# Problem Set _ 14 Oligopolies and

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```					                                     Lecture 10: Oligopoly

Two suspects are arrested and charged with a crime. The police lack sufficient evidence
to convict the suspects, unless at least one confesses. The police hold the suspects in
separate cells and explain the consequences that will follow from the actions they could
take. If neither confesses, then both will be convicted of a minor offense and sentenced
to two years in jail. If both confess, then both will be sentenced to jail for 10 years.
Finally, if one confesses but the other does not, then the confessor will serve only 1 year
but the other will be sentenced to 20 years in jail – 10 for the crime and a further 10 for
obstructing justice. The possible outcomes are depicted below.

Prisoner 2              (Sally)
Confess             Don't Confess
Prisoner 1               Confess                M: -10; S: -10        M: -1; S: -20
(Mike)                Don't Confess            M: -20; S: -1          M: -2; S: -2

What would you do?

A dominant strategy is a strategy that works better than any other strategy regardless of
what the other player is doing.

The Nash Equilibrium is where each player is doing the best that they can given what
the other player is doing.

The Stackelberg Equilibrium is the Nash Equilibrium to a sequential move game.

The dominant strategy in the prisoner’s dilemma is to confess. The Nash Equilibrium is
for both players to confess.

If each player has a dominant strategy, then the dominant strategies will indicate the Nash
Equilibrium. However, it is possible to have a Nash Equilibrium where the players have
no dominant strategies. Now, show this as a simultaneous move game tree.
Prisoner 1

Confess                                 Not Confess

Prisoner 2

Confess                    Not Confess    Confess                    Not Confess

(prisoner 1):   10 years                 1 year           20 years                   2 years
(prisoner 2):   10 years                 20 years         1 year                     2 years
David Wessel, in his 2002 Wall Street Journal article, “The Civilizing Effect of
the Market,” attempts to determine whether those in market economies are more
generous. To do this, he sets up an experiment with two consumers. He gives the first
consumer \$100.00. The first consumer must decide how much of the \$100.00 to give to
consumer two. Then, consumer two either accepts or rejects the offer. If consumer two
rejects the offer, then both consumers get nothing. However, if consumer two accepts
consumer one’s offer, then they both receive money according to consumer one’s
proposed division. Let the following payoff matrix depict the game. In this setup,
consumer one may give consumer two either \$1.00 or \$50.00. Then, consumer two may
either accept or reject the offer.
Player one
Offer \$1.00             Offer \$50.00

Accept    one\$99two\$1         one\$two\$

Player two
Reject     one\$two\$one\$two\$

a) Does consumer one have a dominant strategy? If so, what is it?

b) Does consumer two have a dominant strategy? If so, what is it?

c) Assume consumer one moves first. Identify the Stackelberg Equilibrium.

Surprisingly, Wessel found that those in capitalism were more generous than those in
Marxist economies.

Characteristics of Oligopoly:
1. Many buyers and a few sellers
2. Homogeneous product
3. There may or may not be entry
4. Imperfect Information

Suppose the demand for oil is D: P = 100-QD, and that there are two oil companies:
Ewing and Wesstar Oil. Let MC be constant with no fixed costs so that MC=ATC=
\$20.00. These two oil companies first have the option of producing either 20 or 25 units
of oil.
Ewing
Oil
20         25
E: 800 E: 875
20     W: 800 W: 700
E: 700 E: 750
Wesstar Oil      25     W: 875 W: 750

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The dominant strategies are for each company to produce 25 barrels of oil. This strategy
works better than producing 20 barrels of oil regardless of what the rival firm does. The
Nash Equilibrium is for each company to produce 25 barrels of oil. In this box, neither
player wants to change their behavior given what their rival firm is doing.

Now suppose that they have a third option: to produce 40 units of oil.

Ewing
Oil
20     25     40
E: 800 E: 875 E: 800
20      W: 800 W: 700 W: 400
E: 700 E: 750 E: 600
Wesstar Oil      25      W: 875 W: 750 W: 375
E: 400 E: 375  E: 0
40      W: 800 W: 600 W: 0

The companies no longer have dominant strategies. What works best hinges on what the
rival firm does. However, the Nash Equilibrium is for each company to produce 25
barrels of oil.

If this were a sequential move game where Ewing Oil moves first, then the Stackelberg
Equilibrium would be for Ewing to produce 40 barrels and Wesstar to produce 20 barrels.
Note that it is an advantage to move first. By moving first, Ewing Oil has higher profits
than if they moved second or simultaneously with Wesstar.

