# Social surplus and profitability under different spatial pricing policies.

I. 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.

II. 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 parameter [Mu] conveys the degree of dispersion of these tastes across consumers. For [Mu] [right arrow] 0, we recover the case of homogeneous tastes (or homogeneous sellers).

Following de Palma et al. [14], the probability of an individual at x purchasing the product from firm i is given by the binomial logit formulation(5) as (3) [Mathematical Expression Omitted] so that clearly [P.sub.1] (x) + [P.sub.2] (x) = 1. Note that sellers are not differentiated (i.e., they are perfect substitutes) for [Mu] [right arrow] 0. [P.sub.i] (x) is a continuously decreasing function of [p.sub.i] (x), which is concave (convex) for [p.sub.i] (x) [is less than or equal to] ([is greater than or equal to]) [p.sub.j] (x Thus it has the standard shape associated with a well-behaved unimodal density function.

Costs

We assume each firm produces with constant and identical marginal costs, which can therefore be set equal to zero without loss of generality. Transport costs per unit shipped are assumed linear in distance and invariant to volume. The transportation rate will be normalized to unity by appropriate choice of unit of account. We consider three types of alternative spatial pricing policy.

Profits and the Equilibrium Concept

Firms are assumed to be risk neutral. Each firm's expected profits under the three alternative pricing policies are (4) [Mathematical Expression Omitted] where [p.sup.i] (x) is constrained by the pricing policy under consideration and specified below. Notice than an increase in both firms' delivered prices by one dollar at all locations raises total profits ([[Pi].sub.1] + [[Pi].sub.2]) by one dollar. This results from the property of (3) that the purchase probabilities depend only upon price differences.

The market equilibrium we consider for each of the alternative pricing policies is the equilibrium to a two-stage game. Locations are chosen at the first stage and are predicted upon the knowledge of the second-stage pricing equilibrium. Hence we analyze a sub-game perfect Nash equilibrium. This concept is now made precise.

The solution is recursive. We first solve the second-stage pricing sub-game. For given firm locations ([x.sub.1], [x.sub.2]), a price-schedule equilibrium [Mathematical Expression Omitted] is defined by (5) [Mathematical Expression Omitted] for all allowable price schedules {[p.sub.i] (x)}, for all x [Epsilon] [0, 1]; i, j = 1, 2, i [is not equal to] j. For both mill and uniform pricing policies, the allowable price schedules are defined in terms of a single variable, [Mathematical Expression Omitted] (the mill price) and [Mathematical Expression Omitted] (the uniform delivered price respectively.

Now consider the first stage location game. Define [[Pi] [tilde].sub.i] ([x.sub.i], [x.sub.j]) as i's profit evaluated at the second stage price schedule equilibrium. Equilibrium to the full two-stage game is then defined by a location pair, [Mathematical Expression Omitted] such that (6) [Mathematical Expression Omitted]

Consumer Surplus

One criterion for comparison is aggregate consumer surplus, defined as the integral of net benefits of consumers in the market. Once we allow for heterogeneity across products, it will no longer be the case that consumers patronize the firm with the lower delivered price. Hence market segments will overlap. The consumer surplus measure therefore must also include the benefits from product variety, as expressed by [Mu], along with the delivered price paid. The former element is absent from standard location models with homogeneous tastes.(6) For the demand system (3), consumer surplus is given (up to a positive constant) by (7) [Mathematical Expression Omitted] (see Anderson, de Palma, and Thisse [3] and Small and Rosen [35]). Note that a one dollar increase in both delivered prices at all locations causes aggregate consumer surplus to fall by one dollar (this can be seen by using [p.sub.i] = [p.sub.i] + 1, i = 1, 2 in (7)).

Social Surplus and the Optimum

We define social surplus as total consumer surplus (as defined in Equation (7) plus profits of both firms (as given by Equation (4)). Social surplus (or "welfare") is given by (8) W = CS + [[Pi].sub.1] + [[Pi].sub.2]. The optimal solution entails choice of both spatial price schedules and locations. It is therefore the solution to (9) [Mathematical Expression Omitted] where W(.) is given by (8). The level of social surplus depends only upon price differences, although the distribution of that surplus depends on price levels: as noted previously, one dollar rise in both prices at all locations is simply a transfer of surplus from consumer to firms. This has important implications for the analysis below. In particular, any situation which differs from another only by the absolute level of prices (and not by their difference) entails the same allocation of consumers to outlets and hence the same total social surplus.

Having defined the model and criteria for comparison, we now turn to the location results. It should be stressed at this point that all location equilibria and the optimum can be expressed in terms of a single parameter. This parameter is [Mu], the measure of consumer taste heterogeneity across retailers.(7)

III. Equilibrium and Optimum Locations under Alternative Spatial Pricing Policies

We shall see below that taking explicit account of taste variations (as per equations (1)-(3) can restore the existence of equilibrium providing the taste variations are sufficiently large, thus allowing us to compare the price policies.

Mill Pricing

Under this pricing policy, each firm charges a single mill price to all consumers, and all transport costs are passed on to consumers. The delivered price paid by a consumer at x and purchasing from firm i at [x.sub.i] is (10) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the mill price.

We have not been able to fully characterize algebraically the equilibrium locations under mill pricing for all values of [Mu]. We have, however, simulated the market equilibrium under this pricing policy. These simulations show a large variety of possibilities. The results are illustrated in Figure 1. As [Mu] rises from zero, there is first no equilibrium; then there is a region where there is a unique equilibrium, at which firms are separated. As [Mu] continues to rise, there is next a region where there are two equilibria; finally, there is a unique equilibrium which involves central agglomeration.

The reason for the initial non-existence of equilibrium for [Mu] < 0.062 is similar to that given in d'Aspremont, Gabszewicz, Thisse [13] for the case of homogenous products I that case each firm's profits rise as it approaches is rival's location. However this central tendency eventually leads firms into the situation where there exists no price equilibrium because each wishes to undercut its rival's mill price at any candidate solution to the pricing first-order conditions. That is, there is a fundamental failure of quasiconcavity (in own price) in the profit functions. By continuity, this explains why there is a region of [Mu]-values for which there is no equilibrium. When [Mu] is high enough, the failure of quasiconcavity becomes non-critical since the relative benefit of mill price undercutting is reduced because demand is less responsive to small price differences (consumers care more about other aspects of the products). In other words, introducing preference heterogeneity smoothes the profit functions so that they become well behaved. Hence for [Mu] large enough, there exists an equilibrium to the second-stage game so that the first-stage location game can be defined. The latter then also has an equilibrium, as illustrated in Figure 1.

For 0.062 [is less than or equal to] [Mu] < 1.47, there is a symmetric dispersed equilibrium which initially entails increasing spatial separation of firms as [Mu] rises. This effect can be ascribed to a tendency for firms to consolidate their monopoly power as the influence of the rival becomes more far-reaching (in the sense that rising [Mu] implies a greater invasion of a firm's hinterland). For larger [Mu] (around 0.3), the firms start to move together again with increasing [Mu]. This effect is due to the increased desirability of a central location given that, ceteris paribus, the demand addressed to a firm becomes increasingly even over space as [Mu] rises. Hence the equilibrium can be seen as the result of two opposing forces, with the agglomerative force eventually outweighing he deglomerative one.

