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

Rothschild, R.

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] + [alpha][rho][[pi].sup.M]

[[[pi].sup.M].sub.1] = [[[integral].sup..5[L.sub.1] + .5[L.sub.2]].sub.0] ([L.sub.2] - x)t dx - [[[integral].sup.[L.sub.1]].sub.0] ([L.sub.1] - x)t dx - [[[integral].sup..5[L.sub.1] + .5[L.sub.2]].sub.[L.sub.1]] (x - [L.sub.1])t dx

+ [alpha][rho]{[[[integral].sup..5[L.sub.1]] + .5[L.sub.2]].sub.0] [([L.sub.3] - x) - ([L.sub.2] - x)]t dx + [[[integral].sup..5[L.sub.1] + .5[L.sub.3]].sub..5[L.sub.1] + .5[L.sub.2]] [([L.sub.3] - x) - (x - [L.sub.1])]t dx}, (3)

where [L.sub.3] = [L.sub.3]([L.sub.2]) as given in Equation 2 and

[[[pi].sup.M].sub.2] = [[pi].sub.2] + (1 - [alpha])[pho][[pi].sup.M]

[[[pi].sup.M].sub.2] = [[[integral].sup..5[L.sub.1] + .5[L.sub.3]].sub..5[L.sub.1] + .5[L.sub.2]] (x - [L.sub.1])t dx + [[[integral].sup..5[L.sub.2] + .5[L.sub.3]].sub..5[L.sub.1] + .5[L.sub.3]] ([L.sub.3] - x)t dx - [[[integral].sup.[L.sub.2]].sub..5[L.sub.1] + .5[L.sub.2]] (x - [L.sub.2])t dx

- [[[integral of].sup..5[L.sub.2] + .5[L.sub.3].sub.[L.sub.2] ([L.sub.2] - x)t dx

+ (1 - [alpha])[rho]{[[[integral].sup..5[L.sub.1] + .5[L.sub.2]].sub.0] [([L.sub.3] - x) - ([L.sub.2] - x)]t dx + [[[integral].sup..5[L.sub.1] + .5[L.sub.3]].sub..5[L.sub.1] + .5[L.sub.2]] [([L.sub.3] - x) - (x - [L.sub.1])]t dx}, (4)

where, again, [L.sub.3] = [L.sub.3]([L.sub.2]) from Equation 2.

Each firm maximizes profit with respect to its own location, [partial][[[pi].sup.M].sub.1]/[partial][L.sub.1] = 0 and [partial][[[pi].sup.M].sub.2]/[partial][L.sub.2] = 0. These conditions generate reaction functions [L.sub.1]([alpha], [rho], N, [L.sub.2]) and [L.sub.2]([alpha], [rho], N, [L.sub.1]), which are jointly solved to yield [[L.sup.*].sub.1]([alpha], [rho], N), [[L.sup.*].sub.2]([alpha], [rho], N) and, from Equation 1, [[L.sup.*].sub.i]([alpha], [rho], N). The general expressions are presented in Appendix A, but the case when merger is certain, [rho] = 1, is shown here: [[L.sup.*].sub.1]([rho] = 1) = (3 - 2N + 6[alpha] - 3[[alpha].sup.2] - 4[alpha]N + 2[[alpha].sup.2]N)/(-3 + 2N)(6 - 6N - 6[alpha] + 4[alpha]N - [[alpha].sup.2])

[[L.sup.*].sub.2]([rho] = 1) = (-3 - [[alpha].sup.2])/(6 - 6N - 6[alpha] + 4[alpha]N - [[alpha].sup.2]). (5)

Table 1 sets out the case in which [rho] = 1 and N = 5. The location and profits of all firms are shown for [alpha] between zero and one. The case with no opportunity for merger ([rho] = 0) is shown in the first line.

We derive several comparative statics using the locations from Appendix A in which 0 [alpha] [rho] [less than or equal to] 1. The first two are also illustrated in Table 1.

PROPOSITION 3.1. [partial][[L.sup.*].sub.1]/[partial][alpha] [greater than] 0, [partial][[L.sup.*].sub.2]/[partial][alpha] [greater than] 0 [forall] [alpha], [rho], N.

PROOF. Sign the derivatives of [[L.sup.*].sub.1] and [[L.sup.*].sub.2] from Appendix A.

As the share of profit captured by firm 1 increases, both firms move to the right. Firm 1 has a fixed market edge on the left and expands both market and profit by moving right, an action accommodated by firm 2.