Let demand be D: P = 200 – Q and MC = ATC = 10 with four firms:

   Ewing Oil
   Wesstar Oil
   Barnes-Wentworth Oil
   People’s Oil

Maximize profits by picking the best output level across 3 periods.

Ewing           Wesstar          Barnes-Wentworth          Peoples
Period 1
Period 2
Period 3

Now consider an example of entry. Suppose Boeing is the only airplane maker. Demand
for airplanes is D: P = 200 – Q with constant MC = 20. Boeing can produce 45, which
will maximize profits, but Planes Inc., may enter the industry and also produce 45. To
keep Planes, Inc. out, Boeing is also considering producing 140. Thus, Planes Inc.,’s
other option is not to enter (produce 0).

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Boeing
45     140
B: 6075 B: 5600
0      P: 0    P: 0
B: 4050 B: -700
Planes Inc.        45     P: 4050 P: -225

What is each firm’s dominant strategy?

What is the Nash Equilibrium?

Now suppose that Boeing moves first in a sequential move game. Find the Stackelberg
Equilibrium.
Boeing

45                                    140

Planes, Inc.

0                         45               0                45

(Boeing):     \$6,075                  \$4,050            \$5,600            \$-700
(Planes):     \$0                      \$4,050            \$0                \$-225

Now consider an example of cooperation. In this example, suppose Iraq and Kuwait are
the only producers of oil. To maximize profits, the two oil-producing nations agree to
maximize joint profits as a monopoly would and then divide such profits. That is, Iraq
and Kuwait agree to form a cartel.

D: P = 100 – ½ Q and MC = ATC = 10.
Iraq
45       50
I: 2025 I: 2125
45       K: 2025 K: 1912.5
I: 1912.5 I: 2000
Kuwait       50       K: 2125 K: 2000

Find the Nash Equilibrium.

Is the cartel agreement sustainable?

Second Game:

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Five firms must pick a high (cartel) price or a low price. The game is played for three
periods. Firms may discuss strategies. Company with the most profits gets the extra
credit. In the case of a tie, divide the extra credit among the winners.
Number of                     Number of               Profit for Each         Profit for Each
High-Priced Firms             Low-Priced Firms           High-Priced Firm        Low-Priced Firm
0                           5                           -                      50
1                           4                          20                      70
2                           3                          40                      90
3                           2                          60                     110
4                           1                          80                     130
5                           0                         100                       -

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Problem Set 10: Oligopoly

1.      Let Procter and Gamble (P&G) and SmithKline Beecham (SKB) be the only two
firms that produce toothpaste, making this industry a duopoly. Proctor and Gamble and
SmithKline Beecham may each produce either 40 or 50 units of output. The following
payoff matrix shows the profits each firm would earn for these levels of output.

SmithKline Beecham
QSK B= 40               QSK B =50

QP&G=40 SKBPGSKBPG

Proctor and Gamble

QP&G=50 SKBPGSKBPG

a) What is Procter and Gamble's dominant strategy?
b) What is SmithKline Beecham's dominant strategy?
c) What is the Nash Equilibrium of this game for this duopoly?
d) Depict this game with a decision tree below. Then, identify the terminal node that
correctly represents the quantity each firm in the duopoly will produce in equilibrium.

2.      Suppose Iran and Iraq each produce crude oil and both want a high market price
for crude oil to create larger profits. To keep the market price for oil high, Iran and Iraq
have agreed to keep production low: Iran and Iraq have each agreed to only produce 35
units of crude oil. Now, Iran and Iraq must each decide whether to honor this agreement
and cooperate. Honoring the agreement means producing 35 units of crude oil.
However, each country also has the option of secretly breaking the agreement and
producing 50 units of output to earn higher profits. Suppose Iran and Iraq must each
decide how much crude oil to produce simultaneously. Thus, when Iran decides how
much to produce, it doesn’t know whether Iraq is honoring the agreement. Similarly,
Iraq must decide how much to produce without knowing how much Iran is producing.
This situation is depicted using a payoff matrix below.

Iran
Q = 35                           Q =50

Q=35      IRANiRAQIRANIRAQ

Iraq

Q=50       IRANIRAQIRANIRAQ


a)     Does Iran have a dominant strategy? If so, what is it?
b)     Does Iraq have a dominant strategy? If so, what is it?
c)     Find the Nash Equilibrium for this game.

