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Chapter 28 NAME
Oligopoly
Introduction. In this chapter you will solve problems for firm and indus-
try outcomes when the firms engage in Cournot competition, Stackelberg
competition, and other sorts of oligopoly behavior. In Cournot competi-
tion, each firm chooses its own output to maximize its profits given the
output that it expects the other firm to produce. The industry price de-
pends on the industry output, say, qA+qB, where A and B are the firms.
To maximize profits, firm A sets its marginal revenue (which depends on
the output of firm A and the expected output of firm B since the expected
industry price depends on the sum of these outputs) equal to its marginal
cost. Solving this equation for firm A’s output as a function of firm B’s
expected output gives you one reaction function; analogous steps give you
firm B’s reaction function. Solve these two equations simultaneously to
get the Cournot equilibrium outputs of the two firms.
Example: In Heifer’s Breath, Wisconsin, there are two bakers, Anderson
and Carlson. Anderson’s bread tastes just like Carlson’s—nobody can
Let us find Anderson’s Cournot reaction function. If Carlson bakes
qCloaves, then if Anderson bakes qAloaves, total output will be qA+
Therefore if Carlson is going to bake qCunits, then Anderson will choose
qAto maximize 6qA−.01q2
A−.01qCqA−qA. This expression is maximized
when 6 −.02qA−.01qC= 1. (You can find this out either by setting
A’s marginal revenue equal to his marginal cost or directly by setting
We can find Carlson’s reaction function in the same way. If Carlson
knows that Anderson is going to produce qAunits, then Carlson’s profits
will be p(qA+qC)−2qC=(6−.01qA−.01qC)qC−2qC=6qC−.01qAqC−
.01q2
336 OLIGOPOLY (Ch. 28)
Let us denote the Cournot equilibrium quantities by ¯qAand ¯qC.The
Cournot equilibrium conditions are that ¯qA=RA(¯qC)and¯qC=RC(¯qA).
Solving these two equations in two unknowns we find that ¯qA= 200 and
¯qC= 100. Now we can also solve for the Cournot equilibrium price and for
the profits of each baker. The Cournot equilibrium price is 6 −.01(200 +
100) = $3. Then in Cournot equilibrium, Anderson makes a profit of $2
on each of 200 loaves and Carlson makes $1 on each of 100 loaves.
In Stackelberg competition, the follower’s profit-maximizing output
choice depends on the amount of output that he expects the leader to
produce. His reaction function, RF(qL), is constructed in the same way
as for a Cournot competitor. The leader knows the reaction function of
the follower and gets to choose her own output, qL, first. So the leader
knows that the industry price depends on the sum of her own output and
the follower’s output, that is, on qL+RF(qL). Since the industry price
can be expressed as a function of qLonly, so can the leader’s marginal
revenue. So once you get the follower’s reaction function and substitute it
into the inverse demand function, you can write down an expression that
depends on just qLand that says marginal revenue equals marginal cost
for the leader. You can solve this expression for the leader’s Stackelberg
output and plug in to the follower’s reaction function to get the follower’s
Stackelberg output.
Example: Suppose that one of the bakers of Heifer’s Breath plays the role
of Stackelberg leader. Perhaps this is because Carlson always gets up an
hour earlier than Anderson and has his bread in the oven before Anderson
gets started. If Anderson always finds out how much bread Carlson has
in his oven and if Carlson knows that Anderson knows this, then Carlson
can act like a Stackelberg leader. Carlson knows that Anderson’s reaction
function is RA(qC) = 250−.5qc. Therefore Carlson knows that if he bakes
qCloaves of bread, then the total amount of bread that will be baked in
28.1 (0) Carl and Simon are two rival pumpkin growers who sell their
pumpkins at the Farmers’ Market in Lake Witchisit, Minnesota. They are
the only sellers of pumpkins at the market, where the demand function
for pumpkins is q=3,200 −1,600p. The total number of pumpkins sold
at the market is q=qC+qS,whereqCis the number that Carl sells
NAME 337
and qSis the number that Simon sells. The cost of producing pumpkins
for either farmer is $.50 per pumpkin no matter how many pumpkins he
produces.
