In this chapter you will solve problems for firm and industry outcomes
when the firms engage in Cournot competition, Stackelberg competition,
and other sorts of oligopoly behavior. In Cournot competition, each firm
chooses its own output to maximize its profits given the output that it
expects the other firm to produce. The industry price depends 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.
In Heifer’s Breath, Wisconsin, there are two bakers, Anderson and Carlson.
Anderson’s bread tastes just like Carlson’s—nobody can tell the
difference. Anderson has constant marginal costs of $1 per loaf of bread.
Carlson has constant marginal costs of $2 per loaf. Fixed costs are zero for
both of them. The inverse demand function for bread in Heifer’s Breath
is p(q) = 6 − .01q, where q is the total number of loaves sold per day.
Let us find Anderson’s Cournot reaction function. If Carlson bakes
qC loaves, then if Anderson bakes qA loaves, total output will be qA +
qC and price will be 6 − .01(qA + qC). For Anderson, the total cost of
producing qA units of bread is just qA, so his profits are
pqA − qA = (6 − .01qA − .01qC)qA − qA
= 6qA − .01q2
A − .01qCqA − qA.
Therefore if Carlson is going to bake qC units, then Anderson will choose
qA to 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
the derivative of profits with respect to qA equal to zero.) Anderson’s
reaction function, RA(qC) tells us Anderson’s best output if he knows
that Carlson is going to bake qC. We solve from the previous equation to
find RA(qC) = (5 − .01qC)/.02 = 250 − .5qC.
We can find Carlson’s reaction function in the same way. If Carlson
knows that Anderson is going to produce qA units, then Carlson’s profits
will be p(qA +qC)−2qC = (6−.01qA −.01qC)qC −2qC = 6qC −.01qAqC −
.01q2
C −2qC. Carlson’s profits will be maximized if he chooses qC to satisfy
the equation 6−.01qA −.02qC = 2. Therefore Carlson’s reaction function
is RC(qA) = (4 − .01qA)/.02 = 200 − .5qA.
Let us denote the Cournot equilibrium quantities by ¯qA and ¯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 qL only, 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 qL and 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.
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 qC loaves of bread, then the total amount of bread that will
be baked in Heifer’s Breath will be qC + RA(qC) = qC + 250 − .5qC =
250 + .5qC. Since Carlson’s production decision determines total production
and hence the price of bread, we can write Carlson’s profit simply
as a function of his own output. Carlson will choose the quantity
that maximizes this profit. If Carlson bakes qC loaves, the price will be
p = 6 − .01(250 + .5qC) = 3.5 − .005qC. Then Carlson’s profits will be
pqC − 2qC = (3.5 − .005qC)qC − 2qC = 1.5qC − .005q2
C. His profits are
maximized when qC = 150. (Find this either by setting marginal revenue
equal to marginal cost or directly by setting the derivative of profits to
zero and solving for qC.) If Carlson produces 150 loaves, then Anderson
will produce 250 − .5 × 150 = 175 loaves. The price of bread will be
6 − .01(175 + 150) = 2.75. Carlson will now make $.75 per loaf on each
of 150 loaves and Anderson will make $1.75 on each of 175 loaves.
27.1 (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, where qC is the number that Carl sells
and qS is 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
p = a − b(qC + qS), where a = and b = The
marginal cost of producing a pumpkin for either farmer 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? If
Simon sold qt−1
S pumpkins in year t − 1, then in the spring of year t, Carl
thinks that if he, Carl, sells qt
C pumpkins this year, the price of pumpkins
this year will be .
(c) If Simon sold 400 pumpkins last year, Carl believes that if he sells
qt
C pumpkins 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
S pumpkins last year,
then Carl believes that if he, himself, sells qt
C pumpkins this year, his
marginal revenue this year will be .
(d) Carl believes that Simon will never change the amount of pumpkins
that he produces from the amount qt−1
S that 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.
.
(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 ) = (Hint:
This is a linear expression of the form a − bqt−1
S . You have to find the
constants a and b.)
(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 ) = (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 produced
1,000 pumpkins. In year 2, how many would Carl produce?
How many would Simon produce? In year 3,
how many would Carl produce? How many would Simon
produce? 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 to be converging? How about Simon’s? .
(h) Write down two simultaneous equations that could be solved to find
outputs qS and qC such that, if Carl is producing qC and Simon is producing
qS, then they will both want to produce the same amount in the
next period. (Hint: Use the reaction functions.)