For 0.76 [is less than or equal to] [Mu] < 1.47, there is an agglomerated equilibrium at the centre along with the dispersed one. The phenomenon of multiple equilibria can be understood in the following manner. Equilibrium prices are relatively high at the non-agglomerated equilibrium locations. Any move toward the center increases price competition substantially and is avoided on those grounds. On the other hand, equilibrium prices are relatively low at the agglomerated equilibrium in the center. Moving away reduces competitive pressure (and enables higher prices) but also reduces market share so much as to reduce profits. When the central equilibrium co-exists with the dispersed one, it is the latter which is the stable equilibrium. Equilibrium profits are also higher at the dispersed equilibrium.

Finally, for [Mu] [is greater than or equal to] 1.47, the only equilibrium is the central one, which is then stable. As [Mu] becomes large, the two firms start to behave as monopolies facing elastic demands and locate at the market center.

Uniform Delivered Pricing

Under this pricing policy, each firm charges a single delivered price to all consumers it serves. The firm will refuse to supply consumers which could only be served at a loss. We can therefore describe this policy by (11) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the uniform price.

Proposition 1. For the duopoly model described in section II, there is a unique (pure-strategy) two-stage location-price equilibrium at [Mathematical Expression Omitted] under uniform delivered pricing for [Mu] [is greater than or equal to] [Mu] [bar] [approximately equal to] .286. For [Mu] < [Mu] [bar], there exists no equilibrium.

Proof. First note that no equilibrium can exist with a firm refusing to serve a part of the market. If one firm sets a price such that it is not profitable for it to serve the whole market, then its rival has the incentive to increase its price over the remainder of the market. When a firm does serve the whole market, the expected demand to it is equal at all points in space. Hence we can write its expected profit as (12) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the average transport cost paid by firm i in serving its customers. It is readily shown that each firm's equilibrium profit is a strictly decreasing function of [c.sub.i], the average transport cost.[8] Hence the best locational response of each firm to any possible location of a rival is the center of the market. This establishes that the central agglomeration is the only possible equilibrium.

Now consider the existence problem. For the center to be well defined as a two-stage location-price equilibrium, it is necessary that equilibrium profits be defined (and lower) for alternative locations of one firm given the other remains at the center. That is, there must exist a price equilibrium for the second-stage game for all [x.sub.1] [Epsilon] [0, 1/2] given [Mathematical Expression Omitted]. The feature that may jeopardize existence of a price equilibrium is the possibility that a firm may not wish to serve the whole market at the solution to the first-order conditions corresponding to [12]. It is straightforward to show that this possibility is the more likely to arise the further is firm 1 from the center.(9) According we can find (from the solutions to the first-order conditions) the critical value of [Mu] such that [Mathematical Expression Omitted], the value such that firm 1 just wishes to serve the whole market at the candidate equilibrium prices. This value is approximately 0.286, as stated in the proposition.[10] For all greater values of [Mu], the center is ensured as the equilibrium. Note that the center is at least a local equilibrium for lower values of [Mu] in the sense that profits are maximized there for the whole range of locations for which a price equilibrium exists. The center ceases to be a local equilibrium for [Mu] < 1/8, which value is the solution to [Mathematical Expression Omitted] i.e., this is the lowest value of [Mu] for which there exists a price equilibrium at the center.

Spatial Discriminatory Pricing

Under this policy, each firm sets a location-specific delivered price at each point in space, [Mathematical Expression Omitted]. This pricing policy was initially described in Hoover [26], and further analyzed in Lederer and Hurter [28] for the case of homogeneous consumer tastes.

Proposition 2. (Anderson and de Palma). For the duopoly model of section II, there exists a two-stage location-price equilibrium under spatial discriminatory pricing. The equilibrium locations are given by:

a) [Mathematical Expression Omitted]

b) [Mathematical Expression Omitted] where [Theta] [bar] solves ln [Theta] [bar] + 2 [Theta] [bar] + 1 = 1/ [Mu], and [Mathematical Expression Omitted]. For [Mu] [Epsilon] [1/6, 1/3]) there are two location-price equilibria.

The equilibrium price schedules corresponding to these location-price equilibria are described in Anderson and de Palma [2], where the result is proved. Over the interval outside the firm locations, both firms' delivered prices rise linearly with distance at the same rate as the transport cost. Firms can therefore be seen as using mill pricing policies over these intervals: even though firms have the ability to price discriminate by location, they choose not to do so over this range in equilibrium.(11) Over the inter-firm interval, prices never rise as fast as the transport cost rate. Indeed, for small enough values of [Mu], delivered prices actually fall initially away from each firm towards its rival. (This is seen, for example, in the limit case [Mu] [right arrow] 0, where each firm's delivered price follows its rival's transport cost scheduled.) For higher values of [Mu] they always rise over this interval as the product heterogeneity effect becomes more important.

The equilibrium locations described in Proposition 2 are illustrated in Figure 1. The pattern is very similar to that observed for mill pricing, although the maximum separation of firms is less pronounced. Again there can be two equilibria for some values of [Mu], although now equilibrium, existence is ensured throughout. The intuition underlying these results is qualitatively similar to that given for the mill pricing case. Note that for [Mu] [Epsilon] [1/6, 1/3]) the central equilibrium is unstable, but becomes stable for [Mu] [is greater than or equal to] 1/3. The dispersed equilibrium is stable wherever it exists (i.e., for [Mu] < 1/3).

Just as for the mill pricing case, when there are multiple equilibria, the central agglomeration is less profitable than the disagglomerated one, and is therefore dominated. As noted above, the central agglomeration is also unstable in such cases. For these reasons, we shall only describe the stable equilibria in section IV.

The Optimum

The analysis above has considered equilibrium locations. We now analyze the full social optimum where the planner maximizes the welfare function (8) by choice of both locations and spatial delivered prices for the two firms.

Proposition 3. (Anderson and de Palma). For the model of section II, the optimum pricing policy is mill pricing, and the optimal locations are given implicitly by [Mathematical Expression Omitted] and [Mathematical Expression Omitted] where [Mathematical Expression Omitted] if and only if [Mu] [is greater than or equal to] 1/2.

This result is also proved in Anderson and de Palma [2]. The fact that the optimal pricing policy is mill pricing follows directly from the spatial application of marginal cost pricing. It is illuminating to compare the optimal locations to those arising under a mill pricing equilibrium (see Figure 1). Even though the same price policy is used at the optimum and in equilibrium, the planner's location solution may be very different from the equilibrium one. Whereas the optimal location, [x.sub.1], is monotonically increasing n [Mu], the equilibrium initially exhibits the opposite tendency. Indeed, for sufficiently low values of [Mu] (such that equilibrium nevertheless exists) we observe too little spatial differentiation of products, whereas for [Mu] > [Mu] [Tilde] [approximately equal to] .100 there is excess spatial differentiation and indeed the equilibrium and optimum locations are diverging. Eventually the mill-pricing equilibrium converges and reaches the market center. It is interesting to note that for [Mu] = [Mu] [Tilde] (where the two loci cross) the full social optimum is attained at the market equilibrium for mill pricing.

Exactly the opposite qualitative picture under perfect spatial price discrimination. Except for [Mu] = 0, where the optimum and equilibrium coincide, the equilibrium initially involves excess spatial product differentiation, whereas for higher values of [Mu] > [Mu] [caret] [approximately equal to] .099 there is too little differentiation. For [Mu] = [Mu] [caret] (where these two loci cross), although the equilibrium and optimum locations coincide, the equilibrium does not replicate the optimum since the former does not involve mill pricing.

