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```ECON191 (Spring 2011)
11 & 13.5.2011 (Tutorial 11)
Chapter 12 Monopolistic Competition and Oligopoly (Part 2)
Price competition with homogenous products: The Bertrand duopoly model
[Simultaneous move price setting duopoly]

Bertrand model: Oligopoly model in which firms produce a homogeneous good, each
firm treats the price of its competitors as fixed, and all firms decide
simultaneously what price to charge.

Each firm assumes its rivals will keep their price level constant when it changes its own
price.
The model
 Residual demands for Firm 1:
D1 ( p1 , p 2 )  D( p1 ) if p1  p 2
D1 ( p1 , p 2 )  12 D( p1 ) if p1  p 2
D1 ( p1 , p 2 )  0 if p1  p 2

Residual demands for Firm 2:
D2 ( p1 , p2 )  D( p2 ) if p1  p 2
D2 ( p1 , p2 )  12 D( p2 ) if p1  p 2
D2 ( p1 , p2 )  0 if p1  p 2
The Bertrand equilibrium
 Bertrand equilibrium: if the two firms have identical marginal cost equal to c, then the
Bertrand equilibrium price is equal to c.
 p1 = p2 = c (stable equilibrium)
 Suppose p1 > p2 > c
 Firm 2 captures the whole market and earns profits
 Firm 1 will match or undercut firm 2’s price
 p2 > p1’ > c
 Firm 1 captures the whole market and earns profits
 Firm 2 will match or undercut firm 1’s price
 p1’ > p2’ > c
 Firm 1 will match or undercut firm 2’s price
 The adjustment will continue until p1 = p2 = c
 Any pair of unequal prices cannot be the Bertrand equilibrium
 The situation p1 = p2 > c is not stable.
 As long as there is profit for the one who can capture the whole market, both firms will
have incentive to undercut the other firm’s price.
 The outcome in Bertrand equilibrium is the same as the perfectly competitive market.
 No firm can earn profit.
1
Representing the Bertrand duopoly game by algebra
Market demand:
p  a  (q1  q 2 )
Cost functions:
c(q1) = cq1, c(q2) = cq2
As pB = c in the equilibrium,
 Marginal cost = c for both firms
p B  a  (q1  q2 )
q1  q2  a  p B
q1  q 2  a  c
Since the two firms will share the market when their prices are equal, therefore,
ac
B
B
q1  q 2 
2
B
B
1   2  0
Price competition with differentiated products
 Market shares are now determined not just by prices, but by differences in the design,
performance, and durability of each firm’s product.
 In these markets, more likely to compete using price instead of quantity
Example:
 Duopoly with fixed costs of \$20 but zero variable costs. Firms face the same demand
curves:
Firm 1’s demand: q1  12  2 p1  p2
Firm 2’s demand: q2  12  2 p2  p1
 Quantity that each firm can sell decreases when it raises its own price but increases when
its competitor charges a higher price
Firm 1’s maximizing problem:
Max  1  12 p1  2 p12  p1 p2  20
0 p1 
FOC: 12  4 p1  p2  0
p1* 
Firm 2’s maximizing problem:
12  p 2*
(Firm 1’s reaction function)
4
Max  2  12 p2  2 p22  p1 p2  20
0 p2  
FOC: 12  4 p2  p1  0
p 2* 
12  p1*
(Firm 1’s reaction function)
4
Solving the two reaction functions for the equilibrium, it yields,
p1*  p2*  4
q1*  q2*  8
q1*  q2*  16
Profit for Firm 1 = Profit for Firm 2 = 12
2
 Collusion in price competition
 They both decide to charge the same price that maximized both of their profits
 Firms will charge \$6 and will be better off colluding since they will earn a profit of \$16
(Try to verify!)
Representing price competition (differentiated products) by algebra
The two firms are facing the same demand curve: q1 = a – bp1 + cp2 and q2 = a – bp2 + cp1
Also assume MC = 0
Firm 1’s maximizing problem: Max  1  ap1  bp12  cp1 p2
0 p1 
FOC: a  2bp1  cp2  0
a  cp 2*
p 
(Firm 1’s reaction function)
2b
*
1
Firm 2’s maximizing problem: Max  2  ap2  bp22  cp1 p2
0 p2 
FOC: a  2bp2  cp1  0
a  cp1*
p 
(Firm 1’s reaction function)
2b
*
2
Solving the two reaction functions for the equilibrium, it yields,
a
ab
2ab
, q1*  q2* 
, q1*  q2* 
p1*  p2* 
2b  c
2b  c
2b  c
Representing the Collusion model in price competition
2
2
 Total profit:  12  p1q1  p2 q2  ap1  bp1  cp1 p2  ap2  bp2  cp1 p2
 The maximizing problems of the two firms are as follows:
Max  12  p1q1  p2 q2  ap1  bp12  cp1 p2  ap2  bp22  cp1 p2
0 p1 
Firm 1:
FOC: a  2bp1  cp2  cp2  0
Max  12  p1q1  p2 q2  ap1  bp12  cp1 p2  ap2  bp22  cp1 p2
0 p2 
Firm 2:
FOC: a  2bp2  cp1  cp1  0
a
a
 Solving for the equilibrium, it yields p1*  p2* 
, q1*  q2*  , q1*  q2*  a
2
2(b  c)
3
Competition VS Collusion: The Prisoner’s Dilemma
 Dominate strategy: when one strategy is best for a player no matter what strategy the
other player uses.
 We will explain these concepts with the classic example of Prisoner’s Dilemma.
Example: Prisoner’s Dilemma
The story:
Ann and Bob have been caught stealing a car. The police suspect that they have also robbed
the bank, a more serious crime. The police has no evidence for the robbery, and needs at least
one person to confess to get a conviction.
Ann and Bob are separated and each told:
(i)
If each confesses, then each will get a 10 year sentence.
(ii)
If one confesses, but the other denies, then he will get 2 year and his accomplice will
get 12 yrs.
(iii)
If neither confesses, then each will get a 3 year sentence for auto theft.
 We will represent the prisoner’s dilemma with normal form.
Bob
Ann
Confess
Deny
Confess
-10, -10
-12, -2
Deny
-2, -12
-3, -3