.
(i) Solve the two equations you wrote down in the last part for an equilibrium
output for each farmer. Each farmer, in Cournot equilibrium, produces
units of output. The total amount of pumpkins brought to
the Farmers’ Market in Lake Witchisit is The price of pumpkins
in that market is How much profit does each farmer make?
.
27.2 (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.
(a) If Carl plants enough pumpkins to yield qt
C this 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.
.
(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
and the total output of the two producers will be
Therefore Carl knows that if his own output is qC, the
price of pumpkins in the market will be .
(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 t as a function of his own output, qt
C.
Write an expression for Carl’s marginal revenue in year t as a
function of qt
C. .
(d) Find the profit-maximizing output for Carl. Find the
profit-maximizing output for Simon. Find the equilibrium
price of pumpkins in the Lake Witchisit Farmers’ Market.
How much profit does Carl make? How much profit does
Simon make? An equilibrium of the type we discuss here is
known as a equilibrium.
(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 equilibrium?)
.
27.3 (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? How much does each
one of them produce? How much profit does each one of them
make? .
27.4 (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 The
change in price for a one-unit increase in output is equal to .
(b) If the bean-sprout industry were perfectly competitive, the industry
output would be , and the industry price would be .
(c) Suppose that two Cournot firms operated in the market. The reaction
function for Firm 1 would be (Reminder: Unlike
the example in your textbook, the marginal cost is not zero here.) The
reaction function of Firm 2 would be If the firms were
operating at the Cournot equilibrium point, industry output would be
, each firm would produce , and the market price
would be .
(d) For the Cournot case, draw the two reaction curves and indicate the
equilibrium point on the graph below.
y2
24
18
12
6
0 6 12 18 24
y1
(e) If the two firms decided to collude, industry output would be
and the market price would equal .
(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
own profits if it increased its output by one unit?
(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
written as .
Solving this problem results in the leader producing an output of
and the follower producing This implies an industry
output of and price of .
27.5 (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, where p is the price per gallon
and q is the number of gallons sold.
(a) Write down an expression for profits as a function of q: Π(q) =
Find the profit-maximizing choice of q for Grinch.
.
(b) What price does Grinch get per gallon of mineral water if he produces
the profit-maximizing quantity? How much profit does he
make? .
(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 output
for Grubb? What is the price in the Cournot equilibrium?
.
27.6 (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, where q is 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.
AC, MR, MC
80
60
40
20
0 20 40 60 80
q
(b) Calculate the profit-maximizing price and quantity and total daily
profits for Albatross Airlines. p = , q = , π =
.
(c) If the interest rate is 10% per year, how much would someone be willing
to pay to own Albatross Airlines’s monopoly on the Dubuque-Peoria
route. (Assuming that demand and cost conditions remain unchanged
forever.) .
(d) If another firm with the same costs as Albatross Airlines were to enter
the Dubuque-Peoria market and if the industry then became a Cournot
duopoly, would the new entrant make a profit?
.
(e) Suppose that the throbbing night life in Peoria and Dubuque becomes
widely known and in consequence the population of both places doubles.
As a result, the demand for airplane trips between the two places doubles
to become q = 320−4p. Suppose that the original airplane had a capacity
of 80 passengers. If AA must stick with this single plane and if no other
airline enters the market, what price should it charge to maximize its
output and how much profit would it make? p = , π = .
(f) Let us assume that the overhead costs per plane are constant regardless
of the number of planes. If AA added a second plane with the same costs
and capacity as the first plane, what price would it charge?
How many tickets would it sell? How much would its profits
be? If AA could prevent entry by another competitor, would
it choose to add a second plane? .
(g) Suppose that AA stuck with one plane and another firm entered the
market with a plane of its own. If the second firm has the same cost
function as the first and if the two firms act as Cournot oligopolists, what
will be the price, , quantities, , and profits? .
27.7 (0) Alex and Anna are the only sellers of kangaroos in Sydney,
Australia. Anna chooses her profit-maximizing number of kangaroos to
sell, q1, based on the number of kangaroos that she expects Alex to sell.
Alex knows how Anna will react and chooses the number of kangaroos that
she herself will sell, q2, after taking this information into account. The
inverse demand function for kangaroos is P(q1 + q2) = 2, 000 − 2(q1 + q2).
