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

**1. ** Use the neoclassical theory of distribution to predict the impact on the real wage and the real rental price of capital of each of the following events:

A wave of immigration increases the labor force.

An earthquake destroys some of the capital stock.

A technological advance improves the production function.

High inflation doubles the prices of all factors and outputs in the economy.

**2. **
• Suppose the production function in medieval Europe is *Y* = *K*^{0.5}*L*^{0.5}, where *K* is the amount of land and *L* is the amount of labor. The economy begins with 100 units of land and 100 units of labor. Use a calculator and equations in the chapter to find a numerical answer to each of the following questions.

How much output does the economy produce?

What are the wage and the rental price of land?

What share of output does labor receive?

If a plague kills half the population, what is the new level of output?

What is the new wage and rental price of land?

What share of output does labor receive now?

**3. ** If a 10 percent increase in both capital and labor causes output to increase by less than 10 percent, the production function is said to exhibit *decreasing returns to scale*. If it causes output to increase by more than 10 percent, the production function is said to exhibit *increasing returns to scale*. Why might a production function exhibit decreasing or increasing returns to scale?

**4. ** Suppose that an economy’s production function is Cobb–*α* = 0.3.

What fractions of income do capital and labor receive?

Suppose that immigration increases the labor force by 10 percent. What happens to total output (in percent)? The rental price of capital? The real wage?

Suppose that a gift of capital from abroad raises the capital stock by 10 percent. What happens to total output (in percent)? The rental price of capital? The real wage?

Suppose that a technological advance raises the value of the parameter

*A*by 10 percent. What happens to total output (in percent)? The rental price of capital? The real wage?

**5. ** Figure 3-5 shows that in U.S. data, labor’s share of total income is approximately a constant over time. Table 3-1 shows that the trend in the real wage closely tracks the trend in labor productivity. How are these facts related? Could the first fact be true without the second also being true? Use the mathematical expression for labor’s share to justify your answer.

**6. ** According to the neoclassical theory of distribution, a worker’s real wage reflects her productivity. Let’s use this insight to examine the incomes of two groups of workers: farmers and barbers. Let *W*_{f} and *W*_{b} be the nominal wages of farmers and barbers, *P*_{f} and *P*_{b} be the prices of food and haircuts, and *A*_{f} and *A*_{b} be the marginal productivity of farmers and barbers.

For each of the six variables defined above, state as precisely as you can the units in which they are measured. (

*Hint*: Each answer takes the form “*X*per unit of*Y*.”)Over the past century, the productivity of farmers

*A*_{f}has risen substantially because of technological progress. According to the neoclassical theory, what should have happened to farmers’ real wage,*W*_{f}/*P*_{f}? In what units is this real wage measured?Over the same period, the productivity of barbers

*A*_{b}has remained constant. What should have happened to barbers’ real wage,*W*_{b}/*P*_{b}? In what units is this real wage measured?79

Suppose that, in the long run, workers can move freely between being farmers and being barbers. What does this mobility imply for the nominal wages of farmers and barbers,

*W*_{f}and*W*_{b}?What do your previous answers imply for the price of haircuts relative to the price of food,

*P*_{b}/*P*_{f}?Suppose that barbers and farmers consume the same basket of goods and services. Who benefits more from technological progress in farming—

farmers or barbers? Explain how your answer is consistent with the results on real wages in parts (b) and (c).

**7. ** (This problem requires the use of calculus.) Consider a Cobb–*K* is capital (the number of machines), *L* is labor (the number of workers), and *H* is human capital (the number of college degrees among the workers). The production function is

*Y* = *K*^{1/3} *L*^{1/3} *H*^{1/3}.

Derive an expression for the marginal product of labor. How does an increase in the amount of human capital affect the marginal product of labor?

Derive an expression for the marginal product of human capital. How does an increase in the amount of human capital affect the marginal product of human capital?

What is the income share paid to labor? What is the income share paid to human capital? In the national income accounts of this economy, what share of total income do you think workers would appear to receive? (

*Hint*: Consider where the return to human capital shows up.)An unskilled worker earns the marginal product of labor, whereas a skilled worker earns the marginal product of labor plus the marginal product of human capital. Using your answers to parts (a) and (b), find the ratio of the skilled wage to the unskilled wage. How does an increase in the amount of human capital affect this ratio? Explain.

Some people advocate government funding of college scholarships as a way of creating a more egalitarian society. Others argue that scholarships help only those who are able to go to college. Do your answers to the preceding questions shed light on this debate?

**8. ** The government raises taxes by $100 billion. If the marginal propensity to consume is 0.6, what happens to the following? Do they rise or fall? By what amounts?

Public saving

Private saving

National saving

Investment

**9. ** 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?

**10. **
• Consider an economy described as follows:

*Y* = *C* + *I* + *G*.

*Y* = 8,000.

*G* = 2,500.

*T* = 2,000.

*C* = 1,000 + *2/3*(*Y* — *T*).

