Chapter 5: Answers to Questions and Problems
Q1. A firm can manufacture a product according to the production function
Q = F(K, L) = K3/4L1/4
a. Calculate the average product of labor, APL, when the level of capital is fixed at
16 units and the firm uses 16 units of labor. How does the average product of labor
change when the firm uses 81 units of labor?
b. Find an expression for the marginal product of labor, MPL, when the amount of
capital is fixed at 16 units. Then, illustrate that the marginal product of labor
depends on the amount of labor hired by calculating the marginal product of labor
for 16 and 81 units of labor.
c. Suppose capital is fixed at 16 units. If the firm can sell its output at a price of
$100 per unit and can hire labor at $25 per unit, how many units of labor should the
firm hire in order to maximize profits?
Answer 3.
0.75
0.25
a) When K = 16 and L = 16, Q = (16 ) (16 ) = 16
Thus, APL = Q/L = 16/16 = 1. When K = 16 and L = 81,
0.75
0.25
Q = (16 ) (81) = (8 )( 3) = 24 . Thus, APL = 24/81 = 8/27.
b) The marginal product of labor is MPL = 2 ( L )
When
L
MPL = 2 (81)
=
−3 4
16,
MPL = 2 (16 )
−3 4
−3 4
= 1/ 4 .
When
L
=
81,
= 2 / 27 .
Thus, as the number of units of labor hired increases, the marginal product
of labor decreases MPL (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, VMPL = ( P )( MPL ) = ( $100 ) 2 ( L )
−3/ 4
) = 200 ( L )
−3/ 4
= 25 .
Setting this equal to the wage of $25 gives 200 ( L )
Solving for L, the optimal quantity of labor is L = 16.
−3/ 4
Q2. Explain the difference between the law of diminishing marginal returns and the law
of diminishing marginal rate of technical substitution?
Answer 2.
➢ The law of diminishing 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
the rate at which a firm can substitute among different inputs while
maintaining the same level of output.
AbuTurki
This study source was downloaded by 100000796483839 from CourseHero.com on 09-24-2022 10:39:59 GMT -05:00
https://www.coursehero.com/file/81238121/chapter-5-Economic-solutionspdf/
Q3.
An economist estimated that the cost function of a single-product firm is
C(Q) = 50 +25Q +30Q2 + 5Q3
Based on this information, determine:
a. The fixed cost of producing 10 units of output.
b. The variable cost of producing 10 units of output.
c. The total cost of producing 10 units of output.
d. The average fixed cost of producing 10 units of output.
e. The average variable cost of producing 10 units of output.
f. The average total cost of producing 10 units of output.
g. The marginal cost when Q = 10.
Answer 3.
d. FC = 50
2
3
e. VC (10 ) = 25 (10 ) + 30 (10 ) + 5 (10 ) = $8, 250 .
C (10) = 50 + 25(10) + 30(10) + 5(10) = $8,300 .
$50
= $5 .
g. AFC (10 ) =
10
VC (10 ) $8, 250
h. AVC (10 ) =
=
= $825 .
10
10
i. ATC (10 ) = AFC (10 ) + AVC (10 ) = $830 .
f.
j.
2
3
MC (10) = 25 + 60(10) + 15(10) = $2,125 .
2
Q4.
Explain the difference between fixed costs, sunk costs, and variable costs. Provide an
example that illustrates that these costs are, in general, different.
Answer 4.
➢ 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.
AbuTurki
~2~
This study source was downloaded by 100000796483839 from CourseHero.com on 09-24-2022 10:39:59 GMT -05:00
https://www.coursehero.com/file/81238121/chapter-5-Economic-solutionspdf/
Powered by TCPDF (www.tcpdf.org)