It costs $400 to raise a kangaroo to sell.
(a) Alex and Anna are Stackelberg competitors. is the leader
and is the follower.
(b) If Anna expects Alex to sell q2 kangaroos, what will her own marginal
revenue be if she herself sells q1 kangaroos? .
(c) What is Anna’s reaction function, R(q2)? .
(d) Now if Alex sells q2 kangaroos, what is the total number of kangaroos
that will be sold? What will be the market price as a
function of q2 only? .
(e) What is Alex’s marginal revenue as a function of q2 only?
How many kangaroos will Alex sell? How
many kangaroos will Anna sell? What will the industry price
be? .
27.8 (0) Consider an industry with the following structure. There are
50 firms that behave in a competitive manner and have identical cost
functions given by c(y) = y2
/2. There is one monopolist that has 0
marginal costs. The demand curve for the product is given by
D(p) = 1, 000 − 50p.
(a) What is the supply curve of one of the competitive firms?
The total supply from the competitive sector at price p is S(p) =
.
(b) If the monopolist sets a price p, the amount that it can sell is Dm(p) =
.
(c) The monopolist’s profit-maximizing output is ym = What
is the monopolist’s profit-maximizing price? .
(d) How much output will the competitive sector provide at this price?
What will be the total amount of output sold in this
industry? .
27.9 (0) Consider a market with one large firm and many small firms.
The supply curve of the small firms taken together is
S(p) = 100 + p.
The demand curve for the product is
D(p) = 200 − p.
The cost function for the one large firm is
c(y) = 25y.
(a) Suppose that the large firm is forced to operate at a zero level of
output. What will be the equilibrium price? What will be the
equilibrium quantity? .
(b) Suppose now that the large firm attempts to exploit its market power
and set a profit-maximizing price. In order to model this we assume that
customers always go first to the competitive firms and buy as much as they
are able to and then go to the large firm. In this situation, the equilibrium
price will be The quantity supplied by the large firm will be
and the equilibrium quantity supplied by the competitive firms
will be .
(c) What will be the large firm’s profits? .
(d) Finally suppose that the large firm could force the competitive firms
out of the business and behave as a real monopolist. What will be the
equilibrium price? What will be the equilibrium quantity?
What will be the large firm’s profits? .
27.10 (2) In a remote area of the American Midwest before the railroads
arrived, cast iron cookstoves were much desired, but people lived far apart,
roads were poor, and heavy stoves were expensive to transport. Stoves
could be shipped by river boat to the town of Bouncing Springs, Missouri.
Ben Kinmore was the only stove dealer in Bouncing Springs. He could
buy as many stoves as he wished for $20 each, delivered to his store. Ben’s
only customers were farmers who lived along a road that ran east and west
through town. There were no other stove dealers along the road in either
direction. No farmers lived in Bouncing Springs, but along the road, in
either direction, there was one farm every mile. The cost of hauling a
stove was $1 per mile. The owners of every farm had a reservation price
of $120 for a cast iron cookstove. That is, any of them would be willing to
pay up to $120 to have a stove rather than to not have one. Nobody had
use for more than one stove. Ben Kinmore charged a base price of $p for
stoves and added to the price the cost of delivery. For example, if the base
price of stoves was $40 and you lived 45 miles west of Bouncing Springs,
you would have to pay $85 to get a stove, $40 base price plus a hauling
charge of $45. Since the reservation price of every farmer was $120, it
follows that if the base price were $40, any farmer who lived within 80
miles of Bouncing Springs would be willing to pay $40 plus the price of
delivery to have a cookstove. Therefore at a base price of $40, Ben could
sell 80 cookstoves to the farmers living west of him. Similarly, if his base
price is $40, he could sell 80 cookstoves to the farmers living within 80
miles to his east, for a total of 160 cookstoves.
(a) If Ben set a base price of $p for cookstoves where p < 120, and if he
charged $1 a mile for delivering them, what would be the total number of
cookstoves he could sell? (Remember to count the ones
he could sell to his east as well as to his west.) Assume that Ben has no
other costs than buying the stoves and delivering them. Then Ben would
make a profit of p−20 per stove. Write Ben’s total profit as a function of
the base price, $p, that he charges: .
(b) Ben’s profit-maximizing base price is (Hint: You just
wrote profits as a function of prices. Now differentiate this expression
for profits with respect to p.) Ben’s most distant customer would be
located at a distance of miles from him. Ben would sell
cookstoves and make a total profit of .
