Chapter 09 - Basic Oligopoly Models
Chapter 9: Answers to Questions and Problems
1.
a. D2.
b. D1.
c.
i. $20.
ii. 0 units.
iii. $20 to $50.
2.
a. 𝑄1 =
𝑄2 =
𝑎−𝑐1
2𝑏
𝑎−𝑐2
2𝑏
1
200−26
2
1
2(3)
200−32
− 𝑄2 =
− 2 𝑄1 =
2(3)
1
− 𝑄2 = 29 − 0.5𝑄2 and
2
1
− 2 𝑄1 = 28 − 0.5𝑄1
b. Q1 = 20; Q2 = 18.
c. P = 200 – 3(38) = $86.
d. Π1 = $1200; Π2 = $972.
3.
a.
b.
c.
d.
e.
125 units.
100 units each.
The leader produces 150 units and the follower produces 75 units.
150 units.
i. 75 units.
ii. About 112.5 units.
4.
a. 𝑄𝐹 =
𝑎−𝑐𝐹
2𝑏
1
− 2 𝑄𝐿 =
16,000−6,000
2(4)
1
− 2 𝑄𝐿 = 1,250 − 0.5𝑄𝐿
b. QL = 1750; QF = 375.
c. P = 16,000 – 4(2125) = $7,500.
d. ΠL = $6.125 million; ΠF = $562,500.
5.
a. Set P = MC to get 800 – 4Q = $260. Solving yields Q = 135 units.
b. P = MC = $260.
c. Each firm earns zero economic profits.
6.
a. Oil production. Each firm produces output independently and the market price is
determined by the total amount produced.
b. Diamond production. DeBeers is the leader that sets diamond production, and
smaller firms follow with their own levels of production.
c. Competitive bidding by identical contractors. In this case, the contractor bidding
the lowest fee will win the contract.
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Chapter 09 - Basic Oligopoly Models
7.
Model
Cournot
Stackelberg
Bertrand
Collusion
Output
Q1 = Q2 = 33.33
QL = 50; QF = 25
Market output = 100 units
Market output = 50 units
Profits
π1 = π2 = $3,333.33
πL = $3,750; πF = $1,875
Zero
Industry Profits = $7,500
8.
a. Firm 1’s output and profit would increase. Firm 2’s output and profits would
decrease.
b. For small changes in costs, there would be no change in output or profits.
9.
a.
b.
c.
d.
Cournot duopoly.
Bertrand duopoly.
Stackelberg duopoly.
Sweezy duopoly.
10.
The equilibrium price will equal marginal cost, so P = 4. The profits will be P*Q –
C(Q) = 4*Q – 4*Q = 0.
11.
This would positively impact sales and the firm’s bottom line if Ford is the only
company to offer such a program. However, one would expect rivals (such as GM) to
respond with a similar plan. This would reduce the impact of Ford’s program on your
sales and bottom line. Indeed, GM did quickly respond with its Drive America
program.
12.
This market is a homogeneous product Cournot oligopoly. Using the given
information about demand and costs, each firm has a reaction function of 𝑄𝑖 =
5900−800
1
−
𝑄 = 2550 − 0.5𝑄𝑗 . Solving for equilibrium quantities results in each
2(1)
2 𝑗
firm producing 1700 units. This means the market price is P = 5900 – (3400) =
$2,500. Since BlackSpot’s marginal cost is $800, it follows that its profit gross of
fixed costs is (P – MC)Qi = ($2500 - $800)(1700) = $2,890,000. (Since profits net of
fixed costs are only $890,000, it follows that BlackSpot’s fixed costs are $2 million).
When marginal cost for BlackSpot falls to $500 (but Condensed Computers’ marginal
5900−500
cost remains at $800), BlackSpot’s new reaction function becomes 𝑄𝑖 = 2(1) −
1
𝑄𝑗 = 2700 − 0.5𝑄𝑗 . Solving for equilibrium quantities using this new reaction
function for BlackSpot (but the old one for Condensed Computers) gives us 1900
units produced by BlackSpot and 1600 units produced by Condensed Computers. So,
the market price is P = 5900 – (1900 + 1600) = $2,400. BlackSpot’s profit becomes
($2,400 – $500)(1900) – $2,000,000 = $1,610,000. So, profit increases by $720,000.
2
9-2
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Chapter 09 - Basic Oligopoly Models
13.
In this homogeneous product Bertrand oligopoly, the equilibrium price equals
marginal cost of $2.20. The total market quantity sold is Q = 80 – 6(2.20) = 66.8
units. If one station is self-serve while the other is full-service, the differentiated
nature of the products may permit each firm to charge a price above $2.20. This will
result in fewer units of gasoline being sold, but the firms will enjoy higher profits.
14.
No. A quota may actually increase the profits of the follower.
15.
OPEC must reduce its output when rivals (such as Russia, Omar, Mexico, Norway,
and other non-OPEC countries) increase their output. This reduction in output lowers
their profits, making it difficult to accomplish.