(a) The inverse demand function for pumpkins at the Farmers’ Market is
(b) Every spring, each of the farmers decides how many pumpkins to
grow. They both know the local demand function and they each know
how many pumpkins were sold by the other farmer last year. In fact,
each farmer assumes that the other farmer will sell the same number this
year as he sold last year. So, for example, if Simon sold 400 pumpkins
last year, Carl believes that Simon will sell 400 pumpkins again this year.
If Simon sold 400 pumpkins last year, what does Carl think the price of
pumpkins will be if Carl sells 1,200 pumpkins this year? 1. If
Simon sold qt−1
Spumpkins in year t−1, then in the spring of year t, Carl
thinks that if he, Carl, sells qt
Cpumpkins this year, the price of pumpkins
S+qt
(c) If Simon sold 400 pumpkins last year, Carl believes that if he sells
qt
Cpumpkins this year then the inverse demand function that he faces is
p=2−400/1,600 −qt
C/1,600 = 1.75 −qt
C/1,600. Therefore if Simon
sold 400 pumpkins last year, Carl’s marginal revenue this year will be
1.75 −qt
C/800. More generally, if Simon sold qt−1
Spumpkins last year,
then Carl believes that if he, himself, sells qt
Cpumpkins this year, his
marginal revenue this year will be 2−qt−1
(d) Carl believes that Simon will never change the amount of pumpkins
that he produces from the amount qt−1
Sthat he sold last year. Therefore
Carl plants enough pumpkins this year so that he can sell the amount
that maximizes his profits this year. To maximize this profit, he chooses
the output this year that sets his marginal revenue this year equal to
his marginal cost. This means that to find Carl’s output this year when
Simon’s output last year was qt−1
S, Carl solves the following equation.
2−qt−1
(e) Carl’s Cournot reaction function, Rt
C(qt−1
S), is a function that tells us
what Carl’s profit-maximizing output this year would be as a function of
Simon’s output last year. Use the equation you wrote in the last answer to
find Carl’s reaction function, Rt
C(qt−1
S/2.(Hint:
This is a linear expression of the form a−bqt−1
S. You have to find the
constants aand b.)
338 OLIGOPOLY (Ch. 28)
(f) Suppose that Simon makes his decisions in the same way that Carl
does. Notice that the problem is completely symmetric in the roles played
by Carl and Simon. Therefore without even calculating it, we can guess
that Simon’s reaction function is Rt
S(qt−1
C/2.(Of
course, if you don’t like to guess, you could work this out by following
similar steps to the ones you used to find Carl’s reaction function.)
(g) Suppose that in year 1, Carl produced 200 pumpkins and Simon pro-
duced 1,000 pumpkins. In year 2, how many would Carl produce?
700. How many would Simon produce? 1,100. In year 3, how
850. Use a calculator or pen and paper to work out several more
terms in this series. To what level of output does Carl’s output appear
(h) Write down two simultaneous equations that could be solved to find
outputs qSand qCsuch that, if Carl is producing qCand Simon is produc-
ing qS, then they will both want to produce the same amount in the next
(i) Solve the two equations you wrote down in the last part for an equi-
librium output for each farmer. Each farmer, in Cournot equilibrium,
produces 800 units of output. The total amount of pumpkins brought
to the Farmers’ Market in Lake Witchisit is 1,600. The price of
pumpkins in that market is $1. How much profit does each farmer
28.2 (0) Suppose that the pumpkin market in Lake Witchisit is as
we described it in the last problem except for one detail. Every spring,
the snow thaws off of Carl’s pumpkin field a week before it thaws off of
Simon’s. Therefore Carl can plant his pumpkins one week earlier than
Simon can. Now Simon lives just down the road from Carl, and he can
tell by looking at Carl’s fields how many pumpkins Carl planted and how
many Carl will harvest in the fall. (Suppose also that Carl will sell every
pumpkin that he produces.) Therefore instead of assuming that Carl will
sell the same amount of pumpkins that he did last year, Simon sees how
many Carl is actually going to sell this year. Simon has this information
before he makes his own decision about how many to plant.
NAME 339
(a) If Carl plants enough pumpkins to yield qt
Cthis year, then Simon
knows that the profit-maximizing amount to produce this year is qt
S=
Hint: Remember the reaction functions you found in the last problem.
