Social surplus and profitability under different spatial pricing policies.

• Author: Anderson, Simon P.

1. Introduction

   One of the most controversial parts of the U.S. antitrust laws is the Robinson-Patman Act. Posner has argued it should be discarded entirely [34]. Bork has called it "the misshapen progeny of intolerable draughtsmanship coupled to wholly mistaken economic theory" [10, 382]. The Robinson-Patman Act is an amendment, passed in 1936, to Section 2 of the Clayton Act. In the words of its co-sponsor, its purpose was to "prevent discrimination between competing customers of a seller" so that any "price differentials should be limited to the sound economic differences in costs" [33,5].

   Because of its concern with price discrimination, it is the Robinson-Patman Act that is relevant to the spatial pricing policies of firms. However, the precise limitations the law imposes on spatial prices are not exactly well defined. One point is relatively clear: that "no question of unlawful [price] discrimination would arise so long as the f.o.b. price is (uniform and (2) available all customers on nondiscriminatory basis. No legal requirement exists that the alternative f.o.b price be of any particular amount or computed in any particular way" ((85 F.T.C. 1174, 1176 [1, 49]). While f.o.b. mill pricing is legally unassailable, the legal status of uniform delivered pricing--where firms charge the same price at all points of sale--is somewhat ambivalent (see Dunn [16] for a discussion). What is certain is that firms charging neither mill or uniform prices may be open to successful prosecution, especially if they set price schedules which have bump-dip changes. For example, in the Utah Pie (see Breit and Elzinga [12] for more details), firms were found guilty of price discrimination since their delivered prices were lower in the local Utah market than at points closer to their plants.

   One obvious reason for the amibiguity in the letter and the application of the law is the lack of a sound economic analysis of spatial pricing. The pricing policies we consider in this paper are mill pricing (where each firm charges a f.o.b. pricing schedule so that all transport costs are passed on to consumers), uniform delivered pricing (each firm contracts to charge all consumers served the same price, irrespective of their locations) and spatial price discrimination (firms set location-specific delivered prices to consumers). These price policies capture a large part of actual firm pricing practices--the results of a survey of 241 firms in West Germany, Japan and the U.S. are presented in Greenhut [19]. One quarter of the firms surveyed used only uniform pricing; a further 29% used only mill pricing.(1)

   Clearly, mill pricing is the first-best optimal pricing policy with the mill price equal to marginal cost. However, firms that are spatially separated typically enjoy some degree of market power (imparted by their spatial advantage over consumers located close by) so that, even if they employ mill pricing, the mill price typically exceeds marginal cost. When demand is not completely inelastic this will be distortionary. Indeed, some other (discriminatory) pricing policy may yield higher social welfare given the constraint of monopoly or oligopoly pricing. Such results have been established for the monopoly case [6; 21; 23]. The oligopoly problem has been treated in Hobbs [22] and Holahan and Schuler [24]. These latter papers assume linear demand an that firms are equidistantly located along an infinite line, the interfirm distance being determined by free entry and exit. In all previous analyses the problem of firm locations per se has not been treated directly.

   In this paper we directly consider the location decisions of firms by analyzing the equilibrium locations in a linear bounded market. The use of a particular price policy by firms will tend to affect location decisions (see Greenhut [18] for an early recognition of this point). Our objective is to look at the locational inefficiencies induced by pricing policies alone, independently of whatever may be the deadweight loss associated with pricing above marginal cost. In order to separate the distortions in pricing above marginal costs from those which are due to purely locational effects, we shall specify a model in which mill pricing is the (first best) optimal pricing policy (regardless of the absolute level of the mill price) for any fixed (symmetric) pair of locations. That is, we shall set up a model where any distortions are due solely to locational tendencies.

   The importance of the locational effect is highlighted when the present results are compared with our previous ones [4]. In that paper we confined ourselves to fixed symmetric firm locations. Mill pricing therefore yields highest welfare in that context. Once we account for endogenous locations this finding is overturned. Because mill pricing yields equilibrium locations way outside the social optimum ones it causes welfare to be lower than that arising under pricing policies which are not in themselves optimal.

