978-0393123982 Chapter 28 Solution Manual Part 1

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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

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

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

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

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

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

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

MR, MC