Anti-competitive versus Pro-competitiveMergers¤

Sven-Olof Fridolfsson

December 13, 2000

Abstract

In a framework where mergers are mutually excluding, I show that…rms pursue anti-competitive rather than (alternative) pro-competitivemergers. Potential outsiders to anti-competitive mergers refrain frompursuing pro-competitive mergers if the positive externalities fromanti-competitive mergers are strong enough. Potential outsiders topro-competitive mergers pursue anti-competitive mergers (thereby pre-empting pro-competitive mergers) if the negative externalities fromthe pro-competitive mergers are strong enough. Potential participantsin anti-competitive mergers are cheap to buy due to the risk of be-coming outsiders to pro-competitive mergers. Firms may even pursuean unpro…table and anti-competitive merger when alternative mergersare pro…table and pro-competitive.

Key Words: anti- and pro-competitive mergers, consumers’ welfare,coalition formation, endogenous split of surplus.

¤This work has been much improved due to the guidance of my supervisor JohanStennek. Comments from Jonas Björnestedt, Helen Jakobsson, Lars Persson, GiancarloSpagnolo and Andreas Westermark, as well as from participants in workshops at theStockholm University and at IUI, are also gratefully acknowledged. The author thanksKonkurrensverket for …nancial support. He is a¢liated to the Research Institute of Indus-trial Economics (IUI), Box 5501, S-11485 Sweden and to the Department of Economics,Stockholm University.

1

1 Introduction

Today, competitive forces drive a merger wave of historical proportions. The

total global value of mergers and acquisitions (M&As) exceeded 3.4 trillion

US dollars in 1999 (The Economist, 2000). While many of the M&As in

the current wave appear to be motivated by a legitimate response to fast

changing business conditions such as global competition, deregulation, and

over capacity, a larger share than in the recent past seems to involve direct

competitors (Pitofsky, Chairman of the FTC, 1997). As a result, one may

legitimately fear that many of the recent mergers have increased …rms’ market

power and thereby have reduced consumers’ welfare.

This concern has been the main motivation for ruling some horizontal

mergers illegal. For example, the European Commission recently blocked

the merger between the two Swedish truck manufacturers Volvo and Sca-

nia on the ground that the merger would nearly eliminate all competition

in the Scandinavian market while reducing it signi…cantly in Irland and the

United Kingdom (European Commission, 2000a). Interestingly, the prohi-

bition of this merger also induced Volvo to acquire Renault Véhicules In-

dustriels (RVI).1 Unlike the attempted merger between Volvo and Scania,

the latter merger was not blocked by the Commission (European Commis-

sion, 2000b). According to the Commission, this merger would not increase

market concentration signi…cantly in any geographical market and conse-

quently should not hurt the consumers. One may even hypothesize that the

Volvo-RVI merger will bene…t the consumers; if a merger does not reduce

competition and its only impact is to save on costs, some of the associated

bene…ts should spill over to the consumers.1Similarly, Volkswagen purchased a large minority stake in Scania. However, this pur-

chase was not investigated by the Commission, since it was not classi…ed as a merger.

2

These events raise the concern that mergers harming consumers, that is

anti-competitive mergers, may preempt pro-competitive mergers bene…tting

them. The main …nding of this paper is that this concern is legitimate. While

the market sometimes select the most desirable merger from the consumers’

point of view, the subsequent analysis highlights several mechanisms that

lead …rms to pursue anti- rather than pro-competitive mergers.

The starting point of this analysis is a robust …nding in the theoretical

literature on mergers, that the “competitive” nature of mergers is linked

to their impact on the pro…tability of outsiders (competitors). While anti-

competitive mergers typically bene…t outsiders, the opposite is true for pro-

competitive mergers.2 In turn, the signs and magnitudes of these external

e¤ects on outsiders favor anti- rather than pro-competitive mergers.

To be more precise, external e¤ects have a direct in‡uence on the …rms’

merger decisions which, depending on their sign, materializes into di¤erent

incentives for potential outsiders. First, consider anti-competitive mergers.