Is there any dominated strategy for Ann and Bob?







Let’s consider Ann,
If Ann expects Bob to confess, then Ann should confess. (–10  –12)
If Ann expects Bob to deny, then Ann should confess. (–2  –3)
Ann gets a higher payoff with confess than deny no matter what she expects Bob to do.
If Ann is rational, she will confess.
Formally, we say that deny is strictly dominated by confess.
Or we say that confess is a dominant strategy for Ann.

By the same way, we can find that confess is a dominant strategy and deny is dominated
strategy for Bob.

In the prisoner’s dilemma, if both players are rational, they will choose to use their
dominant strategies, Confess. The Nash equilibrium for this game is (Confess, Confess)
with a payoff of –10 for each player.

We find that the payoff for both players will be much better {–3, –3} if they both choose
deny, however in the prisoner’s dilemma the NE is (Confess, Confess).

Individual rationality does not imply socially optimal outcome in the prisoner’s dilemma
4
Nash equilibrium
 Nash equilibrium: a collection of strategies, one for each player, such that no player can
improve his situation by choosing a different strategy that is available
to him, given that all other players stay put.

In other words, the strategy (s1*, s2*) constitutes a NE if given player 1’s strategy s1*,
player 2 finds it optimal to choose s2*, and given player 2’s strategy s2*, player 1 finds it
optimal to choose s1*. (Best response)
 When the NE is reached, there is no incentive for any player to deviate from it. No player
can benefit or increase his/her payoff by deviating from the NE.
 For example, Ann would not deviate if given Bob uses his dominate strategy confess.
Deviation would lower her payoff to –12 given Bob stay puts.
Example: Cartel/ Collusion (Please refer to T10)
Firm A
Honor Agreement
Break Agreement
Firm B
Honor Agreement
Break Agreement
72, 72
54, 81
81, 54
64, 64
 In the duopoly model, price is lower than the monopoly price.
 Incentive for the two firms to collude
 They get into agreement to set a higher price, and produce less (monopoly output) in
order to have higher profit
 In this duopoly game, Break Agreement is a dominant strategy for both firms.
 NE: (Break Agreement, Break Agreement) with profit of 64 to each firm.
 This game is a prisoner dilemma
 They can both get a higher profit of 72 by following the cartel.
 The cartel is not stable, they will have incentive to cheat and deviate from the agreement
 Both firms will cheat and ends up in the Cournot equilibrium
5
Chapter 12: Problem 2
Consider two firms facing the demand curve P = 50 - 5Q, where Q = Q1 + Q2. The firms’ cost
functions are C1(Q1) = 20 + 10Q1 and C2(Q2) = 10 + 12Q2.
a.
Suppose both firms have entered the industry. What is the joint profit-maximizing level
of output? How much will each firm produce? How would your answer change if the
firms have not yet entered the industry?
If both firms enter the market, and they collude, they will set MR = MC1 to
determine the profit-maximizing output. (MC1 < MC2)
MR = 50 - 10Q = 10 = MC1  Q = 4, P = \$30
The question now is how the firms will divide the total output of 4 among
themselves.
Since the two firms have different cost functions, it will not be optimal for them
to split the output evenly between them. The profit maximizing solution is for
firm 1 to produce all of the output so that
The profit for Firm 1 will be: 1 = (30)(4) - (20 + (10)(4)) = \$60
The profit for Firm 2 will be: 2 = (30)(0) - (10 + (12)(0)) = -\$10
Total industry profit will be: T = 1 + 2 = 60 - 10 = \$50.
If they split the output evenly between them then total profit would be \$46 (\$20
for firm 1 and \$26 for firm 2). If firm 2 preferred to earn a profit of \$26 as
opposed to \$25 (\$50/2) then firm 1 could give \$1 to firm 2 and it would still
have profit of \$24, which is higher than the \$20 it would earn if they split
output.
Note that if firm 2 supplied all the output then it would set marginal revenue
equal to its marginal cost or 12 and earn a profit of 62.2. In this case, firm 1
would earn a profit of –20, so that total industry profit would be 42.2.
If Firm 1 were the only entrant, its profits would be \$60 and Firm 2’s would be
0
If Firm 2 were the only entrant, then it would equate marginal revenue with its
marginal cost to determine its profit-maximizing quantity: 50 - 10Q2 = 12, or
Q2 = 3.8. P = 50 – 5*3.8 = \$31
The profits for Firm 2 will be: 2 = (31)(3.8) - (10 + (12)(3.8)) = \$62.20
b.
What is each firm’s equilibrium output and profit if they behave noncooperatively?
Use the Cournot model. Draw the firms’ reaction curves and show the equilibrium.
In the Cournot model, Firm 1 takes Firm 2’s output as given and maximizes
profits. The profit function derived in 2.a becomes
2
1 = (50 - 5Q1 - 5Q2 )Q1 - (20 + 10Q1 ), or   40Q1  5Q1  5Q1Q2  20.