PROPOSITION 3.2. ([partial][[L.sup.*].sub.2]/[partial][alpha] - [partial][[L.sup.*].sub.1]/[partial][alpha]) [greater than] 0 [forall] [alpha], [rho], N.

PROOF. Compare the magnitudes calculated for Proposition 3.1.

As the profit share for firm 1 increases, firm 2 moves further to the right than firm 1, thereby increasing the distance between them. This follows from the fact that firm 2 moves right against accommodating firms but firm 1 moves left against a fixed market edge. [4]

PROPOSITION 3.3. [partial][[L.sup.*].sub.1]/[partial][rho] [less than over greater than] 0 as [alpha] [less than over greater than] [[alpha].sub.1], [partial][[L.sup.*].sub.2]/[partial][rho] [less than over greater than] 0 as [alpha] [less than over greater than] [[alpha].sub.2] [forall] N.

PROOF. From Appendix A, [partial][[L.sup.*].sub.i]([alpha], [rho], N)/[alpha][rho] is derived, set equal to zero and the value of [[alpha].sub.i], solved out. [5] The value [[alpha].sub.i] depends on [rho] but not on N. The sign of [pratial][[L.sup.*].sub.i]/[pratial][rho] is unambiguous when [alpha] is either above or below [[alpha].sub.i].

As [rho] goes toward zero, locations tend toward those adopted when [rho] = 0. If [[L.sup.*].sub.i] with merger is to the right of that without (true for high values of [alpha]), a reduction in [rho] generally causes the firm to move left. If [[L.sup.*].sub.i] with merger is to the left of that without (true for low values of [alpha]), a reduction in p generally causes the firm to move right. In particular, [[alpha].sub.2] varies from .860 when [rho] = 0 to .889 when [rho] = 1.

Profits, Efficiency, and Competitive Harm

The profit of the excluded firms declines whenever the possibility of merger has firm 2 moving to the right of the location that would have been chosen in the absence of merger, [rho] = 0. The symmetrical location of the excluded firms guarantees that any decline in profit is equally shared.

PROPOSITION 3.4. The excluded firms will suffer reduced profit whenever [alpha] [greater than] [[alpha].sub.2]([rho]), .873[\.sub.[rho]=1] [greater than or equal to] [[alpha].sub.2] [greater than or equal to] .858[\.sub.[rho]=[epsilon]] as [epsilon] [right arrow] 0, [forall] N.

PROOF. Set [[L.sup.*].sub.2]([alpha], [rho], N) = [L.sub.2]([rho] = 0) = 3/(2N) and solve for [alpha]. [6] The resulting function, [[alpha].sub.2]([rho]), depends on [rho] but not N (see Appendix A). From Proposition 3.1, [alpha] [greater than] [[alpha].sub.2] implies [[L.sup.*].sub.2] [greater than] [L.sub.2]([rho]) = 0).

Thus, whenever firm 1's profit share exceeds a critical value that falls within a narrow band and depends on the possibility of merger, the excluded firms are hurt. Firm 1 pushes firm 2 sufficiently far right to reduce the market of the excluded firms.

The cost of transport measures efficiency and provides the basis for comparing the merger and no merger outcomes.

PROPOSITION 3.5. Mergers reduce social welfare for all N, p [greater than] 0 and [alpha].

PROOF. When [rho] = 0, the locations uniquely minimize transport cost (Lederer and Hurter 1986). If [[L.sup.*].sub.2]([alpha], [rho], N) = [L.sub.2]([rho] = 0), then from Proposition 3.4, [alpha] = [[alpha].sub.2]([rho]). Yet, [[L.sup.*].sub.1]([[alpha].sub.2]([rho]), [rho], N) [greater than] [L.sub.1]([rho] = 0). Thus, there exist no N, [alpha], and [rho] [greater than] 0 yielding the cost-minimizing locations.

Mergers result in inefficient location choices and increased transport cost. When N = 5, the smallest possible cost associated with a merger occurs when [alpha] = .8327 and is .0525t. The cost associated with the no merger outcome is .0500t.

With certain merger, [rho] = 1, the combined profit of the merging firms exceeds that which occurs when there is no possibility of merger, [rho] = 0.

PROPOSITION 3.6. [[[pi].sup.M].sub.1]([rho] = 1) + [[[pi].sup.M].sub.2]([rho] = 1) [greater than] [[pi].sub.1]([rho] = 0) + [[pi].sub.2]([rho] = 0) [forall] [alpha], N.

PROOF. From Propositions 3.1 and 3.2 the merged firms have smallest market and so profit when [alpha] = 0. This profit can be directly compared with that associated with the no merger case.

