Note: All illegible answers will be marked incorrect, so please write legibly. To receive full credit, do not leave any part of the question unanswered. If you type and submit the print out of your homework, you will receive 5 points extra credit. You may work in a group of two/three. Submit only one of your works. Do not forget to write names of your team members.
1. A firm can manufacture a product according to the production function
Q = F(K,L)= K1/4L1/4
a. Calculate the average product of labor, APL, when the level of capital is fixed at 16 units and the firm uses 16 units of labor. How does the average product of labor change when the firm uses 81 units of labor?
b. Find the marginal product of labor, MPL, when the amount of capital is fixed at 16 units. How does the marginal product of labor change when the firm increases the amount of labors to 81 units?
c. Suppose the capital is fixed at 16 units. If the firm can sell its output at a price of $100 per unit and can hire labor at $25 per unit, how many units of labor should the firm hire in order to maximize the profits?
2. The manager of a national retailing outlet recently hired an economist to estimate the firm's production function. Based on the economist's report, the manager now knows that the firm's production function is given by Q = K1/2L1/2 and that capital is fixed at I unit.
a. Calculate the average product of labor when 9 units of labor are utilized.
b. Calculate the marginal product of labor when 9 units of labor are utilized.
c. Suppose the firm can hire labor at a wage of $10 per hour and output can be sold at a price of $100 per unit. Determine the profit-maximizing levels of labor and output.
d. What is the maximum price of capital at which the firm will still make nonnegative profits?
3. You are a manager for Herman Miller - a major manufacturer of office furniture. You recently hired an economist to work with engineering and operations experts to estimate the production function for a particular line of office chairs. The report from these experts indicates that the relevant production function is
Q = F(K,L) = 2(K)1/2(L)1/2
Where K represents capital equipment and L is labor. Your company has already spent a total of $10,000 on the 4 unit of capital equipment it owns. Due to current economic conditions, the company does not have the flexibility needed to acquire additional equipment. If workers at the firm are paid a competitive wage of $100 and chairs can be sold for $200 each, what is your profit-maximizing level of output and labor usage? What is your maximum profit?
4. You were recently hired to replace the manager of the Roller Division at a major conveyor-manufacturing firm, despite the manager's strong external sales record. Roller manufacturing is relatively simple, requiring only labor and a machine that cuts the crimps rollers. As you begin reviewing the company's production information, you learn that labor is paid $8 per hour and the last worker hired produced 100 rollers per hour. The company rents roller cutters and crimping machines for $16 per hour, and the marginal product of capital is 100 rollers per hour. What do you think the previous manager could have done to keep his job?
5. A firm's product sells for $2 per unit in a highly competitive market. The firm produces output using capital (which it rents at $75 per hour) and labor (which is paid a wage of $15 per hour under a contract for 20 hours of labor services). Complete the following table and use that information to answer the questions that follow:
|
K
|
L
|
Q
|
MPK
|
API(
|
API.
|
VMPK
|
|
0
|
20
|
0
|
|
|
|
|
|
1
|
20
|
50
|
|
|
|
|
|
2
|
20
|
150
|
|
|
|
|
|
3
|
20
|
300
|
|
|
|
|
|
4
|
20
|
400
|
|
|
|
|
|
5
|
20
|
450
|
|
|
|
|
|
6
|
20
|
475
|
|
|
|
|
|
7
|
20
|
475
|
|
|
|
|
|
8
|
20
|
450
|
|
|
|
|
|
9
|
20
|
400
|
|
|
|
|
|
10
|
20
|
300
|
|
|
|
|
|
11
|
20
|
150
|
|
|
|
|
a. Identify the fixed and variable inputs.
b. What are the firm's fixed costs?
c. What is the variable cost of producing 475 units of output?
d. How many units of the variable input should be used to maximize profits?
e. What are the maximum profits in this firm can earn?
f. Over what range of the variable input usage do increasing marginal returns exist?
g. Over what range of the variable input usage do decreasing marginal returns exist?
h. Over what range of input usage do negative returns exist?
6. 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 C(Q)= 100 +10Q + Q2, 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?
7. Your firm produces two products, Q1 and Q2. An economic consulting firm has estimated your cost function to be C(Q1, Q2)= 100 + Q1Q2 + Q1Q2 + Q12 + Q22
a. Are there economies of scope?
b. Are there cost complementarities?
c. Your market for Qi if not very good, and an overseas firm has offered to buy the division of your company that produces Qi. What will happen to your marginal cost of producing Q2 if you sell the division?
8. Determine whether the following statements are true, false or uncertain. Give reasons for your answer.
i. The short run is defined as the time period that is less than one year.
ii. The marginal cost curve intersects the average fixed cost curve at its minimum point.
iii. When AC is below MC, MC is falling and when AC is above MC, Mc is rising.
iv. When AC is falling marginal product of an input must be rising.
v. Marginal rate of technical substitution implies that isoquants are straight lines.