A multiple discrete–continuous extreme value model: formulation and application to discretionary time-use decisions

Several consumer demand choices are characterized by the choice of multiple alternatives simultaneously. An example of such a choice situation in activity-travel analysis is the type of discretionary (or leisure) activity to participate in and the duration of time investment of the participation. In this context, within a given temporal period (say a day or a week), an individual may decide to participate in multiple types of activities (for example, in-home social activities, out-of-home social activities, in-home recreational activities, out-of-home recreational activities, and out-of-home non-maintenance shopping activities). In this paper, we derive and formulate a utility theory-based model for discrete/continuous choice that assumes diminishing marginal utility as the level of consumption of any particular alternative increases (i.e., satiation). This assumption yields a multiple discreteness model (i.e., choice of multiple alternatives can occur simultaneously). This is in contrast to the standard discrete choice model that is based on assuming the absence of any diminishing marginal utility as the level of consumption of any alternative increases (i.e., no satiation), leading to the case of strictly single discreteness. The econometric model formulated here, which we refer to as the multiple discrete-continuous extreme value (MDCEV) model, has a surprisingly simple and elegant closed form expression for the discrete-continuous probability of not consuming certain alternatives and consuming given levels of the remaining alternatives. To our knowledge, we are the first to develop such a simple and powerful closed-form model for multiple discreteness in the literature. This formulation should constitute an important milestone in the area of multiple discreteness, just as the multinomial logit (MNL) represented an important milestone in the area of single discreteness. Further, the MDCEV model formulated here has the appealing property that it collapses to the familiar multinomial logit (MNL) choice model in the case of single discreteness. Finally, heteroscedasticity and/or correlation in unobserved characteristics affecting the demand of different alternatives can be easily incorporated within the MDCEV model framework using a mixing approach. The MDCEV model and its mixed variant are applied to analyze time-use allocation decisions among a variety of discretionary activities on weekends using data from the 2000 San Francisco Bay Area survey.