# Dynamic price competition with capacity constraints

I have a dynamic price (Bertrand) competition with 2 players with the same capacity constraints. Grim trigger strategy is to set monopoly price at t=0 or if the monopoly price was set by both players at t-1, otherwise set the NE price. I have to find a discount factor such that the collusion can be sustained.

If players collude, each gets half of the monopoly quantity (which satisfies the capacity constraints), hence half of the monopoly profit.

If one of the players deviates at time t, at t+1 they return to NE (prices such that maximize quantities are set, quantities are equal to the capacity constraints).

Suppose player 1 deviates at time t. In such a game it is logical for him to do so by setting a price lower than the one set by player 2, i. e. the monopoly one. If the firms set different prices, the demand of each firm is calculated according to the efficient rationing rule. Then player 1 will sell the quantity equal to his capacity constraint, the player 2 will sell according to the residual demand D(p2)-q1.

I wonder, whether my idea of deviation in such a game is correct or not and how it is possible to find a discount factor such that the collusion can be sustained if we don’t know the exact price set by the deviating player.

I’m currently reading through an article by Biglaiser & Vettas (2004) but their results are quite different due to the different rationing rule.