Some theoretical aspects of destination choice and their relevance to production-constrained gravity models

Aspects of destination choice which concern relationships between destinations are explored in the context of a production-constrained gravity model. It is shown that, if competition exists between destinations or, alternatively, if agglomeration effects are present, the gravity model is misspecifled and estimated distance-decay parameters obtained from the model are related to spatial structure. Introduction The misspecification of gravity models has recently been posited, based on an alternative theory of destination choice (Fotheringham, 1983). The presence of this misspecification appears to produce a relationship between spatial structure and estimated distance-decay parameters [see Fotheringham (1981) for a review of this relationship]. The destination-choice theory that is invoked to explain the potential misspecification of gravity models concerns the perception of destinations in groups. As the number of destinations within a group increases, interaction with that group is likely to increase, but the relationship may not be a linear one as is implied by the gravity-model formulation. Until now, this hypothesis has only been demonstrated graphically in a relatively simple manner (Fotheringham, 1983). A more detailed discussion of the hypothesis is undertaken here. The discussion also concerns the elasticity of interaction to alternative destinations and a new interpretation of the estimated distance-decay parameter in production-constrained gravity models is derived in terms of this elasticity. Frameworks of analysis The potential misspecification of the production-constrained gravity model is demonstrated by the behaviour of the model in the spatial systems described in figures 1(a) and 1(b). In both systems the destinations are assumed to be equally attractive in terms of their size, and each destination lies at the same distance from the origin. Consequently, the use of a production-constrained gravity model would Figure 1. Two spatial arrangements of destinations 1-7. t This work was supported by a grant (number SES82-08339) from the National Science Foundation. 1122 A S Fotheringham lead one to the conclusion that the volume of interaction terminating at each destination is a constant, and that this volume is invariant between the two systems in figure 1. That is, the model takes no account of any effect the grouping of destinations may have on interaction patterns. For instance, the volume of interaction predicted to terminate at destination 1 would be the same in both systems, even though in figure 1 (a) destination 1 is relatively isolated from other destinations, whereas in figure 1(b) it is located in a group of destinations. Similarly, the predicted volume of interaction terminating at destinations 1 and 5 would be the same in both systems despite the obvious differences in the relative locations of the two destinations. The existence of a relationship between actual interaction patterns and the clustering of destinations is examined here, and the implications of such a relationship to gravity modelling are then discussed with particular reference to parameter estimation in a productionconstrained gravity model. Consider figure 2 which describes the set of possible relationships between the perceived attractiveness of a group of destinations and the number of destinations within that group. Assume that all destinations are of equal size so that the attractiveness of the group is increased solely by the addition of destinations to the group. A linear relationship is produced when the addition of a destination to a group of destinations increases the perceived attractiveness of the group by exactly the attractiveness of the individual destination. This is the assumption inherent in the production-constrained gravity model. A consequence of such behavior is that the interaction volume terminating at any destination is independent of the location of that destination with respect to other destinations. However, other relationships besides this linear one are feasible. For example, competition may exist between destinations so that interaction to an individual destination decreases as the destination is located in increasing proximity to other destinations, ceteris paribus. That is, interaction to a destination is lower when that destination is part of a group of destinations than when it is isolated, ceteris paribus. Grocery shopping, for example, may follow such a pattern: grocery stores may compete with each other for sales to increasing agglomeration effects increasing I competition * effects Number of destinations in group Figure 2. Theoretical relationships between the perceived attractiveness of a group of destinations and the number of destinations within the group. ^ For example, in either figure 1(a) or 1(b), if 0,represents the total outflow of interaction from origin z, then the predicted interaction volume terminating at any one destination is 0;/l or, more generally, it is Orfn, where n is the number of destinations in the system. Consequently, in a spatial system consisting of 100 destinations and where everything else is equal, if one destination is added to a group of, say, three destinations and to a group of, say, fifty destinations, the predicted increase in interaction terminating at each of the two groups will be 0,-/l 00. The increase in predicted interaction terminating at a group of destinations which results from the addition of destinations to the group is thus linear. Destination choice and production-constrained gravity models 1123 the extent that a grocery store in relative isolation may be able to capture a larger market than if it were in close proximity to similar stores. Alternatively, agglomeration effects may be present. Interaction to an individual destination would then increase as the destination is located in increasing proximity to other destinations, ceteris paribus. That is, interaction to a destination is greater when it is part of a group than when it is isolated, ceteris paribus. Consumer goods shopping, for example, may follow such a pattern. Consumer goods stores may generate a greater volume of sales when they are clustered than when they are dispersed. If either competition or agglomeration effects are present, then gravity models are misspecified. In figure 2, the horizontal axis denotes the situation where competition effects are a maximum. In this instance, the perceived attractiveness of a group of destinations is constant regardless of the number of destinations in the group. Under conditions of maximum competition, any increase in the attractiveness of the group produced by the addition of an extra destination is offset entirely by increased competition between the destinations. This is the situation discussed by Fotheringham (1983). The vertical axis of figure 2 denotes conditions of maximum agglomeration effects. Under such conditions, all interactions will terminate in the largest group of destinations, irrespective of the location of that group. The presence of either competition or agglomeration effects will produce misspecification errors in gravity models. However, the two effects have different implications for parameter estimation in such models. Consider a spatial system consisting of accessible and inaccessible origins all of equal size. The presence of strong competition effects implies that the relationship between interaction and distance would be very similar amongst origins of different accessibility. That is, the arrangement or grouping of destinations in space would have little effect on the relationship between interaction and distance. The presence of strong agglomeration effects would produce the opposite result. The arrangement of destinations in space would be the prime determinant of interaction patterns. Most interactions would terminate in the largest groupings of destinations irrespective of the location of the grouping. To examine more precisely the implications of competition or agglomeration effects for destination choice and for parameter estimation in gravity modelling, assume that in a spatial system consisting of accessible and inaccessible origins, competition effects are at a maximum. This assumption is not critical to the subsequent theory and it is later relaxed. Initially, however, it is a useful assumption to make for purposes of exposition, since it implies that the relationship between interaction and distance is constant for all origins. Also assume that in the spatial system under consideration the perception of distance as a deterrent to interaction is constant in each origin. This perception is measured in a gravity model by the distance-decay parameter, ft. It is then shown that under these conditions, if a production-constrained gravity model is calibrated independently for each origin in this spatial system, it is impossible for ft, the estimated value of ft, to be constant. Consequently, since ft is constant, the spatial variation of ft in this instance has nothing to do with variations in the perceptions of distance as a deterrent to interaction. Rather, any variation must be produced by variations in spatial structure— the arrangement of destinations in space. This hypothesis is now examined further. () In figure 2 it is unlikely that the perceived attractiveness of a group of destinations would continue to increase exponentially as destinations are added to the group. However, such a relationship may occur over a limited range of additional destinations, and thereafter the perceived attractiveness may level off. Hence, it may be more accurate to consider the agglomeration effect described in figure 2 as part of a logistic curve. 1124 A S Fotheringham Competition effects and destination choice The calibrated form of the origin-specific production-constrained gravity model can be written as: Iik = OtDkdfc\ tofdif 9 (1)