L]1–α, where K is capital, L is the labour force, and u is the natural rate of unemployment. The national saving rate is s, the labour force grows at rate n, and capital depreciates at rate δ.

248

1. Express output per worker (y = Y/L) as a function of capital per worker (k = K/L) and the natural rate of unemployment. Describe the steady state of this economy.

2. Suppose that some change in government policy reduces the natural rate of unemployment. Describe how this change affects output both immediately and over time. Is the steady-state effect on output larger or smaller than the immediate effect? Explain.

9. Choose two countries that interest you—one rich and one poor. What is the income per person in each country? Find some data on country characteristics that might help explain the difference in income: investment rates, population growth rates, educational attainment, and so on. (Hint: The website of the World Bank, www.worldbank.org, is one place to find such data.) How might you figure out which of these factors is most responsible for the observed income difference?