CHAPTER 20: Profit Maximization
TRUE/FALSE
1. The weak axiom of profit-maximizing behavior states that in a modern mixed economy, firms have
only a weak incentive to maximize profits.
2. A fixed factor is a factor of production that is used in fixed proportion to the level of output.
3. The marginal product of a factor is just the derivative of the production function with respect to the
amount of this factor, holding the amounts of other factor inputs constant.
4. If the value of the marginal product of factor x increases as the quantity of x increases and the value of
the marginal product of x is equal to the wage rate, then the profit-maximizing amount of x is being
used.
5. If the price of the output of a profit-maximizing, competitive firm rises and all other prices stay
constant, then the firm’s output cannot fall.
6. If a profit-maximizing competitive firm has constant returns to scale, then its long-run profits must be
zero.
7. Just as in the theory of utility-maximizing consumers, the theory of profit-maximizing firms allows the
possibility of Giffen factors. These are factors for which a fall in price leads to a fall in demand.
8. If the value of the marginal product of labor exceeds the wage rate, then a competitive,
profit-maximizing firm would want to hire less labor.
9. A firm produces one output with one input and has decreasing returns to scale. The price that it pays
per unit of input and the price it gets per unit of output are independent of the amount that this firm
buys or sells. If the government taxes its net profits at some percentage rate and subsidizes its inputs at
the same percentage rate, the firm’s profit-maximizing output will not change.
MULTIPLE CHOICE
1. A competitive firm produces output using three fixed factors and one variable factor. The firm’s
short-run production function is q = 305x − 2x2, where x is the amount of variable factor used. The
price of the output is $2 per unit and the price of the variable factor is $10 per unit. In the short run,
how many units of x should the firm use?
a.
37
b.
150
c.
21
d.
75
e.
None of the above.
2. A competitive firm produces output using three fixed factors and one variable factor. The firm’s
short-run production function is q = 524x − 4x2, where x is the amount of variable factor used. The
price of the output is $3 per unit and the price of the variable factor is $12 per unit. In the short run,
how many units of x should the firm use?
a.
130
b.
32
c.
25
d.
65
e.
None of the above.
3. A competitive firm produces output using three fixed factors and one variable factor. The firm’s
short-run production function is q = 154x − 5x2, where x is the amount of variable factor used. The
price of the output is $2 per unit and the price of the variable factor is $8 per unit. In the short run, how
many units of x should the firm use?
a.
15
b.
30
c.
17
d.
7
e.
None of the above.
4. A competitive firm produces a single output using several inputs. The price of output rises by $4 per
unit. The price of one of the inputs increases by $2 and the quantity of this input that the firm uses
increases by 8 units. The prices of all other inputs stay unchanged. From the weak axiom of profit
maximization we can tell that
a.
the output of the good must have increased by at least 4 units.
b.
the inputs of the other factors must have stayed constant.
c.
the output of the good must have decreased by at least 2 units.
d.
the inputs of at least one of the other factors must have decreased by at least 8 units.
e.
the inputs of at least one of the other factors must have increased by at least 8 units.
5. A competitive firm produces a single output using several inputs. The price of output rises by $4 per
unit. The price of one of the inputs increases by $4 and the quantity of this input that the firm uses
increases by 16 units. The prices of all other inputs stay unchanged. From the weak axiom of profit
maximization we can tell that
a.
the inputs of the other factors must have stayed constant.
b.
the inputs of at least one of the other factors must have decreased by at least 16 units.
c.
the output of the good must have increased by at least 16 units.
d.
the output of the good must have decreased by at least 8 units.
e.
the inputs of at least one of the other factors must have increased by at least 16 units.
6. A competitive firm produces a single output using several inputs. The price of output rises by $3 per
unit. The price of one of the inputs increases by $6 and the quantity of this input that the firm uses
increases by 12 units. The prices of all other inputs stay unchanged. From the weak axiom of profit
maximization we can tell that
a.
the output of the good must have increased by at least 24 units.
b.
the inputs of the other factors must have stayed constant.
c.
the output of the good must have decreased by at least 12 units.
d.
the inputs of at least one of the other factors must have decreased by at least 12 units.
e.
the inputs of at least one of the other factors must have increased by at least 12 units.
7. If there is perfect certainty, a competitive firm will necessarily
a.
seek to maximize its immediate profits rather than long-run returns because otherwise it
will go broke.
b.
maximize the ratio of the present value of its sales to the present value of its costs.
c.
equalize its profits in all periods.
d.
equalize its sales in all periods.
e.
None of the above.