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3.      Consider the market for baseball caps. You are the CEO of a company, which is
the only producer of baseball caps. Initially, you had been maximizing profits by
producing quantity Q=90. However, a new firm is considering entering the baseball cap
market to get a share of the positive economic profits being earned. The following
decisions must be made: you must decide whether to continue producing quantity Q=90
(the profit maximizing level of production for a monopoly) or to produce Q=150 (you
would produce 150 baseball caps in an effort drive market price down so low that the
new firm would rather not enter the cap making industry); the new firm must decide
whether to enter the market and produce Q=50 or not enter the market and produce
nothing (Q=0). We are therefore assuming that you can produce Q=90 or Q=150 and the
new firm can produce Q=0 or Q=50. This situation is depicted using a payoff matrix.
For parts (a) – (c), assume you and the new firm must decide how much to produce
simultaneously.
You
QY= 90                       QY =150

QNF=0     YNFYNF

New Firm

QNF=50    YNFYNF

a) Do you have a dominant strategy? If so, what is it?
b) Does the new firm have a dominant strategy? If so, what is it?
c) Find the Nash Equilibrium for this game.

Now suppose your firm (being more established and trusted by the consumer) is the
dominant firm in the industry such that you decide how much to produce first and the
new firm decides how much to produce second.

d) Find the Stackelberg Equilibrium for this game. State how much you and the new
firm produce.

4.      The Windows operating system from Microsoft, Inc. runs about 90% of the
world’s personal computers, so it is natural to think that Microsoft has a monopoly in the
market for operating systems. According to economist Richard Schmalensee, an expert
on oligopoly and monopoly, Microsoft’s profit-maximizing monopoly price is between
\$900 and \$2,000. That’s the amount Microsoft would charge if it acted like a regular
monopolist. However, Microsoft has tended to charge only \$99 for Windows. Perhaps
Microsoft is an insecure monopolist and picks a low price to discourage entry and
preserve its monopoly. In other words, if Microsoft charged \$2,000 for its operating
system, there would be an incentive for other firms to develop alternative operating
systems.
Suppose that the market for operating systems is initially a monopoly (Microsoft
is the monopoly) with the potential for a second firm to enter the industry. Suppose
Microsoft can charge a price of either \$99 or \$900 for its operating system. Suppose a
second firm has three options: enter the industry and charge \$99, enter the industry and

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charge \$900, and not enter the industry. The profits that would result from these
outcomes are presented in a payoff matrix below.
Microsoft (M)

P=\$99            P=\$900
Enter, P=\$99                        M=5              M=0
NF=5             NF=10
New Firm (NF)           Enter, P=\$900                      M=10              M=50
NF=0             NF=50
No Entry                        M=10             M=100
NF=0             NF=0

Assume that Microsoft and the New Firm move simultaneously.
a)     Does Microsoft have a dominant strategy? If so, what is it?
b)     Does the New Firm have a dominant strategy? If so, what is it?
c)     Find the Nash Equilibrium for this game. Actually, there are two. Find both
of them.

Now assume that Microsoft moves first and that the New firm moves second.
d)     Find the Stackelberg Equilibrium to a sequential-move game.

5.     Suppose that the market for crude oil is a duopoly comprised of two oil-producing
nations -- Saudi Arabia (SA) and Qatar (Q). Since Saudi Arabia is a much larger oil-
producing nation than Qatar, suppose that Saudi Arabia is the industry leader. Therefore,
Saudi Arabia gets to pick how much to produce first. Then, Qatar finds out how much
Saudi Arabia has produced and picks an output level. Assume that Saudi Arabia’s output
choices are 100 and 200 and that Qatar’s output choices are 3 and 5. This situation is
depicted using a payoff matrix below.
Saudi Arabia
Q = 100                           Q =200

Q=3      SAQSAQ

Qatar

Q=5      SAQSAQ


a)       Find the Stackelberg Equilibrium.

Now assume that Saudi Arabia and Qatar move simultaneously. That is, they decide to
produce without knowing how much each other are producing.
b)     Does Saudi Arabia have a dominant strategy? If so, what is it?
c)     Does Qatar have a dominant strategy? If so, what is it?
d)     Find the Nash Equilibrium for this game.

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Answer Key 10: Oligopoly

Answer to question 1:
SmithKline Beecham
QSK B= 40               QSK B =50

QP&G=40 SKBPGSKBPG

Proctor and Gamble

QP&G=50 SKBPGSKBPG

a)      The dominant strategy is to produce output of 50. This is the dominant
strategy because Proctor and Gamble’s profits will be higher by producing 50
regardless of what SmithKlein Beecham does.
b)      The dominant strategy is to produce output of 50. This is the dominant
strategy because SmithKlein Beecham’s profits will be higher by producing
50 regardless of what Proctor and Gamble does.
c)      The Nash Equilibrium is QSKB = 50 and QPG = 50. This is the only output
level for which both firms do not want to alter their behavior – that the
definition of equilibrium: no more changes. If both firms produced output of
40, then profits would be higher, but his would not be an equilibrium because
each would want to increase production to 50 to increase profits (from 3200 to
3500). Given that Proctor and Gamble produces 50, SmithKlein Beecham
maximizes profits by producing QSKB = 50. There is no pressure for
SmithKlein Beecham to change its behavior. Given that SmithKlein Beecham
is producing 50, Proctor and Gamble maximizes profits by producing 50.
There is no pressure for Proctor and Gamble to change its output level. This
can’t be said for any other output combination, so it’s the Nash Equilibrium.

