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# Chapter 5 - solution manual for managerial economics & business strategy 7th edition Michael

## Chapter 5: Answers to Questions and Problems

1.

a. When *K = 16* and *L = 16*, *Q *= (16 )0.75 (16 )0.25 = 16. Thus, *APL = Q/L = 16/16 = *

*1. *When *K = 16* and *L = 81*, *Q *= (16 )0.75 (81)0.25 = (8 3 )( ) = 24 . Thus, *APL = *

*24/81 = 8/27*.

b. The marginal product of labor is *MP L *= 2( )*L *−3 4 . When *L = 16*, *MPL *= 2 16()=1/ 4 . When *L = 81*, *MPL *= 2 81( ) = 2/ 27. Thus, as the

−3 4 −3 4

number of units of labor hired increases, the marginal product of labor decreases *MP L *(16 ) = 1/ 4 > 2 / 27 = *MPL *(81), holding the level of capital fixed.

c. We must equate the value marginal product of labor equal to the wage and solve for L. Here, *VMP L *= (*P MP *)( *L *) = ($100 2 )( ( )*L *−3/ 4 ) = 200( )*L *−3/ 4 . Setting this

equal to the wage of $25 gives 200 ( )*L *−3/4 = 25. Solving for *L*, the optimal quantity of labor is *L = 16. *

2. See Table 5-1.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

**Capital **

**Labor **

**Output**

**Marginal **

**Product of Capital**

**Average **

**Product of Capital**

**Average **

**Product of Labor**

**Value Marginal **

**Product of Capital**

*K *

*L *

*Q *

*MPK *

*APK *

*APL *

*VMPK*

0

20

0

--

--

--

--

1

20

50

50

50

2.50

100

2

20

150

100

75

7.50

200

3

20

300

150

100

15

300

4

20

400

100

100

20

200

5

20

450

50

90

22.50

100

6

20

475

25

79.17

23.75

50

7

20

475

0

67.86

23.75

0

8

20

450

-25

56.25

22.50

-50

9

20

400

-50

44.44

20

-100

10

20

300

-100

30

15

-200

11

20

150

-150

13.64

7.50

-300

## Table 5-1

a. Labor is the fixed input while capital is the variable input.

b. Fixed costs are 20($15) = $300.

c. To produce 475 units in the least-cost manner requires 6 units of capital, which cost $75 each. Thus, variable costs are ($75)(6) = $450.

d. Using the *VMPK* = *r* rule, *K = 5* maximizes profits.

e. The maximum profits are $2(450) $15(20) $75(5) − − = $225.

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 = 50. *

b. *VC *(10 ) = 25 10 ( )+ 30 10 ( )2 + 5 10 ( )3 = $8, 250 .

c. *C *( )10 = 50 + 25( )10 + 30( )10 2 + 5( )10 3 = $8,300 .

d. *AFC *( )10 = $50 = $5 . 10

e. *AVC *(10 ) = *VC*10(10 ) = $8, 25010 = $825.

f. *ATC *( )10 = *AFC*( )10 + *AVC*( )10 = $830 .

g. *MC *( )10 = 25+ 60( )10 +15( )10 2 = $2,125.

5. Since *MRTSKL *≠ *w *, the firm is not using the cost minimizing combination of labor *r*

and capital. To minimize costs, the firm should use more labor and less capital since

the marginal product per dollar spent is greater for labor: *MP L *= >50 *MPK *= 75 .

*w *6 *r *12

6. See Table 5-2.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

**Quantity**

**Fixed Cost**

**Variable Cost **

**Total Cost**

**Average Fixed Cost**

**Average **

**Variable Cost**

**Average Total Cost**

**Marginal Cost**

*Q*

*FC*

*VC*

*TC*

*AFC*

*AVC*

*ATC*

*MC*

0

10,000

0

10,000

--

--

--

--

100

10,000

10,000

20,000

100

100

200

100

200

10,000

15,000

25,000

50

75

125

50

300

10,000

30,000

40,000

33.33

100

133.33

150

400

10,000

50,000

60,000

25

125

150

200

500

10,000

90,000

100,000

20

180

200

400

600

10,000

140,000

150,000

16.67

233.33

250

500

## Table 5-2

7.

a. For a quadratic multi-product cost function, economies of scope exist if *f *− *aQ*1 *Q*2 > 0 . In this case, *f *=75 and *a *=−0.25, so economies of scope exist

since *f* is fixed cost, which is always nonnegative.

b. Cost complementarities exist since *a *=−0 .25 < 0 .

c. Since *a *=−0 .25 < 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 = 4* units.

b. The cost-minimizing mix of *K* and *L* that produce *Q = 4* 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 = 16.

b. Since the *MRTSKL* is 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 perfectly substitutability between capital and labor means that only input will be utilized. Since

*MP L *= *MP K *⇒ 4 < 2 , the company should hire all capital. *w r *30 10

c. The company should hire only 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 more steep. 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.

