Profitable Mergers in a Cournot Model of Spatial Competition.

George Norman [*]

Lynne Pepall [+]

This paper investigates the profitability and locational effects of mergers when Cournot firms compete in spatially differentiated markets. A two-firm merger is generally profitable because the merged partners can coordinate their location decisions. The merged firm locates its plants outside the market quartiles with distance from the market center being an increasing function of the number of nonmerged firms remaining at the market center. Profitable two-firm mergers reduce competitive pressure, leading to higher prices and reduced consumer surplus. The merger increases total surplus by increased locational efficiency and the increased profits of the merged and nonmerged firms.

1. Introduction

In the wake of an unprecedented wave of merger activity in the U.S. and abroad, there is considerable interest in understanding the market impact of horizontal mergers. A recent report by the Federal Trade Commission, "Anticipating the 21st Century" (FTC 1997), cites the prevention of anticompetitive mergers as one of its four key activities. This goal has focused policymakers' interest on the unilateral competitive effects of horizontal mergers, that is, on those mergers that in the absence of collusion make it profitable for the merging firms to reduce output and to cause market price to increase.

The unilateral effects of mergers are exactly the kind of policy issue where we would expect the theory of industrial organization to make an important contribution. Unfortunately, our theory of horizontal mergers is not yet quite up to task. The standard Cournot model of quantity competition, so widely used in applied theory, is beset by the paradoxical result that the majority of horizontal mergers are simply not profitable. This profitability paradox of horizontal mergers in the Cournot model is particularly associated with the work of Szidarovzky and Yakowitz (1982) and Salant, Switzer, and Reynolds (1983) (also see Davidson and Deneckere 1984). They show, for example, that with linear demand and constant, identical marginal costs, no two-firm merger is profitable unless it is merger to monopoly. More generally, with the same linearity assumptions merger is unprofitable, and so should not occur, unless at least 80% of the firms in the market merge. This is a clear drawback to the use of the standard Cour not model in the formulation of merger policy.

The logic behind the paradox is simply explained. A merger of two firms converts the n-firm premerger game into an n-1-firm postmerger game. The merger is equivalent to the formation of a coalition between the merged firms whose aim is to maximize their joint profit by coordinating their output choices. As a result, the merged firm reduces its combined output. Because firms' outputs are strategic substitutes, this is accompanied by an increase in the outputs of the nonmerged firms, with the result that the merged firm ends up looking just like any other firm in the industry. In other words, the merged firm's market share in the postmerger game is less than the sum of the two firms' premerger market shares, whereas the other firms, who did not merge, increase their market share. The combined impact on output increases price but the loss of market share by the merged firm is so sufficiently great that its profit falls.

It is the combination of strategic substitution and the inability of the merged firm to make a credible commitment to exploit its potentially greater size in the postmerger game that lies at the heart of the Salant, Switzer, and Reynolds merger paradox. [1] Suppose, by contrast, that the strategic variables chosen by the rival firms are strategic complements--the most obvious choice being price--so that the firms are Bertrand competitors, then as long as the firms also produce different products, merger is generally profitable for the merged firms. This should not be particularly surprising. Merger induces the coalition members to increase their prices. As a result of strategic complementarity, this soft action is met by a soft response: an increase in the prices of the rival, nonmerged firms. Deneckere and Davidson (1985) show that merger of Bertrand competitors is profitable by assuming that the firms produce symmetrically differentiated products. Reitzes and Levy (1995) introduce a spatial element to the analysis and show that the merger of price-discriminating Bertrand firms is always profitable, and Pepall, Richards, and Norman (1999) show that the same result holds without price discrimination.

In this paper, we maintain the assumption of strategic substitution--our firms are Cournot competitors--and instead focus on the role of commitment. Our goal in this paper is to provide support for the presumption that a two-firm merger, by combining the firms' assets, should in some sense lead to a "bigger" and perhaps "better" firm than either of the two firms was in the premerger market. In this regard, our paper shares common ground with Daughety (1990), who also recognizes the role of commitment. He does so by changing the timing of the postmerger output game. In his analysis, when two firms merge, they act as Cournot competitors against other merged firms but as Stackelberg leaders with respect to the remaining nonmerged firms. With linear demand and costs, any two-firm merger is profitable. We adopt a less extreme approach to commitment by introducing the assumption that our Cournot firms compete across a set of spatially differentiated markets. In this setting, firms choose not just the quantity of o utput to supply to the spatially differentiated markets but also where to locate their production plants to serve these markets. [2]

Location is a key factor underlying why a merger can lead to a bigger and better firm. Firms that have different locations have different locational advantages in serving the set of spatially separated markets. As a result, in contrast to the standard nonspatial Cournot model, a merger between two firms need not result in one of them effectively being shut down. Rather, a merger between two firms allows them to coordinate their location decisions with the result that the merged firm becomes potentially bigger than its rivals, better adjusted to consumer locations. [3]

Our analysis builds on the work of Anderson and Neven (1990) who show that when demand and costs are such that each Cournot firm wishes to serve the entire set of spatially separated markets, all firms choose to cluster or agglomerate at the center of the market area. This tendency to agglomeration is taken as the starting point of our analysis. We introduce commitment by assuming that merger of two firms confers a leadership advantage on the merging firms in that they can choose first whether and where they wish to relocate their production plants. In making this decision the merged firm is able to anticipate correctly the location choices of the remaining n - 1 nonmerged firms and the outcome of the simultaneous Cournot quantity game that will subsequently be played with these nonmerged rivals.