IV. Firm Profits, Consumer Surplus, and Social Surplus under

Alternative Spatial Pricing Policies

Here we consider the different equilibrium solutions for the three price policies according to the criteria of profitability, consumer surplus and social surplus, which is the sum of the two previous terms.(12) For [Mu] [is greater than or equal to] 1.47 both firms locate at the market center under all three pricing policies. At the center, the equilibrium discriminatory pricing schedules involve mill pricing (see section III), so consumer surplus and profits are the same under both these pricing policies. Given central locations, it is readily verified that [Mathematical Expression Omitted], so that the uniform price equals the average delivered price under mill pricing with [Mathematical Expression Omitted]. Since individual purchase probabilities [3] depend only upon price differences, they equal one half at all points in space and profits are therefore the same under uniform and mill pricing. A similar argument shows that consumer surplus is the same under both policies. Hence, for [Mu] large enough there is always agglomeration at the center and all three surplus measures are independent of the pricing policy. In what follows we concentrate on lower [Mu] values for which equilibria are dispersed for at least one policy.

Firm Profitability

We consider equilibrium profits per firm, gross of any fixed or set-up costs. The results are given in Figure 2, where profit is given as a function of [Mu]. The ranking of the different price policies by profitability is unambiguous, with uniform delivered pricing at the bottom, and mill pricing at the top. Spatial price discrimination has been likened to guerrilla warfare [26]--competition is on many fronts since must be set at all points. On the other hand, mill-pricing is analogous to a single front war--only one price is to be set. Guerrilla warfare involves many local skirmishes which create more attrition: hence the profit superiority of mill pricing over discriminatory pricing. Lastly, uniform pricing is a policy which is much blunter--and as a result less profitable--as a weapon of attack. Since uniform prices do not vary with costs of serving a locality (in contrast to the other policies), they are less well attuned to economic conditions. Profits are therefore lowest under this policy.

Profits eventually rise under all three policies as [Mu] rises: greater market power stems from greater retailers heterogeneity ([Mu]). However, for both mill pricing and discriminatory pricing there is an initial drop in profits with higher [Mu]. This phenomenon can be ascribed to the pro-competitive effect of increased retailer differentiation which initially outweighs the market power effect. For [Mu] [right arrow] 0 there is no market overlap whatsoever. As [Mu] rises, customers begin to be drawn from all over the market spectrum. Competition is enhanced (and hence prices are lower) since a price cut will draw new customers not only from the boundary between firms' markets but also from all other points.

Consumer Surplus

The consumer surplus measure used is given by equation [10] as described in section II. The results of the comparison were computed numerically and are plotted in Figure 3. The surplus values are negative because of the omission of a positive constant in the surplus expression. Hence only relative rankings should be considered and no weight attached to absolute magnitudes. Nor should much attention be focused on the behavior of the schedules as [Mu] rises in this case: higher values of [Mu] represent greater variance of tastes in the population and the changes in consumer welfare that result from this are potentially misleading from the economic point of view.[13] However, as noted above, the rankings for any given [Mu] are still relevant, even if the comparative static results with respect to [Mu] do not have much meaning.

For all values of [Mu] (< 1.470 the ranking of price policies is the opposite of that for the probability criterion. This result can be roughly attributed to the same phenomena that give rise to the profit ranking and indicate how antagonistic firms' and consumers' interests may be in the preference for a price policy. Specifically, prices tend to be lowest under uniform pricing and highest under mill pricing. Even though locations under uniform pricing are far from optimal (for low [Mu]), the intensity of competition under this policy drives prices so low as to yield greatest aggregate consumer surplus

Note finally that the distribution of consumer surplus is very different under the different policies. For uniform pricing, all consumers have the same expected surplus. However, this surplus depends on consumer location for the other two policies.

Social Surplus

Figure 4 describes the social surplus associated with price and location equilibrium for the different policies. For reasons outlined in (b) above, only the ranking (for any given [Mu]) is relevant. For low values of [Mu], mill pricing dominates discriminatory pricing, which in turn dominates uniform pricing. For intermediate values of [Mu] (a blow-up of Figure 4 is given in Figure 5 for [Mu] [Epsilon] [0.2, 0.25]), first uniform pricing comes to dominate discriminatory pricing, and then they both come to dominate mill pricing. The most striking result is the decline of mill-pricing as the welfare-maximizing equilibrium when firms choose locations. This result is due to the extreme locational proclivities of firms under mill-pricing (see Figure 1). Despite the fact that mill-pricing is the socially optimal pricing policy for fixed locations, the equilibrium locations take firms to far astray that the other policies come to dominate in welfare terms.

V. Conclusions

One of the major findings of this paper is the social inefficiency of mill pricing, when compared to other pricing policies, once we account for endogenous location choice by firms and given that firms choose prices non-cooperatively). This result is all the more striking since we have "stacked the cards" in favor of mill pricing by analyzing a demand system for which there is no deadweight loss from a unilateral rise in prices over marginal costs. Indeed, for any fixed (symmetric) location pair, mill pricing is the socially optimal pricing policy regardless of the price level (and as long as mill prices are equal). It is the introduction of endogenous locations that overturns this result. Even a uniform price-location equilibrium may yield higher welfare than the mill price-location equilibrium. This means that policy prescriptions as regards regulation over pricing policies may be very different if the long-run view is taken over the short-run one. That is, the recommendations of regulatory authorities may depend on their perspectives.

Other authors have also cast doubt on the social superiority of mill pricing, although to our knowledge, we are the first to really tackle the location issue.(14) It has previously been shown that spatial price discrimination can lead to higher social surplus than mill pricing in the monopoly case [23]. A similar result is shown in Hobbs[22] for linear demand and under the assumption that firms are located equidistantly on an infinite line market. The result stems from the higher prices arising under mill pricing. Given there is now demand elasticity, this yields deadweight loss and inefficiency. This paper further analyzes the case where the spacing of firms is endogenously determined via a zero profit condition, and the analysis is extended in Holahan and Schuler[24] to the case where firm relocation is prohibitively costly so that entry is not a continuous function of fixed costs but instead occurs in rounds. Under both of these extensions to deal with entry considerations, mill pricing may not be the welfare maximizing policy (depending on parameter values). These results further undermine any suggestion that mill pricing is a practice to be encouraged. Indeed, for a large range of parameter values, spatial price discrimination is socially optimal in the model of this paper. In the models in both Hobbs [22] and Holahan and Schuler [24] spatial price discrimination is also optimal for a large range of parameters. Because these models are significantly different from ours, we may conclude that this result is rather robust.

Our preoccupation with mill pricing stems in part from legal restrictions which may influence or govern firm pricing. According to some authors, mill pricing is "effectively the form of spatial pricing preferred by proponents of the Robinson-Patman Act in the United States, and the only form of spatial pricing that avoids the criticisms of cross-subsidization made by the Price Commission in the United Kingdom" [20, 19]. Whilst the Robinson-Patman Act may be viewed as favoring mill pricing, "the search for complete certainty in such a dynamic field of law and economics is a foregone futility" [36, 37]. Greenhut found that a third of the firms he surveyed in the U.S. used a mill pricing policy and concluded that: "Quite conceivably it is the Robinson-Patman Act that causes the delivered prices of American firms to differ from firms in West Germany and Japan" [19, 84]. Our analysis, in conjunction with the results cited above, casts considerable doubt on the wisdom of such a legal statute as this feature of the Robinson-Patman Act.

