Chapter 5: The Production Process And Costs Answers To Questions And Problems

Chapter 5: The Production Process and Costs Answers to Questions and Problems 1. a. When K = 81 and L = 16, Q = (81)0.75(16)0.25 = 54. Thus, APL = Q/L = 54/16 = 3.375. When K = 81 and L = 256, Q = (81)0.75(256)0.25 = (27)(4) = 108. Thus, APL = 108/81 = 0.422. b. The marginal product of labor is MPL = (1/4) × (81)0.75(L)-3/4 = (27/4)(L)-3/4. When L = 16, MPL = (27/4)(16)-3/4 = 0.844. When L = 81, MPL = (27/4)(81)-3/4 = 0.25. Thus, as the number of units of labor hired increases, the marginal product of labor decreases MPL(16) = 0.844 > 0.25 = MPL(81), holding the level of capital fixed. c. We must equate the value marginal product of labor to the wage and solve for L. Here, VMPL = (P)(MPL) = ($200)(27/4)(L)-3/4=1350(L)-3/4. Setting this equal to the wage of $50 gives 1350(L)-3/4 = 50. Solving for L, the optimal quantity of labor is L = 81.

2. See table below. (1)

(2)

(3)

Capital

Labor

Output

0 1 2 3 4 5 6 7 8 9 10 11

20 20 20 20 20 20 20 20 20 20 20 20

0 50 150 300 400 450 475 475 450 400 300 150

(4) Marginal Product of Capital MPK 50 100 150 100 50 25 0 -25 -50 -100 -150

(5) Average Product of Capital APK 50 75 100 100 90 79.17 67.86 56.25 44.44 30 13.64

(6) Average Product of Labor APL 2.50 7.50 15 20 22.50 23.75 23.75 22.50 20 15 7.50

(7) Value Marginal Product of Capital VMPK 200 400 600 400 200 100 0 -100 -200 -400 -600

a. Labor is the fixed input while capital is the variable input. b. Fixed costs are 20($30) = $600. 1 © 2017 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.

c. To produce 475 units in the least-cost manner requires 6 units of capital, which cost $25 each. Thus, variable costs are ($25)(6) = $150. d. Using the VMPK = r rule, K = 6 maximizes profits. e. The maximum profits are $4(475) - $30(20) - $25(6) = $1,150. f. There are increasing marginal returns when K is between 0 and 3. g. There are decreasing marginal returns when K is between 3 and 11. h. There are negative marginal returns when K is greater than 7.

3.

The law of diminishing marginal returns is the decline in marginal productivity experienced when input usage increases, holding all other inputs constant. In contrast, the law of diminishing marginal rate of technical substitution is a property of a production function stating that as less of one input is used, increasing amounts of another input must be employed to produce the same level of output.

4. a. FC = $100. b. VC(10) = 20(10) + 15(10)2 + 10(10)3 = $11,700. c. C(10) = 100 + 20(10) + 15(10)2 + 10(10)3 = $11,800. d. 𝐴𝐹𝐶(10) = e. 𝐴𝑉𝐶(10) =

$100 10

= $10.

𝑉𝐶(10) 10

=

$11,700 10

= $1,170.

f. ATC(10) = AFC(10) + AVC(10) = $1,180. g. MC(10) = 20 + 30(10) + 30(10)2 = $3,320.

5.

𝑤

Since 𝑀𝑅𝑇𝑆𝐾𝐿 ≠ 𝑟 , the firm is not using the cost minimizing combination of labor and capital. To minimize costs, the firm should increase capital (and decrease labor) since the marginal product per dollar spent is greater for capital: 𝑀𝑃𝐾 𝑟

=

45 8

>

𝑀𝑃𝐿 𝑤

=

60

.

12

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6. See table below. (1)

(2)

(3)

(4)

Quantity Q

Fixed Cost FC

Variable Cost VC

Total Cost TC

0 100 200 300 400 500 600

15,000 15,000 15,000 15,000 15,000 15,000 15,000

0 15,000 25,000 37,500 75,000 147,500 225,000

15,000 30,000 40,000 52,500 90,000 162,500 240,000

(5) Average Fixed Cost AFC 150 75 50 37.50 30 25

(6) Average Variable Cost AVC 150 125 125 187.50 295 375

(7) Average Total Cost ATC 300 200 175 225 325 400

(8) Marginal Cost MC 150 100 125 375 725 775

7. a. For a quadratic multi-product cost function, economies of scope exist if f – aQ1Q2 > 0. In this case, f = 90 and a = -0.5, so economies of scope exist since f is fixed cost, which is always nonnegative. b. Cost complementarities exist since a = -0.5 < 0, which holds in this case. c. Since a = -0.5 < 0, the marginal cost of producing product 1 will increase if the division that produces product 2 is sold.

