International Choice Modelling Conference, International Choice Modelling Conference 2015

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A Discrete-Continuous Choice Model with Perfect and Imperfect Substitutes
Abdul Pinjari, Vijayaraghavan Sivaraman

Last modified: 18 May 2015


Many consumer choice situations involve decisions of “what to choose” from a set of discrete goods (or alternatives) along with the decisions of “how much to consume” of the chosen good(s). Such discrete-continuous choices are pervasive in consumer decisions and of interest in a variety of social sciences, including transportation, economics, and marketing. Examples include households’ choice of vacation destinations and corresponding time allocation, and grocery shoppers’ choice of brand choice and purchase quantity.

A variety of approaches have been used in the literature to model discrete-continuous choices. Among these, a particularly attractive approach is based on the classical microeconomic consumer theory of random utility maximization (RUM). Specifically, consumers are assumed to optimize a direct utility function over a bundle of non-negative consumption quantities (of the choice alternatives) subject to a budget constraint on the resources used for consumption (e.g., time and/or money). In such direct-utility based RUM models, the function form of the utility specification determines the properties of the optimal consumption bundles. For example, if the utility function is such that the marginal utility of consumption at the point of zero consumption (i.e., baseline marginal utility) is either infinity or indeterminate for each and every good, it results in interior solution (positive consumption) for all goods. In this case, the indifference curves are asymptotic to the consumption axes precluding the possibility of corner solutions (zero consumption). On the other hand, goods with a finite value of baseline marginal utility result in indifference curves that meet the consumption axes with a finite slope and allow corner solutions.

A special case of discrete-continuous choices is the single discrete-continuous (SDC) choice, where consumers choose a single discrete alternative along with the corresponding continuous quantity decision. In such situations, the choice alternatives can be viewed as perfect substitutes where the choice of one alternative precludes the choice of other alternatives. Situations with perfect substitutes can be modeled using linear functional forms (linear with respect to consumption) for the utility function, as in Hanemann (1984).

Another case of discrete-continuous choices is when the choice alternatives are imperfect substitutes where the choice of one alternative does not necessarily preclude the choice of other alternatives. In such situations, consumers can potentially choose multiple discrete alternatives, along with the corresponding continuous quantity choice decisions. For example, a household might choose to visit multiple vacation destinations over a given time frame. Such multiple discrete-continuous (MDC) choices are being increasingly recognized and modeled in the recent literature (e.g., Bhat, 2008). Such studies employ non-linear utility functions (non-linear with respect to consumption) that allow for diminishing marginal utility and corner solutions to treat the choice alternatives as imperfect substitutes and to allow for “multiple discreteness”.

A more general case of discrete-continuous choices includes both SDC and MDC choices, where consumers choose at most a single alternative from a subset of alternatives and potentially multiple alternatives from the remaining alternatives. Such situations arise when the choice set comprises a mix of both perfect substitutes (from which no more than a single alternative could be consumed) and imperfect substitutes (from which potentially multiple alternatives could be consumed). The vast majority of choice modeling literature has been focused on analyzing SDC choices, while there recently has been growing interest in analyzing MDC choices. Not much exists in the literature on modeling consumer behavior involving both SDC and MDC choices from choice sets that comprise a mix of perfect and imperfect substitutes (except, for example, Bhat 2006, whose formulation does not consider price variation across choice alternatives). To fill this gap, the current paper formulates a unified random utility maximization (RUM) framework that can be used as a joint MDC-SDC modeling framework to analyze discrete-continuous choices from a combination of perfect and imperfect substitutable choice alternatives. The formulation also recognizes price variation across choice alternatives. The key to this formulation is a utility form that is linear with respect to consumption across perfectly substitutable alternatives and non-linear with respect to consumption across imperfectly substitutable alternatives. To do so, the paper brings together the utility formulations of Hanemann (1984) and Bhat (2008) to form a unified utility function that allows both perfect and imperfect substitutes. In addition to the RUM formulation, the paper presents a procedure to apply the proposed framework for forecasting purposes. Furthermore, the formulation is extended to accommodate multiple linear budget constraints, as opposed to a single budget constraint.

As an empirical demonstration, the proposed framework is applied to develop a joint model of annual, long-distance vacation destination and mode choices to simultaneously analyze the vacation destinations that a household visits over an entire year, along with the time allocation and the travel mode to each of the visited destinations. The empirical framework recognizes that the vacation destinations are imperfect substitutes in that a household can potentially choose to visit multiple destinations over a year, while the travel mode alternatives to a destination are perfect substitutes in that only one primary mode is chosen to travel to a destination. Also, the framework recognizes that households operate under annual time and monetary budget constraints in making their vacation choices. Empirical models are estimated using household-level, annual long-distance travel data from the 1995 American Travel Survey (ATS) collected in the United States.

A disadvantage of the above formulation is it does not recognize that visiting a destination requires fixed travel time and travel cost expenditures that do not vary with the time allocated to the destination. Most formulations and applications of RUM-based discrete-continuous models in the literature also do not distinguish such fixed costs from variable costs of consuming choice alternatives. The paper will explore the ramifications of not recognizing travel times and travel costs (to the destinations) as fixed expenditures in our formulation. In addition, alternative approaches for dealing with fixed time and cost expenditures will be discussed. The authors hope to arrive at a practically feasible approach to explicitly recognize fixed time and cost expenditures by the time of the conference.


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