(c) Suppose that instead of setting a single base price and making all
buyers pay for the cost of transportation, Ben offers free delivery of
cookstoves. He sets a price $p and promises to deliver for free to any
farmer who lives within p − 20 miles of him. (He won’t deliver to anyone
who lives further than that, because it then costs him more than
$p to buy a stove and deliver it.) If he is going to price in this way,
how high should he set p? How many cookstoves would
Ben deliver? How much would his total revenue be?
How much would his total costs be, including the cost of deliveries
and the cost of buying the stoves? (Hint: For any
n, the sum of the series 1 + 2 + . . . + n is equal to n(n + 1)/2.) How
much profit would he make? Can you explain why it is
more profitable for Ben to use this pricing scheme where he pays the cost
of delivery himself rather than the scheme where the farmers pay for
their own deliveries?
.
27.11 (2) Perhaps you wondered what Ben Kinmore, who lives off in
the woods quietly collecting his monopoly profits, is doing in this chapter
on oligopoly. Well, unfortunately for Ben, before he got around to selling
any stoves, the railroad built a track to the town of Deep Furrow, just 40
miles down the road, west of Bouncing Springs. The storekeeper in Deep
Furrow, Huey Sunshine, was also able to get cookstoves delivered by train
to his store for $20 each. Huey and Ben were the only stove dealers on
the road. Let us concentrate our attention on how they would compete
for the customers who lived between them. We can do this, because Ben
can charge different base prices for the cookstoves he ships east and the
cookstoves he ships west. So can Huey.
Suppose that Ben sets a base price, pB, for stoves he sends west
and adds a charge of $1 per mile for delivery. Suppose that Huey sets
a base price, pH, for stoves he sends east and adds a charge of $1 per
mile for delivery. Farmers who live between Ben and Huey would buy
from the seller who is willing to deliver most cheaply to them (so long as
the delivered price does not exceed $120). If Ben’s base price is pB and
Huey’s base price is pH, somebody who lives x miles west of Ben would
have to pay a total of pB + x to have a stove delivered from Ben and
pH + (40 − x) to have a stove delivered by Huey.
(a) If Ben’s base price is pB and Huey’s is pH, write down an equation
that could be solved for the distance x∗
to the west of Bouncing Springs
that Ben’s market extends. If Ben’s base price
is pB and Huey’s is pH, then Ben will sell cookstoves
and Huey will sell cookstoves.
(b) Recalling that Ben makes a profit of pB − 20 on every cookstove that
he sells, Ben’s profits can be expressed as the following function of pB
and pH. .
(c) If Ben thinks that Huey’s price will stay at pH, no matter what price
Ben chooses, what choice of pB will maximize Ben’s profits?
(Hint: Set the derivative of Ben’s profits with respect to
his price equal to zero.) Suppose that Huey thinks that Ben’s price will
stay at pB, no matter what price Huey chooses, what choice of pH will
maximize Huey’s profits? (Hint: Use the symmetry
of the problem and the answer to the last question.)
(d) Can you find a base price for Ben and a base price for Huey such that
each is a profit-maximizing choice given what the other guy is doing?
(Hint: Find prices pB and pH that simultaneously solve the last two
equations.) How many cookstoves does Ben sell to
farmers living west of him? How much profit does he make on
these sales? .
(e) Suppose that Ben and Huey decided to compete for the customers
who live between them by price discriminating. Suppose that Ben offers
to deliver a stove to a farmer who lives x miles west of him for a price
equal to the maximum of Ben’s total cost of delivering a stove to that
farmer and Huey’s total cost of delivering to the same farmer less 1 penny.
Suppose that Huey offers to deliver a stove to a farmer who lives x miles
west of Ben for a price equal to the maximum of Huey’s own total cost of
delivering to this farmer and Ben’s total cost of delivering to him less a
penny. For example, if a farmer lives 10 miles west of Ben, Ben’s total cost
of delivering to him is $30, $20 to get the stove and $10 for hauling it 10
miles west. Huey’s total cost of delivering it to him is $50, $20 to get the
stove and $30 to haul it 30 miles east. Ben will charge the maximum of
his own cost, which is $30, and Huey’s cost less a penny, which is $49.99.