We show that a two-firm merger results in the merged firm relocating its plants away from the market center, whereas the nonmerged firms remain at the center. Being a leader or Stackelberg first mover in location choice and coordinating the two plants' locations to serve certain segments of the market more efficiently lead to a merged firm with a larger overall market share than its nonmerged rivals. This explains why, in sharp contrast to the standard Cournot model, a two-firm merger can be profitable even in relatively unconcentrated markets.

A two-firm merger not only improves locational efficiency, but it also increases market concentration and reduces competitive pressure across the set of markets. The net effect is that it generally leads to higher prices across the set of consumer markets. However, and again in sharp contrast to nonspatial analysis, the increased profit of both the merged firm and the nonmerged firms, together with improved locational efficiency, is such that the merger increases total surplus. In other words, the introduction of commitment through location leadership means that merger can indeed lead to a bigger firm, which so far as total welfare is concerned serves the market "better." [4]

The remainder of the paper is organized as follows. In section 2, we present the Cournot model of spatial competition with all firms located at the center of the market and derive the equilibrium conditions that characterize a two-firm merger. In section 3, we derive the optimal distance from the center that the merged firm locates its plants and identify the conditions under which such a merger is profitable. In section 4, we evaluate the welfare effects of a two-firm merger in the spatial model. Concluding remarks are presented in the final section.

2. The Model

We begin by describing a general model of Cournot competition in which firms supply output across a set of spatially differentiated markets. Specifically, suppose there are n identical firms, each with a single production plant located on a Hotelling line whose length is normalized to unity. The location of firm i on the line is denoted by [x.sub.i], i = 1, ..., n. Each firm produces a homogeneous product at constant marginal cost, which without loss of generality is set equal to zero. We ignore any set-up costs that the firms might incur. Demand at each consumer location x on the line is identical and is described by the linear inverse demand function p(x) = v - Q(x) where p(x) is the product price and Q(x) is aggregate output in the local market x, and v is the maximum consumer reservation price in each local market. Firms' costs of supplying consumers at different market locations on the line may differ. Each firm i incurs shipping costs in supplying each consumer location that we assume are linear in dis tance and quantity shipped. Specifically, firm i's cost of shipping one unit of output to consumer location x is t(/[x.sub.i] - x/) = t/[x.sub.i] - x/, i = 1, ..., n. We eliminate the transport cost parameter by working with the transport-cost adjusted reservation price v = v/t. [5] In the product-differentiation analogy of this model, v can be interpreted as an inverse measure of the extent to which consumer tastes are strongly localized.

The n firms compete at each location x by playing a Cournot game in quantities. There is no scope for consumer arbitrage between local markets as a result of which the firms can and will price discriminate across markets. Competition between the firms results in a set of independent Cournot equilibria, one for each location x on the line.

Anderson and Neven (1990) consider a two-stage game in this framework. In the first stage, the n firms simultaneously choose their locations knowing that in the second stage the firms will compete in quantities in the manner we have described above. Their main result is to show that the subgame perfect location equilibrium has all n firms located at the market center. The intuition behind this agglomeration result is quite straightforward. Every firm is assumed to be able to sell profitably to all locations in the market no matter the location it chooses. [6] Moreover, a Cournot market game softens, at least to some extent, competition between the firms in each local market. As a result, every firm has a strong incentive to adopt a central location and to sell to all locations in the market from that location.

We take agglomeration at the center of the market as the starting point in our analysis. That is, we assume that the n firms are initially located at the center of the market, or [x.sub.i] = 1/2, i = 1, n. In the Appendix, we how that if v [greater than] n/2, then no firm individually has an incentive to deviate and change its location by moving from the center, given that the remaining n - 1 firms are located at the center. We confine our attention throughout the analysis, therefore, to parameters that satisfy the condition v [greater than] n/2.

Given a starting point with all n firms agglomerated at the center of the line, standard analysis gives the Cournot-Nash equilibrium firm output, aggregate output, and price in each localized market x:

[[q.sup.c].sub.i](x) = v - /x - 1/2// n + 1 (1a)

[[Q.sup.c].sub.i](x) = n. v - /x - 1/2//n + 1 (1b)

[p.sup.c](x) = v - n /x - 1/2//n + 1. (1c)

From Equations la and lc, firm i's profit from supplying the entire set of markets is

[[[pi].sup.C].sub.i] = 2 [[[integral of].sup.1/2].sub.0] [(v - /x - 1/2//n+1)].sup.2] dx = 12[v.sup.2] - 6v + 1/12[(n+1).sup.2].

Consider now a merger between two of the n firms. Of course, if the two merging firms do not change their locations, then such a merger is subject to the merger paradox discussed in the introduction. If n [greater than or equal to] 3 and two firms merge but do not change their locations, then the postmerger profit of the merged firm in serving every local market on the line is less than the sum of the two firms' premerger profit from serving these markets. A simple revealed preference argument suggests that such a merger cannot be expected to occur.

