DISCRETE SPATIAL CHOICE MODELS FOR AGGREGATE DESTINATIONS*

ABSTRACT. In problems of spatial choice, the choice set is typically more aggregated than the one considered by decision-makers, often because choice data are available only at the aggregate level. These aggregate choice units will exhibit heterogeneity in utility and in size. To be consistent with utility maximization, a choice model must estimate choice probabilities on the basis of the maximum utility within heterogeneous aggregates. The ordinary multinomial logit model applied to aggregate choice units fails this criterion as it is estimated on the basis of average utility. In this paper, we derive and discuss a model which utilizes the theory underlying the nested logit model to estimate the appropriate maximum utilities of aggregates. We also demonstrate that the aggregate alternative error terms are asymptotically Gumbel, thereby relaxing the assumption of extreme value distributed error terms. This is accomplished with help from the asymptotic theory of extremes.     

ELASTICITIES OF CONSTRAINED GRAVITY MODELS*

ABSTRACT Doubly constrained gravity models require a balancing procedure to make estimated origin and destination totals consistent. We show through a series of examples that not just the magnitudes but even the signs of derivatives and elasticities of gravity model flows depend on the specific balancing procedure used. Hence, if such elasticities are regarded as meaningful, then the balancing of origin and destination totals must be regarded as an integral part of the gravity theory itself.     

ADDITIVE-INTERACTION MODELS OF SPATIAL AGGLOMERATION*

A wide variety of existing models of spatial agglomeration postulate additive-interaction effects among agents. In this paper, a synthesis of such models is achieved by establishing certain mathematical equivalences between them. In particular, it is shown that Rockafellar's conjugacy theory of concave functions yields a symmetric one-to-one correspondence between three classes of existing models, here designated as spatial-accessibility models, endogenous-contact models, and fixed-contact models. These correspondences not only allow the transference of results between models, but also suggest new economic interpretations of each model in terms of its conjugate model. A series of examples are drawn from the literature to illustrate these results.     

POISSON GRAVITY MODELS OF SPATIAL FLOWS*

ABSTRACT The purpose of this paper is to bring together a number of results on spatial flow models, and to reformulate them within a unified probability framework based on Poisson frequencies. To do so, a class of discrete stochastic processes, designated as independent flow processes, is developed which not only yields a complete characterization of Poisson flow frequencies, but also allows a wide range of gravity-type flow models to be formulated within this distributional framework. In particular, a hierarchical classification of 12 model types is developed, and each model type is shown to be characterizable directly in terms of behavioral axioms on independent flow processes.     

THE ECONOMICS OF AIR TRAVEL GRAVITY MODELS*

Starting with the aggregate demand model of economics, a model of demand for intercity air travel is developed which contains the gravity model as a less general submodel. The more general model is referred to as the alternative opportunities model since it takes account of alternative destinations open to travelers, not just origin and destination as does the gravity model. The demand model approach has the virtue of providing a theoretical basis for understanding and analyzing the gravity model. The gains from treating alternative locations and demand motivation variables are a substantial increase in explanatory power over that yielded by the gravity model, the identification of statistically significant determinants of air travel, and better measurement of the coefficients of population and distance by taking account of these other variables and somewhat better forecasts. A shortcoming of procedures used here is aggregation of air trips with different purposes and thus lack of clear specification of the size of effects of different variables on different types of travel. Overcoming this difficulty must await origin-destination data listed by trip purpose.