*I* = 1,200 — 100*r*.

In this economy, compute private saving, public saving, and national saving.

Find the equilibrium interest rate.

Now suppose that

*G*is reduced by 500. Compute private saving, public saving, and national saving.Find the new equilibrium interest rate.

**11. ** 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-

80

**12. ** When the government subsidizes investment, such as with an investment tax credit, the subsidy often applies to only some types of investment. This question asks you to consider the effect of such a change. Suppose there are two types of investment in the economy: business investment and residential investment. The interest rate adjusts to equilibrate national saving and total investment, which is the sum of business investment and residential investment. Now suppose that the government institutes an investment tax credit only for business investment.

How does this policy affect the demand curve for business investment? The demand curve for residential investment?

Draw the economy’s supply and demand curves for loanable funds. How does this policy affect the supply and demand for loanable funds? What happens to the equilibrium interest rate?

Compare the old and the new equilibria. How does this policy affect the total quantity of investment? The quantity of business investment? The quantity of residential investment?

**13. ** Suppose that consumption depends on the interest rate. How, if at all, does this alter the conclusions reached in the chapter about the impact of an increase in government purchases on investment, consumption, national saving, and the interest rate?

**14. ** Macroeconomic data do not show a strong correlation between investment and interest rates. Let’s examine why this might be so. Use our model in which the interest rate adjusts to equilibrate the supply of loanable funds (which is upward sloping) and the demand for loanable funds (which is downward sloping).

Suppose the demand for loanable funds is stable but the supply fluctuates from year to year. What might cause these fluctuations in supply? In this case, what correlation between investment and interest rates would you find?

Suppose the supply of loanable funds is stable but the demand fluctuates from year to year. What might cause these fluctuations in demand? In this case, what correlation between investment and interest rates would you find now?

Suppose that both supply and demand in this market fluctuate over time. If you were to construct a scatterplot of investment and the interest rate, what would you find?

Which of the above three cases seems most empirically realistic to you? Why?

1 This is a simplification. In the real world, the ownership of capital is indirect because firms own capital and households own the firms. That is, real firms have two functions: owning capital and producing output. To help us understand how the factors of production are compensated, however, we assume that firms only produce output and that households own capital directly.

2 *Mathematical note*: To prove Euler’s theorem, we need to use some multivariate calculus. Begin with the definition of constant returns to scale: *zY* = *F*(*zK, zL*). Now differentiate with respect to *z* to obtain:

*Y* = *F*_{1}(*zK*, *zL*) *K* + *F*_{2}(*zK*, *zL*) *L*,

where *F*_{1} and *F*_{2} denote partial derivatives with respect to the first and second arguments of the function. Evaluating this expression at *z* = 1, and noting that the partial derivatives equal the marginal products, yields Euler’s theorem.

3 Carlo M. Cipolla, *Before the Industrial Revolution: European Society and Economy, 1000– 1700*, 2nd ed. (New York: Norton, 1980), 200–

4 *Mathematical note*: To prove that the Cobb–*z*:

*F*(*zK, zL*) = *A*(*zK*)* ^{α}*(

Expanding terms on the right,

*F*(*zK, zL*) = *Az ^{α} K^{α}z*

Rearranging to bring like terms together, we get

*F*(*zK, zL*) = *Az ^{α} z*

Since *z ^{α} z*

*F*(*zK, zL*) = *z A K ^{α}L*

But *A K ^{α}L*

*F*(*zK, zL*) = *zF*(*K,L*) = *zY*.

Hence, the amount of output *Y* increases by the same factor *z*, which implies that this production function has constant returns to scale.

5 *Mathematical note:* Obtaining the formulas for the marginal products from the production function requires a bit of calculus. To find the *MPL*, differentiate the production function with respect to *L*. This is done by multiplying by the exponent (1 − *α*) and then subtracting 1 from the old exponent to obtain the new exponent, −*α*. Similarly, to obtain the *MPK*, differentiate the production function with respect to *K*.

6 *Mathematical note:* To check these expressions for the marginal products, substitute in the production function for *Y* to show that these expressions are equivalent to the earlier formulas for the marginal products.

7 For recent papers examining this phenomenon, see Michael W. L. Elsby, Bart Hobijn, and Aysegül Sahin, “The Decline of the U.S. Labor Share,” *Brookings Papers on Economic Activity*, Fall 2013: 1–*Quarterly Journal of Economics* 129, no. 1 (February 2014): 61–

8 *Mathematical note:* The Gini coefficient can be interpreted as follows. If you randomly select two incomes from the population, the absolute value of their difference, as a share of the population’s average income, is expected to be twice the Gini coefficient.

9 Claudia Goldin and Lawrence F. Katz, *The Race Between Education and Technology* (Cambridge, MA: Belknap Press, 2011). See also David H. Autor, “Skills, Education, and the Rise of Earnings Inequality Among the ‘Other 99 Percent,’” *Science* 344, no. 6186 (May 23, 2014): 843–