Discrete Choice and Variable Returns to Scale

Abstract

In previous studies of the probabilistic theory of qualitative choice (e.g., Marschak [1960] and McFadden [1974]), it has been assumed that an individual chooses among various alternatives with their utility levels independent of the number of individuals who choose each alternative. There are, however, many examples in which the utility of a choice depends on the number of individuals who make the same choice. For instance, the utility of a given spatial choice may depend upon the number of others who choose neighboring locations: neighborhood utility may decrease (crowding) or it may increase (agglomerative economies) with population size. A second example is the automobile traveler's choice of routes. The benefit of one route choice over another, due to congestion, depends on the number of drivers who choose the same route. Both of the examples suggested above deal with spatial choices of a particular kind, but there are even examples from the more traditional macroeconomic theory. One such example is of a product produced under variable returns to scale by a number of different sellers. The price that each seller charges will depend upon the number of his customers. If a buyer must choose to purchase from only one seller, the price paid and, thus, utility, will depend upon the total number of buyers choosing the same seller. In all of the examples cited, a reasonable model of choice is one that postulates uncertainty about the actual utility of each alternative. The point of view is that a chooser's notional utility about each choice is a random variable. In this case it is not certain which alternative will be chosen, but the chosen alternative is the one with the greatest perceived utility. It follows, therefore, that the probability that a given alternative is chosen is the probability that the utility perceived to be associated with it is larger than the utility perceived to be associated with any other alternative.