SPATIAL COMPETITION WITH CONSISTENT CONJECTURES*

ABSTRACT Equilibrium in spatial models invariably depends on firms' conjectures about how competitors will react to their price changes. This paper analyzes spatial price and location equilibrium when firms hold consistent (i.e. correct) conjectures. Most spatial models assume an exogenous conjecture. Consistent conjectures are one method, albeit a controversial one, for endogenizing the conjecture. We show that the consistent conjecture about a competitor's reaction to a price change in the simplest case is 1/3. When demand is elastic the consistent conjecture is a decreasing function of the radius. It is always below 1/3 and can be negative. In the third model, we show that the consistent conjecture declines as the number of dimensions and the number of competitors increases.     

MILL PRICING AND SPATIAL PRICE DISCRIMINATION MONOPOLY PERFORMANCE AND LOCATION WITH SPATIAL RETAIL MARKETS*

ABSTRACT. Most of the monopoly spatial price discrimination literature explicitly assumes uniform population density over space. It also implicitly assumes that firms (plants) are spatially isolated from each other with production and retail points that coincide in location. While departures from these assumptions have been explored separately in the literature, it remains to examine performance and location when these assumptions are relaxed simultaneously. What emerges in this paper is a model where density functions approximate a pair of cities isolated from other cities. Each city has its own retail market, while the location of a single production or wholesale point is determined by characteristics of the two markets. Comparisons of mill pricing and spatial price discrimination found in the spatial monopoly literature can be interpreted as special cases of the more general framework provided here.     

LÖSCHIAN COMPETITION VERSUS SPATIAL COLLUSION: PRICE AND WELFARE COMPARISONS*

ABSTRACT. Löschian duopoly under heterogeneous cost conditions is examined to show that it is not equivalent, contra past findings, to spatial collusion. Moreover, within the confines of the assumed demand and cost conditions spatial collusion is shown to be superior to Loschian competition in terms of both (aggregate) consumer surplus and producer surplus, which implies a possible welfare gain from collusion. A general, if not the general, prices-and-welfare comparison of alternative pricing schema including collusive, Löschian, and optimal pricing is summarily presented in a table.     

UNCERTAIN INCOME AND HOUSING PRICE AND LOCATION CHOICE*

ABSTRACT. In this paper, we adapt recent developments in uncertainty modeling to the location choice paradigm. In particular, we analyze the impact of income and housing price uncertainty on housing demand and location demand characteristics in a partial equilibrium framework. We begin by recognizing that housing consumption cannot readily be altered in response to changes in income and price. We find that income and house price risk affect housing and location demands in different ways. Additionally, the spatial characteristics of price risk also affect consumer housing and location demands. For example, if housing price risk is lower farther from the central city, housing demand can be greater in those locations even with the higher transportation cost. Thus, over some locations, the expected price gradient could be positive.     

SPATIAL PRICE COMPETITION WITH CONSUIMERS ON A PLANE,AT INTERSECTIONS,AND ALONG MAIN ROADWAYS*

ABSTRACT. This paper examines two-dimensional price competition on a plane, with a block metric and a square grid of main roadways. One store is located at each intersection of main roadways. Consumer locations include a uniform distribution over the plane, linear concentrations along main roadways, and point concentrations at intersections. Bertrmd-Nash mill price competition is examined first. The equilibrium price depends on the relative numbers of consumers in the three types of locations (and on travel costs per mile and the spacing between stores). If too many consumers are in each point concentration, then the price equilibrium is undermined by a high-price strategy or by mill-price undercutting. Spatial competition with price discrimination is examined next, and compared to Bertrand-Nash mill price competition.     

SPATIAL COMPETITION À LA COURNOT: PRICE DISCRIMINATION BY QUANTITY-SETTING OLIGOPOLISTS*

In this paper, we consider oligopolistic competition in a spatial model when firms take care of goods' delivery and discriminate among consumers. Firms compete by setting quantity schedules independently over space. We show that under general conditions a Nash equilibrium in this game exists and is unique. In equilibrium, firms’ markets overlap, a feature which accords with intuition and empirical observations. Over the interval between two firms, the equilibrium spatial price schedule is quasi-concave (quasi-convex) when transport costs are concave (convex). With linear transport costs, the model predicts uniform delivered pricing. Uniform pricing could moreover be obtained by a combination of increasing returns to volume in transportation together with concavity of unit transport costs in distance.     

SPATIAL COMPETITION AND THE JOINT CONSUMPTION OF SERVICES*

In Hotelling location models, an implicit assumption has been made that a customer uses the service provided by the firm, independent of other customers. However, for firms that supply meeting rooms, wedding halls, tennis courts, and golf links it is essential to incorporate the fact that the customer only uses the service in conjunction with other customers. The objective of this paper is to formulate the Hotelling location model n consideration of the interdependence among customers in one-dimensional space, and to characterize the state of equilibrium. In our model, each group of members within a given distance enjoys a fixed amount of service of the firms, under the assumption that the travel cost incurred by each group is defined as the travel cost of the farthest customer in the group.     

SPATIAL COMPETITION WITH LOCATION-DEPENDENT COSTS*

This paper investigates the behavior of spatially-competitive firms whose costs depend on their locations. Necessary and sufficient conditions for optimum locations are derived. Comparative-statistics analysis of the equilibrium shows that the responses of firms to changes in input and output transport rates depend on the properties of consumer demand curves or are ambiguous in sign.     

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.     

COMPUTATIONAL COMPARISONS OF SPATIAL PRICE EQUILIBRIUM METHODS*

ABSTRACT. In this paper we propose new algorithms for the solution of both general and standard spatial price equilibrium problems, and test their performance with existing algorithms on randomly generated problems. For the standard problem, we propose decomposition schemes based on the concept of “equilibration operator” and compare their performance with the Frank-Wolfe method. For the general problem, we present alternative variational inequality formulations defined over Cartesian products of sets and then exploit these formulations to construct Gauss-Seidel-type serial decomposition methods. We then compare their performance with the projection method. Our computational tests suggest that the new schemes are substantially more efficient than earlier ones.     

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 OUTPUT EFFECTS OF SPATIAL PRICE DISCRIMINATION REVISITED*

ABSTRACT Some prior findings on spatial price discrimination are extended. Under certain spatially defined market conditions, discriminatory pricing is known to yield greater output than does nondiscriminatory f.o.b. mill pricing. However, this seemingly surprising result is based on the form of the basic demand function which is relatively less convex than a certain standard curve. The present paper makes this prior result more general by permitting the assumed basic demand to be relatively more convex.     

SPATIAL DEPENDENCE AND SPATIAL STRUCTURAL INSTABILITY IN APPLIED REGRESSION ANALYSIS*

The stability of regression coefficients over the observation set (“regional homogeneity”) is typically assessed by means of a Chow test or within a seemingly unrelated regression (SUR) framework. When spatial error autocorrelation is present in cross-sectional equations the traditional tests are no longer applicable. I evaluate this both in formal terms as well as empirically. I introduce a taxonomy of spatial effects in models for structural instability, and discuss its implication for testing. I compare the performance of traditional tests, robust approaches, maximum-likelihood procedures and pretest techniques by means of a series of simple Monte Carlo experiments.     

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.