Optimal Pricing to Retard Entry

In analogy to the notion of delayed gratification, economists have long been aware that profit maximization entails taking account of the entire planning horizon with a mind to balance immediate profits against long term profits. As applied to the oligopolist, Bain (1949) noted that the threat of competition might cause established firms to sacrifice current profit in order to preclude entry and the concomitant competition. However, in defining the " limit price " as the highest price the established firms can set without inducing entry, it is clearly revealed that Bain viewed entry as a deterministic event. The first probabilistic approaches to the entry issue were suggested by Williamson (1963) and Stigler (1968) where it was noted that the probability of entry is an increasing function of the product price and that retarding rather than precluding entry is associated with the profit maximizing strategy. More recently, Kamien and Schwartz (1971), Baron (1973), Deshmukh and Chikte (1976), and Deshmukh and Winston (1979) have treated models with probabilistic entry. In the model presented here the industry is composed of (nearly) identical firms selling the same product and experiencing the same constant average cost of production, with industry demand divided amongst the established firms. A product price must be selected at each instant of time during the time horizon. Entry occurs according to a Poisson process with the arrival rate at any given point in time an increasing function of the product price at that time; thus, any number of firms can enter during the time horizon. The profit rate is a function of both the current product price and the current industry size. The crucial feature of this model as well as in those cited above is the fact that there is a dominant firm or price leader or, equivalently, the established firms (industry) act collectively as a cartel to maximize industry profits. Competition appears not within the industry but rather with potential entrants. The problem is to find the prices, termed optimal, that maximize the expected discounted value of the price leader's profits over the time horizon as a function of the industry size (which increases with the passage of time). Our model is nearly that of Deshmukh and Chitke (1976) and Deshmukh and Winston (1979). The fundamental difference between our model and Baron's discrete time model is that there the immediate profit is a function of the product price only if there is no entry during the period. The basic model is presented in Section 2, whereas the fundamental result, one not found elsewhere in the literature, regarding the nature of the optimal price as a function of the industry size is presented in Section 3. There we demonstrate that the optimal price increases with the number of firms in the industry and in the limit converges to the monopolistic price. That is, in the presence of a price leader, a decrease in industry concentration leads to an increase in the product price rather than to a more " competitive " price. Of course, this result depends upon the price leadership being maintained even as the industry grows. (See Telser (1972, Chapter 5) for a discussion of the problems associated with cartel maintenance.) When we allow the entry rate to be a decreasing function of the industry size as well as the price, we can show only that the optimal price converges to the