8.

Fixed costs are associated with fixed inputs, and do not change when output changes. Variable costs are costs associated with variable inputs, and do change when output changes. Sunk costs are costs that are forever lost once they have been paid.

9. a. When K = 2 and L = 3, Q = 8 units. b. The cost-minimizing mix of K and L that produce Q = 8 is K = 2, L = 1. c. Since K and L are perfect complements in the production process, the costminimizing levels of K and L do not depend on the rental rates of K and L. Therefore, the cost-minimizing levels of K and L do not change with changes in the relative rental rates.

10. a. With K = 2 and L = 3, Q = 4(2) + 8(3) = 32. b. Since the MRTSKL is 8/4 = 2, that means a company can trade two units of capital for every one unit of labor. This production function does not exhibit diminishing marginal rate of technical substitution. The perfect substitutability between capital 𝑀𝑃 8 4 𝑀𝑃 and labor means that only one input will be utilized. Since 𝑤 𝐿 = 60 < 20 = 𝑟 𝐾 , the 3 © 2017 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.

company should hire all capital. To get 32 units of output, this will require 8 units of capital. 𝑀𝑃

8

4

𝑀𝑃

c. Here, we have 𝑤 𝐿 = 20 > 20 = 𝑟 𝐾 , so the company should hire only labor. To get 32 units of output, this will require 4 units of labor.

11.

An investment tax credit would reduce the relative price of capital to labor. Other 𝑤 things equal, this would increase , thereby making the isocost line steeper. This 𝑟 means that the cost-minimizing input mix will now involve more capital and less labor, as firms substitute toward capital. Labor unions are likely to oppose the investment tax credit since the higher capital-to-labor ratio will translate into lost jobs. You might counter this argument by noting that, while some jobs will be lost due to substituting capital for labor, many workers will retain their jobs. Absent the plan, automakers have an incentive to substitute cheaper foreign labor for U.S. labor. The result of this substitution would be a movement of plants abroad, resulting in the complete loss of U.S. jobs.

12.

Since 𝑀𝑅𝑇𝑆𝐾𝐿 ≠ 𝑟 , the firm was not using the cost minimizing combination of labor and capital. To achieve the cost minimizing combination of inputs, the previous 𝑀𝑃 manager should have used more units of capital and fewer units of labor, since 𝑤 𝐿 =

𝑤

80 12

13.

<

𝑀𝑃𝐾 𝑟

=

110 15

.

The profit-maximizing level of labor and output is achieved where VMPL = w. Here, 1 𝑉𝑀𝑃𝐿 = (2) 2($400)(9)1/2 (𝐿)−1/2 = $1200(𝐿)−1/2 and w = $120 per day. Solving

yields L = 100. The profit-maximizing level of output is 𝑄 = 2(9)1/2 (100)1/2 = 60 units. The firm’s fixed costs are $8,000, its variable costs are $120(100) = $12,000, and its total revenues are $400(60) = $24,000. Profits are $24,000 – $12,000 – $8,000 = $4,000.

14.

The higher wage rate in Europe induces Airbus to employ a more capital intensive input mix than Boeing. Since Airbus optimally uses fewer workers than Boeing, and profit-maximization entails input usage in the range of diminishing marginal product, it follows that the lower quantity of labor used by Airbus translates into a higher marginal product of labor at Airbus than at Boeing.

15.

The following table provides some useful information for making your decision. According to the VMPL = w rule, you should hire six units of labor and produce 95 units of output to maximize profits. Your fixed costs are ($20)(8) = $160, your 4

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variable costs are ($60)(6) = $360, and your revenues are ($12)(95) = $1,140. Thus, your maximum profits are $1,140 – $360 – $160 = $620. (1)

(2)

(3)

Labor L

Capital K

Output Q

0 1 2 3 4 5 6 7 8 9 10 11

8 8 8 8 8 8 8 8 8 8 8 8

0 10 30 60 80 90 95 95 90 80 60 30

(4)

(5)

(6)

(7) Value Marginal Average Average Marginal Product of Product of Product of Product of Labor Labor Capital Labor MPL APL APK VMPL 10 10 1.3 120 20 15 3.8 240 30 20 7.5 360 20 20 10 240 10 18 11.3 120 5 15.8 11.9 60 0 13.6 11.9 0 -5 11.3 11.3 -60 -10 8.9 10 -120 -20 6 7.5 -240 -30 2.7 3.8 -360

16.