However, suppose instead that the two merging firms can change their production plant locations. In particular, suppose that the merging firms can coordinate their location decisions by acting as location leaders in a two-stage game. In the first stage, the merged firm locates what are now its two divisions a distance d from the market center with, of course, one branch to the left and the other to the right of the market center. It follows from the symmetry of the model that the left-hand division supplies all consumer markets in the interval [0, 1/2] and the right-hand division all consumer markets in [1/2, 1]. In the second stage, the merged firm and the nonmerged firms simultaneously choose the outputs they supply to each local market. [7]

Denote the merged firm by a subscript m and the remaining firms by a subscript i. Again by symmetry, we need only concentrate on consumer markets in the left-hand half of the set of markets. Given the merged firm's decision to relocate to x = 1/2 - d, we can identify the new Cournot-Nash equilibrium quantities for the left side of the market line in the second stage of the game. They are (see the Appendix for details)

[[q.sup.c].sub.m](v , n, x, d) = {v - (n-1)(1/2 - d - x) + (n - 2)(1/2 - x)/n for x [less than] 1/2 - d v - (n - 1)(x - 1/2 + d) + (n - 2)(1/2 - x)/n for x [greater than] 1/2 - d

x [epsilon] [0, 1/2) (2a)

[[q.sup.C].sub.i](v , n, x, d) = {v - 2(1/2 - x) + (1/2 - d - x)/n for x [less than] 1/2 - d v - 2(1/2 -x) + (x - 1/2 + d)/n for x [greater than] 1/2 - d

x [epsilon] [0, 1/2), i = 1,...., (n - 2). (2b)

The postmerger product price at each location x on the line is

[P.sup.M](v , n, x, d) = {v + 1/2 - d - x) + (n - 2)(1/2 - x)/n for x [less than] 1/2 - d v + (x - 1/2 + d) + (n - 2)(1/2 - x)/n for x [greater than] 1/2 - d. (3)

Note that the equilibrium price in each local market x falls as we move from the market periphery toward the merged firm's location of its branch plant. If n = 3, the consumer markets in the interval (1/2 - d, 1/2) are charged a uniform delivered price. However, for n [greater than] 3, the price in that interval also continues to fall as we move from the branch location toward the market center. [8]

From Equations 2 and 3, the profit earned by the left-hand branch of the merged firm and the profit of each nonmerged firm from supplying this side of the market are respectively

[[[pi].sup.c].sub.m](v, n, d) = [[[integral of].sup.1/2].sub.0]] [[(q.sup.c].sub.m][(v, n, x, d)).sup.2] dx (4a)

[[[pi].sup.c].sub.i](v, n, d) = [[[integral of].sup.1/2].sub.0] ([[q.sup.c].sub.i][(v, n, x, d)).sup.2] dx (i = 1,..., n - 2). (4b)

3. Location Equilibrium for a Single Two-Firm Merger

From Equations 2a and 4a, we can identify the subgame perfect Nash equilibrium location choice d of the left-hand branch of the merged firm, given by the solution to the first-order profit-maximizing condition [9]:

[delta][[[pi].sup.c].sub.m](v, n, d)/[delta]d = 2(n - 1)/n [[integral of].sup.1/2-d].sub.0] [[q.sup.c].sub.m](v, n, x, d) dx - [[integral of].sup.1/2].sub.1/2-d] [[q.sup.c].sub.m](v, n, x, d) dx]. (5)

Equation 5 tells that the optimal distance [d.sup.*] satisfying the first-order condition for profit-maximization is such that the following median location result holds:

THEOREM 1. With linear demand and transport costs, a two-firm merger results in the merged firm locating its two branch plants such that for each branch plant aggregate sales to the left of the branch equals aggregate sales to the right of the branch.

Solving Equation 5 for the profit-maximizing location [d.sup.*](v, n) gives the following:

THEOREM 2. With linear demand and transport costs, a two-firm merger leads the merged firm to locate its two branches a distance [d.sup.*](v, n) from the market center, where

[d.sup.*](v, n) = 1/4 + [square root of][(4v - 1/4(n - 2)).sup.2] + 1/16 - 4v - 1/4(n - 2) (6)

It follows immediately from Equation 6 that [d.sup.*](v, n) [greater than] 1/4 for n [greater than or equal to] 3. The merged firm relocates its two branch plants nearer to the market periphery than to the market center. Furthermore, [[d.sup.*].sub.v](v, n) [less than]0 and [[d.sup.*].sub.n] (v, n) [greater than] 0. In other words, [d.sup.*](v, n) is a decreasing function of the maximum reservation price v and an increasing function of the initial number of firms n. Table 1 gives [d.sup.*](v, n) for a range of values of v and n, satisfying the condition v [greater than] n/2.

The intuition behind these results is straightforward. The left-hand branch of the merged firm faces much stronger competition for consumers to its right--that is, consumers located between itself and the nonmerged firms at the market center--than it does for consumers to its left. Examination of Equation 2a indicates that aggregate sales of the branch fall much more slowly with distance for consumer markets located to its left than they do with distance for consumer markets located to its right. Therefore, because of the stronger competition near the market center, the branch is willing to forego more sales in these local markets and to locate outside the market quartile.