The maximum of these two numbers is . Huey will charge the
maximum of his own total cost of delivering to this farmer, which is $50,
and Ben’s cost less a penny, which is $29.99. Therefore Huey will charge
to deliver to this farmer. This farmer will buy from
whose price to him is cheaper by one penny. When the two merchants
have this pricing policy, all farmers who live within miles of
Ben will buy from Ben and all farmers who live within miles
of Huey will buy from Huey. A farmer who lives x miles west of Ben
and buys from Ben must pay dollars to have a cookstove
delivered to him. A farmer who lives x miles east of Huey and buys from
Huey must pay for delivery of a stove. On the graph
below, use blue ink to graph the cost to Ben of delivering to a farmer who
lives x miles west of him. Use red ink to graph the total cost to Huey
of delivering a cookstove to a farmer who lives x miles west of Ben. Use
pencil to mark the lowest price available to a farmer as a function of how
far west he lives from Ben.
Dollars
80
60
40
20
0 10 20 30 40
Miles west of Ben
(f) With the pricing policies you just graphed, which farmers get stoves
delivered most cheaply, those who live closest to the merchants or those
who live midway between them?
On the graph you made, shade in the area representing each merchant’s
profits. How much profits does each merchant make? If Ben
and Huey are pricing in this way, is there any way for either of them to
increase his profits by changing the price he charges to some farmers?
.
In this introduction we offer two examples of two-person games. The first
game has a dominant strategy equilibrium. The second game is a zerosum
game that has a Nash equilibrium in pure strategies that is not a
dominant strategy equilibrium.
Albert and Victoria are roommates. Each of them prefers a clean room
to a dirty room, but neither likes housecleaning. If both clean the room,
they each get a payoff of 5. If one cleans and the other doesn’t clean,
the person who does the cleaning has a utility of 2, and the person who
doesn’t clean has a utility of 6. If neither cleans, the room stays a mess
and each has a utility of 3. The payoffs from the strategies “Clean” and
“Don’t Clean” are shown in the box below.
Clean Room–Dirty Room
Albert
Victoria
Clean Don’t Clean
Clean 5, 5 2, 6
Don’t Clean 6, 2 3, 3
In this game, whether or not Victoria chooses to clean, Albert will get
a higher payoff if he doesn’t clean than if he does clean. Therefore “Don’t
Clean” is a dominant strategy for Albert. Similar reasoning shows that
no matter what Albert chooses to do, Victoria is better off if she chooses
“Don’t Clean.” Therefore the outcome where both roommates choose
“Don’t Clean” is a dominant strategy equilibrium. This is true despite
the fact that both persons would be better off if they both chose to clean
the room.
This game is set in the South Pacific in 1943. Admiral Imamura must
transport Japanese troops from the port of Rabaul in New Britain, across
the Bismarck Sea to New Guinea. The Japanese fleet could either travel
north of New Britain, where it is likely to be foggy, or south of New
Britain, where the weather is likely to be clear. U.S. Admiral Kenney
hopes to bomb the troop ships. Kenney has to choose whether to concentrate
his reconnaissance aircraft on the Northern or the Southern route.
Once he finds the convoy, he can bomb it until its arrival in New Guinea.
Kenney’s staff has estimated the number of days of bombing time for each
of the outcomes. The payoffs to Kenney and Imamura from each outcome
are shown in the box below. The game is modeled as a “zero-sum game:”
for each outcome, Imamura’s payoff is the negative of Kenney’s payoff.
The Battle of the Bismarck Sea
Kenney
Imamura
North South
North 2, −2 2, −2
South 1, −1 3, −3
This game does not have a dominant strategy equilibrium, since there
is no dominant strategy for Kenney. His best choice depends on what Imamura
does. The only Nash equilibrium for this game is where Imamura
chooses the northern route and Kenney concentrates his search on the
northern route. To check this, notice that if Imamura goes North, then
Kenney gets an expected two days of bombing if he (Kenney) chooses
North and only one day if he (Kenney) chooses South. Furthermore, if
Kenney concentrates on the north, Imamura is indifferent between going
north or south, since he can be expected to be bombed for two days
either way. Therefore if both choose “North,” then neither has an incentive
to act differently. You can verify that for any other combination of
choices, one admiral or the other would want to change. As things actually
worked out, Imamura chose the Northern route and Kenney concentrated
his search on the North. After about a day’s search the Americans found
the Japanese fleet and inflicted heavy damage on it.∗
28.1 (0) This problem is designed to give you practice in reading a game
matrix and to check that you understand the definition of a dominant
strategy. Consider the following game matrix.
