Chapter 20 NAME
Profit Maximization
Introduction. A firm in a competitive industry cannot charge more than
the market price for its output. If it also must compete for its inputs, then
it has to pay the market price for inputs as well. Suppose that a profit-
maximizing competitive firm can vary the amount of only one factor and
that the marginal product of this factor decreases as its quantity increases.
Then the firm will maximize its profits by hiring enough of the variable
factor so that the value of its marginal product is equal to the wage. Even
if a firm uses several factors, only some of them may be variable in the
short run.
Example: A firm has the production function f(x1,x
2)=x1/2
1x1/2
2. Sup-
pose that this firm is using 16 units of factor 2 and is unable to vary this
quantity in the short run. In the short run, the only thing that is left for
the firm to choose is the amount of factor 1. Let the price of the firm’s
output be p, and let the price it pays per unit of factor 1 be w1.We
want to find the amount of x1that the firm will use and the amount of
In the long run, a firm is able to vary all of its inputs. Consider
the case of a competitive firm that uses two inputs. Then if the firm is
maximizing its profits, it must be that the value of the marginal product
of each of the two factors is equal to its wage. This gives two equations in
In the problems on the weak axiom of profit maximization, you are
asked to determine whether the observed behavior of firms is consistent
with profit-maximizing behavior. To do this you will need to plot some of
the firm’s isoprofit lines. An isoprofit line relates all of the input-output
combinations that yield the same amount of profit for some given input
252 PROFIT MAXIMIZATION (Ch. 20)
20.1 (0) The short-run production function of a competitive firm is
given by f(L)=6L2/3,whereLis the amount of labor it uses. (For
those who do not know calculus—if total output is aLb,whereaand b
are constants, and where Lis the amount of some factor of production,
then the marginal product of Lis given by the formula abLb−1.) The cost
per unit of labor is w= 6 and the price per unit of output is p=3.
(a) Plot a few points on the graph of this firm’s production function and
sketch the graph of the production function, using blue ink. Use black
ink to draw the isoprofit line that passes through the point (0,12), the
isoprofit line that passes through (0,8), and the isoprofit line that passes
through the point (0,4). What is the slope of each of the isoprofit lines?
(b) How many units of labor will the firm hire? 8. How much
0 8 12 16 20
Labour input
12
24
36
48
Output
424
8
4
Black lines
Blue curve
Squiggly line
13.3
Red line
NAME 253
(c) Suppose that the wage of labor falls to 4, and the price of output
remains at p. On the graph, use red ink to draw the new isoprofit line
for the firm that passes through its old choice of input and output. Will
the firm increase its output at the new price? Yes. Explain why,
Calculus 20.2 (0) A Los Angeles firm uses a single input to produce a recreational
commodity according to a production function f(x)=4
√x,wherexis
the number of units of input. The commodity sells for $100 per unit. The
input costs $50 per unit.
(a) Write down a function that states the firm’s profit as a function of
(b) What is the profit-maximizing amount of input? 16. of output?
(c) Suppose that the firm is taxed $20 per unit of its output and the price
of its input is subsidized by $10. What is its new input level? 16.
firm’s profit as a function of its input and solve for the profit-maximizing
amount of input.)
(d) Suppose that instead of these taxes and subsidies, the firm is taxed
at 50% of its profits. Write down its after-tax profits as a function of the
20.3 (0) Brother Jed takes heathens and reforms them into righteous
individuals. There are two inputs needed in this process: heathens (who
are widely available) and preaching. The production function has the
following form: rp= min{h, p},whererpis the number of righteous
254 PROFIT MAXIMIZATION (Ch. 20)
persons produced, his the number of heathens who attend Jed’s sermons,
and pis the number of hours of preaching. For every person converted,
Jed receives a payment of sfrom the grateful convert. Sad to say, heathens
do not flock to Jed’s sermons of their own accord. Jed must offer heathens
apaymentofwto attract them to his sermons. Suppose the amount of
preaching is fixed at ¯pand that Jed is a profit-maximizing prophet.
(a) If h<¯p, what is the marginal product of heathens? 1. What
(b) If h>¯p, what is the marginal product of heathens? 0. What
(c) Sketch the shape of this production function in the graph below. Label
the axes, and indicate the amount of the input where h=¯p.
r
h
p
p
_
20.4 (0) Allie’s Apples, Inc. purchases apples in bulk and sells two prod-
ucts, boxes of apples and jugs of cider. Allie’s has capacity limitations of
three kinds: warehouse space, crating facilities, and pressing facilities. A
box of apples requires 6 units of warehouse space, 2 units of crating facili-
ties, and no pressing facilities. A jug of cider requires 3 units of warehouse
space, 2 units of crating facilities, and 1 unit of pressing facilities. The
total amounts available each day are: 1,200 units of warehouse space, 600
units of crating facilities, and 250 units of pressing facilities.