An increase in n increases the degree of competition the merged firm faces from the nonmerged firms, which strengthens its desire to distance itself from the market center. By contrast, an increase in the maximum consumer reservation price v has a greater proportionate effect on sales at local markets where sales are relatively low (locations near the market center) than where sales are relatively high. An increase in v, therefore, makes a location nearer to the quartile more desirable.

In deriving Theorems I and 2, we have assumed that the nonmerged firms do not change their locations postmerger. It is straightforward to show (see Appendix) that no nonmerged firm wishes to change its location from 1/2 in response to the two-firm merger relocating its divisions. This result is, of course, not surprising given that the two-firm merger does not disturb the underlying symmetry of the market.

Although we have shown that a two-firm merger induces the merged firm to locate its branches away from the center, we have not as yet established whether merger is in fact profitable. For merging to be of interest to the two firms, the profit of the merged firm in the postmerger game should be at least as great as the sum of the profits of the two firms in the premerger game. The question is whether the advantage of coordinating locations to serve the market in a more cost-effective way is sufficiently strong to outweigh the disadvantage of the standard merger paradox in each local market. In each local market, there are n competitors premerger, two of which are the merging firms, whereas postmerger there are n -- 1 competitors, one of which is a division of the merged firm.

The complex nature of Equations 1, 4, and 6 precludes the derivation of any analytical results on merger profitability. However, it is clear from these equations that the profitability of a two-firm merger depends solely on the two parameters of the model, the maximum reservation price, v, and the number of firms, n. As a result, numerical simulations are sufficient to give us considerable insight into the potential profitability of a two-firm merger in our Cournot model of spatial competition. [10] This is particularly the case since we can confine our attention to values of v satisfying the equilibrium condition v [greater than or equal to] n/2 and n is an integer. Table 2 illustrates the impact of a two-firm merger on the merged firm's profit, 2([[[pi].sup.C].sub.m] (v, n, [d.sup.*](v, n)) - [[[pi].sup.C].sub.i]), for a range of values of v and n.

Profits for both the merged and nonmerged firms are increasing in the consumer willingness to pay v. However, because a higher v makes distance less of an issue and hence locational advantage less important, a two-firm merger that confers locational advantage becomes less attractive as v increases. In other words, it is possible to identify an upper limit v(n) on the maximum consumer reservation price such that if the consumer reservation price is less than this maximum, or v [less than] v(n), then a two-firm merger is profitable, however, if v [greater than(n), a two firm merger is unprofitable. Our numerical simulations indicate that the profitability condition v [less than] v(n) and the equilibrium condition v [greater than] n/2 can both be satisfied provided that the premerger market contains no more than 8 firms. Table 3 gives v(n) and the optimal distance [d.sup.*](v(n), n), for values of n between 3 and 10.

What impact does the merger have on the profits of the nonmerged firms? There are two conflicting forces at work as a result of the merger. On the one hand, there are now only (n - 1) firms in direct competition with each other at each local market x, and this reduction in the degree of competition over the entire market has a positive effect on each nonmerged firm's profitability. On the other hand, for all consumer markets located at a distance greater than [d.sup.*](v, n)/2 from the market center, the nonmerged firms are at a cost disadvantage relative to the merged firm's divisions, and this has a negative impact on their profitability. The former effect outweighs the latter. A profitable two-firm merger in this model always increases the profits of the firms outside the merger. We should not, therefore, expect the rivals of the two merger partners to contest their decision to merge.

The final question we consider in this section is the impact of the merger on the relative profits of the merged and nonmerged firms. When we compare the profit of a single division of the merged firm with the profit of a nonmerged firm, we find that the latter is greater. This is not especially surprising for two reasons. First, we have already noted that the nonmerged firms enjoy a positive externality from the merger. Second, the merged firm explicitly limits each division to supply only half the market. However, as Table 4 indicates, the aggregate profit of the merged firm is greater than the profit of each nonmerged firm throughout the parameter range for which a two-firm merger is profitable.

A merger that confers upon the two firms the ability to coordinate their location decisions, correctly anticipating the outcome of the subsequent quantity competition with their nonmerged rivals, can give the merged firm sufficient commitment power to overcome the merger paradox of the standard Cournot model. By contrast with the standard model, a merger results in a bigger firm, which has combined the assets of the two merging firms. [11] However, there is a countervailing force: competition from the nonmerged firms. It is not surprising, therefore, that a two-firm merger is likely to be profitable only if the industry is not too unconcentrated: in our model, for industries such that n [less than or equal to] 8.

4. Welfare Effects of a Two-Firm Merger

Prices

In the spaceless Cournot model, a two-firm merger always results in increased consumer prices in the absence of what Farrell and Shapiro (1990) refer to as "cost synergies." A two-firm merger in the spatial Cournot model exhibits some such cost synergies because the branches of the merged firm move closer to consumer markets on the market periphery, reducing the costs of the merged firm in supplying those consumer markets. The important question is whether these cost synergies are strong enough to offset the reduced degree of competition at each local market x that results from the merger.