8. A firm produces one output using one input. When the cost of the input was $3 and the price of the
output was $3, the firm used 6 units of input to produce 18 units of output. Later, when the cost of the
input was $7 and the price of the output was $4, the firm used 5 units of input to produce 20 units of
output. This behavior
a.
is consistent with WAPM.
b.
is not consistent with WAPM.
c.
is impossible no matter what the firm is trying to do.
d.
suggests the presence of increasing returns to scale.
e.
suggests the presence of decreasing returns to scale.
9. A profit-maximizing competitive firm uses just one input, x. Its production function is q = 4x1/2. The
price of output is $28 and the factor price is $7. The amount of the factor that the firm demands is
a.
8.
b.
16.
c.
64.
d.
60.
e.
None of the above.
10. A profit-maximizing competitive firm uses just one input, x. Its production function is q = 4x1/2. The
price of output is $12 and the factor price is $3. The amount of the factor that the firm demands is
a.
64.
b.
16.
c.
60.
d.
8.
e.
None of the above.
11. A profit-maximizing competitive firm uses just one input, x. Its production function is q = 8x1/2. The
price of output is $24 and the factor price is $8. The amount of the factor that the firm demands is
a.
11.
b.
128.
c.
144.
d.
27.71.
e.
None of the above.
12. A competitive, profit-maximizing firm uses two inputs a and b. Its production function is
F(a, b) = a1/2 + a1/2. Its output sells for $5 per unit. The price of input a is $1 per unit. If the price of
output rises to $6 per unit but factor prices do not change.
a.
it will increase its purchases of factor a by 11/4 units.
b.
it will increase its purchases of factor a by 9/4 units.
c.
it will increase its purchases of factor a by 3/4 units.
d.
we need to know the price of factor b to be able to determine the change in demand for a.
e.
None of the above.
13. A competitive firm’s production function is f(x1, x2) = 12x1/21 + 4x1/22. The price of factor 1 is $1 and
the price of factor 2 is $2. The price of output is $4. What is the profit-maximizing quantity of output?
a.
304
b.
608
c.
300
d.
612
e.
292
14. A competitive firm’s production function is f(x1, x2) = 6x1/21 + 8x1/22. The price of factor 1 is $1 and the
price of factor 2 is $4. The price of output is $8. What is the profit-maximizing quantity of output?
a.
416
b.
208
c.
204
d.
419
e.
196
15. A competitive firm’s production function is f(x1, x2) = 8x1/21 + 8x1/22. The price of factor 1 is $1 and the
price of factor 2 is $3. The price of output is $6. What is the profit-maximizing quantity of output?
a.
256
b.
512
c.
252
d.
516
e.
244
16. Jiffy-Pol Consultants is paid $1,000,000 for each percentage of the vote that Senator Sleaze receives in
the upcoming election. Sleaze’s share of the vote is determined by the number of slanderous campaign
ads run by Jiffy-Pol according to the function S = 100N/(N + 1), where N is the number of ads. If each
ad costs $4,900 approximately how many ads should Jiffy-Pol buy in order to maximize its profits?
a.
2,853
b.
1,428
c.
98
d.
1,477
e.
714
17. Jiffy-Pol Consultants is paid $1,000,000 for each percentage of the vote that Senator Sleaze receives in
the upcoming election. Sleaze’s share of the vote is determined by the number of slanderous campaign
ads run by Jiffy-Pol according to the function S = 100N/(N + 1), where N is the number of ads. If each
ad costs $10,000 approximately how many ads should Jiffy-Pol buy in order to maximize its profits?
a.
200
b.
1,995
c.
999
d.
1,099
e.
500
18. Jiffy-Pol Consultants is paid $1,000,000 for each percentage of the vote that Senator Sleaze receives in
the upcoming election. Sleaze’s share of the vote is determined by the number of slanderous campaign
ads run by Jiffy-Pol according to the function S = 100N/(N + 1), where N is the number of ads. If each
ad costs $3,600 approximately how many ads should Jiffy-pol buy in order to maximize its profits?
a.
3,329
b.
72
c.
1,666
d.
1,702
e.
833
19. A competitive firm uses a single input x to produce its output y. The firm’s production function is
given by y = x3/2 for quantities of x between 0 and 4. For quantities of a greater than 4, the firm’s
output is y = 4 + x. If the price of the output y is $1 and the price of the input x is $3, how much x
should the firm use to maximize its profit?
a.
16/9
b.
4
c.
0
d.
4/9
e.
9/2
20. The production function is given by F(L) = 6L2/3. Suppose that the cost per unit of labor is $16 and the
price of output is $8. How many units of labor will the firm hire?
a.
16
b.
8
c.
4
d.
24
e.
None of the above.
21. The production function is given by F(L) = 6L2/3. Suppose that the cost per unit of labor is $16 and the
price of output is $16. How many units of labor will the firm hire?
a.
