## On the Integrability Conditions for Discrete Travel Choice

### Authors

J N Ibáñez, R P Batley, ITS, University of Leeds, UK

### Description

We repeat McFadden?s (1981) derivation of the integrability conditions for discrete choice, relaxing several restrictions within a more general analysis. This serves to define the subset of discrete choice models which are faithful to integrability.

### Abstract

A key issue in establishing the validity of travel choice models for economic appraisal is whether or not they adhere to the so-called ?integrability? conditions. These conditions ensure that, for any system of demand functions involving a symmetric negative semi-definite substitution matrix, there necessarily exists an underlying utility function from which the demand functions can be derived. In short, the integrability conditions ensure that a given observed pattern of demand is consistent with economic theory.

Conventionally, these integrability conditions exploit ?continuous? demand theory, wherein preferences are defined on a continuous commodity space. Indeed the integrability conditions are based on the partial derivatives of Hicksian demand functions with respect to price and income, and thus appeal to smooth and continuous demand functions. Travel choice models may be seen as special case of continuous demand theory, such that choice is restricted to a finite and exhaustive subset of the commodity space, and this provokes some challenges in translating the conventional integrability conditions.

The definitive contribution in this regard is McFadden (1981), who considers the applicability of the integrability conditions to a random utility maximisation (RUM) framework. Subsequently, a small number of authors have pursued similar interests (e.g. Kockelman, 1998; Koning & Ridder, 2003; Bates, 2003; and Jara-Díaz, 2007), although it is notable that Bates described McFadden?s analysis as ??path-breaking though relatively inaccessible ?? (p19). Our own paper seeks to promote deeper understanding of McFadden?s analysis by repeating his derivation from first principles, and annotating this derivation with commentary throughout. We then extend the analysis beyond the scope of McFadden?s, in the following respects.

First, in performing this derivation, we reveal a number of important, and possibly restrictive, properties of McFadden?s analysis. For example, implicit within McFadden?s model is an assumption that direct utility is not functional on either the price of discrete choice alternatives or income. Whilst such an assumption might on first inspection appear uncontroversial (and indeed attractive), it is important to note that this assumption is violated by specifications of RUM routinely employed in practical travel choice modelling. The implications of the additive income form describing McFadden?s model are also discussed.

Second, a particularly obscure element of McFadden?s derivation is his progression from discrete deterministic choice to discrete probabilistic choice, via the mechanism of probability measures. In this regard, we significantly expand upon the detail of McFadden?s derivation, presenting the clearest and fullest account hitherto. This reveals important restrictions on the manner by which utilities can be aggregated across a sample of individual decision-makers; interestingly this issue again follows from the dependence of direct utility on price and income.

Third, on concluding our derivation we review Koning & Ridder?s (2003) recent treatment of the integrability conditions for discrete choice, which exploited McFadden?s (1981) notion of a representative agent model to propose a discrete choice analogue to the integrability conditions. We demonstrate that Koning & Ridder?s treatment is based on a fundamental misunderstanding concerning the necessary and sufficient conditions that give rise to RUM.

Fourth, convention within the small literature dealing with the integrability conditions for discrete choice is to omit consideration of the random error terms inherent in the (truncated and conditional) indirect utility functions of RUM. We consider the significance of this omission, showing that this depends on whether the random error terms are (e.g. multinomial logit), or are not (e.g. mixed multinomial logit), independent of the deterministic part of the utilities.

Last but not least, McFadden imposes only the constraint of monetary budget on choice, whereas travel choice is subject to both income and time budgets. Responding to this need, we extend McFadden?s analysis to include the constraint of time, deriving definitive integrability conditions for travel choice. Specifically, we apply the Karush-Kuhn-Tucker conditions and the envelope theorem to a portfolio of RUM models subject to both income and time budgets, considering the implications of these constraints for Roy?s identity and Shephard´s Lemma. Moreover, we find that some members of the RUM class comply with the integrability conditions for travel choice, whilst others do not. The validity of applying the latter subset of models to economic appraisal is thus exposed to challenge.

#### Publisher

Association for European Transport