Another important conclusion of this paper concerns the question of whether the market provides excessive or insufficient product variety (where variety is measured by locations). Under uniform pricing, there is no spatial differentiation of firms in equilibrium so that product differentiation is too little (for low values of [Mu]) when compared to the optimum locations. For mill and discriminatory pricing the results are more involved. Under discriminatory pricing there can be either too much or too little spatial differentiation when compared to the optimum. In complete contrast to the discriminatory pricing case, the market solution for mill pricing provides insufficient diversity for low values of product heterogeneity (low [Mu]) and excessive diversity for higher product heterogeneity. Once [Mu] is sufficiently large, the market provides the right amount of diversity under all three pricing policies. These results highlight the fact that the question of optimal vs. equilibrium diversity is a complex issue with many facets. (1)Almost a third of firms used mixed pricing, 15% used only discriminatory pricing other than uniform pricing [19]. (2)Mixed strategy equilibrium in the mill pricing model have been considered in Osborne and Pitchik [32]. However, the complexity of the solution would seem to preclude using this approach as a basis for comparison. (3)Other papers in which the logit model has been used in the context of spatial competition include Besanko and Perry [9] and Braid [11]. (4)An alternative interpretation of the term [e.sub.i](x) is that the products sold by firms are themselves not homogenous. Under the present interpretation the product is the same but the characteristics of retailers are different, and are evaluated differently by different consumers. Clearly a combination of product and retailer heterogeneity is consistent with the analysis in the text. (5)Despite some serious limitations due to the property of independence from irrelevant (or IIA--see Ben-Akiva and Lerman [8, 108-111] for discussion of this property) there are several good reasons for using the logit formulation. First, IIA itself is not restrictive in the present duopoly context. Second, the logit has been used extensively in transportation studies and has proved a powerful tool in explaining travel demand [8; 15]. Third, the logit can be given a sound axiomatic basis in the probabilistic choice context (see especially Luce [29] and Yellott [37]). Last, the logit model admits a closed-form for the choice probabilities which is easy to work with. See also McFadden [31] for further discussion of the derivation of the logit model. (6)The homogeneous model is obtained as the limit case where [Mu] = 0, and CS = max[--[p.sub.1] (x), -- [p.sub.2] (x)], which can be shown to be the limit case of (7) as [Mu] [right arrow] 0. (7)Without normalization, this parameter is [Mu]/ cl, where c is the transport rate per unit per mile and l is the length of the market. (8)The first-order conditions corresponding to (3) are [Mathematical Expression Omitted]. Totally differentiating these equations [Mathematical Expression Omitted]. Equilibrium profits for firm 1 are (using the first-order conditions): [Mathematical Expression Omitted]. Hence [Mathematical Expression Omitted] as claimed. (9)The profit per unit earned by firm one from the consumers at the far end of the market, x = 1, i given by [Mathematical Expression Omitted] so that the derivative of this expression is [Mathematical Expression Omitted]. We therefore wish to show that [Mathematical Expression Omitted] The expression for [Mathematical Expression Omitted] is given in the preceding footnote. Also, by definition of [Mathematical Expression Omitted]. Combining these two expressions yields the condition desired as [Mathematical Expression Omitted], which is obviously true for [x.sub.1] [is greater than or equal to] 0. (10)This value is calculated from the first-order conditions, [Mathematical Expression Omitted] and [c.sub.2] = 1/4, which are the cost values associated with [x.sub.1] = 0 and [x.sub.2] = 1/2. (11)This result stems from the assumed linearity of transport costs and that demand depends only on price differences. It is not an artificial of the logit, and holds for all models with these properties [2]. (12)In the diagrams that follow, the surpluses are shown for uniform pricing as long as there is at least a local equilibrium ([Mu] [is greater than or equal to] 1/8). Recall from Proposition 1 that a global equilibrium is ensured for [Mu] [is greater than or equal to] 0.286. (13)As an illustration of this sort of issue in a different context, consider the Cobb-Douglas utility function [Mathematical Expression Omitted] where [x.sub.1] is consumption of good i, i = 1, 2. A rise in the parameter [Alpha] represents a shift in tastes, but it is not meaningful to say that such a shift causes utility to rise. Note that in the profitability analysis of the preceding section this sort of problem does not arise. The comparative static exercise there is interpreted as an increase in retailer differentiation. (14)In other papers of which we are aware in this area, a circular of infinitely long market is assumed and a symmetry condition imposed on firm locations so all interfirm distances are equal. The location problem per se can scarcely be addressed in such frameworks. By contrast, the introduction of market boundaries, as in our model, renders this problem non-trivial. Although we take no account of entry, we focus explicitly on locations.

References

[1]American Bar Association. Section of Antitrust Law of the (1979) Antitrust Law Developments, Second Supplement, American Bar Association. [2]Anderson, Simon P. and Andre de Palma, "Spatial Price Discrimination with Heterogeneous Products." Review of Economic Studies, October 1988, 573-92. [3]--, --, and Jacques-Francois Thisse. "A Representative Consumer Theory of the Logit Model." International Economic Review, August 1988, 461-66. [4]--, --, and --, "Spatial Price Policies Reconsidered." Journal of Industrial Economics, September 1989, 1-18. [5]--, --, and --, "Demand for Differentiated Products, Discrete Choice Models, and the Characteristics Approach." Review of Economic Studies, January 1989, 21-35. [6]Beckmann, Martin J., "Spatial Price Policies Revisited." Bell Journal of Economics, Autumn 1976, 619-30. [7]-- and Jacques-Francois Thisse. "The Location of Production Activities," in Handbook of Regional and Urban Economics, volume 1, edited by P. Nijkamp. Amsterdam: North-Holland, 1986. [8]Ben-Akiva, Moshe and Steven R. Lerman. Discrete Choice Analysis Theory and Application to Travel Demand. Cambridge, Massachusetts: MIT Press, 1985. [9]Besanko, David and Martin K. Perry. "Exclusive Dealing in a Spatial Model of Retail Competition." Mimeo, Bellcore, New Jersey, 1988. [10]Bork, Robert H. The Antitrust Paradox. New York: Basic Books Inc., 1978. [11]Braid, Ralph, "Heterogeneous Preferences and Non-Central Agglomeration of Firms." Regional Science and Urban Economics, February 1988, 57-68. [12]Breit, William and Kenneth G. Elzinga. The Antitrust Casebook, 2nd ed. New York: The Dryden Press, 1989. [13]d'Aspremont, Claude, Jean Jaskold Gabszewicz and Jacques-Francois Thisse, "On Hotelling's |Stability in Competition'." Econometrica, 1979, 1145-50. [14]de Palma, Andre, Victor Ginsburgh, Yorgo Y. Papageorgiou and Jacques-Francois Thisse, "The Principle of Minimum Differentiation holds under Sufficient Heterogeneity." July Econometrica, 1985, 767-81. [15]Domencich, Thomas A. and Daniel McFadden. Urban Travel Demand: A Behavioral Analysis. Amsterdam: North-Holland, 1975. [16]Dunn, Charles Wesley. "The Validity of a Uniform Delivered Price," in The Robinson Patman Act Symmposium. Commerce Clearing House, Inc., 1947. [17]Gabszewicz, Jean Jaskold and Jacques-Francois Thisse. "Location," in Handbook of Game Theory with Economic Applications, edited by R. J. Aumann and S. Hart. Amsterdam: North-Holland, 1992. [18]Greenhut, Melvin L. Plant Location in Theory and in Practice. Chapel Hill: University of North Carolina Press, 1956. [19]--, "Spatial Pricing in the U.S., West Germany and Japan." February Economica, 1981, 79-86. [20]--, George Norman and Chao-shun Hung. The Economics of Imperfect Competition. A Spatial Approach. Cambridge: Cambridge University Press. 1987. [21]-- and Hiroshi Ohta, "Monopoly Output under Alternative Spatial Pricing Techniques." American Economic Review, September 1972-13. [22]Hobbs, Benjamin F., "Mill-Pricing versus Spatial Price Discrimination under Bertrand and Cournot Spatial Competition." Journal of Industrial Economics, December 1986, 173-91. [23]Holahan, William L., "The Welfare Effects of Spatial Price Discrimination." American Economic Review, June 1975, 498-503. [24]-- and Robert E. Schuler. "Imperfect Competition in a Spatial Economy: Pricing Policies and Economic Welfare." CORE Discussion Paper 8821, Universite Catholique de Louvain, Belgium, 1988. [25]Hoover, Edgar M., "Spatial Price Discrimination." Review of Economic Studies, June 1937, 182-92. [26]--. The Location of Economic Activity. New York: McGraw-Hill, 1948. [27]Hotelling, Harold, "Stability in Competition." Economic Journal, 1929, 41-57. [28]Lederer, Philip J. and Arthur P. Hurter, "Competition of Firms: Discriminatory Pricing and Location." Econometrica, May 1986, 623-41. [29]Luce, R. Duncan. Individual Choice Behavior: A Theoretical Analysis New York: Wiley, 1959. [30]McFadden, Daniel. "Conditional Logit Analysis of Qualitative Choice Behavior," in Frontiers of Econometrics, edited by P. Zarembka, New York: Academic Press, 1973. [31]--. "Econometric Models of Probabilistic Choice," in Structural Analysis of Discrete Data with Economic Applications, edited by Charles F. Manski and Daniel McFadden. Cambridge, Massachusetts: MIT Press, 1981. [32]Osborne, Martin J. and Carolyn Pitchik, "Equilibria in Hotelling's Model of Spatial Competition." Econometrica, July 1987, 9911-22. [33]Patman, Wright. The Robinson-Patman Act. New York: The Ronald Press Company, 1938. [34]Posner, Richard A. The Robinson-Patman Act. Federal Regulation of Price Differences. Washington, D.C.: American Enterprise Institute for Public Policy Research, 1976. [35]Small, Kenneth A. and Harvey S. Rosen, "Applied Welfare Economics with Discrete Choice Models." Econometrica, January 1981, 105-30. [36]Wooden, Walter B. "The Delivered Price Generally," in The Robinson Patman Act Symposium. Commerce Clearing House, Inc., 1947. [37]Yellott, John I., "The Relationship between Luce's Choice Axiom, Thurstone's Theory of Comparative Judgment, and the Double Exponential Distribution." Journal of Mathematical Psychology, April 1977, 109-44.