Potential outsiders to such mergers refrain from pursuing pro-competitive

mergers if the positive external e¤ect from the anti-competitive merger is

large enough. This lack of incentives for merging is referred to as the “induce-

ment mechanism.” Second, consider pro-competitive mergers with negative

external e¤ects. Potential outsiders to such mergers pursue anti-competitive

mergers to preempt the pro-competitive merger that would hurt them. This

incentive for merging is referred to as the “preemption mechanism.”

Furthermore, external e¤ects also have an indirect in‡uence on …rms’2Intuitively, the merging …rms in an anti-competitive merger restrict their output rel-

ative to their combined pre-merger output in order to increase the equilibrium price. Asa result, the external e¤ect of the merger is positive, since the outsider bene…ts from thehigher price without bearing the cost of reducing its own output (Salant, Switzer andReynolds, 1983; Perry and Porter, 1985; Farrell and Shapiro, 1990). Throughout the pa-per, I use the sign of the external e¤ects accordingly in order to identify wether a mergeris anti- or pro-competitive.

3

merger decisions. Since …rms’ pre-merger values incorporate the risk of be-

coming an outsider, potential outsiders to anti-competitive mergers with pos-

itive external e¤ects have high pre-merger values. As a result, such …rms tend

to be expensive to buy. Conversely, potential outsiders to pro-competitive

mergers tend to be cheap to buy. In turn, other …rms, including …rms that

are potential participants in pro-competitive mergers, tend to …nd it prof-

itable to buy potential outsiders to pro-competitive mergers. Thereby, they

preempt the pro-competitive merger and instead induce an anti-competitive

one. This incentive for buying …rms that lose as outsiders is referred to as

the “valuation mechanism.”

To illustrate these mechanisms, I extend the model in Fridolfsson and

Stennek (1999) to asymmetric …rms. Unlike other models of endogenous

merger formation, this model predicts how the merging …rms split the surplus.

In the present context, such a prediction is crucial. Indeed, …rms pursue

anti- rather than pro-competitive mergers, since the split of the surplus in

the former type of merger is more favorable to bidding …rms. Thus, I am

able to identify the valuation mechanism precisely because the merging …rms

split the surplus endogenously.

Previous merger analysis, starting with Stigler (1950), has mainly fo-

cused on the question whether the process of merger formation leads to the

most desirable level of concentration.3 In contrast, the present paper asks the

question whether the process of merger formation induces the most desirable

merger for a given level of concentration. This issue was …rst addressed in

a full-‡edged model of endogenous merger formation by Horn and Persson

(1996).4 They propose a cooperative model of endogenous merger forma-3More recent contributions along these lines are Kamien and Zang (1990), (1991),

(1993), Fridolfsson and Stennek (1999), (2000a), (2000b), Gowrisankaran and Holmes(2000) and Nocke (2000).

4A number of other papers treat related questions. Barros (1998) studies whether the

4

tion which captures the inducement and preemption mechanisms. However

their model does not predict endogenously the split of the surplus among

the merging …rms and is therefore not suitable for identifying the valuation

mechanism. Moreover, they do not explicitly analyze cases where mergers

are pro-competitive.

The paper is organized as follows. Section 2 introduces the model. To

focus on the competitive nature of mergers, Section 3 considers cases where

the pro…tability of mergers (the internal e¤ects of mergers) is small relative to

their external e¤ects. As a result, merger incentives are determined to a large

extent by the external e¤ects of mergers. If one merger is anti-competitive

(has a positive external e¤ect) while an alternative merger is pro-competitive

(has a negative external e¤ect), the …rms tend to pursue the former merger.

Furthermore, the market may fail to select the most desirable merger, also

when all mergers bene…t the consumers (are pro-competitive). Section 4

brie‡y discusses cases when internal e¤ects are larger than external e¤ects.

