Microeconomics | EE 311 | Faculty of Economic, Thammasat University
Oligopoly and Game Theory
1.
A monopolist can produce at constant average (marginal) cost of AC = MC = 5. It faces a market demand curve given by Q = 53 – P. a) Calculate the profit-maximizing price and quantity for this monopolist. Also calculate its profits. b) Suppose the second firm enters the market. Let Q
1
be the output of the first firm and Q
2
be the output of the second. Market demand is now given by
Q
1
+ Q
2
= 53 – P.
Assuming that this second firm has the same costs as the first, write the profits of each firm as functions of Q
1
and Q
2
. c) Suppose (as in the Cournot model) that each firm chooses its profit-maximizing level of output on the assumption that its competitor’s output is fixed. Find each firm’s reaction curve. d) Calculate the Cournot equilibrium. What are the resulting market price and profits of each firm. e) Find the Stackleberg equilibrium when the first firm is the leader and the new entrant is the follower.
2.
Two computer firms, A and B, are planning to market network systems for office information management. Each firm can develop either a fast, high-quality system
(High), or a slower, low quality system (Low). Market research indicates that the resulting profits to each firm for the alternative strategies are given by the following payoff matrix:
Firm A
High
High 30 , 30
Low 40 , 60
Firm B
Low
50 , 35
20 , 20 a) If both firms make their decisions at the same time and follow maximin (low-risk) strategies, what will the outcome be? b) Suppose both firms try to maximize profits, but Firm A has a head start in planning and can commit first. Now what will the outcome be? What will the outcome be if Firm B has the head start in planning and can commit first? c) Getting a head start costs money (you have to gear up a large engineering team).
Now consider the two-stage game in which first, each firm decides how much money to spend to speed up its planning, and second, it announces which product
(H or L) it will produce. Which firm will spend more to speed up its planning?
How much will it spend? Should the other firm spend anything to speed up its planning? Explain.
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11/4/20
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Microeconomics | EE 311 | Faculty of Economic, Thammasat University
3.
Two firms are in the chocolate market. Each can choose to go for the high end of the market (high quality) or the low end (low quality). Resulting profits are given by the following payoff matrix:
Firm 2
Low High
Firm 1 Low -20 , -30 900 , 600
High 100 , 800 50 , 50 a) What outcomes, if any, are Nash equilibria? b) If the managers of both firms are conservative and each follows a maximin strategy, what is the outcome? c) What is the cooperative outcome? Which firm benefits most from the cooperative outcome? How much would that firm need to offer the other to persuade it to collude?
Chayun Tantivasadakarn |
11/4/20
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Microeconomics | EE 311 | Faculty of Economic, Thammasat University
Solution:
1.a) The monopolist’s profits are maximized when MC =
MR. Since TR = 53Q – Q 2 , then MR = 53 – 2Q. Set this equal to MC = 5 and solve to get Q
M
= (53 – 5)/2 = 24 and
P
M
= 53 – 24 = 29. Hence the profits = (P
M
– AC)Q
M
= (29
– 5)24 = 576.
1.b) Profits of the firm i for a given output level Q j
is given by  i

( P i

Q i

53 i

Q i
2 
Q i
Q j

5 Q i

48 Q i

Q i
2 
Q i
Q
1.c) Under Cournot competition, each firm i maximize its j profits, treating the rival’s output as fixed. Differentiate
 i with respect to Q i
and set it to zero gives the reaction function of i in terms of Q j
:
48
Q i


2 Q i
24


Q j
Q j

0
2
for i, j = 1, 2.
1.d) Substitute Q i
into the other firm’s reaction function and solve will give: Q
1
= Q
2
= 16. Thus the price = 53 – 2(16)
= 21 and the profits will be the same and equal to (21 –
5)16 = 256 for each firm.
1.e) For Stackelberg model, we have to start from the follower’s reaction function of firm 2 which is given by
Chayun Tantivasadakarn |
11/4/20
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Microeconomics | EE 311 | Faculty of Economic, Thammasat University
Q
2
 24  Q
1
2
. Substitute this into the profit function of the leader to get

1
 53 Q
1
 Q
2
1
 Q
1

24  Q
1
/ 2

 5 Q
1
 24 Q
1
 Q
2
1
/ 2
Differentiate 
1 with respect to Q
1
and set it to zero gives
Q
1
= 24. Substitute this back into the reaction function of 2 yields Q
2
= 12. Therefore the total output Q = 24 + 12 = 36 and the price P = 53 – 36 = 17. Finally, profit for each firm is:

1

( 17

5 ) 24
 
2

( 17

5 ) 12

144
2.
From the payoff matrix
2.a) Using Maximin strategies the equilibrium is “High”,
“High” since:
Firm A’s worse payoff for “High” strategy is 30 while the worse payoff for “Low” strategy is 20. Thus the maximum of the minimum is 30 and A should choose
“High” strategy.
Firm B’s worse payoff for “High” strategy is 30 while the worse payoff for “Low” strategy is 20. Thus the maximum of the minimum is 30 and A should choose
“High” strategy.
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11/4/20
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Microeconomics | EE 311 | Faculty of Economic, Thammasat University
2.b) This is a sequential game where Firm A moves first. The game tree is as shown below. We use backward induction to solve the game by starting at the second stage of the follower. Firm B will choose “Low” if it knows that Firm A chooses “High” since the payoff for “Low” is 35 which is higher than 30 of “High”.
Similarly, Firm B will choose “High” if it knows that Firm
A chooses “Low” since the payoff for “High” is 60 which is higher than 20 of “High”. At the first stage problem for the leader, Firm A compare the two alternative and will conclude that “High” will give a higher payoff (50 compared to 40), so it will choose “High”. The equilibrium, therefore, will be “High”, “Low” with payoffs
= (50,35) if Firm A moves first. Similarly, if Firm B move first, the equilibrium will be “Low”, “High” with payoffs = (40,60).
A
Hig h
B
Hi gh
Lo
30,30
50,35
B
40,60
Lo w gh
Lo w
20,20
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11/4/20
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Microeconomics | EE 311 | Faculty of Economic, Thammasat University
2.c) The results from 2.b) suggest that Firm A will gain
= 10 (= 50 – 40) if it has the head start. So it will not pay more than 10 to get the head start. Firm B, on the other hand will gain = 25 = (60 – 35) which is larger.
Therefore Firm B will be willing to spend more to speed things up. Since the information is known to both firms,
Firm B should only spend enough to discourage Firm A which would be an amount slightly more than 10and
Firm A will not spend it.
3.
From the payoff matrix
3.a) There are two Nash equilibria: “Low”, “High” with payoffs = (900,600) and “High”, “Low” with payoffs =
(100,800).
3.b) The Maximin equilibrium is “High”, “High” with payoffs = (50,50). The explanation is similar to those given in 2, so it will not be repeated again.
3.c) The cooperative outcome should maximize the summation of the two firms’ payoffs which is = 1,500 (=
900 + 600) when Firm 1 uses “Low” and Firm 2 uses
“High”. If both firm play a non-cooperative game, the result will be either one of the Nash equilibria given in
3.a). Note that Firm 1 will gain = 800 (= 900 – 100) if both firms cooperate while Firm 2 will lose = –200 (=
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11/4/20
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Microeconomics | EE 311 | Faculty of Economic, Thammasat University
600 – 800). Hence, Firm 1 needs to offer at least 200 to persuade Firm 2 to cooperate. And since Firm 1 stands to gain more than what it needs to pay (800 >
200), Firm 1 will have the incentive to do so.
Chayun Tantivasadakarn |
11/4/20
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