A Game Matrix
Player A
Player B
Left Right
Top a, b c, d
Bottom e, f g, h
∗
This example is discussed in R. Duncan Luce and Howard Raiffa’s
Games and Decisions, John Wiley, 1957, or Dover, 1989. We recommend
this book to anyone interested in reading more about game theory.
(a) If (top, left) is a dominant strategy equilibrium, then we know that
a > , b > , > g, and > h.
(b) If (top, left) is a Nash equilibrium, then which of the above inequalities
must be satisfied? .
(c) If (top, left) is a dominant strategy equilibrium must it be a
Nash equilibrium? Why?
.
28.2 (0) In order to learn how people actually play in game situations,
economists and other social scientists frequently conduct experiments in
which subjects play games for money. One such game is known as the
voluntary public goods game. This game is chosen to represent situations
in which individuals can take actions that are costly to themselves but
that are beneficial to an entire community.
In this problem we will deal with a two-player version of the voluntary
public goods game. Two players are put in separate rooms. Each player
is given $10. The player can use this money in either of two ways. He can
keep it or he can contribute it to a “public fund.” Money that goes into
the public fund gets multiplied by 1.6 and then divided equally between
the two players. If both contribute their $10, then each gets back $20 ×
1.6/2 = $16. If one contributes and the other does not, each gets back
$10 × 1.6/2 = $8 from the public fund so that the contributor has $8
at the end of the game and the non-contributor has $18–his original $10
plus $8 back from the public fund. If neither contributes, both have their
original $10. The payoff matrix for this game is:
Voluntary Public Goods Game
Player A
Player B
Contribute Keep
Contribute $16, $16 $8, $18
Keep $18, $8 $10, $10
(a) If the other player keeps, what is your payoff if you keep?
If the other player keeps, what is your payoff if you contribute? .
(b) If the other player contributes, what is your payoff if you keep?
If the other player contributes, what is your payoff if you
contribute? .
(c) Does this game have a dominant strategy equilibrium? If
so, what is it? .
28.3 (1) Let us consider a more general version of the voluntary public
goods game described in the previous question. This game has N players,
each of whom can contribute either $10 or nothing to the public fund.
All money that is contributed to the public fund gets multiplied by some
number B > 1 and then divided equally among all players in the game
(including those who do not contribute.) Thus if all N players contribute
$10 to the fund, the amount of money available to be divided among the
N players will be $10BN and each player will get $10BN/N = $10B back
from the public fund.
(a) If B > 1, which of the following outcomes gives the higher payoff to
each player? a) All players contribute their $10 or b) all players keep their
$10. .
(b) Suppose that exactly K of the other players contribute. If you keep
your $10, you will have this $10 plus your share of the public fund contributed
by others. What will your payoff be in this case?
. If you contribute your $10, what will be the total number of
contributors? What will be your payoff? .
(c) If B = 3 and N = 5, what is the dominant strategy equilibrium for this
game? Explain your answer.
.
(d) In general, what relationship between B and N must hold for “Keep”
to be a dominant strategy?
.
(e) Sometimes the action that maximizes a player’s absolute payoff, does
not maximize his relative payoff. Consider the example of a voluntary
public goods game as described above, where B = 6 and N = 5. Suppose
that four of the five players in the group contribute their $10, while the
fifth player keeps his $10. What is the payoff of each of the four contributors?
What is the payoff of the player who keeps his
$10? Who has the highest payoff in the group?
What would be the payoff to the fifth
player if instead of keeping his $10, he contributes, so that all five players
contribute. If the other four players contribute, what
should the fifth player to maximize his absolute payoff?
What should he do to maximize his payoff relative to that of the other
players? .
(f) If B = 6 and N = 5, what is the dominant strategy equilibrium for this
game? Explain your answer.
.
28.4 (1) The Stag Hunt game is based on a story told by Jean Jacques
Rousseau in his book Discourses on the Origin and Foundation of Inequality
Among Men (1754). The story goes something like this: “Two
hunters set out to kill a stag. One has agreed to drive the stag through
the forest, and the other to post at a place where the stag must pass. If
both faithfully perform their assigned stag-hunting tasks, they will surely
kill the stag and each will get an equal share of this large animal. During
the course of the hunt, each hunter has an opportunity to abandon the
stag hunt and to pursue a hare. If a hunter pursues the hare instead of the
stag he is certain to catch the hare and the stag is certain to escape. Each
hunter would rather share half of a stag than have a hare to himself.”
