Symmetry, Aggregation, and the Practical Specification of Travel Choice Models
R Batley, J N Ibáñez, ITS, University of Leeds, UK
A fundamental 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 compensated demand functions involving a symmetric negative semi-definite substitution matrix, there exists an underlying utility function from which demand can be derived. Conventionally, these integrability conditions exploit ?continuous? demand theory, wherein preferences are defined on a continuous commodity or attribute space. 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/attribute space, and this provokes some challenges in translating the integrability conditions. The purpose of our paper is to articulate the manner in which the aforementioned symmetry of substitution, often referred to as ?Slutsky symmetry? (Slutsky, 1915), applies to travel choice models, and the implications that follow for the practical specification of such models. To this end, we consider two alternative representations of the discrete-continuous interface, as follows.
First, and following McFadden (1981) and Small & Rosen (1981), we consider a problem of joint discrete-continuous demand, such that the demand for continuous consumption is conditional on the discrete choice. An example of this problem would be where a household makes a discrete choice of whether to purchase a particular car (whether as an additional or replacement car), and contingent on the discrete choice decides the quantity of mileage that will be ?consumed?. Starting from Small & Rosen?s statement of the Slutsky equation (#3.18) for the continuous component of the joint demand, we extend this to consider cross-price effects, thereby deriving conditions for Slutsky symmetry. A number of practical implications follow from this derivation. In particular, if we adopt the popular ?additive income? (AIRUM) specification of discrete choice then, in most instances, symmetry of the continuous demands implies that we must specify different marginal utilities of price for different discrete choice alternatives (i.e. each car would need to have an alternative-specific cost coefficient).
Second, and following Train (2001), we conceptualise discrete choice as an end in itself, such that the demand for the ?continuous? consumption might simply be seen as the sample enumeration of discrete choices. An example of this problem would be where a commuter chooses the mode by which he/she will travel to work, and the total number of commuting trips (say over the course of a year) is readily predictable. We again derive the conditions for Slutsky symmetry, and draw implications for the practical specification of travel choice models. In contrast to the first problem, we now find that, under AIRUM, we must specify common marginal utilities of price across the different discrete choice alternatives (i.e. different modes would need to have a generic cost coefficient).
Having considered individual-level demand, which was the focus of Slutsky?s (1915) analysis, we repeat our derivation of Slutsky symmetry for aggregate demand, which is the perspective more commonly adopted by discrete choice analysts. In principle, there is no analogous requirement for aggregate demand to observe symmetry. In practice, however, discrete choice analysts frequently employ the aggregate demand as the basis for appraisal, and this provokes a requirement for symmetry to be observed at the aggregate level. Following Gorman (1953) and Diewert (1980), a common resolution is to formulate the aggregate demand in terms of a ?representative consumer?. Noting that, for the representative consumer, symmetry implies homothetic preferences across the population, which itself implies a zero income effect, the paper arrives at similar findings to the individual-level under AIRUM.
In summary, our paper provokes an important question regarding the interface between discrete and ?continuous? demands; how can we best represent the aggregation/separability of demands within and across individuals, so as to ensure that models of travel choice are valid for economic appraisal? The answer to this question may well depend on the study context, and the perspective of the analysis. In the case of the car purchase example, an analytical interest in car use would call for one particular specification of indirect utility for the discrete choice (with alternative-specific cost coefficients), whilst an interest in simply car purchase choices would call for a different specification (with a generic cost coefficient). Our analysis reveals a clear message for practice, and one that must be heeded if the validity of economic appraisals from discrete choice models is to be defended.
Association for European Transport