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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 High price

Firm 1 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

1 2 2

M

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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: 1

C

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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)

1

12 1

MC

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:

1 1C NC

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3.- Bertrand’s supergame

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:

( ) es the market demand

( ) is the supply of the competitive fringe

D p

F p MgC

( ) ( )EDD D p F p

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4.- Fringe of competition model

Dominant firm behaviour:

( ) ( ) ( ) ( )maxp

p D p F p C D p F p

C.P.O. ( ) ( ) 0d D F C D F

D p F p pdp p p q p p

0C D F

D F pq p p

0D F

p MgCD Fp p

0D F

p MgCD p D F p Fp D p p F P

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4.- Fringe of competition model

0D F

p CMg pD p F p

D Fp D p F

Price elasticity of demand D

D pp D

Price elasticity of supply of the competitive fringe F

F pp F

And then we define:

1 //D F D F

p MgC D F F Dp D F F D

1 F

D F F

p MgC sPCM

p s

We define the market share of the fringe of competitve firms as sF=F/D

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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

DFiMgC F MgC

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

p

q

pA

pB

D

F=∑MgCi

DDF

MgRDF

pDF MgCDF

qDF

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4.- Fringe of competition model

p

q

pA

pM

pB

D

DDF

F=∑MgCi

MgRMgRDF

MgCDF

qDF

Comparison with the monopoly

pDF < pM

qDF < qM

pDF

qM