The matrix below shows payoffs in a stag hunt game. If both hunters
hunt stag, each gets a payoff of 4. If both hunt hare, each gets 3. If one
hunts stag and the other hunts hare, the stag hunter gets 0 and the hare
hunter gets 3.
The Stag Hunt Game
Hunter A
Hunter B
Hunt Stag Hunt Hare
Hunt Stag 4, 4 0, 3
Hunt Hare 3, 0 3, 3
(a) If you are sure that the other hunter will hunt stag, what is the best
thing for you to do? .
(b) If you are sure that the other hunter will hunt hare, what is the best
thing for you to do? .
(c) Does either hunter have a dominant strategy in this game? If
so, what is it? If not explain why not.
.
(d) This game has two pure strategy Nash equilibria. What are they?
.
(e) Is one Nash equilibrium better for both hunters than the other?
If so, which is the better equilibrium? .
(f) If a hunter believes that with probability 1/2 the other hunter will
hunt stag and with probability 1/2 he will hunt hare, what should this
hunter do to maximize his expected payoff? .
28.5 (1) Evangeline and Gabriel met at a freshman mixer. They want
desperately to meet each other again, but they forgot to exchange names
or phone numbers when they met the first time. There are two possible
strategies available for each of them. These are Go to the Big Party or
Stay Home and Study. They will surely meet if they both go to the party,
and they will surely not otherwise. The payoff to meeting is 1,000 for
each of them. The payoff to not meeting is zero for both of them. The
payoffs are described by the matrix below.
Close Encounters of the Second Kind
Evangeline
Gabriel
Go to Party Stay Home
Go to Party 1000, 1000 0, 0
Stay Home 0, 0 0, 0
(a) A strategy is said to be a weakly dominant strategy for a player if
the payoff from using this strategy is at least as high as the payoff from
using any other strategy. Is there any outcome in this game where both
players are using weakly dominant strategies?
.
(b) Find all of the pure-strategy Nash equilibria for this game.
.
(c) Do any of the pure Nash equilibria that you found seem more reasonable
than others? Why or why not?
.
(d) Let us change the game a little bit. Evangeline and Gabriel are still
desperate to find each other. But now there are two parties that they
can go to. There is a little party at which they would be sure to meet
if they both went there and a huge party at which they might never see
each other. The expected payoff to each of them is 1,000 if they both go
to the little party. Since there is only a 50-50 chance that they would find
each other at the huge party, the expected payoff to each of them is only
500. If they go to different parties, the payoff to both of them is zero.
The payoff matrix for this game is:
More Close Encounters
Evangeline
Gabriel
Little Party Big Party
Little Party 1000, 1000 0, 0
Big Party 0, 0 500, 500
(e) Does this game have a dominant strategy equilibrium? What
are the two Nash equilibria in pure strategies?
.
(f) One of the Nash equilibria is Pareto superior to the other. Suppose
that each person thought that there was some slight chance that the
other would go to the little party. Would that be enough to convince
them both to attend the little party? Can you think of any reason
why the Pareto superior equilibrium might emerge if both players
understand the game matrix, if both know that the other understands it,
and each knows that the other knows that he or she understands the
game matrix?
.
28.6 (1) The introduction to this chapter of Workouts, recounted the
sad tale of roommates Victoria and Albert and their dirty room. The
payoff matrix for their relationship was given as follows.
Domestic Life with Victoria and Albert
Albert
Victoria
Clean Don’t Clean
Clean 5, 5 2, 6
Don’t Clean 6, 2 3, 3
Suppose that we add a second stage to this game in which Victoria
and Albert each have a chance to punish the other. Imagine that at the
end of the day, Victoria and Albert are each able to see whether the
other has done any housecleaning. After seeing what the other has done,
each has the option of starting a quarrel. A quarrel hurts both of them,
regardless of who started it. Thus we will assume that if either or both of
them starts a quarrel, the day’s payoff for each of them is reduced by 2.
(For example if Victoria cleans and Albert doesn’t clean and if Victoria,
on seeing this result, starts a quarrel, Albert’s payoff will be 6 − 2 = 4
and Victoria’s will be 2 − 2 = 0.)