(a) If the only capacity limitations were on warehouse facilities, and if all
warehouse space were used for the production of apples, how many boxes
could be produced each day if, instead, all warehouse space were used in
NAME 255
the production of cider and there were no other capacity constraints?
(b) Following the same reasoning, draw a red line to represent the con-
straints on output to limitations on crating capacity. How many boxes of
apples could Allie produce if he only had to worry about crating capacity?
(c) Finally draw a black line to represent constraints on output combina-
tions due to limitations on pressing facilities. How many boxes of apples
could Allie produce if he only had to worry about the pressing capacity
(d) Now shade the area that represents feasible combinations of daily
production of apples and cider for Allie’s Apples.
0 300 400 500
100
200
300
400
500
600
100
Cider
200
Apples
600
Blue line
Black revenue line
Red line
Black line
(e) Allie’s can sell apples for $5 per box of apples and cider for $2 per
jug. Draw a black line to show the combinations of sales of apples and
cider that would generate a revenue of $1,000 per day. At the profit-
256 PROFIT MAXIMIZATION (Ch. 20)
20.5 (0) A profit-maximizing firm produces one output, y, and uses one
input, x, to produce it. The price per unit of the factor is denoted by
wand the price of the output is denoted by p. You observe the firm’s
behavior over three periods and find the following:
Period y x w p
1 1 1 1 1
22.5 3.5 1
3 4 8 .25 1
(a) Write an equation that gives the firm’s profits, π, as a function of the
amount of input xit uses, the amount of output yit produces, the per-unit
cost of the input w, and the price of output p.π=py −wx.
(b) In the diagram below, draw an isoprofit line for each of the three
periods, showing combinations of input and output that would yield the
same profits that period as the combination actually chosen. What are
the equations for these three lines? y=x,y=1+.5x,
y=2+.25x.Using the theory of revealed profitability, shade in
the region on the graph that represents input-output combinations that
could be feasible as far as one can tell from the evidence that is available.
06810
2
4
6
8
10
12
2
Output
4
Input
12
Period 3
Period 2
Period 1
NAME 257
20.6 (0) T-bone Pickens is a corporate raider. This means that he looks
for companies that are not maximizing profits, buys them, and then tries
to operate them at higher profits. T-bone is examining the financial
records of two refineries that he might buy, the Shill Oil Company and
the Golf Oil Company. Each of these companies buys oil and produces
gasoline. During the time period covered by these records, the price of
gasoline fluctuated significantly, while the cost of oil remained constant
at $10 a barrel. For simplicity, we assume that oil is the only input to
gasoline production.
Shill Oil produced 1 million barrels of gasoline using 1 million barrels
of oil when the price of gasoline was $10 a barrel. When the price of
gasoline was $20 a barrel, Shill produced 3 million barrels of gasoline
using 4 million barrels of oil. Finally, when the price of gasoline was $40
a barrel, Shill used 10 million barrels of oil to produce 5 million barrels
of gasoline.
Golf Oil (which is managed by Martin E. Lunch III) did exactly the
same when the price of gasoline was $10 and $20, but when the price
of gasoline hit $40, Golf produced 3.5 million barrels of gasoline using 8
million barrels of oil.
(a) Using black ink, plot Shill Oil’s isoprofit lines and choices for the three
different periods. Label them 10, 20, and 40. Using red ink draw Golf
Oil’s isoprofit line and production choice. Label it with a 40 in red ink.
06810
2
4
6
8
10
12
2
Million barrels of gasoline
4
Million barrels of oil
12
10
20
40
Red 40
258 PROFIT MAXIMIZATION (Ch. 20)
(b) How much profits could Golf Oil have made when the price of gasoline
was $40 a barrel if it had chosen to produce the same amount that it did
(c) Is there any evidence that Shill Oil is not maximizing profits? Explain.
(d) Is there any evidence that Golf Oil is not maximizing profits? Explain.
20.7 (0) After carefully studying Shill Oil, T-bone Pickens decides that
it has probably been maximizing its profits. But he still is very interested
in buying Shill Oil. He wants to use the gasoline they produce to fuel his
delivery fleet for his chicken farms, Capon Truckin’. In order to do this
Shill Oil would have to be able to produce 5 million barrels of gasoline
from 8 million barrels of oil. Mark this point on your graph. Assuming
that Shill always maximizes profits, would it be technologically feasible
for it to produce this input-output combination? Why or why not?
20.8 (0) Suppose that firms operate in a competitive market, attempt to
maximize profits, and only use one factor of production. Then we know
that for any changes in the input and output price, the input choice and
the output choice must obey the Weak Axiom of Profit Maximization,
ΔpΔy−ΔwΔx≥0.
Which of the following propositions can be proven by the Weak Ax-
iom of Profit Maximizing Behavior (WAPM)? Respond yes or no, and
give a short argument.