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.

II. 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 parameter [Mu] conveys the degree of dispersion of these tastes across consumers. For [Mu] [right arrow] 0, we recover the case of homogeneous tastes (or homogeneous sellers).

Following de Palma et al. [14], the probability of an individual at x purchasing the product from firm i is given by the binomial logit formulation(5) as (3) [Mathematical Expression Omitted] so that clearly [P.sub.1] (x) + [P.sub.2] (x) = 1. Note that sellers are not differentiated (i.e., they are perfect substitutes) for [Mu] [right arrow] 0. [P.sub.i] (x) is a continuously decreasing function of [p.sub.i] (x), which is concave (convex) for [p.sub.i] (x) [is less than or equal to] ([is greater than or equal to]) [p.sub.j] (x Thus it has the standard shape associated with a well-behaved unimodal density function.

Costs

We assume each firm produces with constant and identical marginal costs, which can therefore be set equal to zero without loss of generality. Transport costs per unit shipped are assumed linear in distance and invariant to volume. The transportation rate will be normalized to unity by appropriate choice of unit of account. We consider three types of alternative spatial pricing policy.

Profits and the Equilibrium Concept

Firms are assumed to be risk neutral. Each firm's expected profits under the three alternative pricing policies are (4) [Mathematical Expression Omitted] where [p.sup.i] (x) is constrained by the pricing policy under consideration and specified below. Notice than an increase in both firms' delivered prices by one dollar at all locations raises total profits ([[Pi].sub.1] + [[Pi].sub.2]) by one dollar. This results from the property of (3) that the purchase probabilities depend only upon price differences.

The market equilibrium we consider for each of the alternative pricing policies is the equilibrium to a two-stage game. Locations are chosen at the first stage and are predicted upon the knowledge of the second-stage pricing equilibrium. Hence we analyze a sub-game perfect Nash equilibrium. This concept is now made precise.

The solution is recursive. We first solve the second-stage pricing sub-game. For given firm locations ([x.sub.1], [x.sub.2]), a price-schedule equilibrium [Mathematical Expression Omitted] is defined by (5) [Mathematical Expression Omitted] for all allowable price schedules {[p.sub.i] (x)}, for all x [Epsilon] [0, 1]; i, j = 1, 2, i [is not equal to] j. For both mill and uniform pricing policies, the allowable price schedules are defined in terms of a single variable, [Mathematical Expression Omitted] (the mill price) and [Mathematical Expression Omitted] (the uniform delivered price respectively.

Now consider the first stage location game. Define [[Pi] [tilde].sub.i] ([x.sub.i], [x.sub.j]) as i's profit evaluated at the second stage price schedule equilibrium. Equilibrium to the full two-stage game is then defined by a location pair, [Mathematical Expression Omitted] such that (6) [Mathematical Expression Omitted]

Consumer Surplus

One criterion for comparison is aggregate consumer surplus, defined as the integral of net benefits of consumers in the market. Once we allow for heterogeneity across products, it will no longer be the case that consumers patronize the firm with the lower delivered price. Hence market segments will overlap. The consumer surplus measure therefore must also include the benefits from product variety, as expressed by [Mu], along with the delivered price paid. The former element is absent from standard location models with homogeneous tastes.(6) For the demand system (3), consumer surplus is given (up to a positive constant) by (7) [Mathematical Expression Omitted] (see Anderson, de Palma, and Thisse [3] and Small and Rosen [35]). Note that a one dollar increase in both delivered prices at all locations causes aggregate consumer surplus to fall by one dollar (this can be seen by using [p.sub.i] = [p.sub.i] + 1, i = 1, 2 in (7)).

Social Surplus and the Optimum

We define social surplus as total consumer surplus (as defined in Equation (7) plus profits of both firms (as given by Equation (4)). Social surplus (or "welfare") is given by (8) W = CS + [[Pi].sub.1] + [[Pi].sub.2]. The optimal solution entails choice of both spatial price schedules and locations. It is therefore the solution to (9) [Mathematical Expression Omitted] where W(.) is given by (8). The level of social surplus depends only upon price differences, although the distribution of that surplus depends on price levels: as noted previously, one dollar rise in both prices at all locations is simply a transfer of surplus from consumer to firms. This has important implications for the analysis below. In particular, any situation which differs from another only by the absolute level of prices (and not by their difference) entails the same allocation of consumers to outlets and hence the same total social surplus.

Having defined the model and criteria for comparison, we now turn to the location results. It should be stressed at this point that all location equilibria and the optimum can be expressed in terms of a single parameter. This parameter is [Mu], the measure of consumer taste heterogeneity across retailers.(7)

III. Equilibrium and Optimum Locations under Alternative Spatial Pricing Policies

We shall see below that taking explicit account of taste variations (as per equations (1)-(3) can restore the existence of equilibrium providing the taste variations are sufficiently large, thus allowing us to compare the price policies.

