Spatial Price Discrimination and Merger: The N-Firm Case.

• Author: Heywood, John S.
• Position: Statistical Data Included

John S. Heywood [*]

Kristen Monaco [+]

R. Rothschild [++]

The consequences of merger are analyzed in an N-firm model of spatial price discrimination. The merger occurs with known probability after location decisions have been made. The possibility of merger alters locations, generates inefficiency, and increases the profit of the merging firms. In the case of corner mergers, but never in the case of interior mergers, the possibility of merger may also reduce the profit of the excluded firms.

1. Introduction

   Spatial models often provide unique insights. Using a three-firm spatial model, earlier research shows that mergers can hurt the excluded rival and benefit both merging firms. While this result would not be expected outside the spatial context, it may not be general even within it. This paper presents a complete generalization of two firms merging in an N-firm model of spatial price discrimination. The results show that only mergers of the corner firms have the potential for simultaneously hurting rivals and being in the self-interest of the merging firms. All interior mergers either hurt rivals or increase profits of the merging firms, but not both.

   In Cournot-Nash models outside the spatial context, mergers are profitable only when they capture three-fourths or more of the market. Moreover, the excluded firms typically benefit more than the participants in the merger (on both points, see Salant, Switzer, and Reynolds 1983). Such results are difficult to reconcile with the many voluntary mergers involving smaller market shares and the fact that excluded rivals are the most common source of antitrust complaints regarding mergers (White 1988). As a consequence, merger models involving differentiated products, including spatial models, have received increasing attention (Deneckere and Davidson 1985). This makes sense because the antitrust guidelines recognize that the "closeness of competitors" can be a critical determinant of the welfare consequences of merger. [1]

   Spatial models capture more than just geography. They present a general method of examining markets in which an ordered product characteristic differentiates output (Schmalensee and Thisse 1988). Thus, airline flights between city pairs differ by departure time from early morning to late evening, and the editorial policy of newspapers differ from liberal left to conservative right.

   Mergers in markets with spatial price discrimination have been investigated because such pricing commonly occurs (Thisse and Vives 1988). In these markets, a firm's price is dictated by the delivered cost of its adjacent rivals, and, a consequence, only mergers between adjacent firms influence price. Indeed, when location choices do not anticipate merger, such merger increases the prices and profits of the participants but leaves those of all other firms unchanged (Reitzes and Levy 1995). When a merger is anticipated, Gupta, Heywood, and Pal (1997) show that it influences the location choices of duopolists engaging in spatial price discrimination [2] Rothschild, Heywood, and Monaco (2000) expand on this idea, showing that an anticipated merger of two adjacent firms in a three-firm market generates location choices that can lower the profits of the excluded firm.

   The connection between merger and location choices has not been generally proven. In the earlier work, the number of firms is at most three, with two merging and one excluded. Thus, the adjacent merging firms always have the fixed has not been considered. Moreover, the assumption of only one excluded firm eliminates the possibility that some excluded firms could be hurt by merger while other benefit.

   This paper develops an N-firm model of spatial price discrimination in which the possibility of merger is anticipated prior to location choices being made. This possibility generates inefficient locations and increased profit to the merging firms. The profit of the excluded firms may decrease but only in the case of corner mergers. Interior mergers that increase the profit of the participants always increase the profit of the excluded firms.

   The next section describes the model, and the third and fourth sections present results for the corner and interior cases, respectively. The fifth section concludes and suggests further research.

2. The Model

   The market is a unit line segment with consumers uniformly distributed with density one. Each consumer has inelastic demand for one unit of the good, with reservation price r. Assume that r is sufficiently high that it is profitable, with or without merger, for firms to serve all consumers. If a consumer is offered identical delivered prices from two firms, she buys from the nearer firm.

   We model a three-stage game. In stage 1, N firms enter simultaneously and choose locations. High relocation costs make this choice irreversible for the duration of the game. In stage 2, a pair of adjacent firms consider merger in order to capture the profits that would otherwise be lost through price competition in the later stage. In stage 3, the firms, both those included and those excluded from the merger, engage in spatial discriminatory pricing and announce delivered price schedules. [3]

   This sequence is the same as that adopted by Gupta, Heywood, and Pal (1997) and Rothschild, Monaco, and Heywood (2000) and makes sense because firms must usually make investment and location decisions before becoming involved in mergers. If, as an alternative, firms were to merge before locating, the two plants would always locate so as to minimize costs and maximize profits.

   Let [L.sub.i], i = 1, 2 ... N, denote the location of firm i on the unit line segment, where [L.sub.i] [less than] [L.sub.i+1]. Let x [epsilon] [0, 1] be the location of a consumer. Firms incur no fixed costs of production, and marginal cost is constant and normalized to zero. Each firm transports the good from the point of production to the consumers within its market segment. The cost of transport is a constant t per unit of distance. Thus, the total cost to firm i of supplying all consumers in the line segment g to h is

   [[[integral].sup.h].sub.g] (t/x - [L.sub.i]/) dx.

   Now suppose that two adjacent firms, j and j + 1, consider a horizontal merger. They both possess complete and accurate information and anticipate that the merger will occur with probability p [epsilon] [0, 1]. Should the merger occur, an incremental profit, [II.sup.M], will be generated, and firm j will receive share [lambda] [epsilon] [0, 1] of that profit, and firm j + 1 will receive the remainder, share 1 - [alpha].

   We proceed by recognizing two mutually exclusive and exhaustive alternatives. First, the merging firms may be located against the market edge, with j = 1 or j + 1 = N, the corner case. Second, the merging firms may be in the interior of the market with N - 1 [greater than] j [greater than] 1.

3. Analysis of the "Corner" Case

   The first subsection derives location choices and examines comparative statics, while the second subsection examines the issues of profitability, efficiency, and competitive harm to the excluded rivals.

   Location Choices

   As the market is symmetrical, the two corner cases, j = 1 and j + 1 = N, are identical. Figure 1 illustrates the case of j = 1, showing the profit of firms 1, 2, i and the incremental profit, [II.sup.M], generated by a merger of firms 1 and 2. Note that [II.sup.M] is generated when the pricing constraint on the merged firm becomes the delivered cost of firm 3. We solve for locations by maximizing the profit of each of the N firms, with respect to their own location, as a function of all firm locations. This generates N reaction functions with N unknown locations.

   The excluded firms, 3 to N, locate symmetrically within the market from [L.sub.2] to 1. This follows because spatial price discrimination with simultaneous entry (and no opportunity for merger) results in transport cost minimizing locations along any line segment (Lederer and Hurter 1986). Thus, [L.sub.i](i [greater than] 2) can be expressed as

   [L.sub.i](i [greater than] 2) = [L.sub.2] + (i - 2)(1 - [L.sub.2])[2/(2N - 3)]. (1)

   Given [L.sub.2], the location of all firms i [greater than] 2 are known. The location of firm 3 can be expressed as a function of [L.sub.2]:

   [L.sub.3]([L.sub.2]) = [L.sub.2] + (1 - [L.sub.2])[2/(2N - 3)]. (2)

   The expected profit of firms 1 and 2 is then a function of the parameters [alpha], N, [rho] and the locations [L.sub.1] and [L.sub.2]:

   [[[pi].sup.M].sub.1] = [[pi].sub.1] +...