Cournot Duopoly with Infinitely Repeated Collusion: Cartel Profits and Punishment Strategies

Consider the following infinitely repeated duopoly game. Two firms compete à la Cournot. They face the inverse demand function p(q) = 5 - q, where q = q_1 + q_2. Both firms have a linear cost function C(q) = cq where c = 1. They discount future payoff at a symmetric rate δ∈[0,1].

Calculate the static cartel quantities and profits of the firms when they collude.

When the firms collude, they act as a monopoly and maximize joint profits. The joint profit function is given by π(q_1,q_2) = (5 - q_1 - q_2 - c(q_1+q_2))(q_1+q_2). Taking the first-order conditions, we get:

∂π/∂q_1 = (5 - 2q_1 - q_2 - 2c) + (q_1+q_2)(-2) = 0 ∂π/∂q_2 = (5 - q_1 - 2q_2 - 2c) + (q_1+q_2)(-2) = 0

Solving these equations simultaneously, we get:

q_1 = q_2 = (5 - 2c)/3

Therefore, the static cartel quantity is q = 2(5-2c)/3 and the static cartel price is p(q) = 5 - q = (5 + 2c)/3. The static cartel profit is π(q,q) = (5-q-cq)q = (5-3c)q^2 = 4(5-2c)^2/27.

Note that this result is the same as the monopoly profit in the one-shot game, since the firms are colluding and acting as a single entity. However, in the infinitely repeated game, the firms have an incentive to cheat and deviate from the collusive outcome.

To sustain the cartel, the firms must choose a punishment strategy that deters them from deviating. One such strategy is the trigger strategy, where the firms agree to cooperate as long as neither of them deviates, but if one firm deviates, the other firm punishes by playing the static Nash equilibrium in every period thereafter.

Assuming that δ is high enough to sustain the trigger strategy, the collusive profit is given by:

π_collusion = π(q,q)/(1-δ)

where π(q,q) is the static cartel profit.