With certain merger, individual profits of the merging firms may increase or decrease from their premerger level, depending on [alpha].

PROPOSITION 3.7. There exists a range of a such that [[[pi].sup.M].sub.1]([rho] = 1) [greater than] [[pi].sub.1]([rho] = 0) and [[[pi].sup.M].sub.1]([rho] = 1) [greater than] [[pi].sub.2]([rho] = 0) [forall] N.

PROOF. For any given N and [rho] = 1, solve for the upper and lower bounds of [alpha] from [[[pi].sup.M].sub.1]([rho] = 1) = [[pi].sub.1]([rho] = 0) and [[[pi].sup.M].sub.2]([rho] = 1) = [[pi].sub.2]([rho] = 0) respectively.

When N = 5, the merging firms each earn greater profit whenever .445 [less than] [alpha] [less than] .909. Other ranges can similarly be derived for any given N.

The propositions yield the following conclusion:

COROLLARY 3.1. For any N, there exists a range of a such that a merger of corner firms increases individual profits, harms excluded rivals, and reduces efficiency.

Any [alpha] [greater than] [[alpha].sub.2]([rho]) hurts excluded rivals (by Proposition 3.4), is inefficient (by Proposition 3.5), and increases joint profit (by Proposition 3.6). Individual profits increase if [alpha] also falls in the range identified by Proposition 3.7.

Figure 3 identifies the critical profit regions associated with Corollary 3.1. In region 1, firm 2 earns less profit than it does in the absence of merger. In region 2, Corollary 3.1 holds, with individually profitable mergers hurting excluded rivals. In region 3, all firms, merging and excluded, earn higher profits. In region 4, firm 1 earns less profit than it does in the absence of merger. The figure presents the ranges of [alpha] for all [rho] and also illustrates the role of increasing N. As N increases, the size of regions 2 and 4 grow, while that of regions 1 and 3 shrink. [7]

In region 3, firms 1 and 2 move toward each other to capture their respective shares of the additional profit from merger. This creates a positive externality as the reduced market share of the merged firms increases the profits of the excluded firms. In region 2, firm 1 commits to a location far to the right of that without merger. This commitment is sustainable because there is no firm to the left to occupy the market firm 1 vacates. The result is that firm 2 is forced to the right of its no merger location, and the excluded firms are harmed.

4. Analysis of the Interior Case

Figure 2 illustrates an interior merger. It shows the profit of firms j, j + 1, i and the profit generated from a merger of firms j and j + 1, [[pi].sup.M]. Neither merging firm is at the market corner, and, as will be shown, it is never possible for merging firms to each gain profit and simultaneously harm excluded rivals.

Location Choices

Locations are obtained, as before, by maximizing the individual profit of the N firms, with respect to their own location, as a function of all other firms' locations. The excluded firms, 1 to j - 1 and j + 2 to N locate symmetrically within their respective market segments (Lederer and Hurter 1986). Thus, location [L.sub.i](i [less than] j) can be expressed as

[L.sub.i](i [less than] j) = [L.sub.j] - (j - i)[L.sub.j]/[(j - 1) + (1/2)], (6)

and location [L.sub.i](i [greater than] + 1) can be expressed as

[L.sub.i](i [greater than] j + 1) = [L.sub.j+1] + (i - j + 1)(1 - [L.sub.j+1])/[N - j - (1/2)]. (7)

The [L.sub.j-1] and [L.sub.j+2] depend on the locations of the merging firms:

[L.sub.j-1]([L.sub.j]) = [L.sub.j] - [L.sub.j]/[j - (1/2)] [L.sub.j+2] ([L.sub.j+1]) = [L.sub.j+1] + (1 - [L.sub.j+1])/[(N - j - (1/2)]. (8)

The expressions in Equation 8 are returned to profit functions analogous to Equations 3 and 4. Each merging firm's profit is a function of the other merging firm's location, given [alpha], [rho], N, and j. Maximizing yields [partial][[[pi].sup.M].sub.j]/[partial][L.sub.j] = 0 and [partial][[[pi].sup.M].sub.j+1]/[partial][L.sub.j+1] = 0, which are solved for [L.sub.j] and [L.sub.j+1]. The Reaction functions, [L.sub.j]([alpha], [rho], N, [L.sub.j+1]) and [L.sub.j+1]([alpha], [rho], N, [L.sub.j]), are in Appendix B and yield [[L.sup.*].sub.j] ([alpha], [rho], N), [[L.sup.*].sub.j+1] ([alpha], [rho], N) and, from Equations 6 and 7, [[L.sup.*].sub.i] ([alpha], [rho], N).

