Discrete choice models are suitable for representing a consumer’s micro behavior.  However, typical discrete choice models explain only part of consumer behavior because their use involves making an a priori assumption about the total demand, which is exogenous.  This has been repeatedly argued in the literature, but no clear theoretical foundation for the argument has been developed.

    The purpose of this paper is to formulate a model in which discrete choice models are incorporated consistently into the full utility maximization framework and to establish a theoretical foundation for the discrete choice model that assumes no a priori total demand.  In our framework, the results from discrete choice models are explained consistently whether the total demand is exogenous or endogenous: the case of an exogenous total demand is a special case.  The implications of discrete choice models are also clarified, because the form of the utility function of the representative consumer, the own-price and cross-price elasticities, and the method of measuring welfare are derived in such a way that our results are directly comparable with standard microeconomic utility maximization.

    Focusing on the logit and generalized extreme value (GEV) models and their mixed forms, because it yields analytically closed-form demand functions, we obtain the following main results.

    First, if a representative consumer’s choice is represented by the logit model, his or her demand function for a good has the form of the market demand function for a group of goods multiplied by the choice probability of the good.  (The market demand function for a group of goods is the sum of the market demands across all goods.)  This form of the demand function is obtained when the indirect utility function of the representative consumer follows Gorman’s (1961) framework and incorporates its restriction, and when the log-sum term is incorporated as the price index.  Making the market demand for a group endogenous affects the own-price and cross-price elasticities of the market demand for a good.  The price elasticity of the market demand for a group of goods is added to the usual own-price and cross-price elasticities in ‘classical’ logit models.  Throughout the paper, we use the word ‘classical’ to represent a situation in which a consumer is assumed to make a single selection among a set of mutually exclusive alternatives.  The change in welfare can be measured by using any of the four levels of demand: a consumer’s demand for a good, the market demand for a good, a consumer’s demand for a group of goods, or the market demand for a group of goods.  If we measure the change in welfare by using a consumer’s demand for a good or the market demand for a good, the corresponding price is the price of the good.  If we measure the welfare change by using a consumer’s demand for a group of goods or the market demand for a group of goods, the corresponding price is the log-sum term.  The change in welfare in the classical logit model is typically calculated as the difference in the log-sum term multiplied by the total demand; this method is a special case of our analysis.

    Second, our analysis can easily be extended to cases in which goods are classified into multiple groups.  In this case, we can construct a model in which the choice within each group, such as between food and clothes, is represented by the logit model but the choice among groups is subject to any relationship.  Not restricting relationships between groups is an advantage of our formulation.  For example, the nested-logit model incorporates the grouping of goods, but the relationship between groups is limited to the logit.

    Third, analyzing the mixed logit model requires modification.  In the mixed logit model, each consumer has his or her own parameters.  This implies that the log-sum term, which represents his or her price index, differs among consumers.  Hence, the indirect utility function of the representative consumer must be quasi-linear because all consumers must have the same income coefficient for their Gorman-form indirect utility function.  Similar results are obtained with regard to elasticities: the price elasticity of the market demand for a group is added to the usual own-price and cross-price elasticities.  The change in welfare can be measured by using a consumer’s demand for a good, the market demand for a good, or a consumer’s demand for a group.  A difference from the logit model is that one cannot calculate the change in welfare by using the market demand for a group.  The reason for this is that each consumer has his or her own parameters, the log-sum term differs by consumer; no price index exists that corresponds to the market demand for a group of goods.  As in the case of the logit model, our analysis can easily be extended to the case in which goods are classified into multiple groups and to the case of the mixed GEV model.