1 Topic 2 (continuation): Oligopoly Juan A. Mañez

2 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 priceHigh price Firm 1Low price0,0140, -10 High price-10,140100, 100 Prisoner’s dilemma:

3 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:

4 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).

5 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 p 1 = p 2 = p M such that

6 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 p 1 < p M, then firm 2 sets p 2 =c from round t+1 onwards (and viceversa).  If any firm deviates from p 1 = p 2 = p M it breaks the collusive agreement and from then onwards p=c.

7 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 p 1 = p M in each round, it will obtain Π M /2 profits each round.  If δ is a parameter that represents firms’ preference for the future, total discounted profits from cooperating every period are:

8 3.- Bertrand’s supergame  Total discounted profits from cooperating: the present value of the stream of profits from cooperating is (0<δ<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 p 1 = p M – ε, then the round it deviates it obtains Π 1 ≈ Π M and in every future period, p 1 =p 2 =c y Π 1 =Π 2 =0.  Therefore the present value from the stream of profits from not cooperating is:

9 3.- Bertrand’s supergame  Thus, firm 1 (and by simmetry firm 2) will respect the collusive agreement as long as:

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 p 1 = p 2 = p M  and so the Bertrand Paradox is solved.  The equilibrium solution p 1 = p 2 = p M is only one of the possible solution. Actually, firms could agree in any price between c y p M  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.

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:

Fringe of competition model  Dominant firm behaviour:

Fringe of competition model And then we define: We define the market share of the fringe of competitve firms as s F =F/D

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 s F Inversely related ε F and ε F

Fringe of competition model  Figure:  We make an additional assumption: the dominant firms is more efficient than the competitive fringe firms  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): D ED = D(p) – F(p)  For prices higher than p A the dominant firm demand is 0 (F>D)  For prices lower than p B the dominant firm demand is equal to the market demand (F=0).

Fringe of competition model p q pApA pBpB D F=∑MgCi D DF MgR DF p DF MgC DF q DF

Fringe of competition model p q pApA pMpM pBpB D D DF F=∑MgCi MgR MgR DF MgC DF q DF Comparison with the monopoly p DF < p M q DF < q M p DF qMqM