The general expressions for firm locations when 0 [less than or equal to] [rho] [less than] 1 are complicated, but those for [rho] = 1 are

[[L.sup.*].sub.j]([rho] = 1) = (2j + 2j[alpha] - 1 - [alpha])/(4j[alpha] - 2j - 1 - 2N[alpha] - 2[alpha] + 2[[alpha].sup.2] + 4N)

[[L.sup.*].sub.j+1]([rho] = 1) = (2j[alpha] + 2j + 1 - 3[alpha] + 2[[alpha].sup.2])/(4j[alpha] - 2j - 1- 2N[alpha] - 2[alpha] + 2[[alpha].sup.2] + 4N). (9)

Comparative statics emerge. The first two are illustrated in Table 2, which gives locations and profits for firms 2 and 3 when these merge in a market containing five firms.

PROPOSITION 4.1. ([partial][[L.sup.*].sub.j+1]/[partial][alpha] - [partial][[L.sup.*].sub.j]/[partial][alpha]) [less than] 0 [forall] [alpha], [rho], N and j.

PROFF. Evaluation of the derivatives.

Thus as the share of profit captured by firm j increases, both firms continue to move toward the right.

PROPOSITION 4.2. ([partial]{[L.sup.*].sub.j+1]/[partial][alpha] - [partial][[L.sup.*].sub.j]/[partial][alpha]) [less than] 0 [forall] [alpha] [less than].5, [rho], N, and j ([partial][[L.sup.*].sub.j+1]/[partial][alpha] - [partial][[L.sup.*].sub.j]/[partial][alpha]) [greater than] 0 [forall] [alpha] [greater than] .5, [rho], N, and j.

PROOF. The difference in derivatives is evaluated and minimized at [alpha] = .5.

As [alpha] increases, firm j + 1 initially moves less far to the right than firm j, thereby decreasing the distance between the two firms. Beyond [alpha] = .5, an increase in a induces firm j + 1 to move farther to the right than firm j, thereby increasing the distance between the two firms. Thus, the distance between the two firms is greatest for extreme values of [alpha], near zero or one. The difference between this and the "corner" case arises because each of the merged firms is bordered by other firms rather than by the market boundary.

Finally, we consider the direct consequence of the probability of merger.

Again, as the probability of merger declines, the locations tend to move toward those that exist when there is no possibility of merger.

PROPOSITION 4.3. [partial][[L.sup.*].sub.j]/[[partial].sub.[rho]] [less than over greater than] 0 as [alpha] [less than over greater than] [[alpha].sub.j], [partial][[L.sup.*].sub.j+1]/[[partial].sub.[rho]] [less than over greater than] 0 as [alpha] [less than over greater than] [[alpha].sub.j+1] [forall] N, j.

PROOF. [partial][[L.sup.*].sub.i] ([alpha], [rho], N)/[[partial].sub.[rho]] (j = j, j + 1) is derived, set equal to zero and the value of [alpha], solved out. [8] The value [[alpha].sub.i] depends on [rho] but not N. The sign of [partial][[L.sup.*].sub.i]/[[partial].sub.[rho]] is unambiguous when [alpha] is either above or below [[alpha].sub.i].

In particular, [[alpha].sub.j] varies with [rho] from .331 when [rho] = 1 to .417 when [rho] = 0 and [[alpha].sub.j+1] varies with [rho] from .621 when [rho] = 0 to .735 when [rho] = 1.

Profits, Efficiency, and Competitive Harm

Now consider the consequences of merger for the excluded firms.

PROPOSITION 4.4. When [rho] = 1, excluded firms will suffer reduced profit whenever [alpha] [less than] [[alpha].sub.j](N) or [alpha] [greater than] [[alpha].sub.j+1](N - j).

PROOF. Set [[L.sup.*].sub.j]([rho] = 1) and [[L.sup.*].sub.j+1]([rho] = 1) from Equation 9 equal to [L.sub.j]([rho] = 0) and [L.sub.j+1]([rho] = 0), respectively, and solve for [[alpha].sub.j] and [[alpha].sub.j+1], functions of N and of N - j.