The $8,000 per month that could be earned by renting out the space used for the fitness studio.

17.

Had she not spent the $8,000 on advertising but instead collected the $75,000 refund, her total loss would have been limited to her sunk costs of $15,000. Her decision to spend $8,000 on advertising in an attempt to fetch an extra $5,000 was clearly foolish. However, the $8,000 is a sunk cost and therefore irrelevant in deciding whether to accept the $77,000 offer. She should accept the $77,000 offer because doing so makes her $2,000 better off than obtaining the $75,000 refund.

18.

Facility “L” produces 6 million kilowatt hours of electricity at the lowest average total cost, so this is the optimal facility for South-Florida. Facility “M” produces 2 million kilowatt hours of electricity at the lowest average total cost, so this is the optimal facility for the Panhandle. There are economies of scale up to just below 3 million kilowatts per hour for facility “M,” and diseconomies of scale thereafter. There are economies of scale up to just below 4 million kilowatts per hour for facility “L,” and diseconomies thereafter. Therefore, facility “M” will be operating in the range of economies of scale while facility “L” will be operating in the range of diseconomies of scale.

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19.

To maximize profits the firm should continue adding workers so long as the value marginal product of labor exceeds the wage. The value marginal product of labor is defined as the marginal product of labor times the price of output. Here, output sells for $80 per panel, so the value marginal product of the third worker is $80(290) = $23,200. The table below summarizes the VMPL for each choice of labor. Since the wage is $9,000, the profit maximizing number of workers is 5. Machines 5 5 5 5 5 5 5

20.

Workers Output MPL 0 0 – 1 600 600 2 1,000 400 3 1,290 290 4 1,480 190 5 1,600 120 6 1,680 80

VMPL – $48,000 $32,000 $23,200 $15,200 $9,600 $6,400

Wage – $9,000 $9,000 $9,000 $9,000 $9,000 $9,000

The rental rate of capital is ¥960,000, computed as r = MPK × P = 0.8 × 1,200,000 = 960,000. Therefore, the marginal product of labor is 0.0015 cars per hour, which is found by solving

𝑀𝑃𝐿 1,800

=

0.8 960,000

. Costs are minimized when the marginal rate of

technical substitution is 0.001875 =

𝑤 𝑟

=

1,800 960,000

.

21.

Given the tightly woven marine engine and shipbuilding divisions, economies of scope and cost complementarities are likely to exist. Eliminating the unprofitable marine engine division may actually raise the shipbuilding division’s costs and cause that division to become unprofitable. For this argument to withstand criticism, you must show the CEO that the quadratic multi-product cost function exhibits cost complementarities and economies of scope, which occurs when a  0 and f  aQ1Q2  0 , respectively, and compare profitability under the different scenarios.

22.

Taking into account both implicit and explicit costs, the total fixed cost from operating the kiosk is $7,500; the $2,500 in rent plus the $5,000 in forgone earnings. Total variable costs are $1.34 times the number of gallons. The cost function is C(Q) = 7,500 + 1.34Q. The marginal cost is the cost of the last unit produced, which is constant in this case: MC(Q) = $1.34; the wholesale price. The average variable cost is 𝐴𝑉𝐶(𝑄) =

𝐴𝐹𝐶(𝑄) =

$7,500 𝑄

𝐶(𝑄) 𝑄

=

1.34𝑄 𝑄

= $1.34. The average fixed cost is

. The entrepreneur will earn a profit when revenues exceed costs,

which occurs when 2.25Q > 7,500 + 1.34Q. Solving for Q implies the entrepreneur earns a profit when she sells Q > 8,242 gallons (rounding up). The average fixed cost of selling Q = 8,242 is 𝐴𝐹𝐶(8,242) =

$7,500 8,242

= $0.91.

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23.

Assuming that the optimal mix of unskilled and semi-skilled labor were being utilized at the time the legislation passed, in the short run, a higher minimum wage paid to unskilled labor implies that to minimize costs the retailer should increase its use of semi-skilled worker and decrease its use or unskilled workers. In the longer run, the retailer may want to consider substituting capital for labor (invest in some machines to automate a portion of your boxing needs). Obviously, additional information would be required to conduct a net present value analysis for these long-run investments, but it is probably worth getting this information and running some numbers.

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