Price and Capacity Competition of Make-To-Order

Abstract

In this paper, we consider the (non-cooperative) competition of a duopoly of make-to-order firms, each modeled by an M/M/1 queue. The firms post the prices they charge for service. Knowing the expected reward and the expected waiting time at each firm, potential customer choose either to receive service at one of the firm or to seek an alternative opportunity by maximizing their expected net rewards. We consider both the short-run competition and the long-run competition. In the short run, the service capacities of both firms are given, and the firms maximize their expected revenues by choosing prices only. In the long run, the firms maximize their expected profits by choosing both prices and service capacities. In the short run, a Nash equilibrium may not exist, and when it exits, it may not be unique. Given all else being the same, a firm with a large capacity, or a higher value of service or a lower cost of waiting can charge a price premium and take a larger market share. The firms may either raise or cut their optimal prices, in response to an increase in their service capacity, even though a higher service capacity always leads to a shorter expected waiting time. The firms may have either a higher or a lower optimal price for a higher unit cost of waiting. In the special case of homogeneous ( or identical ) firms, the equilibrium always exists. In the short run, not all potential customer are served, but in the long run, all potential customers will be served unless none of the firms choose to enter the market.