Section 5 shows that the signs and magnitudes of external e¤ects may be

crucial to predict the likelihood of speci…c mergers even though pro…tability

considerations clearly favor speci…c mergers. Indeed, …rms may pursue an un-

pro…table and anti-competitive merger, even though other mergers are prof-

itable and pro-competitive. The welfare e¤ects may also be perverse; …rms

may pursue an unpro…table merger that reduces both the consumers’ and pro-

ducers’ surpluses, even though an alternative and pro…table merger increases

both these surpluses. The Concluding Remarks discuss some prospects to

test the prediction that more anti-competitive mergers preempt less anti-

competitive ones as well as policy implications of this …nding.

merger formation process eliminates ine¢cient rather than e¢cient …rms. Persson (1998)formalizes the failing …rm defense as an auction and …nds that the worst buyer from theconsumers’ point of view, often acquires the failing …rm. Horn and Persson (2000) analyzewhether …rms pursue national rather than cross border mergers.

5

2 The Model

Time is in…nite and continuous but divided into short periods of length ¢.

Each period is divided into two phases. In the …rst phase, there is an acqui-

sition game where nature, with equal probability, selects a …rm as bidder.5

The selected …rm chooses then whether or not to bid, the identity of the

target …rm and the size of the bid. A …rm receiving a bid can only accept or

reject it; if it rejects, it can give a (counter) o¤er in some future period when

selected by nature. It is assumed that no time elapses during the acquisition

game, although it is described as a sequential game.

I consider an industry which initially consists of three …rms: two identical

…rms, labelled x1 and x2, and one other …rm, labelled y. Mergers to monopoly

are not allowed, i.e. it is implicitly assumed that mergers for monopoly are

blocked by competition authorities. Consequently, …rms can submit bids for

one other …rm at a time only.

In the second phase, there is a market game. Rather than specifying an

explicit oligopoly model, I take the pro…t levels of each …rm in each market

structure as exogenous. In the triopoly, a …rm xi earns pro…t ‡ow ¼x (3)

and …rm y earns pro…t ‡ow ¼y (3). After the xx-merger, that is the merger

between …rms x1 and x2, the merged …rm earns pro…t ‡ow ¼xx (2), and the

outsider (i.e. …rm y) earns ¼y (2). Similarly, after a xy-merger, that is a

merger between say …rm xi and …rm y, the merged …rm earns pro…t ‡ow

¼xy (2), and the outsider (i.e. …rm xj) earns ¼x (2).

A …rm’s strategy describes the …rm’s behavior in the acquisition game:

whether to bid (if selected by nature), the identity of the target …rm, how5This speci…cation di¤ers from the one in Fridolfsson and Stennek (1999). They assume

that all …rms can bid in the same period but that only one bid is transmitted (each withequal probability) if more than one …rm bid in the same period. I adopt the speci…cationwhere a bidder in each period is selected randomly by nature, since it is simpler.

6

much to bid, and a reservation price at which to accept an o¤er. The strategy

speci…es the behavior for all periods, and for all possible histories. I restrict

attention to Markov strategies and to symmetric equilibria. In the present

context, symmetry means that …rm y treats …rms x1 and x2 identically, and

that …rms x1 and x2 behave identically. With a slight abuse of notation, let

the subscripts yx, xx and xy denote the events that …rm y submits a bid

to …rm x1 or x2, …rm xi submits a bid to …rm xj and …rm xi submits a bid

to …rm y respectively. Firm y’s strategy consists of the triple (pyx; byx; ay),

where pyx 2 [0; 1=2] denotes the probability that …rm y bids for a speci…c

…rm xi in a given period, byx denotes the size of this bid and ay denotes the

lowest bid that …rm y accepts. Firm xi’s strategy consists of the quintuple

(pxx; bxx; pxy; bxy; ax), where (pxx; pxy) 2©[0; 1]2 : pxx + pxy 2 [0; 1]

