Chapter 05 - The Production Process and Costs
Chapter 5: 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 5-1.
(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
Table 5-1
a. Labor is the fixed input while capital is the variable input.
b. Fixed costs are 20($30) = $600.
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.
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Chapter 05 - The Production Process and Costs
d.
e.
f.
g.
h.
3.
Using the VMPK = r rule, K = 6 maximizes profits.
The maximum profits are $4(475) - $30(20) - $25(6) = $1,150.
There are increasing marginal returns when K is between 0 and 3.
There are decreasing marginal returns when K is between 3 and 11.
There are negative marginal returns when K is greater than 7.
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.
b.
c.
d.
FC = $100.
VC(10) = 20(10) + 15(10)2 + 10(10)3 = $11,700.
C(10) = 100 + 20(10) + 15(10)2 + 10(10)3 = $11,800.
$100
𝐴𝐹𝐶(10) = 10 = $10.
𝑉𝐶(10)
5.
𝑀𝑃𝐾
𝑟
6.
$11,700
e. 𝐴𝑉𝐶(10) = 10 = 10 = $1,170.
f. ATC(10) = AFC(10) + AVC(10) = $1,180.
g. MC(10) = 20 + 30(10) + 30(10)2 = $3,320.
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:
=
45
8
>
𝑀𝑃𝐿
𝑤
=
60
.
12
See Table 5-2.
(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
37.5
30
25
(6)
Average
Variable
Cost
AVC
150
125
125
187.5
295
375
(7)
Average
Total
Cost
ATC
300
200
175
225
325
400
(8)
Marginal
Cost
MC
150
100
125
375
725
775
Table 5-2
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Chapter 05 - The Production Process and Costs
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.
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 = 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
and labor means that only one input will be utilized. Since
𝑀𝑃𝐿
𝑤
8
4
= 60 < 20 =
𝑀𝑃𝐾
𝑟
,
the company should hire all capital.
c. Here, we have
11.
𝑀𝑃𝐿
𝑤
8
4
= 20 > 20 =
𝑀𝑃𝐾
𝑟
, so the company should hire only labor.
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.
5-3
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Chapter 05 - The Production Process and Costs
The result of this substitution would be a movement of plants abroad, resulting in the
complete loss of U.S. jobs.
12.
𝑤
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 =
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.
Table 5-3 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 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
8
8
8
8
8
8
8
8
8
8
0
10
30
60
80
90
95
95
90
80
(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
5-4
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Chapter 05 - The Production Process and Costs
10
11
8
8
60
30
-20
-30
Table 5-3
6
2.7
7.5
3.8
-240
-360
16.
The $1,500 per month that could be earned by renting out the excess rental space.
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.
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. Table 5-4 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
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
Table 5-4
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Chapter 05 - The Production Process and Costs
20.
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 > 8242 gallons (rounding up). The average fixed cost
of selling Q = 8242 is 𝐴𝐹𝐶(8242) =
23.
$7,500
8242
= $0.91.
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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