For any problem marked with
, there is a Work It Out online tutorial available for a similar problem online. To access these interactive, step-*Macroeconomics*, 9e at www.macmillanhighered.com/

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

*Y* = *F*(*K*, *L*) = *K*^{1/3}*L*^{2/3}.

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

What is the per-

worker production function, *y*=*f*(*k*)?Assume that neither country experiences population growth or technological progress and that 20 percent of capital depreciates each year. Assume further that country A saves 10 percent of output each year and country B saves 30 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. Suppose that both countries start off with a capital stock per worker of 1. 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, better yet, 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 labor force. Assume the economy was in a steady state before the war, the saving rate is unchanged, and the rate of population growth after the war is the same as it was before.

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

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 it was before the war?

**3. **
• Consider an economy described by the production function: *Y* = *F*(*K*, *L*) = *K*^{0.4}*L*^{0.6}.

What is the per-

worker production function? 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. Assume that the depreciation rate is 15 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 might find it easiest to use a computer spreadsheet.) What saving rate maximizes output per worker? What saving rate maximizes consumption per worker? Use information from

**Chapter 3**to find the marginal product of capital. Add to your table from part (c) the marginal product of capital net of depreciation for each of the saving rates. What does your table show about the relationship between the net marginal product of capital and steady-state consumption?

**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, using the Solow model.

**5. ** Draw a well-

A change in consumer preferences increases the saving rate.

A change in weather patterns increases the depreciation rate.

Better birth-

control methods reduce the rate of population growth. A one-

time, permanent improvement in technology increases the amount of output that can be produced from any given amount of capital and labor.

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**6. ** Many demographers predict that the United States will have zero population growth in the coming decades, in contrast to the historical average population growth of about 1 percent per year. 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-**Chapter 3**, “Problems and Applications,” Problem 3.)

**8. ** Consider how unemployment would affect the Solow growth model. Suppose that output is produced according to the production function *Y* = *K ^{α}*[(1 −

Express output per worker (

*y*=*Y*/*L*) as a function of capital per worker (*k*=*K*/*L*) and the natural rate of unemployment (*u*).Write an equation that describes the steady state of this economy. Illustrate the steady state graphically, as we did in this chapter for the standard Solow model.

Suppose that some change in government policy reduces the natural rate of unemployment. Using the graph you drew in part (b), 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.

1 The Solow growth model is named after economist Robert Solow and was developed in the 1950s and 1960s. In 1987 Solow won the Nobel Prize in economics for his work on economic growth. The model was introduced in Robert M. Solow, “A Contribution to the Theory of Economic Growth,” *Quarterly Journal of Economics* (February 1956): 65–

2 Edmund Phelps, “The Golden Rule of Accumulation: A Fable for Growthmen,” *American Economic Review* 51 (September 1961): 638–

3 *Mathematical note:* Another way to derive the condition for the Golden Rule uses a bit of calculus. Recall that *c** = *f*(*k**) − *δk**. To find the *k** that maximizes *c**, differentiate to find *dc**/*dk** = *f*′(*k**) − *δ* and set this derivative equal to zero. Noting that *f*′(*k**) is the marginal product of capital, we obtain the Golden Rule condition in the text.

4 *Mathematical note:* To derive this formula, note that the marginal product of capital is the derivative of the production function with respect to *k*.

5 *Mathematical note:* Formally deriving the equation for the change in *k* requires a bit of calculus. Note that the change in *k* per unit of time is *dk/dt* = *d*(*K/L*)*/dt*. After applying the standard rules of calculus, we can write this as *dk/dt* = (1*/L*)(*dK/dt*) − (*K/L*2)(*dL/dt*). Now use the following facts to substitute in this equation: *dK/dt* = I − *δk* and (*dL/dt*)*/L* = *n*. After a bit of manipulation, this produces the equation in the text.

6 For modern analyses of the Malthusian model, see Oded Galor and David N. Weil, “Population, Technology, and Growth: From Malthusian Stagnation to the Demographic Transition and Beyond,” *American Economic Review* 90 (September 2000): 806–*American Economic Review* 92 (September 2002): 1205–

7 Michael Kremer, “Population Growth and Technological Change: One Million B.C. to 1990,” *Quarterly Journal of Economics* 108 (August 1993): 681–