SPATIAL PRICING IN INTERDEPENDENT MARHETS*

ABSTRACT. Conditions for spatial price equilibrium are derived for a set of firms in oligopolistic spatial competition, distributed at fixed locations in a heterogeneous region where consumer purchasing patterns are a probabilistic function of the price distribution rather than a deterministic function of proximity to firms. The resulting prices vary with accessibility to consumers or with the degree of local spatial monopoly, and result in non-zero profits for firms. Conditions describing the existence and stability properties of this spatial price equilibrium are defined, and are shown to be equivalent for two different hypotheses concerning disequilibrium pricing behavior: a partial price adjustment model and a Bertrand game. For two different profit goals, total profit maximization and profit rate maximization, it is shown that a spatial price equilibrium exists and is at least locally quasi-stable.     

A SPATIAL COMPETITION APPROACH TO CENTRAL PLACE THEORY: SOME BASIC PRINCIPLES*

ABSTRACT. This paper investigates the potential offered by the model of spatial competition for the study of central place theory. We consider n firms selling m substitutable or complementary goods to a continuum of consumers evenly distributed along a linear segment. Consumers have the same income and the same utility function which is quadratic in the goods supplied by the firms and linear in the numeraire. The main results are as follows. (1) In any location equilibrium in which all goods are consumed everywhere, each good supplied by a single firm is sold at the market center. In Christaller's terminology, this means that when the exhaustive principle holds in equilibrium, highest-order goods are made available at the center. (2) When all goods (excluding the numéraire) are complements to each other and each good is sold by a single firm, there always exists an equilibrium in which all the firms locate coincidentally. (3) If the stores selling a given good are under the control of a single owner then, in any equilibrium for which the exhaustive principle holds, the stores are located in a way such that the total transport cost (borne by consumers) is minimized.     

The Max-Min-Min Principle of Product Differentiation

We analyze two and three-dimensional variants of Hotelling's model of differentiated products. In our setup, consumers can place different importances on each product attribute; these are measured by weights on the disutility of distance in each dimension. Two firms play a two-stage game; they choose locations in stage 1 and prices in stage 2. We seek subgame-perfect equilibria. We find that all such equilibria have maximal differentiation in one dimension only; in all other dimensions they have minimum differentiation. An equilibrium with maximal differentiation in a certain dimension occurs when consumers place sufficient importance (weight) on that attribute. Thus, depending on the importance consumers place on each attribute, in two dimensions there is a max-min equilibrium, a min-max equilibrium, or both. In three dimensions, depending on the weights, there can be a max-min-min equilibrium, a min-max-min equilibrium, a min-min-max equilibrium, any two of these, or all three.     

Price Variation in Spatial Oligopolies

The “geography of price” is being given renewed attention by geographers and economists. This paper examines spatial price variation in both unbounded (circular) and bounded (linear) one-dimensional markets. Firms compete for consumers in the short run by adjusting price until the Bertrand equilibrium is reached in the market. While firms act as spatial oligopolists in specific market segments, the profit-maximizing price of any given firm depends directly and indirectly upon the spatial-economic properties (locations, marginal costs) of all other firms in the market.     

Duopoly prices under congested access*

Abstract. Consider two firms, at different locations, supplying a homogenous good at constant marginal production cost. Consumers incur travel costs to the firm for each unit purchased, and the travel costs increase with the amount of travel to each firm (congestion). When all traffic and all congestion are generated by travel to a duopolist, both the Nash–Bertrand equilibrium prices and the Nash–Cournot equilibrium prices exceed the sum of the marginal production cost and the marginal external travel cost. However, when the road is shared by travelers to the duopolists' facilities and travelers in competitive markets, the Nash–Bertrand duopoly price equals the competitive price and the Nash–Cournot price contains a markup.     

Uniform Pricing Versus Mill Pricing

This paper analyzes the coexistence of different pricing strategies. The purpose is to discuss how firms that are limited to uniform pricing affect the outcome of price competition among mill–price–setting firms. Price competition among (three) firms that are restricted to mill pricing is analyzed within the classic Hotelling framework and uniform–price–setting firms are considered as first movers. If uniform–price–setting firms deliver any good, they effectively separate mill–price–setting firms from each other. Finally, it is shown that price competition among first movers strengthens the effects of cross–type price competition.     

THE WELFARE ASPECTS OF SPATIAL PRICING POLICIES RECONSIDERED FOR A MONOPOLY CASE*

ABSTRACT. We reexamine the price structures and their welfare implications in three pricing regimes (mill, uniform and discriminatory) for a monopoly. We show that spatial price discrimination could provide the highest social welfare and, when consumers tastes are heterogeneous enough, also the highest consumer surplus. The superiority of spatial price discrimination is partially due to the larger output produced and partially due to differential treatment for consumers with heterogeneous tastes.     

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.     

PRODUCT DIFFERENTIATION,PRICING POLICY AND EQUILIBRIUM*

ABSTRACT. This paper studies the price-location equilibrium of duopolists supplying differentiated goods and competing in a spatial market with elastic demand. We show that a price-location equilibrium exists under all three pricing policies traditionally considered by the literature: f.o.b. mill, uniform delivered, and spatially discriminatory pricing. We also show that firms always cluster at the market center. The second part of the paper studies the endogenous choice of pricing policy. A surprising feature of the resulting equilibrium is asymmetry. The greater the extent to which the goods are substitutes, the more likely is it that one firm will choose f.o.b. pricing and the other price discrimination. Finally, the welfare consequences of the analysis show some interesting trade-offs.     