   The equilibrium concept we use is a standard one, a two-stage game with locations as the first stage and price-setting at the second. The justification for this set-up is that prices tend to be relatively flexible vis-a-vis locations. Hence we consider a game where locations are chosen first, bearing in mind the anticipated equilibrium in the subsequent pricing game.

   There are good reasons why the comparison we propose has not previously been analyzed: spatial models are plagued by non-existence of equilibrium in pure strategies. A two-stage location-price equilibrium will exist only under stringent conditions for mill pricing policies (see Gabszewicz and Thisse [17] for further discussion). For uniform delivered pricing, equilibrium will not usually exist at all [7]. Several modifications have since been proposed to deal with the non-existence problem.(2) We shall adopt one of these; specifically, we allow for heterogeneity of consumer preferences over the sellers of products (in the standard model consumer tastes are assumed to be homogeneous). This is the approach introduced in [14], where it was shown that equilibrium will exist in the mill pricing model for a sufficiently large degree of consumer heterogeneity. (Note however that this paper considers a one-stage game where prices and locations are chosen simultaneously--in the present paper we consider the location then price game).

   The idea behind the model developed here is that most sellers are inherently differentiated by a multitude of factors which are valued differently by different consumers. In addition to the difference in spatial locations of retailers, consumers may have a preference for one over another "because he is a fellow Elk or Baptist, or on account of some difference in service or quality, or for a combination of reasons" [27, 44]. Given that individual consumer tastes over the many attributes of sellers are typically unobservable, the best firms can do is to make estimates of them. Hence firms look at the probability that a given consumer will choose its product. We shall use the terms consumer taste heterogeneity and retailer heterogeneity interchangeably throughout the paper: retailers are only differentiated from each other because consumers view them as such.

   The precise model we use to characterize the diversity of individual consumer tastes is the logit model.(3) We have shown elsewhere [3; 5] that the logit demand model can be derived from consumer preference foundations other than the traditional probabilistic choice ones. Specifically, the approach we shall use in this paper is consistent with both the representative consumer and the address (or characteristics) approaches to modelling taste heterogeneity.

   In the next section, we present the model and describe the different pricing policies and the equilibrium concept. In section III, we analyzed price and location equilibria, as well as the optimum. Section IV compares the equilibrium outcomes in terms of profits, consumer surplus and total social surplus. Section V concludes with a discussion of pricing policies and regulation.

2. Framework of Analysis

   We assume there is a uniform distribution of consumers (with unit density) over a linear market normalized (without loss of generality) to [0, 1]. There are two firms, each with a single outlet. Their locations are denoted [x.sub.1] and [x.sub.2] with [x.sub.i] [Epsilon] [0, 1]; i = 1, 2, and [x.sub.1] and [x.sub.2].

   Consumer Behavior

   Each consumer is assumed to purchase one unit of product per period according to a decision rule (1) [Mathematical Expression Omitted] where [P.sub.i] (x) is the delivered price charged by firm i at location x [Epsilon] [0, 1], and [e.sub.i] (x) is the consumer-specific evaluation of the seller of good i by the individual at x.(4) If a tie occurs ([U.sub.1] (x) = [U.sub.2] (x)), the individual is assumed to purchase from each firm with probability one half. Whenever [e.sub.i] (x) is constant for all x [Epsilon] [0, 1], the model reverts to the case of homogeneous sellers, which is the standard assumption in the literature on spatial pricing. He we consider the case where [e.sub.i] (x) is not constrained to be constant. In accord with discrete choice theory, the precise value of [e.sub.i] (x) is assumed to be not observed by the firm, so that the firm must form an estimate of the probability that the consumer at x prefers to do business with it. In particular, we shall assume that [e.sub.i] (x) is distributed in the consumer population according to: (2) [e.sub.i] (x) = [[Mu] [[Epsilon].sub.i]], [Mu] [is greater than or equal to] 0; i = 1, 2, where the [[Epsilon].sub.i] are independent random variables, with zero mean and unit variance, which are identically distributed according to the double exponential distribution. The terms [[Mu] [Epsilon].sub.i]] therefore reflect idiosyncratic tastes (independently of consumer locations), and the...