Collusion with capacity constraints under a sales maximization rationing rule

In this paper, we study full collusion (total payoff maximization) in the repeated Bertrand duopoly with capacity constraints. Instead of a standard rationing rule, Efficient rule (E rule), we introduce a sales maximization rationing rule. Under this rule, when the demand of a firm with a lower price exceeds its capacity, the consumers who are willing to buy at that price are rationed to that firm according to their unwillingness to buy. Then, we investigate whether the full collusion can be sustained or not by an equilibrium under our rule. We have four main results. First, we find that unless each firm's capacity is too large, an asymmetric price pair maximizes one shot total payoffs and the maximum total payoff is strictly greater than the one under E rule. Second, we explicitly find a minimum discount factor under which the full collusion can be sustained along a simple path such that the firms alternate two asymmetric price pairs. Third, we find that there exists a range of capacity constraints within which the minimum discount factors above which the full collusion can be sustained are lower under our rule than under E rule. This implies that the payoff of the full collusion, which is greater than under E rule, can be sustained within a wider range of discount factors rather than under E rule. Fourth and finally, we show that there exists the interior optimal capacity which maximizes the total payoffs of the full collusion, and the total payoff is strictly greater than the profit of a monopolist with aggregate capacities. This implies that sufficiently patient rms intend to reduce their capacities to just the optimal level when they have extra capacities, and that each middle-size firm prefers to be independent, instead of being horizontally integrated.