When [alpha] [less than] [[alpha].sub.j](N), firm j moves to the left of its no-merger location, reducing the profits of the excluded rivals to its left. When [alpha] [greater than] [[alpha].sub.j+1](N - j), firm j + 1 moves to the right of its no-merger location, reducing the profit of the excluded rivals to its right. Symmetrical location of the excluded firms guarantees that any decline in profit will be equally shared. The relationship between N, j, and the range of critical values of [alpha] is presented in Appendix B and can be summarized

[[alpha].sub.j](N - j = 2) = .321, [[alpha].sub.j](N - j [right arrow] [infinity]) = .500 and [[alpha].sub.j+1](J = 2) = .679,

[[alpha].sub.j+1](j [right arrow] [infinity]) = .500. (10)

Since the critical values never overlap, either excluded rivals on the left are hurt or those on the right are hurt, but never both. As N increases, the range of [alpha] for which some excluded rivals are hurt becomes larger. As N approaches [infinity], virtually the entire range of [alpha] results in harm to rivals. [9]

The cost of transport remains the measure of efficiency.

PROPOSITION 4.5. Mergers reduce social welfare for all N, [rho] = 1 and [alpha].

PROOF. When [rho] = 0, the locations uniquely minimize transport cost (Lederer and Hurter 1986). If [[L.sup.*].sub.j+1]([alpha], [rho], N) = [L.sub.j+1]([rho] = 0), then from Proposition 4.4, [alpha] [[alpha].sub.j+1]([rho]). Yet, [[L.sup.*].sub.j]([[alpha].sub.j+1]([rho]), [rho], N) [greater than] [L.sub.j]([rho] = 0). Thus, there exist no N, [alpha], and [rho] = 1 yielding the cost-minimizing locations.

The cost associated with a merger always exceeds the cost when no merger is possible. [10]

The merged firms will obtain increased joint profit.

PROPOSITION 4.6. [[[pi].sup.M].sub.j]([rho] = 1) + [[[pi].sup.M].sub.j+1]([rho] = 1) [greater than] [[pi].sub.j]([rho] = 0) + [[pi].sub.j+1]([rho] = 0) [forall] [alpha], N, and j.

PROOF. From Propositions 4.2 and 4.3, [[[pi].sup.M].sub.j]([rho] = 1) + [[[pi].sup.M].sub.j+1]([rho] = 1) is minimized when [alpha] = .5. This profit exceeds that associated with [rho] = 0, t/[n.sup.2].

The merger may increase or decrease the individual profits of the merging firms, depending on [alpha].

PROPOSITION 4.7. [[[pi].sup.M].sub.j] [greater than] [[pi].sub.j]([rho] = 0) and [[[pi].sup.M].sub.j+1] [greater than] [[pi].sub.j+1]([rho] = 0) i.f.f. [[alpha].sub.j] [less than] [alpha] [less than] [[alpha].sub.j+1] [forall] N and j.

PROOF. [[L.sup.*].sub.j] and [[L.sup.*].sub.j+1] from Equation 9 and [[L.sup.*].sub.j-1] and [[L.sup.*].sub.j+2] from Equation 8 are placed in profit expressions analogous to Equations 3 and 4. [[[pi].sup.M].sub.j] ([rho] = 1) and [[[pi].sup.M].sub.j+1] ([rho] = 1)are functions of N, j, and [alpha] and are Set equal to [[pi].sub.j] ([rho] = 0) = [[pi].sub.j+1] ([rho] = 0) = t/2[N.sup.2]. The equalities are solved for [alpha], yielding exactly [[alpha].sub.j] and [[alpha].sub.j+1] as shown in Appendix B.

The foregoing propositions yield the following conclusion, which throws into sharp relief the distinction between the corner and interior cases:

COROLLARY 4.1. It is impossible for two interior firms to merge, increase their individual profits, and simultaneously harm excluded rivals.

The range of [alpha] in which both firms individually increase profit (Proposition 4.7) is identical to that in which neither firm moves outside of its no-merger location. Thus, individually rational interior mergers will be inefficient but will always benefit rivals. As an illustration, when N = 5 and firms 2 and 3 merge, all firms, merging and excluded, earn greater profit whenever .378 [less than] [alpha] [less than] .679. For [alpha] outside this range, merger is not individually rational.

Returning to Figure 3, we observe that there exists no equivalent to region 2 for interior mergers. Instead, the externality exists for all individually rational mergers. To capture their respective shares of the profit from merger, the participants move toward each other. This increases the market share for the excluded firms and so also their profits. In this respect, the result for interior mergers parallels the case of Cournot-Nash competitors outside of the spatial context (Salant, Switzer, and Reynolds 1983).

