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The repeated Cournot game &amp; Oligopoly Practice question

# The repeated Cournot game &amp; Oligopoly Practice...

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The repeated Cournot game Equilibrium of the Cournot game exists where the firms collude to produce the monopolist level of output. Firms do not cheat due to the threat of the other firm not colluding in future repetitions of the game. Cooperation is achieved under the threat of future punishment. q A =64 q A =48 q U =64 q U =48

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Strategy 1- Eternal Punishment: Cooperate in the first round. Then, Cooperate if the other firm Cooperated in the past. Otherwise, do Not Cooperate. C = cooperate collude, producing ½ monopoly quantity q = 48 NC = not cooperate, cheat not collude, producing Cournot q q=64 In other words, this strategy says to collude and produce ½ monopoly quantity as long as the other firm does so. However, if one firm ever cheats and produces more than ½ monopoly q, then the other firm will never collude again and produces q=64, the Cournot quantity, in all subsequent periods. Then we could have the following stream of strategies if both firms cooperate (collude for q=48) in all periods: Period 1 Period 2 Period 3 Period 4 Period 5 Etc…… (C,C) (C,C) (C,C) (C,C) (C,C) (C,C) COOPERATION PAYOFF: And the payoffs for each firm would be: 4.6 + δ4.6 + δ 2 4.6 + δ 3 4.6 + δ 4 4.6 + δ 5 4.6 + …. . (continuing infinitely) where δ is the discount factor per period (δ <1) representing the fact that future income is worth less to a person than current income. This can be re-written as: 4.6(1 + δ + δ 2 + δ 3 + δ 4 + δ 5 + …. .) = 4.6/(1- δ) using infinite sums from calculus
deviated from this strategy, and cheated in the first period by producing q = 64 while United cooperated and produced q = 48. Then we could have the following stream of strategies: Period 1 Period 2 Period 3 Period 4 Period 5 Etc…… (NC,C) (NC,NC) (NC,NC) (NC,NC) (NC,NC) (NC,NC) CHEATING PAYOFF: American would have the following stream of payoffs: 5.1 + δ4.1 + δ 2 4.1 + δ 3 4.1 + δ 4 4.1 + δ 5 4.1 + …. . = 5.1 + δ4.1(1 + δ + δ 2 + δ 3 + δ 4 + δ 5 + …. .) = 5.1 + δ4.1/(1- δ) When American cheats, United would have the following stream of payoffs: 3.8 + δ4.1 + δ 2 4.1 + δ 3 4.1 + δ 4 4.1 + δ 5 4.1 + … ***When will American decide to cooperate/collude, and never cheat/not cooperate? Collusion can be sustained when the payoff to always cooperating (C,C,C,…) is greater than the payoff to cheating (NC), or: 4.6/(1- δ) ≥ 5.1 + δ4.1/(1- δ) Rearranging the expression above, cooperation/collusion can be sustained when: δ ≥ 0.5 Cooperating is better than deviating when δ ≥ 0.5. δ tells us how much someone values the future. So δ closer to 1 means that a lot of value placed on the future, so that the future isn’t discounted much. People are patient when δ is close to one.

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The repeated Cournot game &amp; Oligopoly Practice...

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