Once again, by symmetry of the model, we need only consider prices in local markets in the interval [0, 1/2]. Comparison of Equations 1 and 3 indicates that the impact of a two-firm merger on consumer prices in this interval, as measured by the difference between the postmerger and the premerger price, is

[p.sup.M](v, n, x, d) - [p.sup.C](x) = {2v + 2x - (2dn + 2d + 1)/2n(n + 1) for x [less than] 1/2 - d 2v + 2x(2n + 3) - (3 - 2d) - 2n(1 - d)/2n(n + 1) for x [greater than] 1/2 - d. (7)

Equation 7 indicates that a two-firm merger is most likely to reduce prices at low values of v and x. In other words, the consumer markets that are most likely to benefit from a two-firm merger are those on the market periphery. When we substitute the optimal distance [d.sup.*](v, n) into Equation 7 and take account of the constraint that v [equal to or greater than] n/2, we find that the only case in which a two-firm merger might lead to some consumers being charged lower prices is if n

= 3. With n = 3 and v = 1.5, the consumers that are charged lower prices as a result of the merger are those in the intervals [0, 0.1] and [0.9, 1]. For any consumers at all to be charged lower prices, we must have n = 3 and v [less than or equal to] 1.592. Note also that when we substitute x = 1/2 into Equation 7, the numerator becomes 2(v + 2d(1 + n)) [greater than] 0. A two-firm merger always leads to higher prices being charged to consumers located at the market center.

Output

Aggregating the output supplied in each local market x over the relevant market areas (integrating Eqns. 1 and 2) lets us calculate the impact of a two-firm merger on the total output supplied by the merged and the nonmerged firms. Denote by [[q.sup.c].sub.i] the aggregate output of each firm in the premerger game. For the postmerger game, denote by [[q.sup.c].sub.m](d) the aggregate output of each branch of the merged firm and by [[q.sup.c].sub.i](d) the aggregate output of each nonmerged firm. We find that the differences in output supplied in the postmerger to that supplied in the premerger game are as follows:

{(a) [[2q.sup.c].sub.m](d) - [[q.sup.c].sub.i](d) = d(1 - 2d) [greater than] 0

(b) [[q.sup.c].sub.m](d) - [[q.sup.c].sub.i](d) = - F(v, n, d)/8n [less than] 0

(c) [[q.sup.c].sub.m](d) - [[q.sup.c].sub.i] = - (n - 1)F(v, n, d)/8n(n + 1) [less than] 0

(d) [[q.sup.c].sub.i](d) - [[q.sup.c].sub.i] = F(v, n, d)/4n(n + 1) [greater than] 0, (8)

where F(v, n, d) = 4v - 4d(1 - 2d)(n + 1) - 1 and F(*) [greater than] 0 in the range of the parameters v and n for which we find that a two-firm merger is profitable.

In contrast to the standard Cournot model, the merged firm in the spatial model does have a greater total output in the postmerger game than its nonmerged competitors (see Eqn. 8a). Unlike a merger in the spaceless Cournot model, the merged firm in the spatial model can, by relocating its divisions, exploit its potentially greater size in the postmerger game. Note, however, that each branch of the merged firm produces less aggregate output than do each of the nonmerged firms (see Eqn. 8b). Moreover, each nonmerged firm produces more total output in the postmerger output game than it produced prior to the merger (see Eqn. 8d). Similarly each branch of the merged firm produces less total output than either of the two firms did in the premerger game (see Eqn. 8c).

Surplus

In a spaceless model, we have the strong result that "in the absence of fixed costs, all mergers in quantity competition are socially undesirable." (Gaudet and Salant 1992, p. 153) This is not the case once we introduce spatial competition and give the merged firms the power to exercise location leadership.

We noted in this section that a narrow subset of consumers are likely to be charged lower prices as a result of a merger. It is not surprising, therefore, that a two-firm merger reduces aggregate consumer surplus. Furthermore, the loss in consumer surplus increases as both the consumer reservation price v and the number of firms n increase. Since it is reasonable to confine our attention to two-firm mergers that are profitable, it is obvious that a two-firm merger leads to an overall increase in aggregate profit of the industry. Aggregate profit is also an increasing function of the maximum reservation price.

While consumers in the aggregate lose and firms individually and in the aggregate gain from a profitable two-firm merger, the profit-increasing effect of the merger is almost always sufficient to offset the consumer surplus reducing effect. As a result, such mergers are generally socially efficient. Figure 1 illustrates the effect of a profitable two-firm merger on total surplus. This figure and our numerical analysis indicate that a profitable two-firm merger increases total surplus even if there are no savings in fixed costs: if n = 3 and v [less than] 2.75 and if n [greater than or equal to] 4.

