c. Compute the demand schedule showing the number of workers hired for all wages from zero to \$100 a day. d. Graph the firm's demand curve. e. What happens to this demand curve if the price of output rises from \$10 to \$12 per unit? Yi Chun L. Washington University in St Louis ### Problem 4 Smiling Cow Dairy can sell all the milk it wants for$\$$4 a gallon, and it can rent all the robots it wants to milk the cows at a capital rental price of \$$100 a day. It faces the following production schedule:

a. In what kind of market structure does the firm sell its output? How can you tell?
b. In what kind of market structure does the firm rent robots? How can you tell?
c. Calculate the marginal product and the value of the marginal product for each additional robot.
d. How many robots should the firm rent?
Explain. Yi Chun L.
Washington University in St Louis

### Problem 5

The nation of Ectenia has 20 competitive apple orchards, all of which sell apples at the world price of \$2 per apple. The following equations describe the production function and the marginal product of labor in each orchard: $$Q = 100L 2 - L^2$$ $$MPL = 100 - 2L,$$ where Q is the number of apples produced in a day, L is the number of workers, and MPL is the marginal product of labor. a. What is each orchard's labor demand as a function of the daily wage W? What is the market's labor demand? b. Ectenia has 200 workers who supply their labor inelastically. Solve for the wage W. How many workers does each orchard hire? How much profit does each orchard owner make? c. Calculate what happens to the income of workers and orchard owners if the world price doubles to \$4 per apple.
d. Now suppose the price is back at \$2 per apple, but a hurricane destroys half the orchards. Calculate how the hurricane affects the income of each worker and of each remaining orchard owner. What happens to the income of Ectenia as a whole? Yi Chun L. Washington University in St Louis ### Problem 6 Your enterprising uncle opens a sandwich shop that employs 7 people. The employees are paid \$12 per hour, and a sandwich sells for \$6. If your uncle is maximizing his profit, what is the value of the marginal product of the last worker he hired? What is that worker's marginal product? Kaylee M. Numerade Educator ### Problem 7 Leadbelly Co. sells pencils in a perfectly competitive product market and hires workers in a perfectly competitive labor market. Assume that the market wage rate for workers is \$150 per day.

a. What rule should Leadbelly follow to hire the profit-maximizing amount of labor?
b. At the profit-maximizing level of output, the marginal product of the last worker hired is 30 boxes
of pencils per day. Calculate the price of a box of pencils.
c. Draw a diagram of the labor market for pencil workers (as in Figure 4 of this chapter) next to a diagram of the labor supply and demand for Leadbelly Co. (as in Figure 3). Label the equilibrium wage and quantity of labor for both the market and the firm. How are these diagrams related?
d. Suppose some pencil workers switch to jobs in the growing computer industry. On the side-by-side diagrams from part (c), show how this change affects the equilibrium wage and quantity of labor for both the pencil market and for Leadbelly. How does this change affect the marginal product of labor at Leadbelly? Kaylee M.
a. Suppose that a law required firms to give each worker \$3 of fringe benefits for every hour that the worker is employed by the firm. How does this law affect the marginal profit that a firm earns from each worker at a given cash wage? How does the law affect the demand curve for labor? Draw your answer on a graph with the cash wage on the vertical axis. b. If there is no change in labor supply, how would this law affect employment and wages? c. Why might the labor-supply curve shift in response to this law? Would this shift in labor supply raise or lower the impact of the law on wages and employment? d. As discussed in Chapter 6, the wages of some workers, particularly the unskilled and inexperienced, are kept above the equilibrium level by minimum-wage laws. What effect would a fringe-benefit mandate have for these workers? Kaylee M. Numerade Educator ### Problem 9 Some economists believe that the U.S. economy as a whole can be modeled with the following production function, called the$\textit{Cobb$-$Douglas production function}$: $$Y=AK^{1/3}L^{2/3},$$ where$Y$is the amount of output,$K$is the amount of capital,$L$is the amount of labor, and$A$is a parameter that measures the state of technology. For this production function, the marginal product of labor is $$MPL=(2/3) A(K/L)^{1/3}.$$ Suppose that the price of output$P$is 2,$A$is 3,$K$is 1,000,000, and$L$is 1,000. The labor market is competitive, so labor is paid the value of its marginal product. a. Calculate the amount of output produced$Y$and the dollar value of output$PY$. b. Calculate the wage$W$and the real wage$W/P$. (Note: The wage is labor compensation measured in dollars, whereas the real wage is labor compensation measured in units of output.) c. Calculate the labor share (the fraction of the value of output that is paid to labor), which is$(WL)/(PY)$. d. Calculate what happens to output$Y$, the wage$W$, the real wage$W/P$, and the labor share$(WL)/(PY)$in each of the following scenarios: i. Inflation increases$P$from 2 to 3. ii. Technological progress increases$A$from 3 to 9. iii. Capital accumulation increases$K$from 1,000,000 to 8,000,000. iv. A plague decreases$L$from 1,000 to 125. e. Despite many changes in the U.S. economy over time, the labor share has been relatively stable. Is this observation consistent with the Cobb$-\$Douglas production function? Explain. 