Mill Pricing

Under this pricing policy, each firm charges a single mill price to all consumers, and all transport costs are passed on to consumers. The delivered price paid by a consumer at x and purchasing from firm i at [x.sub.i] is (10) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the mill price.

We have not been able to fully characterize algebraically the equilibrium locations under mill pricing for all values of [Mu]. We have, however, simulated the market equilibrium under this pricing policy. These simulations show a large variety of possibilities. The results are illustrated in Figure 1. As [Mu] rises from zero, there is first no equilibrium; then there is a region where there is a unique equilibrium, at which firms are separated. As [Mu] continues to rise, there is next a region where there are two equilibria; finally, there is a unique equilibrium which involves central agglomeration.

The reason for the initial non-existence of equilibrium for [Mu] < 0.062 is similar to that given in d'Aspremont, Gabszewicz, Thisse [13] for the case of homogenous products I that case each firm's profits rise as it approaches is rival's location. However this central tendency eventually leads firms into the situation where there exists no price equilibrium because each wishes to undercut its rival's mill price at any candidate solution to the pricing first-order conditions. That is, there is a fundamental failure of quasiconcavity (in own price) in the profit functions. By continuity, this explains why there is a region of [Mu]-values for which there is no equilibrium. When [Mu] is high enough, the failure of quasiconcavity becomes non-critical since the relative benefit of mill price undercutting is reduced because demand is less responsive to small price differences (consumers care more about other aspects of the products). In other words, introducing preference heterogeneity smoothes the profit functions so that they become well behaved. Hence for [Mu] large enough, there exists an equilibrium to the second-stage game so that the first-stage location game can be defined. The latter then also has an equilibrium, as illustrated in Figure 1.

For 0.062 [is less than or equal to] [Mu] < 1.47, there is a symmetric dispersed equilibrium which initially entails increasing spatial separation of firms as [Mu] rises. This effect can be ascribed to a tendency for firms to consolidate their monopoly power as the influence of the rival becomes more far-reaching (in the sense that rising [Mu] implies a greater invasion of a firm's hinterland). For larger [Mu] (around 0.3), the firms start to move together again with increasing [Mu]. This effect is due to the increased desirability of a central location given that, ceteris paribus, the demand addressed to a firm becomes increasingly even over space as [Mu] rises. Hence the equilibrium can be seen as the result of two opposing forces, with the agglomerative force eventually outweighing he deglomerative one.

For 0.76 [is less than or equal to] [Mu] < 1.47, there is an agglomerated equilibrium at the centre along with the dispersed one. The phenomenon of multiple equilibria can be understood in the following manner. Equilibrium prices are relatively high at the non-agglomerated equilibrium locations. Any move toward the center increases price competition substantially and is avoided on those grounds. On the other hand, equilibrium prices are relatively low at the agglomerated equilibrium in the center. Moving away reduces competitive pressure (and enables higher prices) but also reduces market share so much as to reduce profits. When the central equilibrium co-exists with the dispersed one, it is the latter which is the stable equilibrium. Equilibrium profits are also higher at the dispersed equilibrium.

Finally, for [Mu] [is greater than or equal to] 1.47, the only equilibrium is the central one, which is then stable. As [Mu] becomes large, the two firms start to behave as monopolies facing elastic demands and locate at the market center.

Uniform Delivered Pricing

Under this pricing policy, each firm charges a single delivered price to all consumers it serves. The firm will refuse to supply consumers which could only be served at a loss. We can therefore describe this policy by (11) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the uniform price.

Proposition 1. For the duopoly model described in section II, there is a unique (pure-strategy) two-stage location-price equilibrium at [Mathematical Expression Omitted] under uniform delivered pricing for [Mu] [is greater than or equal to] [Mu] [bar] [approximately equal to] .286. For [Mu] < [Mu] [bar], there exists no equilibrium.

Proof. First note that no equilibrium can exist with a firm refusing to serve a part of the market. If one firm sets a price such that it is not profitable for it to serve the whole market, then its rival has the incentive to increase its price over the remainder of the market. When a firm does serve the whole market, the expected demand to it is equal at all points in space. Hence we can write its expected profit as (12) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the average transport cost paid by firm i in serving its customers. It is readily shown that each firm's equilibrium profit is a strictly decreasing function of [c.sub.i], the average transport cost.[8] Hence the best locational response of each firm to any possible location of a rival is the center of the market. This establishes that the central agglomeration is the only possible equilibrium.

Now consider the existence problem. For the center to be well defined as a two-stage location-price equilibrium, it is necessary that equilibrium profits be defined (and lower) for alternative locations of one firm given the other remains at the center. That is, there must exist a price equilibrium for the second-stage game for all [x.sub.1] [Epsilon] [0, 1/2] given [Mathematical Expression Omitted]. The feature that may jeopardize existence of a price equilibrium is the possibility that a firm may not wish to serve the whole market at the solution to the first-order conditions corresponding to [12]. It is straightforward to show that this possibility is the more likely to arise the further is firm 1 from the center.(9) According we can find (from the solutions to the first-order conditions) the critical value of [Mu] such that [Mathematical Expression Omitted], the value such that firm 1 just wishes to serve the whole market at the candidate equilibrium prices. This value is approximately 0.286, as stated in the proposition.[10] For all greater values of [Mu], the center is ensured as the equilibrium. Note that the center is at least a local equilibrium for lower values of [Mu] in the sense that profits are maximized there for the whole range of locations for which a price equilibrium exists. The center ceases to be a local equilibrium for [Mu] < 1/8, which value is the solution to [Mathematical Expression Omitted] i.e., this is the lowest value of [Mu] for which there exists a price equilibrium at the center.

Spatial Discriminatory Pricing

Under this policy, each firm sets a location-specific delivered price at each point in space, [Mathematical Expression Omitted]. This pricing policy was initially described in Hoover [26], and further analyzed in Lederer and Hurter [28] for the case of homogeneous consumer tastes.

Proposition 2. (Anderson and de Palma). For the duopoly model of section II, there exists a two-stage location-price equilibrium under spatial discriminatory pricing. The equilibrium locations are given by:

a) [Mathematical Expression Omitted]

b) [Mathematical Expression Omitted] where [Theta] [bar] solves ln [Theta] [bar] + 2 [Theta] [bar] + 1 = 1/ [Mu], and [Mathematical Expression Omitted]. For [Mu] [Epsilon] [1/6, 1/3]) there are two location-price equilibria.

The equilibrium price schedules corresponding to these location-price equilibria are described in Anderson and de Palma [2], where the result is proved. Over the interval outside the firm locations, both firms' delivered prices rise linearly with distance at the same rate as the transport cost. Firms can therefore be seen as using mill pricing policies over these intervals: even though firms have the ability to price discriminate by location, they choose not to do so over this range in equilibrium.(11) Over the inter-firm interval, prices never rise as fast as the transport cost rate. Indeed, for small enough values of [Mu], delivered prices actually fall initially away from each firm towards its rival. (This is seen, for example, in the limit case [Mu] [right arrow] 0, where each firm's delivered price follows its rival's transport cost scheduled.) For higher values of [Mu] they always rise over this interval as the product heterogeneity effect becomes more important.

The equilibrium locations described in Proposition 2 are illustrated in Figure 1. The pattern is very similar to that observed for mill pricing, although the maximum separation of firms is less pronounced. Again there can be two equilibria for some values of [Mu], although now equilibrium, existence is ensured throughout. The intuition underlying these results is qualitatively similar to that given for the mill pricing case. Note that for [Mu] [Epsilon] [1/6, 1/3]) the central equilibrium is unstable, but becomes stable for [Mu] [is greater than or equal to] 1/3. The dispersed equilibrium is stable wherever it exists (i.e., for [Mu] < 1/3).

