Collusion with capacity constraints under a sales maximization rationing rule

Abstract

In the repeated Bertrand duopoly with capacity constraints, we introduce a sales maximization rationing rule, instead of a standard rationing rule, Efficient rule (E rule). Under our new 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. We find that the maximum one-shot total payoff under our rule is not less than that under E rule at any capacities; it is achieved by an asymmetric price pair rather than a symmetric monopoly price pair and it is strictly greater than that under E rule unless each firm’s capacity is too large. Under the one-shot total payoff maximization, total surplus under our rule is also equal to or greater than that under E rule at any capacities. Hence, our rule is also appealing to the social planner, and it is implementable due to information technology which collects information about the consumers’ willingness to pay. In the repeated game, patient firms can achieve the total payoff maximization by an equilibrium under our rule. Moreover, a range of discount factors within which the total payoff maximization can be sustained under our rule is wider than that under E rule at some capacities.