Homework #8 Spring 2024

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University of North Carolina, Chapel Hill *

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410

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Economics

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May 24, 2024

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ECON 410: Homework #8 Directions: 1. This assignment is due by 11:59pm on Thursday, April 4. An answer key will be posted on Sakai under Resources in the Homework Assignments folder at 12:15pm Friday. It’s your responsibility to allow enough time before submission to safeguard against technical glitches. 2. Submit your answers through Gradescope. ✓ Click here for Gradescope submission instructions. ✓ Upload a clear scan or image for each long form question. You will not receive any credit for illegible answers. I highly recommend using a free app, such as Genius Scan or Scannable, to convert your pictures into a pdf. Apply the black and white or color filter to your pictures to make your answers easier to see!

Multiple Choice Questions (5 points each) 1. A firm’s isoquant and isocost line are shown in the diagram below. The isoquant corresponds to an output of 50 units, and the isocost line corresponds to a cost of $225. Additionally, the price of labor is $10. The MRT of labor for capital equals _______, and at point A the marginal product per dollar spent on capital is ______ the marginal product per dollar spent on labor. a. 9/10; greater than b. 9/10; less than c. 10/9; less than d. 10/9; greater than 2. According to a firm’s technology of production, you can take away 3 units of labor if you add 5 units of capital without changing production. The price of labor is $8, and the price of capital is $6. Assuming the firm’s MRTS is diminishing, this firm a. is minimizing cost at its current output level. b. should use more capital and less labor to lower the cost of producing its current output. c. should use less capital and more labor to lower the cost of producing its current output. d. should use less capital and less labor to lower the cost of producing its current output. e. None of above. A ● capital labor Q 25

3. A firm has a Cobb-Douglas production function for its output using the inputs of capital (K) and labor (L). The firm is currently paying $10 per labor hour and $5 per machine hour and is employing an equal number of machines and workers. If this bundle of inputs is cost minimizing, what we can infer about the marginal productivities of capital and labor at this point? a. MP K = MP L b. MP K = 2MP L c. MP L = 2MP K d. MP L = 5MP K 4. A manufacturing firm’s production function is Q(L,K) = KL + L . Suppose that the price of capital services is equal to $10, and the price of labor services is equal to $60. If the firm wants to produce 24 units of output at the lowest possible cost, how much does it spend? a. $1070 b. $730 c. $680 d. $660 e. $230 5. M ary’s firm uses both capital and labor to produce cooking supplies. Her firm’s production function is given by Q(L,K) = LK 2 . Let w represent the wage and r represent the price of capital. Which of the following represent her long-run capital demand function? a. ? = ( 2𝑤 𝑟 ) 1 3 (?) 1 3 b. ? = ( 𝑤 2𝑟 ) 2 3 (?) 1 3 c. ? = 𝑤 2𝑟 (?) 1 2 d. ? = ( 2𝑤 𝑟 ) 1 2 (?) 1 2

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6. Consider the truthfulness of the following statements. I. The lower is the elasticity of substitution between capital and labor, the more inelastic labor demand will be. II. Suppose capital and labor are perfect substitute inputs for a particular firm. If the price of labor increases, holding the price of capital and the level of output constant, the firm will necessarily lower its costs by substituting towards capital and away from labor. 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. 7. Suppose that a firm’s production function can be specified as 10 Q KL = . Which of the following accurately describes this firm’s long run total cost function? a. LRTC = 10 Q K b. LRTC = 10 Q L c. LRTC = 10 2 rwQ d. LRTC = rw Q 10 2 8. A firm’s production is represented by ?(?, ?) = ? 1/3 + ? 1/3 . Let the price of L equal $16, and the price of K equal $16. What is the firm’s long -run marginal cost function? a. ???𝐶 = 32? b. ???𝐶 = 16? c. ???𝐶 = 12? 2 d. ???𝐶 = 4? 3

9. A plant’s production function is ?(?, ?) = 2? 1/2 ? 1/2 . The price of labor services, w, is $25, and the price of capital services, r, is $100. In the short- run, the plant’s capital is fixed at K = 36. The firm must produce 240 units of output. How much money is it sacrificing by not having the ability to choose its level of capital optimally? That is, how much more does it cost to produce 240 units of output in the short-run compared to the long-run? a. $1,600 b. $12,000 c. $25 d. $11,900 e. $3,700 10. Consider a production process for which labor and capital are perfect substitutes and where 1 unit of labor can always be substituted for 3 units of capital. Additionally, the price of each unit of labor is $10, and the price of each unit of capital is $20. Assuming that labor is on the x-axis and capital is on the y-axis, the expansion path for this firm will: a. be a vertical line. b. be a horizontal line. c. increase at a decreasing rate because we substitute capital for labor as output increases. d. increase at an increasing rate because we substitute capital for labor as output increases. e. The expansion path is undefined.

Long Form Questions 1. A firm has a production function Q = K 1/2 + L . The firm faces a price of labor, w, and a price of capital services, r. a. Derive the long-run input demand functions for L and K . (Assume an interior solution.) (10 points) b. Suppose the firm must produce 100 units of output. What must be true of the relative price of labor in terms of capital for the firm to use a positive amount of labor? (4 points) c. Assuming the relative price of labor meets the condition you specified in part (b), graph the firm ’s expansion path. (5 points) d. Suppose the economy takes a downturn, and the price of labor falls by 50% and is expected to stay at that level for a long time. Show graphically how this change affects the firm’s expansion path. (6 points) 2. At Albert’s Pretzel Company, pretzels (Q) are produced using workers (L) and pretzel-making machines (K) according to the production function Q(L,K) = KL + K. The hourly cost of pretzel machines is $10, and the hourly cost of workers is $40. Where appropriate, round to 2 decimal places. a. Suppose Albert wishes to produce 10,000 pretzels at minimum cost. How many hours of machines and workers should he employ to minimize cost? How much will it cost to produce 10,000 pretzels? (7 points) b. An order arrives, doubling the number of pretzels Albert must produce. Assuming he is unable to purchase more pretzel-making machines in the short-run, how much will it cost to meet the new production level? (4 points) c. In the long-run, Albert will be able to employ more machines as well as workers. If he now produces 20,000 pretzels at minimum cost, how much will it cost? (7 points) d. Use an optimal choice diagram (isoquants and isocost lines) to graphically illustrate your answers to (b) and (c). Use this graph to compare your long-run and short-run input levels as well as your long-run and short-run costs. (7 points)

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