1,200 −qt
C/2.
(b) When Carl plants his pumpkins, he understands how Simon will make
his decision. Therefore Carl knows that the amount that Simon will
produce this year will be determined by the amount that Carl produces.
In particular, if Carl’s output is qt
C, then Simon will produce and sell
(c) In the last part of the problem, you found how the price of pumpkins
this year in the Farmers’ Market is related to the number of pumpkins
that Carl produces this year. Now write an expression for Carl’s total
revenue in year tas a function of his own output, qt
C.1.25qt
C−
(qt
(d) Find the profit-maximizing output for Carl. 1,200. Find the
profit-maximizing output for Simon. 600. Find the equilibrium price
of pumpkins in the Lake Witchisit Farmers’ Market. $7/8. How
(e) If he wanted to, it would be possible for Carl to delay his planting
until the same time that Simon planted so that neither of them would
know the other’s plans for this year when he planted. Would it be in
Carl’s interest to do this? Explain. (Hint: What are Carl’s profits in the
equilibrium above? How do they compare with his profits in Cournot equi-
340 OLIGOPOLY (Ch. 28)
28.3 (0) Suppose that Carl and Simon sign a marketing agreement.
They decide to determine their total output jointly and to each produce
the same number of pumpkins. To maximize their joint profits, how many
pumpkins should they produce in toto? 1,200. How much does each
28.4 (0) The inverse market demand curve for bean sprouts is given by
P(Y) = 100 −2Y, and the total cost function for any firm in the industry
is given by TC(y)=4y.
(a) The marginal cost for any firm in the industry is equal to $4. The
(b) If the bean-sprout industry were perfectly competitive, the industry
(c) Suppose that two Cournot firms operated in the market. The reaction
the example in your textbook, the marginal cost is not zero here.) The
were operating at the Cournot equilibrium point, industry output would
be 32 , each firm would produce 16 ,andthemarketprice
(d) For the Cournot case, draw the two reaction curves and indicate the
equilibrium point on the graph below.
NAME 341
0 6 12 18 24
6
12
18
y1
y2
24
e
Firm 1's reaction
function
Firm 2's
reaction
function
(e) If the two firms decided to collude, industry output would be 24
and the market price would equal $52.
(f) Suppose both of the colluding firms are producing equal amounts of
output. If one of the colluding firms assumes that the other firm would
not react to a change in industry output, what would happen to a firm’s
(g) Suppose one firm acts as a Stackleberg leader and the other firm
behaves as a follower. The maximization problem for the leader can be
Solving this problem results in the leader producing an output of
28.5 (0) Grinch is the sole owner of a mineral water spring that costlessly
burbles forth as much mineral water as Grinch cares to bottle. It costs
Grinch $2 per gallon to bottle this water. The inverse demand curve for
Grinch’s mineral water is p= $20 −.20q,wherepis the price per gallon
and qis the number of gallons sold.
342 OLIGOPOLY (Ch. 28)
(a) Write down an expression for profits as a function of q:Π(q)=
(b) What price does Grinch get per gallon of mineral water if he produces
(c) Suppose, now, that Grinch’s neighbor, Grubb finds a mineral spring
that produces mineral water that is just as good as Grinch’s water, but
that it costs Grubb $6 a bottle to get his water out of the ground and
bottle it. Total market demand for mineral water remains as before.
Suppose that Grinch and Grubb each believe that the other’s quantity
decision is independent of his own. What is the Cournot equilibrium out-
28.6 (1) Albatross Airlines has a monopoly on air travel between Peoria
and Dubuque. If Albatross makes one trip in each direction per day, the
demand schedule for round trips is q= 160 −2p,whereqis the number of
passengers per day. (Assume that nobody makes one-way trips.) There
is an “overhead” fixed cost of $2,000 per day that is necessary to fly the
airplane regardless of the number of passengers. In addition, there is a
marginal cost of $10 per passenger. Thus, total daily costs are $2,000+10q
if the plane flies at all.
(a) On the graph below, sketch and label the marginal revenue curve, and
the average and marginal cost curves.
0204060
80
20
40
60
Q
MR, MC
80
mc
mr
ac