192
b.
64
c.
32
d.
128
e.
None of the above.
22. The production function is given by F(L) = 6L2/3. Suppose that the cost per unit of labor is $16 and the
price of output is $12. How many units of labor will the firm hire?
a.
54
b.
13.50
c.
27
d.
81
e.
None of the above.
23. The production function is given by f(x) = 4x1/2. If the price of the commodity produced is $60 per unit
and the cost of the input is $10 per unit, how much profit will the firm make if it maximizes profits?
a.
$1,440
b.
$718
c.
$2,884
d.
$1,425
e.
$723
24. The production function is given by f(x) = 4x1/2. If the price of the commodity produced is $60 per unit
and the cost of the input is $20 per unit, how much profit will the firm make if it maximizes profits?
a.
$1,444
b.
$705
c.
$720
d.
$358
e.
$363
25. The production function is given by f(x) = 4x1/2. If the price of the commodity produced is $80 per unit
and the cost of the input is $40 per unit, how much profits will the firm make if it maximizes profits?
a.
$318
b.
$1,284
c.
$640
d.
$625
e.
$323
26. The production function is f(x1, x2) = x1/21x1/22. If the price of factor 1 is $12 and the price of factor 2 is
$24, in what proportions should the firm use factors 1 and 2 if it wants to maximize profits?
a.
x1 = x2.
b.
x1 = 0.50x2.
c.
x1 = 2x2.
d.
x1 = 24x2.
e.
We can’t tell without knowing the price of the output.
27. The production function is f(x1, x2) = x1/21x1/22. If the price of factor 1 is $6 and the price of factor 2 is
$12, in what proportions should the firm use factors 1 and 2 if it wants to maximize profits?
a.
x1 = 12x2.
b.
x1 = 0.50x2.
c.
x1 = 2x2.
d.
x1 = x2.
e.
We can’t tell without knowing the price of the output.
28. The production function is f(x1, x2) = x1/21x1/22. If the price of factor 1 is $10 and the price of factor 2 is
$15, in what proportions should the firm use factors 1 and 2 if it wants to maximize profits?
a.
x1 = 1.50x2.
b.
x1 = x2.
c.
x1 = 15x2.
d.
x1 = 0.67x2.
e.
We can’t tell without knowing the price of the output.
29. When Farmer Hoglund applies N pounds of fertilizer per acre, the marginal product of fertilizer is 1
−N/200 bushels of corn. If the price of corn is $4 per bushel and the price of fertilizer is $1.20 per
pound, then how many pounds of fertilizer per acre should Farmer Hoglund use in order to maximize
his profits?
a.
140
b.
280
c.
74
d.
288
e.
200
30. When Farmer Hoglund applies N pounds of fertilizer per acre, the marginal product of fertilizer is 1
−N/200 bushels of corn. If the price of corn is $1 per bushel and the price of fertilizer is $.20 per
pound, then how many pounds of fertilizer per acre should Farmer Hoglund use in order to maximize
his profits?
a.
84
b.
328
c.
320
d.
160
e.
200
31. When Farmer Hoglund applies N pounds of fertilizer per acre, the marginal product of fertilizer is 1
−N/100 bushels of corn. If the price of corn is $2 per bushel and the price of fertilizer is $40 per pound,
then how many pounds of fertilizer per acre should Farmer Hoglund use in order to maximize his
profits?
a.
320
b.
328
c.
160
d.
84
e.
200
32. If the short-run marginal costs of producing a good are $40 for the first 200 units and $50 for each
additional unit beyond 200, then in the short run, if the market price of output is $46, a
profit-maximizing firm will
a.
produce a level of output where marginal revenue equals marginal costs.
b.
produce as much output as possible since there are constant returns to scale.
c.
produce up to the point where average costs equal $46.
d.
not produce at all, since marginal costs are increasing.
e.
produce exactly 200 units.
33. If the short-run marginal costs of producing a good are $20 for the first 400 units and $30 for each
additional unit beyond 400, then in the short run, if the market price of output is $21, a
profit-maximizing firm will
a.
produce a level of output where marginal revenue equals marginal costs.
b.
not produce at all, since marginal costs are increasing.
c.
produce up to the point where average costs equal $21.
d.
produce as much output as possible since there are constant returns to scale.
e.
produce exactly 400 units.
34. If the short-run marginal costs of producing a good are $20 for the first 400 units and $30 for each
additional unit beyond 400, then in the short run, if the market price of output is $24, a
profit-maximizing firm will
a.
not produce at all, since marginal costs are increasing.
b.
produce as much output as possible since there are constant returns to scale.
c.
produce up to the point where average costs equal $24.
d.
produce a level of output where marginal revenue equals marginal costs.
e.
produce exactly 400 units.