5. Conclusions

This research demonstrates the substantial differences between corner and interior mergers. The former may simultaneously harm rivals and be individually rational, while the latter can never simultaneously harm rivals and be individually rational. The corner case is unique among all mergers in that the first merging firm can move to the right without losing any of its previous market. Consequently, the externality so clear in the interior case fades. Only with the corner merger is it possible for the distance between the merging firms to shrink but for the merging firms not to lose market share to the excluded rivals. It is this fact that allows both merging firms to increase profit and move to the right, thereby hurting the excluded rivals. To the extent that markets are not closed, such as on circle, the corner case is irrelevant. Yet, when the market is linear, the corner case is obviously more likely to be relevant when there are only a few firms.

Note the critical role that the sharing rule plays: All interior mergers remain jointly profitable, and the sum of profits from both firms is higher, even when rivals are harmed. Consequently, a side payment after the location decision could generate an interior merger that is individually rational and harms rivals. The difficulty is the timing. If both firms knew a side payment was to be made, their location choices would be influenced. The knowledge of the potential for a side payment would have to be acquired after the location decision is made. However, the location decision would then not be individually rational on the basis of the information available at the time it was taken. It would be so only after the realization of the side payment.

Avenues remain for further research. First, the number of firms in a given merger could be increased beyond the two we consider. Second, the reservation price could be lowered so as to be binding on the price merged firms could charge. Third, fixed costs could be introduced, and the possibility of "shutting down" one of the merged firms could be considered. These further possibilities for research notwithstanding, this paper has demonstrated the importance of considering both the N-firm case and the distinction between corner and interior mergers.

(*.) Department of Economics, University of Wisconsin, Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA; E-mail [email protected]; corresponding author.

(+.) Department of Economics, University of Wisconsin, Eau Claire, Eau Claire, WI 54702, USA.

(++.) Department of Economics, Lancaster University, Lancaster LA1 4YX United Kingdom.

The authors thank Jonathan Hamilton, an anonymous reviewer, and participants in the Graduate Economics Forum at the University of Wisconsin, Milwaukee. Several results were generated using Mathematica, and all programs are available from the authors.

Received December 1998; accepted June 2000.

(1.) The 1984 guidelines were the first to make this explicit.

(2.) This follows earlier work in which Friedman and Thisse (1993) examine the influence of anticipated mergers (and/or collusion) on location choice in a spatial model without price discrimination.

(3.) These schedules indicate the location-specific price for which the firm is willing to produce and deliver the good. Thisse and Vives (1988) make clear that the assumption of delivered pricing rests on the increasingly accepted notion that such pricing is a natural consequence of profit maximization. Moreover, alternative assumptions, such as freight-on-board pricing, often have no equilibrium or can achieve equilibrium only with a restricted transportation cost function.

(4.) Indeed, when [alpha] = 1, the distance between the merging firms is greatest, and the location of firm 2 is exactly twice that of firm 1. When firm 1 obtains all the gains from merger, it locates in the middle of its market, thereby minimizing the cost of serving that market and maximizing its individual profit.

(5.) Multiple roots exist, but only one places a between zero and one.

(6.)Again, there are multiple roots but only one in the zero-to-one range.

(7.) When N = 5 and [rho] = 1, the range for which Corollary 3.1 holds is .873 [less than] [alpha] [less than] .909. When N = 20 and [rho] = 1, the range for which Corollary 3.1 holds is .873 [less than] [alpha] [less than] .947.

(8.) Multiple roots exist, but only one places a between zero and one.

(9.) Nonetheless, a value of [alpha] = .5 might be considered an equilibrium for two interior firms, each of which could always merge with its rival on the other side.

(10.) Alternatively, one could simply note from Equation 10 that [[alpha].sub.j+1] always exceeds [[alpha].sub.j].

References

Deneckere, Raymond, and Carl Davidson. 1985. Incentives to form coalitions with Bertrand competition. Rand Journal of Economics 16:473-86.

Friedman, James, and Jacque Thisse. 1993. Partial collusion fosters minimum product differentiation. Rand Journal of Economics 24:631-45.

Gupta, Barnali, John S. Heywood, and Debashis Pal. 1997. Duopoly, delivered pricing and horizontal mergers. Southern Economic Journal 63:585-93.

Lederer, Philip, and Arthur Hurter. 1986. Competition of firms: Discriminatory pricing and location. Econometrica 54: 623-40.

Reitzes, James, and David Levy. 1995. Price discrimination and mergers. Canadian Journal of Economics 28:427-36.

Rothschild, R., John S. Heywood, and Kristen Monaco. 2000. Spatial price discrimination and the merger paradox. Regional Science and Urban Economics 30:491-506.

