On the integrability conditions for discrete travel choice

Summary: 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: In establishing the validity of discrete choice models for economic analysis, a fundamental issue is whether or not they adhere to the 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. Conventionally, the integrability conditions exploit 'continuous' demand theory, wherein preferences are defined on a continuous commodity space. Indeed the 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. Discrete 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 paper considers a more general, and therefore flexible, model of consumption, arising from the combination of both discrete and continuous choices. In particular, we apply this model to the dual theorem of demand, establishing the integrability conditions applying separately to the discrete and continuous consumptions. The most significant prior work in this vein is McFadden's (1981), although it is useful to note Bates' (2003, p19) description of McFadden's analysis as '…path-breaking though relatively inaccessible …'. 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. In performing this derivation we reveal a number of important, and possibly restrictive, properties of McFadden's analysis. We also discover that a number of common practical specifications of discrete choice models fail to comply with McFadden's integrability conditions; the validity of these specifications for economic analysis is thus exposed to challenge. Last but not least, we review Koning & Ridder's (2003), disputing their conclusion that the integrability conditions on discrete choice consumption are less restrictive than the conditions for random utility maximisation.