Q 
= 40  10 Q1 - 5 Q2 = 0, or Q1 = 4 - 2 .
Firm 1’s reaction function:
 2 
 Q1
6
Q 
Similarly, Firm 2’s reaction function is Q2  3.8  1 .
 2 
Q1 
1 
Solving for the Cournot equilibrium, Q1  4 
and
2 3.8  2 , or Q1  2.8.
Q2 = 2.4.
P = 50 – 5(2.8+2.4) = \$24.
The profits for Firms 1 and 2 are equal to
1 = (24)(2.8) - (20 + (10)(2.8)) = 19.20
2 = (24)(2.4) - (10 + (12)(2.4)) = 18.80
c.
How much should Firm 1 be willing to pay to purchase Firm 2 if collusion is illegal
but the takeover is not?
In order to determine how much Firm 1 will be willing to pay to purchase Firm
2, we must compare Firm 1’s profits in the monopoly situation versus those in
an oligopoly. The difference between the two will be what Firm 1 is willing to
pay for Firm 2. From part a, profit of firm 1 when it set marginal revenue equal
to its marginal cost was \$60. This is what the firm would earn if it was a
monopolist. From part b, profit was \$19.20 for firm 1. Firm 1 would therefore
be willing to pay up to \$40.80 for firm 2.
7
Chapter 12: Problem 6
Suppose that two identical firms produce widgets and that they are the only firms in the
market. Their costs are given by C1 = 60Q1 and C2 = 60Q2, where Q1 is the output of Firm 1
and Q2 the output of Firm 2. Price is determined by the following demand curve:
P = 300 – Q where Q = Q1 + Q2.
a.
Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this
equilibrium.
Firm 1’s profit function: 1  300Q1  Q12  Q1Q2  60Q1  240Q1  Q12  Q1Q2 .
 1
 240  2 Q1  Q2 . = 0
 Q1
Firm 1’s reaction function: Q1 = 120 - 0.5Q2.
Firm 2’s reaction function: Q2 = 120 - 0.5Q1 .
Solving for the Cournot equilibrium, Q1 = 120 - (0.5)(120 - 0.5Q1), or Q1 = 80.
Q2 = 80. P = 300 - 80 - 80 = \$140.
1 = (140)(80) - (60)(80) = \$6,400 and 2 = (140)(80) - (60)(80) = \$6,400.
b.
Suppose the two firms form a cartel to maximize joint profits. How many widgets will
be produced? Calculate each firm’s profit.
Given the demand curve P = 300-Q  MR=300-2Q
MR = 300 – 2Q = 60 = MC
Q = 120, P = 180
Each firm produces 60
Profit for each firm is:  = 180(60)-60(60)=\$7,200.
c.
Suppose Firm 1 were the only firm in the industry. How would the market output and
Firm 1’s profit differ from that found in part (b) above?
MR = 300 – 2Q = 60 = MC
Q = 120, P = 180
Profit = \$14,400.
d.
Returning to the duopoly of part (b), suppose Firm 1 abides by the agreement, but Firm
2 cheats by increasing production. How many widgets will Firm 2 produce? What will
be each firm’s profits?
Assuming their agreement is to split the market equally, Firm 1 produces 60
widgets. Firm 2 cheats by producing its profit-maximizing level, given Q1 = 60.
60
 90.
Given Q1 = 60 into Firm 2’s reaction function: Q2  120 
2
Total industry output, QT, is equal to Q1 plus Q2: QT = 60 + 90 = 150.
P = 300 - 150 = \$150.
1 = (150)(60) - (60)(60) = \$5,400 and 2 = (150)(90) - (60)(90) = \$8,100.
8
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