Salant, Stephen W., Sheldon Switzer, and Robert Reynolds. 1983. Losses from horizontal merger, the effects of an exogenous change in industry structure on Cournot-Nash equilibrium. Quarterly Journal of Economics 98:185-213.

Schmalensee, Richard, and Jacque Thisse. 1988. Perceptual maps and the optimal location of new products: An integrative essay. International Journal of Research in Marketing 5:225-49.

Thisse, Jacque, and Xavier Vives. 1988. On the strategic choice of spatial price policy. American Economic Review 78:122-37.

White, Lawrence. 1988. Private antitrust litigation: New evidence, new Learning. Cambridge, MA: MIT Press.

 Corner Merger with N = 5 [a]
 (Locations) (Profits)
 [L.sub.1] [L.sub.2] [L.sub.3] [L.sub.4] [L.sub.5] [[pi].sub.1]
[rho] = 0 0.1 0.3 0.5 0.7 0.9 0.03
[rho] = 1
[alpha]
 1.0 0.182 0.364 0.545 0.727 0.909 0.099
 0.9 0.163 0.312 0.509 0.705 0.902 0.081
 0.8 0.146 0.271 0.479 0.688 0.896 0.066
 0.7 0.130 0.238 0.455 0.673 0.891 0.053
 0.6 0.115 0.211 0.436 0.662 0.887 0.043
 0.5 0.101 0.188 0.420 0.652 0.884 0.034
 0.4 0.088 0.170 0.407 0.644 0.882 0.027
 0.3 0.076 0.155 0.397 0.638 0.880 0.020
 0.2 0.064 0.143 0.388 0.633 0.878 0.015
 0.1 0.053 0.133 0.381 0.629 0.876 0.010
 0.0 0.042 0.125 0.375 0.625 0.875 0.005
 [[pi].sub.2] [[pi].sub.3] [[pi].sub.4] [[pi].sub.5]
[rho] = 0 0.02 0.02 0.02 0.03
[rho] = 1
[alpha]
 1.0 0.017 0.017 0.017 0.025
 0.9 0.021 0.019 0.019 0.029
 0.8 0.024 0.022 0.022 0.033
 0.7 0.027 0.024 0.024 0.036
 0.6 0.031 0.025 0.025 0.038
 0.5 0.034 0.027 0.027 0.040
 0.4 0.037 0.028 0.028 0.042
 0.3 0.039 0.029 0.029 0.044
 0.2 0.042 0.030 0.030 0.045
 0.1 0.044 0.031 0.031 0.046
 0.0 0.047 0.031 0.031 0.047
(a.)All profits are measured in units of t.
 Profit from Merger: Critical Regions
Region 1: Region 2:
Change in Change in
[[pi].sub.1] [greater than] 0 [[pi].sub.1] [greater than] 0
Change in Change in
[[pi].sub.2] [less than] 0 [[pi].sub.2] [greater than] 0
Change in Change in
[[pi].sub.1] [greater than] 0 [[pi].sub.1] [less than] 0
Region 1: Region 3:
Change in Change in
[[pi].sub.1] [greater than] 0 [[pi].sub.1] [greater than] 0
Change in Change in
[[pi].sub.2] [less than] 0 [[pi].sub.2] [greater than] 0
Change in Change in
[[pi].sub.1] [greater than] 0 [[pi].sub.1] [greater than] 0
Region 1: Region 4:
Change in Change in
[[pi].sub.1] [greater than] 0 [[pi].sub.1] [less than] 0
Change in Change in
[[pi].sub.2] [less than] 0 [[pi].sub.2] [greater than] 0
Change in Change in
[[pi].sub.1] [greater than] 0 [[pi].sub.1] [greater than] 0
 Interior Merger with N =5 [a]
 (Locations) (Profits)
 [L.sub.1] [L.sub.2] [L.sub.3] [L.sub.4] [L.sub.5] [[pi].sub.1]
[rho] = 0 0.1 0.3 0.5 0.7 0.9 0.03
[rho] = 1
[alpha]
 1.0 0.154 0.462 0.615 0.769 0.923 0.071
 0.9 0.146 0.438 0.578 0.747 0.916 0.064
 0.8 0.138 0.413 0.541 0.725 0.908 0.057
 0.7 0.129 0.387 0.507 0.704 0.901 0.050
 0.6 0.120 0.360 0.474 0.685 0.895 0.043
 0.5 0.111 0.333 0.444 0.667 0.889 0.037
 0.4 0.102 0.306 0.418 0.650 0.883 0.031
 0.3 0.093 0.279 0.392 0.635 0.878 0.026
 0.2 0.084 0.252 0.370 0.622 0.874 0.021
 0.1 0.075 0.226 0.350 0.610 0.870 0.017
 0.0 0.067 0.200 0.333 0.600 0.867 0.013
 [[pi].sub.2] [[pi].sub.3] [[pi].sub.4] [[pi].sub.5]
[rho] = 0 0.02 0.02 0.02 0.03
[rho] = 1
[alpha]
 1.0 0.047 0.012 0.012 0.018
 0.9 0.043 0.015 0.014 0.021
 0.8 0.038 0.017 0.017 0.025
 0.7 0.033 0.020 0.019 0.029
 0.6 0.029 0.022 0.022 0.033
 0.5 0.025 0.025 0.025 0.037
 0.4 0.021 0.027 0.027 0.041
 0.3 0.017 0.030 0.030 0.044
 0.2 0.014 0.033 0.032 0.048
 0.1 0.011 0.036 0.034 0.051
 0.0 0.009 0.040 0.036 0.053
(a.)All profits are measured in units of t.

