NAME 343
(b) Calculate the profit-maximizing price and quantity and total daily
profits for Albatross Airlines. p=45 ,q=70 ,π=
(c) If the interest rate is 10% per year, how much would someone be will-
ing to pay to own Albatross Airlines’s monopoly on the Dubuque-Peoria
route. (Assuming that demand and cost conditions remain unchanged
(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
(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 dou-
bles 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 maxi-
(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? $45.
(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
28.7 (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
344 OLIGOPOLY (Ch. 28)
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.
(b) If Anna expects Alex to sell q2kangaroos, what will her own marginal
(d) Now if Alex sells q2kangaroos, what is the total number of kangaroos
that will be sold? 400 + 1/2q2.What will be the market price as
(e) What is Alex’s marginal revenue as a function of q2only?
MR(q2)=1,200 −2q2.How many kangaroos will Alex
sell? 400. How many kangaroos will Anna sell? 200. What will
28.8 (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? y=p.
The total supply from the competitive sector at price pis S(p)= 50p.
(b) If the monopolist sets a price p, the amount that it can sell is Dm(p)=
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(c) The monopolist’s profit-maximizing output is ym=500. What
is the monopolist’s profit-maximizing price? p=5.
(d) How much output will the competitive sector provide at this price?
28.9 (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? 50. What will be the
(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
(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? 225/2.What will be the equilibrium quantity?
346 OLIGOPOLY (Ch. 28)
Calculus 28.10 (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 $pfor
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 $pfor 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? 2(120−p).(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
(b) Ben’s profit-maximizing base price is $70. (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
(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 cook-
stoves. He sets a price $pand promises to deliver for free to any farmer
who lives within p−20 miles of him. (He won’t deliver to anyone who lives
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further than that, because it then costs him more than $pto buy a stove
and deliver it.) If he is going to price in this way, how high should he set p?
$120. How many cookstoves would Ben deliver? 200. How much
would his total revenue be? $24,000 How much would his total
costs be, including the cost of deliveries and the cost of buying the stoves?
$14,100. (Hint: For any n,thesumoftheseries1+2+…+nis
equal to n(n+1)/2.) How much profit would he make? $10,100.
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? If Ben pays
for delivery, he can price-discriminate
Calculus 28.11 (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 pBand
Huey’s base price is pH, somebody who lives xmiles west of Ben would
have to pay a total of pB+xto have a stove delivered from Ben and
pH+(40−x) to have a stove delivered by Huey.
(a) If Ben’s base price is pBand Huey’s is pH, write down an equation that
could be solved for the distance x∗to the west of Bouncing Springs that
348 OLIGOPOLY (Ch. 28)
Ben’s market extends. pB+x∗=pH+(40−x∗).If Ben’s base
price is pBand Huey’s is pH, then Ben will sell 20 + (pH−pB)/2
cookstoves and Huey will sell 20 + (pB−pH)/2cookstoves.
(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.(20 + (pH−pB)/2)(pB−20).
(c) If Ben thinks that Huey’s price will stay at pH, no matter what price
Ben chooses, what choice of pBwill maximize Ben’s profits? pB=
30 + pH/2.(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? pH=30+pB/2.(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 pBand pHthat simultaneously solve the last two
(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 xmiles 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 xmiles
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 $49.99 . 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
$50.00 to deliver to this farmer. This farmer will buy from Ben
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whose price to him is cheaper by one penny. When the two merchants
have this pricing policy, all farmers who live within 20 miles of
Ben will buy from Ben and all farmers who live within 20 miles
of Huey will buy from Huey. A farmer who lives xmiles west of Ben
and buys from Ben must pay 59.99 −xdollars to have a cookstove
delivered to him. A farmer who lives xmiles east of Huey and buys from
Huey must pay 59.99 −xfor delivery of a stove. On the graph
below, use blue ink to graph the cost to Ben of delivering to a farmer who
lives xmiles west of him. Use red ink to graph the total cost to Huey
of delivering a cookstove to a farmer who lives xmiles 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.
0102030
40
20
40
60
Miles west of Ben
Dollars
80
Blue line
Red line
Pencil line
Ben’s profit
Huey’s profit
(f) With the pricing policies you just graphed, which farmers get stoves
make? $400. 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