(a) Suppose that it is evening and Victoria sees that Albert has chosen not
to clean and she thinks that he will not start a quarrel. Which strategy
will give her a higher payoff for the whole day, Quarrel or Not Quarrel?
.
(b) Suppose that Victoria and Albert each believe that the other will try
to take the actions that will maximize his or her total payoff for the day.
Does either believe the other will start a quarrel? Assuming
that each is trying to maximize his or her own payoff, given the actions
of the other, what would you expect each of them to do in the first stage
of the game, clean or not Clean? .
(c) Suppose that Victoria and Albert are governed by emotions that they
cannot control. Neither can avoid getting angry if the other does not
clean. And if either one is angry, they will quarrel so that the payoff of
each is diminished by 2. Given that there is certain to be a quarrel if
either does not clean, the payoff matrix for the game between Victoria
and Albert becomes:
Vengeful Victoria and Angry Albert
Albert
Victoria
Clean Don’t Clean
Clean 5, 5 0, 4
Don’t Clean 4, 0 1, 1
(d) If the other player cleans, is it better to clean or not clean?
If the other player does not clean, is it better to clean or not clean.
Explain
.
(e) Does this game have a dominant strategy? Explain
.
(f) This game has two Nash equilibria. What are they?
.
(g) Explain how it could happen that Albert and Victoria would both
be better off if both are easy to anger than if they are rational about
when to get angry, but it might also happen that they would both
be worse off.
.
(h) Suppose that Albert and Victoria are both aware that Albert will
get angry and start a quarrel if Victoria does not clean, but that Victoria
is level-headed and will not start a quarrel. What would be the
equilibrium outcome?
.
28.7 (1) Maynard’s Cross is a trendy bistro that specializes in carpaccio
and other uncooked substances. Most people who come to Maynard’s
come to see and be seen by other people of the kind who come to Maynard’s.
There is, however, a hard core of 10 customers per evening who
come for the carpaccio and don’t care how many other people come. The
number of additional customers who appear at Maynard’s depends on
how many people they expect to see. In particular, if people expect that
the number of customers at Maynard’s in an evening will be X, then
the number of people who actually come to Maynard’s is Y = 10 + .8X.
In equilibrium, it must be true that the number of people who actually
attend the restaurant is equal to the number who are expected to attend.
(a) What two simultaneous equations must you solve to find the equilibrium
attendance at Maynard’s? .
(b) What is the equilibrium nightly attendance? .
(c) On the following axes, draw the lines that represent each of the
two equations you mentioned in Part (a). Label the equilibrium attendance
level.
y
80
60
40
20
0 20 40 60 80
x
(d) Suppose that one additional carpaccio enthusiast moves to the area.
Like the other 10, he eats at Maynard’s every night no matter how many
others eat there. Write down the new equations determining attendance
at Maynard’s and solve for the new equilibrium number of customers.
.
(e) Use a different color ink to draw a new line representing the equation
that changed. How many additional customers did the new steady
customer attract (besides himself)? .
(f) Suppose that everyone bases expectations about tonight’s attendance
on last night’s attendance and that last night’s attendance is public knowledge.
Then Xt = Yt−1, where Xt is expected attendance on day t and
Yt−1 is actual attendance on day t − 1. At any time t, Yt = 10 + .8Xt.
Suppose that on the first night that Maynard’s is open, attendance is 20.
What will be attendance on the second night? .
(g) What will be the attendance on the third night? .
(h) Attendance will tend toward some limiting value. What is it? .
28.8 (0) Yogi’s Bar and Grill is frequented by unsociable types who hate
crowds. If Yogi’s regular customers expect that the crowd at Yogi’s will
be X, then the number of people who show up at Yogi’s, Y , will be the
larger of the two numbers, 120− 2X and 0. Thus Y = max{120 − 2X, 0}.
(a) Solve for the equilibrium attendance at Yogi’s. Draw a diagram depicting
this equilibrium on the axes below.
y
80
60
40
20
0 20 40 60 80
x
(b) Suppose that people expect the number of customers on any given
night to be the same as the number on the previous night. Suppose that
50 customers show up at Yogi’s on the first day of business. How many
will show up on the second day? The third day? The
fourth day? The fifth day? The sixth day?
The ninety-ninth day? The hundredth day?
.
(c) What would you say is wrong with this model if at least some of Yogi’s
customers have memory spans of more than a day or two?
.