Appendix A: The Corner Case

Propositions 3.1 to 3.3 follow from the relevant derivatives of [[L.sup.*].sub.1]([alpha], [rho], N) and [[L.sup.*].sub.2]([alpha], [rho], N):

[[L.sup.*].sub.1] = (2/3)(2.25 - 1.5N + 2.25[alpha][rho] - 1.5N[alpha][rho] + 2.25[alpha][[rho].sup.2] - 1.5N[alpha][[rho].sup.2] - 2.25[[alpha].sup.2][[rho].sup.2] + 1.5N[[alpha].sup.2][[rho].sup.2])

[divided by] (-4.5 + 3N)(N - 3[rho] + 2N[rho] + 3.5[alpha][rho] - 2N[alpha][rho] - .5[alpha][[rho].sup.2] + .5[[alpha].sup.2][[rho].sup.2])

[[L.sup.*].sub.2] = (1.5 + .5[alpha][rho] - .5[alpha][[rho].sup.2] + .5[[alpha].sup.2][[rho].sup.2])/(N - 3[rho] + 2N[rho] + 3.5[alpha][rho] - 2N[alpha][rho] - .5[alpha][[rho].sup.2] + .5[[alpha].sup.2][[rho].sup.2]).

Proposition 3.4 follows from setting [[L.sup.*].sub.2]([alpha], [rho], N) = [L.sub.2]([rho] = 0) = 3/(2N) and solving for [alpha]. This yields [[alpha].sub.2]([rho]) = (.5/[rho])[-7 + [rho] + [(49 + 10[rho] + [[rho].sup.2]).sup.1/2]].

Appendix B: The Interior Case

Propositions 4.1 to 4.3 follow from the relevant derivatives of [[L.sup.*].sub.j]([alpha], [rho], N), [[L.sup.*].sub.j+1]([alpha], [rho], N). Solving the following reaction functions generates those locations:

[L.sub.j] = 2(j - .5)(.25 + .5j - .5N)[alpha][rho]

+ [-.125 - .5[j.sup.2] + .5N - .5[N.sup.2] - .25[alpha][rho] + .5[alpha][rho] -j(.5 - N + .5[alpha][rho])][L.sub.j+1]/-(.5 + j)[(.5 + j - N).sup.2]

[L.sub.j+1] = (.25 - .5j) + [.125 + .5[j.sup.2] + .25N - .25[rho] + .5N[rho] + .25[alpha][rho] - .5N[alpha][rho] + j(-.5N - .5[rho] + .5[alpha][rho])][L.sub.j]

[divided by] [.125 + .5[j.sup.2] + .25N - .5j(l + N)].

Proposition 4.4 is proved by setting [[L.sup.*].sub.j]([rho] = 1) and [[L.sup.*].sub.j+1]([rho] = 1) equal to [L.sub.j]([rho] = 0) = (2j - l)/(2N) and [L.sub.j+1]([rho] = 0) = (2j + 1)/(2N), respectively, and solving for [[alpha].sub.j] and [[alpha].sub.j+1]. The derived expressions are

[[alpha].sub.j]([rho] = 1) = -j + N + 1/2 + 1/2[(4[j.sup.2] - 8jN + 4[N.sup.2] + 3).sup.1/2] [[alpha].sub.j+1]([rho] = 1) = -j + 1/2 + 1/2[(4[j.sup.2] + 3).sup.1/2].

Note that these are the relevant roots and are identical to the values for which the respective firm earns exactly what it would earn in the absence of merger.