ZONING A CROSS‐BORDER CITY

This paper investigates zoning in a cross‐border linear city that consists of two bordering towns. In each town, a local regulator has a say in the location of the local firm. We find that local regulators may use zoning strategically. The incentive to gain consumers from the other town, or not to lose local consumers, may push regulators to approve only locations for firms close enough to the frontier. When zoning is costly an asymmetric equilibrium may emerge: only one regulator resorts to zoning. In the case of towns of different sizes, the regulator of the larger town is the only one that zones in an asymmetric equilibrium.     

THE OPTIMAL LOCATIONS OF MULTIPLE BRIDGES,CONNECTING FACILITIES,OR PRODUCT VARIETIES*

ABSTRACT This paper considers the optimal locations of two or more facilities, and the optimal number of facilities, when trips are made in pairs. The results are the same as standard models of spatial competition when there is perfect matching, but not when there is random matching. The first interpretation is bridges across a river, with residential locations on one side matched perfectly or randomly to jobs on the other side. The second interpretation is connecting facilities, such as tennis courts or restaurants where pairs of consumers meet. The third interpretation is product differentiation, with husbands and wives jointly choosing from among varieties.     

Hotelling Games with Three,Four, and More Players*

Abstract. This paper extends the interval Hotelling model with quadratic transport costs to the n‐player case. For a large set of locations including potential equilibrium configurations, we show for n > 2 that firms neither maximize differentiation—as in the duopoly model—nor minimize differentiation—as in the multi‐firm game with linear transport cost. Subgame perfect equilibria for games with up to nine players are characterized by a U‐shaped price structure and interior corner firm locations. Results are driven by an asymmetry between firms. Interior firms are weaker competitors than their rivals at the corners. Increasing the number of firms shifts even more power to the corner firms. As a result, there is too much differentiation from the social perspective if n ≤ 3, while adding firms leads to a level of differentiation in equilibrium below the social optimum.     

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 ENTRY DETERRENCE UNDER OLIGOPOLY*

ABSTRACT. This paper examines colluding, oligopolistic firms in a linear market. By assuming that rivals do not compete for consumers at their market boundaries, it is shown that an equilibrium exists without adopting a convex transportation cost function. Two price profiles are derived. The first describes firm prices in the absence of threatened entry. The second details profit-maximizing prices which forestall entrants. Given infinite relocation costs, threatened entry leads to price adjustments by the incumbent firms.     

Competitive Location and Pricing on Networks

In this paper we consider a location and pricing model for a retail firm that wants to enter a spatial market tvhere a competitor firm is already operating as a monopoly with several outlets. The entering firm seeks to determine the optimal uniform mill price and its servers' locations that maximize profits given the reaction in price of the competitor firm to its entrance. A tabu search procedure is presented to solve the model together with computational experience and an example.     

Competitive Location and Pricing on Networks

In this paper we consider a location and pricing model for a retail firm that wants to enter a spatial market where a competitor firm is already operating as a monopoly with several outlets. The entering firm seeks to determine the optimal uniform mill price and its servers' locations that maximize profits given the reaction in price of the competitor firm to its entrance. A tabu search procedure is presented to solve the model together with computational experience and an example.     

TWO-DIMENSIONAL BERTRAND COMPETITION: BLOCK METRIC,EUCLIDEAN METRIC,AND WAVES OF ENTRY*

This paper examines two-dimensional spatial competition, with Bertrand price determination. With a block metric, equilibrium prices are significantly lower when market areas are squares than when they are diamonds (rotated squares) of the same size. If demand density grows, waves of entry occur, and the shapes of market areas change from squares to diamonds and back to squares again. The former change leaves prim unchanged, whereas the latter cuts prices in half. Results are also derived for a Euclidean metric, with square and hexagonal market areas. Optimal waves of entry are examined with the block metric. With either metric, the socially optimal market shape becomes suboptimal if market areas are constrained to be of the zero-profit size.     

LOCATING OUTSIDE A LINEAR CITY CAN BENEFIT CONSUMERS*

ABSTRACT We investigate the effects of restricting the locations of firms in Hotelling duopoly models. In standard location‐price models, the equilibrium distance between firms is too great from the viewpoint of consumer welfare. Thus, restricting the locations of firms and shortening the distance between them improves consumer welfare by reducing prices and transport costs. We introduce strategic reward contracts into location‐price models and find that, in contrast to the above result, restrictions on the locations of firms reduce consumer welfare. These restrictions reduce transport costs but increase prices by changing the strategic commitments of the firms.     

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.     

THE EFFECTS OF ZONING IN SPATIAL COMPETITION

ABSTRACT This paper considers a location model to illustrate the effect of zoning on competition. A planner is in charge of designing a city in a circular space where firms and consumers are located on different sides. With this type of market configuration, equilibrium in location under concave transport costs is proved. Then, a welfare function with different weights attached to consumer and firm surpluses is introduced to highlight zoning regulation as an influential competition policy tool. Depending on the regulator's political profiles and the demand, it is shown that zoning can lead to strong, weak, or moderate competition.