35. Diesel Dan is a contract truck driver. While his revenue is $2.50 per mile driven, the faster he drives,
the greater the risk of a speeding ticket. The cost of driving his truck 1 hour at a speed of S miles per
hour is C(S) = eS − (60/5). To maximize his profit, Dan should drive
a.
60 miles per hour.
b.
60.92 miles per hour.
c.
64.58 miles per hour.
d.
72.63 miles per hour.
e.
79.56 miles per hour.
36. Diesel Dan is a contract truck driver. While his revenue is $2 per mile driven, the faster he drives, the
greater the risk of a speeding ticket. The cost of driving his truck 1 hour at a speed of S miles per hour
is C(S) = eS − (60/5). To maximize his profit, Dan should drive
a.
65.69 miles per hour.
b.
65 miles per hour.
c.
70.38 miles per hour.
d.
67.08 miles per hour.
e.
74.53 miles per hour.
37. Diesel Dan is a contract truck driver. While his revenue is $1.50 per mile driven, the faster he drives,
the greater the risk of a speeding ticket. The cost of driving his truck 1 hour at a speed of S miles per
hour is C(S) = eS − (60/4). To maximize his profit, Dan should drive
a.
60.41 miles per hour.
b.
61.62 miles per hour.
c.
67.17 miles per hour.
d.
60 miles per hour.
e.
72.71 miles per hour.
38. During the height of the pet rock craze in the 1970s, the price elasticity of demand was estimated to be
1.20. Since pet rocks have a marginal cost of zero, a profit-maximizing seller of pet rocks would
a.
increase prices.
b.
decrease prices.
c.
leave prices unchanged.
d.
need more-detailed market information before making any pricing changes.
e.
diversify into selling Karen Carpenter LPs.
39. During the height of the pet rock craze in the 1970s, the price elasticity of demand was estimated to be
1.20. Since pet rocks have a marginal cost of zero, a profit-maximizing seller of pet rocks would
a.
leave prices unchanged.
b.
need more-detailed market information before making any pricing changes.
c.
decrease prices.
d.
increase prices.
e.
diversify into selling Karen Carpenter LPs.
40. During the height of the pet rock craze in the 1970s, the price elasticity of demand was estimated to be
1.80. Since pet rocks have a marginal cost of zero, a profit-maximizing seller of pet rocks would
a.
leave prices unchanged.
b.
decrease prices.
c.
increase prices.
d.
need more-detailed market information before making any pricing changes.
e.
diversify into selling Karen Carpenter LPs.
41. Philip owns and operates a gas station. Philip works 40 hours a week managing the station but doesn’t
draw a salary. He could earn $700 a week doing the same work for Terrance. The station owes the
bank $100,000 and Philip has invested $100,000 of his own money. If Philip’s accounting profits are
$1,000 per week while the interest on his bank debt is $400 per week, the business’s economic profits
are
a.
$0 per week.
b.
−$100 per week.
c.
$600 per week.
d.
$300 per week.
e.
$1,000 per week.
42. Philip owns and operates a gas station. Philip works 40 hours a week managing the station but doesn’t
draw a salary. He could earn $800 a week doing the same work for Terrance. The station owes the
bank $100,000 and Philip has invested $100,000 of his own money. If Philip’s accounting profits are
$1,000 per week while the interest on his bank debt is $300 per week, the business’s economic profits
are
a.
−$100 per week.
b.
$200 per week.
c.
$0 per week.
d.
$700 per week.
e.
$1,000 per week.
43. Philip owns and operates a gas station. Philip works 40 hours a week managing the station but doesn’t
draw a salary. He could earn $600 a week doing the same work for Terrance. The station owes the
bank $100,000 and Philip has invested $100,000 of his own money. If Philip’s accounting profits are
$1,000 per week while the interest on his bank debt is $500 per week, the business’s economic profits
are
a.
$500 per week.
b.
−$100 per week.
c.
$400 per week.
d.
$0 per week.
e.
$1,000 per week.
PROBLEM
1. A competitive firm has a production function described as follows. “Weekly output is the square root
of the minimum of the number of units of capital and the number of units of labor employed per
week.” Suppose that in the short run this firm must use 16 units of capital but can vary its amount of
labor freely.
a. Write down a formula that describes the marginal product of labor in the short run as a function of
the amount of labor used. (Be careful at the boundaries.)
b. If the wage is w = $1 and the price of output is p = $4, how much labor will the firm demand in the
short run?
c. What if w = $1 and p = $10?
d. Write down an equation for the firm’s short-run demand for labor as a function of w and p.