The introduction of strategic location leadership in the merger decision has two important effects that give rise to this outcome. First, a two-firm merger is potentially profitable for the merged firm as well as for the nonmerged firms. Second, while consumer surplus is reduced by the merger, the price-increasing effects of the merger are considerably moderated as compared to what would have happened to price if the merging firms had not been able to change their locations. It is easy to see that the change of location by the merged firms results in all consumers in the intervals [0, (1 - [d.sup.*](v, n))/2] and [(1 + [d.sup.*](v, n))/2, 1] paying lower postmerger prices than they would have paid if the merged firms had instead remained at the market center. [12] As a result, the profit-increasing effect of the merger is almost always sufficient to offset the consumer surplus--reducing effect. The only exception, when n = 3, arises because when the number of premerger firms is this small, the competition-reducing effects of the merger are particularly strong.

5. Conclusions

U.S. policymakers' current interest in the unilateral competitive effects of horizontal mergers has no doubt been motivated by the more than twofold increase in the number of premerger filings over the past five years. It is clearly important to policymakers to understand what market forces underlie this surge in merger activity, for only then can they evaluate the likely impact mergers will have on competition in the market. We have shown how horizontal mergers can confer profitable advantages to firms that compete in quantities across a set of spatially separated markets. The spatial aspect of competition is critically important because it provides the means by which location leadership can give the merged firms commitment power. By having a location choice advantage the merged firms can, by combining the assets of two firms, credibly commit to being a firm that has a larger total market share than its nonmerged rivals.

With respect to location choice, we start from a scenario in which all firms are agglomerated at the center of the market area. A two-firm merger causes the merging firms to disperse their branches, locating its two divisions nearer to the market peripheries than to the market center. The degree to which the merging firms disperse is an increasing function of the initial number of firms in the market, reflecting the fact that dispersal is desired to moderate competition with the nonmerged firms remaining at the market center.

More importantly, we have also shown under what conditions such two-firm mergers are profitable and can therefore be expected to occur. We find that mergers are more likely to occur in markets that are relatively concentrated and in which the consumer reservation price is not too high.

The change in location means that horizontal mergers in spatially differentiated markets generate some cost synergies in that the merged firm's divisions serve the market area in a more cost-effective way. However, these cost synergies are not sufficient to offset the price-increasing effects of the reduction in competitive pressures in the local markets to which the merger gives rise. As a result, the merger reduces consumer surplus. Nevertheless, once we confine our attention to profitable mergers, so that both the merged and nonmerged firms benefit from the merger, we find that a profitable two-firm merger almost always increases total surplus and therefore is efficiency enhancing. In sum, we have found two-firm mergers in a Cournot model that do lead to both a bigger and better firm.

(*.) Department of Economics, Tufts University, Medford, MA 02155, USA; E-mail [e-mail]; corresponding author.

(+.) Department of Economics, Tufts University, Medford, MA 02155, USA; E-mail [e-mail]. We are grateful for the very helpful comments of two referees and the editor. The usual disclaimer applies. Received August 1998; accepted July 1999.

(1.) The importance of assuming constant marginal costs should also be recognized. Perry and Porter (1985) show, for example, that merger can be profitable and lead to firms of different size if marginal costs are increasing and if there is a "tangible asset that the merged firm acquires ... which increases the output it can produce at a given average cost." (Perry and Porter 1985, p. 226).

(2.) It is now well known that the spatial model underlying our analysis has much wider generality as a model of product differentiation. "Space" can be interpreted as referring to the spectrum of possible product characteristics. The location of a firm is now the basic product design chosen by the firm and each firm tailors this basic design to meet specific consumer demand for product variety.

(3.)McAfee, Simons, and Williams (1992) also consider merger of Cournot firms in a spatial setting. However, space actually plays a very limited role in their analysis: A two-firm merger is profitable only if there is a significant difference in the marginal costs of the merging firms (see Norman and Pepall 1998).

(4.)Daughety (1990) also notes the importance of commitment to the social value of merger.

(5.) Rowthom (1992) makes a similar normalization. The consumer reservation price will emerge as being central to the profitability or otherwise of a merger and so cannot be further normalized. By contrast, no generality is lost by normalizing the slope of the inverse demand function to unity since the slope parameter enters as a common multiple in profit calculations and so does not affect comparisons of relative profits.

(6.) To guarantee that each firm is active in each local market irrespective of its location, Anderson and Neven assume that the maximum consumer reservation price v is greater than n, the number of firms. This assumption ensures that even in the case where one firm is located at one end of the market spectrum, x - 0, and the remaining n - 1 firms are located at the other end, x 1, it is still true that the firm located at x = 0 is active in the local market x = 1.

(7.) The ability to coordinate the locations of the merged firma' plants is central to the potential profitability of a merger. It is easy to show, for example, that the equilibrium to the "location leadership" game is also an equilibrium to a simultaneous location game in which the merged firm chooses two locations and the nonmerged firms choose one, provided that the merged firm allocates output efficiently to its two plants so there is no market overlap between them.

It might be thought that we should consider a three-stage game, in the second stage of which the nonmerged firms choose their locations. We do not consider this intermediate stage explicitly because, as we show in the Appendix, the nonmerged firms will not want to change their location decisions in response to the merger.

(8.) See Greenhut and Greenhut (1975) for an analysis of this type of price discrimination.

(9.) Second-order conditions are satisfied for v [greater than] (n - 1)/4, which is always true for v [greater than] n/2.

(10.) Mathematica notebook detailing the simulations is available from the authors on request.