16.
No. The industry is contestable.
17.
You should support the legislation. Absent the legislation, this homogeneous product
Bertrand oligopoly will result in marginal cost pricing and zero profits. Under the
legislation, you will earn a profit of $75 – $60 = $15 on each unit sold. Your 20
percent of the contract amounts to 125 units, so your total profits under the
congresswoman’s plan is $1,875 compared to the $0 you will earn under cutthroat
Bertrand competition.
18.
Yes, it would be profitable to merge. Your current Stackelberg output is 𝑄𝐶 =
1200+120−2(60)
1200−120
=
100.
Your
rival’s
output
is
𝑄
=
− 0.5(100) = 40. So,
𝐹
2(6)
2(6)
price is P = 1200 – 6(100 + 40) = 360. Your profits then are (360 – 60)(100) =
$30,000, and your rival’s are (360 – 120)(40) = $9,600. If you merge, you would not
face any competition. In addition, would be able to produce all output at your own
facilities (which have lower costs). Your post-merger monopoly profits would be
based on your monopoly output, which occurs where MR = MC: 1200 – 12Q = 60, or
Q = 95 units. The monopoly price is $630. Your monopoly profits are ($630 –
$60)(95) = $54,150, so you stand to gain $24,150 by merging. If you offer your rival
$9,600.01 to merge, it is strictly better off and so are you.
19.
Under the status-quo of Cournot oligopoly, your reaction function is Q1 = 39 – 0.5Q2.
Your opponent’s reaction function is the same: Q2 = 39 – 0.5Q1. Solving for
equilibrium, we have each firm producing 26 units. The price is P = 160 – 2(52) =
56. So, your profits are (56 – 4)(26) = $1,352. If costs were the same but you were
the leader in a Stackelberg oligopoly, your equilibrium quantity would be 𝑄𝑙𝑒𝑎𝑑𝑒𝑟 =
160+4−2(4)
= 39. Your opponent would be the follower, and produce 𝑄𝑓𝑜𝑙𝑙𝑜𝑤𝑒𝑟 =
2(2)
160−4
2(2)
− 0.5(39) = 19.5. The price would be P = 160 – 2(58.5) = 43, so your profits
would be (43 – 4)(39) = $1,521. Since this difference is only $169, it would not pay
to spend $200 on the investment: The cost of establishing the first-mover advantage
exceeds the benefits.
9-3
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution
in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 09 - Basic Oligopoly Models
20.
When Ajinomoto was the sole supplier of lysine, it charged the monopoly price of
$1.65 per pound and sold 76 million pounds; found by solving MR = MC. The
dramatic price decline (to marginal cost) in the worldwide market for lysine after
ADM entered the market can be explained by Bertrand competition: the two firms
compete by setting price in the worldwide market driving price to marginal cost. At
this price, 152 million pounds were sold in the worldwide market and each firm sold
76 million pounds (by assumption). The 1993 price increase can be explained by
collusion: ADM and Ajinomoto set the monopoly price and produce an equal share of
the monopoly output. Therefore, price rose to $1.65 per pound and 76 million pounds
are sold on the worldwide market; with each firm supplying 38 million pounds of
lysine.
21.
The inverse demand function for this Sweezy oligopoly is 𝑃 =
900 − 0.5𝑄 𝑖𝑓 𝑄 ≤ 240
{
. The marginal revenue function is
𝑀𝑅 =
1,500 − 3𝑄 𝑖𝑓 𝑄 ≥ 240
900 − 𝑄 𝑖𝑓 𝑄 < 240
{ [60,660] 𝑖𝑓 𝑄 = 240 . Therefore, changes in marginal cost in the range of $60
1,500 − 6𝑄 𝑖𝑓 𝑄 > 240
and $660 will not result in a change in the profit-maximizing level of output.
22.
The 10 percent increase in rent is an increase in both firms’ fixed costs. No matter
whether the Cournot, Stackelberg, Bertrand, or Sweezy model applies to these two
firms’ mode of competition, a change in the firms’ fixed costs should not alter their
strategic decisions regarding quantity or price. Therefore, Jones should not increase
prices by 10 percent.
23.
Since an excise tax is a per-unit tax it effectively increases each firms’ marginal cost.
In a Cournot oligopoly, increases in marginal costs shift each firm’s reaction function
closer to the origin. This results in each firm supplying a lower equilibrium output
and charging a higher market price (including taxes). In a Bertrand oligopoly, where
firms price at marginal cost in equilibrium, firms pass the entire amount of the excise
tax to consumers. In equilibrium, prices rise by the amount of the excise tax and
output declines. Finally, in a Sweezy oligopoly, small changes in marginal cost
(through the excise tax in this case) have no effect on firms’ prices. In equilibrium,
price and output do not change in response to small increases in the excise tax. For
this reason, the Sweezy oligopoly is likely to generate the greatest increase in tax
revenue.
9-4
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution
in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.