Just as for the mill pricing case, when there are multiple equilibria, the central agglomeration is less profitable than the disagglomerated one, and is therefore dominated. As noted above, the central agglomeration is also unstable in such cases. For these reasons, we shall only describe the stable equilibria in section IV.

The Optimum

The analysis above has considered equilibrium locations. We now analyze the full social optimum where the planner maximizes the welfare function (8) by choice of both locations and spatial delivered prices for the two firms.

Proposition 3. (Anderson and de Palma). For the model of section II, the optimum pricing policy is mill pricing, and the optimal locations are given implicitly by [Mathematical Expression Omitted] and [Mathematical Expression Omitted] where [Mathematical Expression Omitted] if and only if [Mu] [is greater than or equal to] 1/2.

This result is also proved in Anderson and de Palma [2]. The fact that the optimal pricing policy is mill pricing follows directly from the spatial application of marginal cost pricing. It is illuminating to compare the optimal locations to those arising under a mill pricing equilibrium (see Figure 1). Even though the same price policy is used at the optimum and in equilibrium, the planner's location solution may be very different from the equilibrium one. Whereas the optimal location, [x.sub.1], is monotonically increasing n [Mu], the equilibrium initially exhibits the opposite tendency. Indeed, for sufficiently low values of [Mu] (such that equilibrium nevertheless exists) we observe too little spatial differentiation of products, whereas for [Mu] > [Mu] [Tilde] [approximately equal to] .100 there is excess spatial differentiation and indeed the equilibrium and optimum locations are diverging. Eventually the mill-pricing equilibrium converges and reaches the market center. It is interesting to note that for [Mu] = [Mu] [Tilde] (where the two loci cross) the full social optimum is attained at the market equilibrium for mill pricing.

Exactly the opposite qualitative picture under perfect spatial price discrimination. Except for [Mu] = 0, where the optimum and equilibrium coincide, the equilibrium initially involves excess spatial product differentiation, whereas for higher values of [Mu] > [Mu] [caret] [approximately equal to] .099 there is too little differentiation. For [Mu] = [Mu] [caret] (where these two loci cross), although the equilibrium and optimum locations coincide, the equilibrium does not replicate the optimum since the former does not involve mill pricing.

IV. Firm Profits, Consumer Surplus, and Social Surplus under

Alternative Spatial Pricing Policies

Here we consider the different equilibrium solutions for the three price policies according to the criteria of profitability, consumer surplus and social surplus, which is the sum of the two previous terms.(12) For [Mu] [is greater than or equal to] 1.47 both firms locate at the market center under all three pricing policies. At the center, the equilibrium discriminatory pricing schedules involve mill pricing (see section III), so consumer surplus and profits are the same under both these pricing policies. Given central locations, it is readily verified that [Mathematical Expression Omitted], so that the uniform price equals the average delivered price under mill pricing with [Mathematical Expression Omitted]. Since individual purchase probabilities [3] depend only upon price differences, they equal one half at all points in space and profits are therefore the same under uniform and mill pricing. A similar argument shows that consumer surplus is the same under both policies. Hence, for [Mu] large enough there is always agglomeration at the center and all three surplus measures are independent of the pricing policy. In what follows we concentrate on lower [Mu] values for which equilibria are dispersed for at least one policy.

Firm Profitability

We consider equilibrium profits per firm, gross of any fixed or set-up costs. The results are given in Figure 2, where profit is given as a function of [Mu]. The ranking of the different price policies by profitability is unambiguous, with uniform delivered pricing at the bottom, and mill pricing at the top. Spatial price discrimination has been likened to guerrilla warfare [26]--competition is on many fronts since must be set at all points. On the other hand, mill-pricing is analogous to a single front war--only one price is to be set. Guerrilla warfare involves many local skirmishes which create more attrition: hence the profit superiority of mill pricing over discriminatory pricing. Lastly, uniform pricing is a policy which is much blunter--and as a result less profitable--as a weapon of attack. Since uniform prices do not vary with costs of serving a locality (in contrast to the other policies), they are less well attuned to economic conditions. Profits are therefore lowest under this policy.

Profits eventually rise under all three policies as [Mu] rises: greater market power stems from greater retailers heterogeneity ([Mu]). However, for both mill pricing and discriminatory pricing there is an initial drop in profits with higher [Mu]. This phenomenon can be ascribed to the pro-competitive effect of increased retailer differentiation which initially outweighs the market power effect. For [Mu] [right arrow] 0 there is no market overlap whatsoever. As [Mu] rises, customers begin to be drawn from all over the market spectrum. Competition is enhanced (and hence prices are lower) since a price cut will draw new customers not only from the boundary between firms' markets but also from all other points.

Consumer Surplus

The consumer surplus measure used is given by equation [10] as described in section II. The results of the comparison were computed numerically and are plotted in Figure 3. The surplus values are negative because of the omission of a positive constant in the surplus expression. Hence only relative rankings should be considered and no weight attached to absolute magnitudes. Nor should much attention be focused on the behavior of the schedules as [Mu] rises in this case: higher values of [Mu] represent greater variance of tastes in the population and the changes in consumer welfare that result from this are potentially misleading from the economic point of view.[13] However, as noted above, the rankings for any given [Mu] are still relevant, even if the comparative static results with respect to [Mu] do not have much meaning.

For all values of [Mu] (< 1.470 the ranking of price policies is the opposite of that for the probability criterion. This result can be roughly attributed to the same phenomena that give rise to the profit ranking and indicate how antagonistic firms' and consumers' interests may be in the preference for a price policy. Specifically, prices tend to be lowest under uniform pricing and highest under mill pricing. Even though locations under uniform pricing are far from optimal (for low [Mu]), the intensity of competition under this policy drives prices so low as to yield greatest aggregate consumer surplus

Note finally that the distribution of consumer surplus is very different under the different policies. For uniform pricing, all consumers have the same expected surplus. However, this surplus depends on consumer location for the other two policies.

Social Surplus

Figure 4 describes the social surplus associated with price and location equilibrium for the different policies. For reasons outlined in (b) above, only the ranking (for any given [Mu]) is relevant. For low values of [Mu], mill pricing dominates discriminatory pricing, which in turn dominates uniform pricing. For intermediate values of [Mu] (a blow-up of Figure 4 is given in Figure 5 for [Mu] [Epsilon] [0.2, 0.25]), first uniform pricing comes to dominate discriminatory pricing, and then they both come to dominate mill pricing. The most striking result is the decline of mill-pricing as the welfare-maximizing equilibrium when firms choose locations. This result is due to the extreme locational proclivities of firms under mill-pricing (see Figure 1). Despite the fact that mill-pricing is the socially optimal pricing policy for fixed locations, the equilibrium locations take firms to far astray that the other policies come to dominate in welfare terms.

V. Conclusions

One of the major findings of this paper is the social inefficiency of mill pricing, when compared to other pricing policies, once we account for endogenous location choice by firms and given that firms choose prices non-cooperatively). This result is all the more striking since we have "stacked the cards" in favor of mill pricing by analyzing a demand system for which there is no deadweight loss from a unilateral rise in prices over marginal costs. Indeed, for any fixed (symmetric) location pair, mill pricing is the socially optimal pricing policy regardless of the price level (and as long as mill prices are equal). It is the introduction of endogenous locations that overturns this result. Even a uniform price-location equilibrium may yield higher welfare than the mill price-location equilibrium. This means that policy prescriptions as regards regulation over pricing policies may be very different if the long-run view is taken over the short-run one. That is, the recommendations of regulatory authorities may depend on their perspectives.

