A dynamic model of duopoly with customer loyalties

Abstract

Consider a static game model of duopoly in which price is the strategic variable, production is costless, capacity is limitless, and all sales are made by the low-price setter according to some fixed, bounded demand function. It is well known that in such a model there is a unique Cournot/Bertrand-Nash equilibrium which involves both sellers setting a price of zero and earning zero l&otits. Neither the model nor the outcome seems to be a good description of reality, and many other models have been proposed in the literature as more realistic and for which more interesting strategic conclusions can be reached (see, for example, Friedman [2] for a discussion of some of these). In this paper I consider a modification of the above model which is motivated by speculations about markets, such as that for automobile insurance, which are characterized by repeat-buying of a homogeneous commodity which is supplied by a few sellers and expenditures on which form a relatively small part of each buyer’s budget in any time period. If expenditures by each buyer are small (or if buyers expect that there is not a great difference between the prices set by different sellers), buyers are not likely to expend many resources shopping for the best price or making complicated strategic decisions. Furthermore, the repeat-purchase aspect suggests that the seller from whom the buyer previously purchased has a natural advantage if information about price (or quality) for the entire market is not free. The model which I propose here reflects these considerations. Specifically, buyers do not optimize. Instead they act according to a simple rule-of-thumb. Each buyer purchases from the same seller from whom he purchased in the last period unless that seller has raised his price, in which case the buyer purchases from the current-period low-price setter. In either case the buyer purchases an amount which is a fixed function of the price he faces (independent of time, the identity of the seller, and history). Assuming such 69 0022-053 l/82/030069-08.$02.00/O