Reformulations and Conclusions
Throughout this article, we have repeatedly encountered the problem of nonexistence of a price and/or location equilibrium. In this last section, we briefly discuss two recent reformulations of the basic model which aim at solving this problem. The first one involves altering the assumption that consumers necessarily buy from the firm with the lowest full price. The second one supposes that firms bear the cost of transporting the product and set discriminatory prices. 67
(i) At the root of several nonexistence results lies the standard assumption that consumers patronize the cheapest firm. Now,
empirical evidence supports the idea that consumers also take variables other than full price into account. Because of the unobservability of these variables, firms can at best determine the shopping behavior of a particular consumer up to a probability distribution. More precisely, following de Palma et al. , it is now assumed that firms model the utility of a consumer at sand purchasing from firm i at Si as a random variable
Uis = -Pi - t(s, Si) + ei' where ei is a random variable with zero mean. Since consumers maximize utility, firms then evaluate the probability P;s that a consumer at S will purchase from firm i as
Pis = pr[uis = .p1ax UjsJ. }-l, ... ,n If we assume that the variables ei are identically, independently Weibull-distributed, then we obtain the logit model (see McFadden ), i.e.,
where (J is the standard-deviation of e (up to a multiplicative constant).68 The function Pis is depicted in Figure 19 for different values of (J in the case of 2 firms selling at the same given price.