Consider the production function 𝑄(𝐿,𝐾) = (𝐿1/2 + 𝐾1/2)2/3, where L denotes labor and K denotes capital. This production function exhibits

ECON 410: Homework #7

Directions: 1. This assignment is due by 11:59pm on Friday, October 29. An answer key will be posted on Sakai under Resources in the Homework Assignments folder at 12:15am Saturday morning. It’s your responsibility to allow enough time before submission to safeguard against technical glitches. Glitches can and do happen! Protect your submission by following the directions below.
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Multiple Choice Questions (5 points each)

1. Consider the production function 𝑄(𝐿,𝐾) = (𝐿1/2 + 𝐾1/2)2/3, where L denotes labor and K denotes capital. This production function exhibits

a. constant returns to scale. b. decreasing returns to scale. c. increasing returns to scale.
(See Besanko 6.19, 6.20, 6.24 and 6.25 and Production Practice Problems #1, 2 and 3.)

2. Suppose the production of paved roadways is represented as Q(L,K) = LK + L. Based on this production function, which of the following statements is (are) TRUE? a. In the short-run when capital is held constant, labor exhibits the law of diminishing returns. b. The isoquants for paved roadways are downward sloping and convex to the origin. c. The production function exhibits constant returns to scale. d. Both a and b are true. e. Both b and c are true. f. None of the above statements are true.
(See Besanko 6.13, 6.14, 6.24 and 6.25 and Production Practice Problems #1, 2 and 3.)

3. A firm’s technology of production can be described by Q(𝐿,𝐾) = 𝐿2𝐾2. Labor exhibits _______ marginal product, and the corresponding isoquants slope downward at _______rate as the firm moves from left to right along an isoquant. a. an increasing; an increasing b. a constant; an increasing c. an increasing; a decreasing d. a decreasing; a decreasing e. none of the above
(See Besanko 4, 6.13, 6.14 and 6.25 and Production Practice Problems #2, 3 and 4.)

4. Consider the following statements: I. If the marginal product of capital is substantially higher than the marginal product of labor, the isoquants are relatively steep when labor is on the horizontal axis and capital is on the vertical axis. II. Diminishing MRTS means a firm must add increasing amounts of labor in order to give up each additional unit of capital while maintaining output. a. Only I is false b. Only II is false

c. Both I and II are false d. Neither I nor II is false
(See Production Practice Problems #2, 3 and 4.)

5. A firm’s production function is represented by Q(M,R) = 4M 3/4R1/3, where Q denotes output, M raw materials, and R robots. The firm is currently using 3 units of raw materials and 8 robots. According to the MRTS, if the firm adds 1 more unit of raw materials, how many robots will it need to give up to maintain output? a. 6 b. 1/6 c. 2/3 d. 3/2 e. 8/3 f. 3/8
(See Production Practice Problems #3 and 5.)

6. Suppose that worker hours (L) and machine hours (K) are perfect substitute inputs in the production of widgets (Q). Moreover, to produce an additional widget, the firm can either employ 50 more worker hours OR 10 more machine hours. Which of the following correctly represents the production of widgets?

a. 𝑄(𝐿,𝐾) = 10𝐿 + 50𝐾 b. 𝑄(𝐿,𝐾) = 50𝐿 + 10𝐾 c. 𝑄(𝐿,𝐾) = 0.1𝐿 + 0.02𝐾 d. 𝑄(𝐿,𝐾) = 0.02𝐿 + 0.1𝐾
(See Production Practice Problem #6.)

7. Joe’s coffee house operates under the production function Q(L,K)= LK + (LK)1/2. In the short- run, capital (K) is fixed while labor (L) is variable. Which statement correctly describes Joe’s short-run total product of labor curve?

a. Total product of labor everywhere increases at a decreasing rate. b. Total product of labor everywhere increases at an increasing rate. c. Total product of labor everywhere increases at a constant rate. d. Total product of labor initially increases at an increasing rate, then increases at a decreasing rate, and eventually decreases. e. Total product of labor initially increases at a decreasing rate and then eventually increases at an increasing rate.
(See Production Practice Problem #7.)

8. The production of customers served (Q) at a convenience store is determined by the function Q(L,K) = LK + K, where L is labor and K is checkout stands. Which statement correctly describes the corresponding short-run average product of labor curve, where the number of checkout stands is fixed?

a. APL decreases at a decreasing rate. b. APL decreases at an increasing rate. c. APL decreases at a constant rate. d. APL increases at a decreasing rate. e. APL increases at an increasing rate.
(See Production Practice Problem #8.)
9. Suppose you observe that MPL > APL and MPL is decreasing but positive as more labor is used. Based on this, you can say that TP is ______ at______ rate, and APL must be ______. a. increasing; a decreasing; increasing b. increasing; a decreasing; decreasing c. increasing; an increasing; decreasing d. decreasing; a decreasing; decreasing e. decreasing; increasing; increasing
(See Besanko 4 and 6.5 and Production Practice Problem #9.)

10. Consider the truthfulness of the following statement. I. When total product increases with labor at a decreasing rate, marginal product of labor is negative. II. If total product of labor is increasing, marginal product of labor must be increasing too.
a. Only statement I is true. b. Only statement II is true. c. Both statements I and II are true. d. Neither statement is true.
(See Besanko 4 and 6.5 and Production Practice Problem #9.)

Long Form Questions
1. Consider a firm that has the following CES production function: Q = f(L,K) = [aLρ + bKρ]1/ρ
where ρ ≤ 1.

a. Derive the MRTS for this production function. Does this production function exhibit a diminishing MRTS? Justify using derivatives and in words. What does this imply about the shape of the corresponding isoquants? (10 points) b. What are the returns to scale for this production function? Show and explain. Explain what will happen to cost if the firm doubles its output. (8 points) c. Explain the difference between the concepts of diminishing marginal product and diminishing marginal rate of technical substitution. (5 points) d. Explain the difference between the concepts of diminishing marginal product and diminishing returns to scale. (5 points)
(See Besanko 4, 6.13 and 6.25 and Production Practice Problems #1, 2 and 3.)

2. To produce cake, you need eggs E and premixed ingredients I. Each cake needs exactly two eggs and one package of ingredients. When you add three eggs to one package of ingredients, you still produce only one cake. Similarly, when you add two packages of ingredients to two eggs, you still produce only one cake. a. Write down the production function that describes the technology of cake production. (5 points) b. Draw an isoquant map for this production function. (5 points) c. A bird flu outbreak has caused an egg shortage. For the short-run, you are stuck with only one dozen eggs. On 3 separate graphs, draw your total product, average product, and marginal product curves. (12 points)
(See Production Practice Problems #6, 7 and 8.)