(11.) Recall that Perry and Porter (1985) also consider mergers that combine the assets of the merged firms, but from a very different perspective.

(12.) We should recall, of course, that if the merged firm could not change locations of its plants, then the merger would not he profitable.

References

Anderson, S. P., and D. J. Neven. 1990. Cournot competition yields spatial agglomeration. International Economic Review 32:793-808.

Daughety, A. F. 1990. Beneficial concentration. American Economic Review 80:1231-7.

Davidson, C., and R. Deneckere. 1984. Horizontal mergers and collusive behavior. International Journal of Industrial Organization 2:117-32.

Deneckere, R., and C. Davidson. 1985. Incentives to form coalitions with Bertrand competition. Rand Journal of Economics 4:473-86.

Farrell, J., and C. Shapiro. 1990. Horizontal mergers: An equilibrium analysis. American Economic Review 80:107-26.

Federal Trade Commission. 1997. Anticipating the 21" Century. Federal Trade Commission, Washington, D.C.

Gaudet, G., and S. W. Salant. 1992. Toward a theory of horizontal mergers. In The new industrial economics, edited by George Norman and Manfredi La Manna, Aldershot, UK: Edward Elgar.

Greenhut, J., and M. L. Greenhut. 1975. Spatial price discrimination, competition and location effects. Economica 42:401-19.

McAfee, R. P., J. J. Simons, and M. A. Williams. 1992. Horizontal mergers in spatially differentiated, noncooperative markets. Journal of Industrial Economics 40:349-58.

Norman, George, and Lynne Pepall. 1998. Horizontal mergers in a spatially differentiated, noncooperative model: A comment. Tufts University Economics Department. Working Paper No. 98-04.

Pepall, Lynne, Daniel Richards, and George Norman. 1999. Industrial organization: Contemporary theory and practice. Cincinnati: South-Western College Publishing.

Perry, Martin K., and Robert H. Porter. 1985. Oligopoly and the incentive for horizontal merger. American Economic Review 75:219-27.

Reitzcs, J. D., and D. T. Levy. 1995. Price discrimination and mergers. Canadian Journal of Economics 28:427-36.

Rowthorn, Robert E. 1992. Intraindustry trade and investment under oligopoly: The role of market size. Economics Journal 102:402-14.

Salant, S., S. Switzer, and R. Reynolds. 1983. Losses from horizontal merger: The effects of an exogenous change in industry structure on Cournot-Nash equilibrium. Quarterly Journal of Economics 98:185-99.

Szidarovzky, F., and S. Yakowitz. 1982. Contributions to Cournot oligopoly theory. Journal of Economic Theory 28:51-70.

                       Equilibrium Distance [d.sup.*](v, n)
            Number of Firms (n)
Reservation
Price (v)            3            4      5      6      7      8      9
1.5               0.2748
2.0               0.2678        0.2850
2.5               0.2638        0.2774 0.2906
3.0               0.2613        0.2725 0.2835 0.2940
3.5               0.2596        0.2691 0.2785 0.2876 0.2964
4.0               0.2583        0.2666 0.2748 0.2828 0.2906 0.2981
4.5               0.2573        0.2647 0.2719 0.2790 0.2860 0.2928 0.2995
                 Impact of Merger on Merged Firms' Profit:
                 2([[[pi].sup.C].sub.m] (v, n, [d.sup.*](v, n)) -
                           [[[pi].sup.C].sub.i])
Reservation Number of Firms (n)
Price(v)             3             4        5        6        7        8
1.5               0.5740
2.0               0.06415        0.04535
2.5               0.06404        0.03343  0.03338
3.0               0.05703        0.01288  0.01406  0.02285
3.5               0.04309       -0.01634 -0.01291  0.00063  0.01385
4.0               0.02222       -0.05427 -0.04758 -0.02798 -0.00925  0.006197
4.5              -0.00558       -0.10090 -0.08998 -0.06304 -0.03767 -0.01685
Reservation
Price(v)       9
1.5
2.0
2.5
3.0
3.5
4.0
4.5         -0.00034
                 Critical Value of Reservation Price for a
                        Profitable Two-Firm Merger
           Upper Limit on
Number of Reservation Price   Distance
Firms (n)      (v(n))       [d.sup.*](v(n), n)
    3          4.4089         0.2575
    4          3.2390         0.2708
    5          3.2782         0.2805
    6          3.5124         0.2875
    7          3.8132         0.2926
    8          4.1443         0.2965
    9          4.4917         0.2995
   10          4.8489         0.3020
                   Impact of Merger on Relative Profits -
               2[[[pi].sup.c].sub m,d*(v, n)) - n, d*(v, n)) -
                [[[pi].sup.c].sub.i]/(v, n, [d.sup.*](v, n))
            Number of Firms (n)
Reservation
 Price (v)         3           4      5      6      7      8      9
    1.5          0.1031
    2.0          0.1451      0.1198
    2.5          0.1869      0.1511 0.1302
    3.0          0.2287      0.1823 0.1550 0.1373
    3.5          0.2704      0.2135 0.1799 0.1579 0.1424
    4.0          0.3122      0.2488 0.2048 0.1785 0.1600 0.1463
    4.5          0.3539      0.2760 0.2297 0.1991 0.1776 0.1616 0.1493

Appendix

CLAIM 1. If v [greater than or equal to] n/2 and if all n firms are located in the center of the market, then no one single firm finds it profitable to deviate from its location at the center of the market, given that the remaining (n - 1) firms are at the center.

