1. (Repeated Bertrand Game) Two firms are playing an infinitely repeated Bertran
ID: 1214905 • Letter: 1
Question
1. (Repeated Bertrand Game) Two firms are playing an infinitely repeated Bertrand game, each with the same marginal cost 20. The market demand function is given by P=150-Q. The firm who charges the lower price wins the whole market. When they charge the same price, each gets 1/2 of the total market. A. In the stage game (only one period), if the firms collude with each other, then what prices will they choose? B. What prices will they choose in the stage Nash equilibrium (only one period)? C. Describe the trigger strategy that can be used to support collusion in every period. D. Suppose that the two firms use the same discount factor. Under what kind of discount factor can the trigger strategy described in (C) actually support collusion in every period?
Explanation / Answer
a) In the stage game (only one period), if the firms collude with each other, then the resultant firm will be a monopoly and produces a quantity at which MR = MC. From the demand equation, MR = 150 - 2Q.
MR = MC
150 - 2Q = 20
130 = 2Q
Q = 65
P = $85
Hence the colluded firm would have charged a price of $85 per unit to produce 65 units
b) In Bertrand Oligopoly, firms engage in price competition. Hence the firms charge a price level equal to their marginal cost. So the price will be $20.
c) A trigger strategy is the one in which a player begins by cooperating but defects in any subsequent periods to cheating for a predefined period of time as a response to a defection by the opponent who initiates this, the trigger. Here both firms begin with the collusive price of $85. In any subsequent period, they will continue to charge this, if in all the previous periods both have chosen $85 per unit as a price. If however, there is a trigger by any of them then they will charge the Nash equilibrium price of $20 per unit
d) The discount factor taken as is a symbol measuring the importance given to the future payoff. Larger the discount factor, Larger will be the importance. Note that the payoff is the profit so the collusion payoff is
(/2)+(/2)+(/2)²+....= /2(1 - ).
Deviation payoffis are given believing that a deviation firm with lower price supplies the whole market.
+ 0* + 0*²+…= .
If each firm chooses collusion in every period, the collusion payoff should be at least equal to the deviation payoff: /2(1-)
1/2
This implies a that a discount factor of greater than 1/2 can the trigger strategy described in (C) actually support collusion in every period.
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