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Topic 2 (continuation): Oligopoly
Juan A. Mañez
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3.- Bertrand’s supergame
 The competitive result of the Bertrand is due to its static nature.
 In the Bertrand equilibrium no firm has an incentive to deviate from
p=MgC.
 However, firms anticipate that they could be better off cooperating,
i.e. they could set a p>MgC , sharing the market and obtaining
positive profits
 An example of the Bertrand model in game form is :
Firm 2
Low price
Firm 1
High price
Low price
0,0
140, -10
High price
-10,140
100, 100
Prisoner’s dilemma:
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3.- Bertrand’s supergame
 Let us consider a supergame (game with a large number of
rounds):
 Firms play the former game a large number of rounds.
 Playing a large number of rounds allows firms to figure out
strategies that ease collusion.
 How is it possible to obtain the cooperative solution in the
Bertrand duopoly?
 We will consider two cases:
1. Finite supergame:
2. Infinite supergame:
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3.- Bertrand’s supergame
1.- Finite supergame: we solve by backwards induction
 Round T (last round of the game): there is no future, there is
no possibility to design strategies that could ease cooperation
and penalize deviations.
 Round T-1: round T has been already played, and there is no
future.
 Round T-2, Round T-3, … Round 1: in each stage, both
firms choose Low-price.
Subgame perfect equilibrium: en each round, firms choose
Low-Price (Bertrand’s result).
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3.- Bertrand’s supergame
2.- Infinite supergame
 The possibility of retaliation in the future makes possible the
existence of a cooperative equilibrium.
 Let us consider that players play the following dynamic game
an infinite number of rounds (times):
 Every round firms set their price simultaneously.
 Firms’ marginal costs is constant an equals c.
 The collusive agreement implies setting a price p1 = p2 = pM
such that
M
1   2 
2
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3.- Bertrand’s supergame
2.- Infinite supergame.
 Let us assume that both firms adopt a trigger strategy:
 Using an example, if in round t firm 1 sets p1 < pM , then
firm 2 sets p2 =c from round t+1 onwards (and viceversa).
 If any firm deviates from p1 = p2 = pM it breaks the
collusive agreement and from then onwards p=c.
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3.- Bertrand’s supergame
2.- Infinite supergame
Which is the best strategy for the firms?
 As both firms strategies are identical, we determine the
best strategy for one of the firms and by symmetry, this
will be also the best strategy for the other firm.
1. Calculation of the profit associated to cooperate:
 If firm 1 cooperates and sets p1 = pM in each round, it will
obtain ΠM/2 profits each round.
 If δ is a parameter that represents firms’ preference for the
future, total discounted profits from cooperating every
period are:
C
1 
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3.- Bertrand’s supergame
 Total discounted profits from cooperating: the present
value of the stream of profits from cooperating is (0<δ<1)
M

C1 
2
 1 
1   


2. Calculation of firm’ profits if it deviates from the
collusive agreement:
 If firm 1 deviates from the collusive agreement and sets a
price p1 = pM – ε , then the round it deviates it obtains Π1≈ ΠM
and in every future period , p1=p2=c y Π1=Π2=0.
Therefore the present value from the stream of profits from
not cooperating is:
1NC 
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3.- Bertrand’s supergame
 Thus, firm 1 (and by simmetry firm 2) will respect the
collusive agreement as long as:
C
NC
1  1
1
 
2
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3.- Bertrand’s supergame
 
1
2
 When this condition is fulfilled (when the firm gives an important
value to future profits) the subgame perfect equilibrium is to
cooperate in every round, i.e. setting p1 = p2 = pM
 and so the Bertrand Paradox is solved.
 The equilibrium solution p1 = p2 = pM is only one of the possible
solution. Actually, firms could agree in any price between c y pM
 FOLK Theorem:
 Conclusion: it will be possible to obtain the cooperative solution in a
Bertrand game when it is an infinite horizon game and firms give
value enough to the future profits.
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4.- Fringe of competition model
 It refers to a market in which there exists a dominant firm
(largest or more efficient) and fringe of small firms
 Assumptions:
1.- Fringe of competition: group of small firms that act as price
takers  they do not any ability to influence in the market price
(p=MgC)
2.- Dominant firm:
 It has ability to set prices
 It takes the strategy of the competitive fringe firms as
given: for any price set by the dominant firm its residual
demand is given by:
DED  D (p )  F (p )
D ( p ) es the market demand
F ( p )   MgC is the supply of the competitive fringe
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4.- Fringe of competition model
 Dominant firm behaviour:
max
p
  p D ( p )  F ( p )   C D ( p )  F ( p ) 
 D F  C
d
 D (p )  F (p )  p 



dp
 p p  q

C   D F 
D  F  p 

0



q   p p 

C.P.O.
p  MgC  
 D F 
 p  p   0


D F
0
D F

p p
D F
p  MgC  
0
D p D F p F

p D p p F P
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4.- Fringe of competition model
p  CMg   p
D F
D p
F p
D
F
p D
p F
And then we define:
Price elasticity of demand  D  
0
D p
p D
Price elasticity of supply of the competitive fringe   F 
F p
p F
p  MgC
D F
1  F /D


p
D D  F F D  F F / D
We define the market share of the fringe of competitve firms as
sF=F/D
PCM 
p  MgC
1  sF

p
D  F s F
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4.- Fringe of competition model
Dominant firm:
Monopoly:
 In the competitive fringe model, the monopoly power
(measured by the price cost margin) of the dominant
firms is smaller than the one of a monopolist
 Market power is smoothed (attenuated) by the existence
of the competitive fringe
 Comparative statics analysis: The dominant firm
market power is :
 Inversely related to sF
 Inversely related εF and εF
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4.- 4.- Fringe of competition model
 Figure:
 We make an additional assumption: the dominant firms is
more efficient than the competitive fringe firms
MgC DF  F  MgC i
 The residual demand for the dominant firm is the
difference between the market demand and the
competitive fringe supply (i.e. that part of the demand
that is not supplied by the competitive fringe):
 DED = D(p) – F(p)
 For prices higher than pA the dominant firm demand is 0
(F>D)
 For prices lower than pB the dominant firm demand is
equal to the market demand (F=0).
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4.- Fringe of competition model
F=∑MgCi
p
D
pA
pDF
DDF
MgCDF
pB
MgRDF
qDF
q
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4.- Fringe of competition model
F=∑MgCi
p
D
Comparison with
the monopoly
pDF < pM
pA
qDF < qM
pM
pDF
DDF
MgCDF
pB
MgRDF MgR
qDF
qM
q
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