Assume that firm 1 locates a distance d from the left-hand side of the market, with 0 [less than] d [less than or equal to] 1/2. We confine explicit analysis to the case in which the deviating firm can supply she entire set of markets along the line. This implies that d [greater than or equal to] [(n + 1) - 2v]/2n, a not particularly restrictive constraint given that v [greater than or equal to] n/2. The proof can easily be generalized to the remaining values of d.

Profit to firm 1 is

[[pi].sub.t](d) = [[[integral of].sup.d].sub.0] [(v - n(d - r) + (n - 1)(1/2 - r)).sup.2]/[(n + 1).sup.2] dr + [[[integral of].sup.1/2].sub.d] [(v - n(r - d) + (n - 1)(1/2 - r)).sup.2]/[(n + 1).sup.2] dr

+ [[[integral of].sup.1].sub.1/2] [(v - n(r - d) + (n - 1)(r - 1/2)).sup.2]/[(n + 1).sup.2] dr.

The first-order condition on d is

[delta][[pi].sub.1](d)/[delta]d = n(1 - 2d)(4v - n - 2d(n - 1) - 1)/2[(n + 1).sub.2].

This is positive for v [greater than or equal to] n/2 and d [less than] 1/2.

Recall that competition is Cournot at every point. Assume that there are N firms and let the cost of firm j be denoted [c.sub.j](j = 1, ..., N). Then standard analysis gives the Cournot-Nash equilibrium output for each firm as

[[q.sup.C].sub.j] = v - (N + 1)[c.sub.j] + [[[sigma].sup.N].sub.k=1] [C.sub.k]/N + 1 (j = 1....., N). (A.1)

The Cournot-Nash equilibrium price is

[p.sup.c] = v + [[[sigma].sup.n].sub.j=1] [c.sub.j]/N + 1.

Without loss of generality, assume that firms 1 and 2 merge to form firm m. If there are n firms premerger, then postmerger we have N n - 1. For the left-hand branch of the merged firm, we have

[C.sub.m] = 1/2 - d - x, [C.sub.x] = 1/2 - x for x [less than]1/2 - d and [C.sub.m] = x - 1/2 + d, [C.sub.k] = 1/2 - x for x [greater than] 1/2 - d.

Substituting in Equation A.1 gives Equation 2a. For a nonmerged firm i, we have

[C.sub.i] = 1/2 - x, [C.sub.k] = 1/2 - d - x for k = m and [C.sub.k] = 1/2 - x for k = 3 ,. . . . n if x [less than] 1/2 - d and

[C.sub.i] = 1/2 - x, [C.sub.k] = x - 1/2 + d for k = m and [C.sub.k] = 1/2 - x for k = 3 ,. . . , n if x [greater than] 1/2 - d.

Substituting in Equation A.1 gives Equation 2b.

CLAIM 2. After the merged firm has relocated its two plants a distance [d.sup.*] from the center, no nonmerged firm has an incentive to locate away from the market center, anticipating the subsequent quantity competition game to be played between itself and all of its rivals, merged and nonmerged.

We deal explicitly with the case in which the potential deviating firm locates somewhere between the center and merged firm's location on the line, which for ease of exposition, we denote here by [d.sup.*]. Once again, the proof can be easily extended to choosing locations between 0 and [d.sup.*]. If the deviating firm 1 locates distance u from the left-hand edge of the market its profit is

[[pi].sub.1](u) = [[[integral of].sup.[d.sup.*]].sub.0] [(v - (n - 1)(u - r) + ([d.sup.*] - r) + (n - 3(1/2 - r)).sup.2]/[n.sup.2] dr

+ [[[integral of].sup.d].sub.[d.sup.*]] [(v - (n - 1)(u - r) + (r - [d.sup.*] + (n - 3)(1/2 - r)).sup.2]/[n.sup.2] dr

+ [[[integral of].sup.1/2].sub.d] [(v - (n - 1)(r - u)+ (r - [d.sup.*]) + (n - 3)(1/2 - r)).sup.2]/[n.sup.2] dr

+ [[[integral of].sup.[1-d.sup.*]].sub.1/2] [(v - (n - 1)(r - u) + (1 - [d.sup.*]) + (n - 3)(r - 1/2)).sup.2]/[n.sup.2] dr

+ [[[integral of].sup.1].sub.[1-d.sup.*]] [(v - (n - 1)(r - u) + ( r - 1 + [d.sup.*]) + (n - 3)(r- 1/2)).sup.2]/[n.sup.2] dr.

The first-order condition on u is

[[delta][[pi].sub.1](u)/[delta]u = (1 - 2u)(n - 1)(4v - n - 2u(n - 4) - [4d.sup.*]/[2n.sup.2]

Since v [greater than or equal to] n/2 and [d.sup.*] [less than] 1/4, this is positive for any u [less than] 1/2.