Other authors have also cast doubt on the social superiority of mill pricing, although to our knowledge, we are the first to really tackle the location issue.(14) It has previously been shown that spatial price discrimination can lead to higher social surplus than mill pricing in the monopoly case [23]. A similar result is shown in Hobbs[22] for linear demand and under the assumption that firms are located equidistantly on an infinite line market. The result stems from the higher prices arising under mill pricing. Given there is now demand elasticity, this yields deadweight loss and inefficiency. This paper further analyzes the case where the spacing of firms is endogenously determined via a zero profit condition, and the analysis is extended in Holahan and Schuler[24] to the case where firm relocation is prohibitively costly so that entry is not a continuous function of fixed costs but instead occurs in rounds. Under both of these extensions to deal with entry considerations, mill pricing may not be the welfare maximizing policy (depending on parameter values). These results further undermine any suggestion that mill pricing is a practice to be encouraged. Indeed, for a large range of parameter values, spatial price discrimination is socially optimal in the model of this paper. In the models in both Hobbs [22] and Holahan and Schuler [24] spatial price discrimination is also optimal for a large range of parameters. Because these models are significantly different from ours, we may conclude that this result is rather robust.

Our preoccupation with mill pricing stems in part from legal restrictions which may influence or govern firm pricing. According to some authors, mill pricing is "effectively the form of spatial pricing preferred by proponents of the Robinson-Patman Act in the United States, and the only form of spatial pricing that avoids the criticisms of cross-subsidization made by the Price Commission in the United Kingdom" [20, 19]. Whilst the Robinson-Patman Act may be viewed as favoring mill pricing, "the search for complete certainty in such a dynamic field of law and economics is a foregone futility" [36, 37]. Greenhut found that a third of the firms he surveyed in the U.S. used a mill pricing policy and concluded that: "Quite conceivably it is the Robinson-Patman Act that causes the delivered prices of American firms to differ from firms in West Germany and Japan" [19, 84]. Our analysis, in conjunction with the results cited above, casts considerable doubt on the wisdom of such a legal statute as this feature of the Robinson-Patman Act.

Another important conclusion of this paper concerns the question of whether the market provides excessive or insufficient product variety (where variety is measured by locations). Under uniform pricing, there is no spatial differentiation of firms in equilibrium so that product differentiation is too little (for low values of [Mu]) when compared to the optimum locations. For mill and discriminatory pricing the results are more involved. Under discriminatory pricing there can be either too much or too little spatial differentiation when compared to the optimum. In complete contrast to the discriminatory pricing case, the market solution for mill pricing provides insufficient diversity for low values of product heterogeneity (low [Mu]) and excessive diversity for higher product heterogeneity. Once [Mu] is sufficiently large, the market provides the right amount of diversity under all three pricing policies. These results highlight the fact that the question of optimal vs. equilibrium diversity is a complex issue with many facets. (1)Almost a third of firms used mixed pricing, 15% used only discriminatory pricing other than uniform pricing [19]. (2)Mixed strategy equilibrium in the mill pricing model have been considered in Osborne and Pitchik [32]. However, the complexity of the solution would seem to preclude using this approach as a basis for comparison. (3)Other papers in which the logit model has been used in the context of spatial competition include Besanko and Perry [9] and Braid [11]. (4)An alternative interpretation of the term [e.sub.i](x) is that the products sold by firms are themselves not homogenous. Under the present interpretation the product is the same but the characteristics of retailers are different, and are evaluated differently by different consumers. Clearly a combination of product and retailer heterogeneity is consistent with the analysis in the text. (5)Despite some serious limitations due to the property of independence from irrelevant (or IIA--see Ben-Akiva and Lerman [8, 108-111] for discussion of this property) there are several good reasons for using the logit formulation. First, IIA itself is not restrictive in the present duopoly context. Second, the logit has been used extensively in transportation studies and has proved a powerful tool in explaining travel demand [8; 15]. Third, the logit can be given a sound axiomatic basis in the probabilistic choice context (see especially Luce [29] and Yellott [37]). Last, the logit model admits a closed-form for the choice probabilities which is easy to work with. See also McFadden [31] for further discussion of the derivation of the logit model. (6)The homogeneous model is obtained as the limit case where [Mu] = 0, and CS = max[--[p.sub.1] (x), -- [p.sub.2] (x)], which can be shown to be the limit case of (7) as [Mu] [right arrow] 0. (7)Without normalization, this parameter is [Mu]/ cl, where c is the transport rate per unit per mile and l is the length of the market. (8)The first-order conditions corresponding to (3) are [Mathematical Expression Omitted]. Totally differentiating these equations [Mathematical Expression Omitted]. Equilibrium profits for firm 1 are (using the first-order conditions): [Mathematical Expression Omitted]. Hence [Mathematical Expression Omitted] as claimed. (9)The profit per unit earned by firm one from the consumers at the far end of the market, x = 1, i given by [Mathematical Expression Omitted] so that the derivative of this expression is [Mathematical Expression Omitted]. We therefore wish to show that [Mathematical Expression Omitted] The expression for [Mathematical Expression Omitted] is given in the preceding footnote. Also, by definition of [Mathematical Expression Omitted]. Combining these two expressions yields the condition desired as [Mathematical Expression Omitted], which is obviously true for [x.sub.1] [is greater than or equal to] 0. (10)This value is calculated from the first-order conditions, [Mathematical Expression Omitted] and [c.sub.2] = 1/4, which are the cost values associated with [x.sub.1] = 0 and [x.sub.2] = 1/2. (11)This result stems from the assumed linearity of transport costs and that demand depends only on price differences. It is not an artificial of the logit, and holds for all models with these properties [2]. (12)In the diagrams that follow, the surpluses are shown for uniform pricing as long as there is at least a local equilibrium ([Mu] [is greater than or equal to] 1/8). Recall from Proposition 1 that a global equilibrium is ensured for [Mu] [is greater than or equal to] 0.286. (13)As an illustration of this sort of issue in a different context, consider the Cobb-Douglas utility function [Mathematical Expression Omitted] where [x.sub.1] is consumption of good i, i = 1, 2. A rise in the parameter [Alpha] represents a shift in tastes, but it is not meaningful to say that such a shift causes utility to rise. Note that in the profitability analysis of the preceding section this sort of problem does not arise. The comparative static exercise there is interpreted as an increase in retailer differentiation. (14)In other papers of which we are aware in this area, a circular of infinitely long market is assumed and a symmetry condition imposed on firm locations so all interfirm distances are equal. The location problem per se can scarcely be addressed in such frameworks. By contrast, the introduction of market boundaries, as in our model, renders this problem non-trivial. Although we take no account of entry, we focus explicitly on locations.

References

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"The Delivered Price Generally," in The Robinson Patman Act Symposium. Commerce Clearing House, Inc., 1947. [37]Yellott, John I., "The Relationship between Luce's Choice Axiom, Thurstone's Theory of Comparative Judgment, and the Double Exponential Distribution." Journal of Mathematical Psychology, April 1977, 109-44.

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Author: | Thisse, Jacques-Francois |
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Publication: | Southern Economic Journal |

Date: | Apr 1, 1992 |

Words: | 7556 |

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