SmithKlein Beecham

40                                50

Proctor and Gamble

40                            50                 40                         50

(PG):         3200                        3500               2800                       3000
(SKB): 3200                      2800              3500                        3000

The dotted lines indicate that this is a simultaneous move game. Proctor and Gamble and
SmithKlein Beecham do not know what each other have selected to produce.

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The Nash Equilibrium is QPG = 50 and QSKB = 50. There, there is no motivation for
either firm to change its behavior.

Answer to question 2:
Iran
Q = 35                               Q =50

Q=35     IRANiRAQIRANIRAQ

Iraq

Q=50     IRANIRAQIRANIRAQ


a)      Iran has a dominant strategy: Iran should produce Q=50 because that will bring
about the largest profits, regardless of what Iraq does.
b)      Iraq also has a dominant strategy: if Iran produces Q=35, then Iraq should
produce Q=50, and if Iran produces 50, then Iraq would optimally produce Q=50. So,
Iraq’s dominant strategy is also to produce Q=50.
c)      The Nash Equilibrium of a simultaneous move game has Iran producing Q=50
and Iraq producing Q=50. At those levels of production each country is doing the very
best that it can, given what the rival country is doing. Neither Iran nor Iraq can change
their behavior to make themselves better off when at the Nash Equilibrium.

So, the cooperation didn’t work: they broke the agreement.

Answer to question 3:

You pick the column and the new firm picks the row.
You
QY= 90                       QY =150

QNF=0      YNFYNF

New Firm

QNF=50     YNFYNF


a)      You have a dominant strategy: produce Q=90 because that will bring about the
largest profits, regardless of what the new firm does.
b)      The new firm does not have a dominant strategy: if you produce Q=90, then the
new firm should produce Q=50; however if you produce 150, then the new firm would
optimally produce Q=0.
c)      The Nash Equilibrium of a simultaneous move game has you producing Q=90
and the new firm producing Q=50. At those levels of production each player is doing the

10
very best that it can, given what its rival firm is doing. Neither you nor the new firm can
change your behavior to make yourself better off when at the Nash Equilibrium.

d)      Now you decide how much to produce first. If you produce Q=90, then the new
firm will pick Q=50 and your profits will be 3,600. If you produce Q=150, then the new
firm will not enter the market, resulting in profits of 4,500 for you. Because 4,500 is
greater than 3,600, you will produce Q=150. The Stackelberg Equilibrium is Q=150 for
you and Q=0 for the new firm.

Notice that the Stackelberg Equilibrium results in greater profits for your firm when
compared to the Nash Equilibrium. From this result we conclude that it is an advantage
to move first.

Answer to question 4.
a)     Microsoft does not have a dominant strategy.
b)     The New Firm does not have a dominant strategy.
c)     There are two Nash Equilibria. They are (i) Microsoft to charge P=\$99
and the New Firm to enter and also charge P=\$99, and (ii) Microsoft to
charge P=\$900 and the New Firm to enter and also charge P=\$900.
d)     The Stackelberg Equilibrium is for Microsoft to charge a price of P=\$900
and for the New Firm then to enter and charge a price of P=\$900. This
gives Microsoft and the New Firm profits of \$50. Conversely, notice that
if Microsoft had moved first by selecting a price of P=\$99 then the New
Firm would have maximized profits by entering and picking a price of
P=\$99. This would have given Microsoft profits of only \$5. So,
Microsoft can reason that if they move first and pick P=\$900, their profits
will be \$50, which is greater than moving first and picking P=\$99 with
profits of \$5. Given that Microsoft picks P=\$900, the New Firm
maximizes profits by moving second and picking P=\$900 too.

Answer to question 5:
a)     The Stackelberg Equilibrium is for Saudi Arabia to produce Q=100 and for
Qatar to produce Q=5.
b)     Saudi Arabia’s dominant strategy is to produce Q=100.
c)     Qatar does not have a dominant strategy.
d)     The Nash Equilibrium is Q=100 for Saudi Arabia and Q=5 for Qatar.

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