Chapter 5: Practice problems
1. You have been hired to replace the manager of a firm that used
only two inputs, capital and labor, to produce output. The firm can
hire as much labor as it wants at a wage of $5 per hour and can rent
as much capital as it wants at a price of $50 per hour. After you
look at the company books, you learn that the company has been
using capital and labor in amounts that imply a marginal product of
labor of 50 and a marginal product of capital of 100. Do you know
why the firm hired you? Explain.
Ans Before the manager is hired, the marginal rate of technical substitution is 1/2. However, the relative input price is 1/10. This means that either more workers or less physical capital should be used. Hence, you are hired to change this input ratio in order to minimize costs.
2. The manager of a meat-packing plant can use either butchers (labor) or meat saws (capital) to prepare packages of sirloin steak. Based on estimates provided by an efficiency expert, the firm's production function for sirloin steak is given by
a. Graph the isoquant corresponding to 5 units of output. b. What is the marginal product of capital and labor? Does the answer depend on how much labor and capital are used?
c. If the price of labor is $2 per hour and the rental price of capital is $3 per hour, how much capital and labor should be used to minimize the cost of production?
Ans a. See Figure 5-3.
b. MP = 1; MP = 1. These marginal products do not depend on how KL
much labor and capital are used.
c. Five hours of labor and zero hours of capital should be used to minimize the cost of producing five units of output.
3. An accountant for a car rental company was recently asked to
report the firm's costs of producing various levels of output. The
accountant knows that the most recent estimate available of the
firm's cost function is , where costs are
measured in thousands of dollars and output is measured in
thousands of hours rented.
a. What is the average fixed cost of producing 2 units of output?
b. What is the average variable cost of producing 2 units of output?
c. What is the average total cost of producing 2 units of output?
d. What is the marginal cost of producing 2 units of output?
e. What is the relation between the answers to (a), (b), and (c)
above? Is this a general property of average cost curves?
Ans a. AFC(2) = 100/2 = $50.
2b. AVC(2) = [(10)(2) + (2)]/2 = $12.
c. ATC(2) = AFC(2) + AVC(2) = $62.
d. MC(2) = 10 + 2(2) = $14.
e. AVC + AFC = ATC. This holds for all output levels, not just Q = 2.
4. Standard Enterprises produces an output that it sells in a highly competitive market at a price of $100 per unit. Its inputs include two machines (which cost the firm $50 each) and workers, who can be hired on an as-needed basis in a labor market at a cost of $2,800 per worker. Based on the following production data, how many workers should the firm employ to maximize its profits?
Ans The relevant production data is as follows:
To maximize profits, the firm should continue adding workers so long as the value marginal product exceeds the wage. The value marginal product is defined as the marginal product times the price of output. Here, output sells for $100 per unit, so the value marginal product of the third worker is $100(29) = $2,900. The table above summarizes the VMP for each possibility. Since the wage is $2,800, the profit L
maximizing number of workers is 3.