*w*

12. Since *MRTS *≠ , the firm was not using the cost minimizing combination of labor

*KL r*

and capital. To achieve the cost minimizing combination of inputs, the previous

manager should have used fewer units of capital and more units of labor, since

*MP L *= 100 > *MPK *= 100 . *w *8 *r *16

13. The profit-maximizing level of labor and output is achieved where *VMPL *= *w*. Here, *VMP L *= 2 $100 4 ( )( ) ( )*L *= $400( )*L* and *w *= $100 per day. Solving yields *L* 1/ 2 −1 2 −1/ 2

= 16. The profit-maximizing level of output is *Q *= 2 4( ) ( )1 2 16 1 2 =16 units. The firm’s fixed costs are $10,000, its variable costs are $100(16) = $1,600, and its total revenues are $200(16) = $3,200. Profits are $3,200 – $11,600 = – $8,400. The firm is suffering a loss, but the loss is lower than the $10,000 that would be lost if the firm shut down its operation.

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. Table 5-3 provides some useful information for making your decision. According to the *VMPL = w* rule, you should hire five units of labor and produce 90 units of output to maximize profits. Your fixed costs are ($10)(5) = $50, your variable costs are ($50)(5) =$250, and your revenues are ($5)(90) = $450. Thus, your maximum profits are $450 - $300 = $150.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

**Labor **

**Capital **

**Output**

**Marginal **

**Product of Labor**

**Average **

**Product of Labor**

**Average **

**Product of Capital**

**Value Marginal **

**Product of Labor**

*L *

*K *

*Q *

*MPL *

*APL *

*APK *

*VMPL*

0

5

0

--

--

--

--

1

5

10

10

10

2

50

2

5

30

20

15

6

100

3

5

60

30

20

12

150

4

5

80

20

20

16

100

5

5

90

10

18

18

50

6

5

95

5

15.8

19

25

7

5

95

0

13.6

19

0

8

5

90

-5

11.3

18

-25

9

5

80

-10

8.9

16

-50

10

5

60

-20

6

12

-100

11

5

30

-30

2.7

6

-150

## Table 5-3

16. The $1,200 per month that could be earned by renting out the excess rental space.

17. Had she not spent the $6,000 on advertising but instead collected the $65,000 refund, her total loss would have been limited to her sunk costs of $10,000. Her decision to spend $6,000 on advertising in an attempt to fetch an extra $5,000 was clearly foolish. However, the $6,000 is a sunk cost and therefore irrelevant in deciding whether to accept the $66,000 offer. She should accept the $66,000 offer because doing so makes her $1,000 better off than obtaining the $65,000 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 about 3 million kilowatts per hour, and diseconomies of scale 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.

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 $50 per panel, so the value marginal product of the third worker is $50(290) = $14,500. Table 5-4 summarizes the *VMPL* for each choice of labor. Since the wage is $7,000, the profit maximizing number of workers is 4.

**Machines **

**Workers **

**Output **

**MPL **

**VMPL **

**Wage **

5

0

0

–

–

–

5

1

600

600

$30,000

$7,000

5

2

1,000

400

$20,000

$7,000

5

3

1,290

290

$14,500

$7,000

5

4

1,480

190

$9,500

$7,000

5

5

1,600

120

$6,000

$7,000

5

6

1,680

80

$4,000

$7,000

## Table 5-4

20. The rental rate of capital is ¥475,000, computed as *r *= *MPK *× *P *= . 5×950,000 = 475,00. Therefore, the marginal product of labor is

*MP L *0.5

0.0014 cars per hour, which is found by solving = . Costs are 1, 330 475,000

minimized when the marginal rate of technical substitution is 0.0028.

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 *< 0and *f *− *aQ*1 *Q*2 > 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 $6,000; the $2,000 in rent plus the $4,000 in forgone earnings. Total variable costs are $1.23 per gallon. The cost function is *C *( )*Q *= 6, 000 +1.23*Q *.

The marginal cost is *MC *( )*Q *= *dC*__( )__*Q *= $1.23; the wholesale price. The average *dQ*

variable cost is *AVC *( )*Q *= *C*__( )__*Q *= 1. 23*Q *= $1.23. The average fixed cost is

*Q Q*

*AFC *( )*Q *= $6000 . The entrepreneur will earn a profit when revenues exceed costs,

*Q*

which occurs when 2*Q * > 6,000 +1.23*Q*. Solving for *Q* implies the entrepreneur earns a profit when she sells *Q* > 8571.43 gallons, or 8572 gallons. The average fixed cost

of selling *Q *= 8572 is *AFC *(8572 )= $6000 = $0.70 .

8572

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.