ª, pxx (pxy)

denotes the probability that …rm xi bids for …rm xj (y) in a given period,

bxx (bxy) denotes the size of this bid and ax denotes the lowest bid that …rm

xi accepts. For simplicity, I also restrict attention to sharp bids, that is bids

accepted in equilibrium. Formally, it implies that bxy ¸ ay and bxx; byx ¸ ax.6

Next, I de…ne the continuation values after a merger, at the date of a

merger, and before a merger. After the xy- (xx-) merger has occurred, the

values of the merged …rm xy (xx) and the outsider x (y) are given by

Wi (2) =¼i (2)r; (1)

for i 2 fxy; x; xx; yg, where r is the common discount rate, and ¼i (2) =r is

the discounted value of all future pro…ts. At the points in time when …rm

y buys …rm xi (event yx), …rm xi buys …rm xj (event xx) and …rm xi buys6This is assumed without loss of generality, since a …rm making a non-sharp bid can

achieve the same outcome by not bidding.

7

…rm y (event xy), the values of the buying and the selling …rms are given by

V buyi = Wi (2) ¡ bi; (2)

V selli = bi; (3)

for i 2 fyx; xx; xyg. Also, at the time of a merger, the value of the outsider

is given by

V outi = Wi (2) ; (4)

for i 2 fy; xg. Finally, …rm y’s pre-merger value, that is its expected value

in the triopoly,is given by

Wy (3) =¼y (3)r

¡1 ¡ e¡r¢

¢+ e¡r¢

£23pyxV

buyyx + 2

3pxyVsellxy + 2

3pxxVouty

+¡1 ¡ 2

3 (pyx + pxy + pxx)¢Wy (3)

¤:(5)

The …rst term is the value generated by …rm y in the current period. The

second term is the discounted expected value of all future pro…ts, that is the

values for …rm y of being a buyer, seller, outsider and triopolist in the next

period, multiplied by the respective probabilities of becoming a buyer, seller,

outsider and triopolist. For example, the probability for …rm y of being a

buyer in the next period is 23pyx, since …rm y is selected by nature with prob-

ability 13 and then buys each x-…rm with probability pyx. Moreover, given the

probabilities for …rm y of being a buyer, a seller and an outsider in the next

period, the probability of remaining in the triopoly is 1¡ 23 (pyx + pxy + pxx).

In particular, note that …rm y’s pre-merger value incorporates the risk of

becoming an outsider in the next period, that is 23pxxV

outy . Similarly, a …rm

8

xi’s pre-merger value, that is its expected value in the triopoly, is given by

Wx (3) =¼x (3)r

¡1 ¡ e¡r¢

¢

+e¡r¢£13

¡pxxV buyxx + pxyV buyxy

¢+ 1

3

¡pxxV sellxx + pyxV sellyx

¢

+13 (pxy + pyx)V

outx +

¡1 ¡ 2

3 (pyx + pxy + pxx)¢Wx (3)

¤:(6)

Three types of equilibrium conditions complete the model. First, by

subgame perfection, an o¤er is accepted if, and only if, the bid is at least as

high as the value of the …rm, that is, for i 2 fx; yg,

ai = Wi (3) : (7)

Second, for the bidders to maximize their value, it is necessary that

bxy = ay =Wy (3) ;

bxx = byx = ax =Wx (3) :(8)

The third type of equilibrium condition is that …rms, when selected by nature,

submit a bid if, and only if, it is pro…table to do so. Once …rm y is selected by

nature, it can either choose to buy a x-…rm which is worth V buyyx , or choose to

remain in the triopoly which is worth Wy (3). Hence, by subgame perfection

it is necessary that8>>><>>>:

pyx = 0 only if V buyyx ·Wy (3) ;pyx = 1

2 only if V buyyx ¸Wy (3) ;pyx 2

¡0; 12

¢only if V buyyx = Wy (3) :

(9)

Similarly, …rm xi can choose between remaining triopolist and buying …rm

9

xj or …rm y. Hence, by subgame perfection it is necessary that8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

(pxx; pxy) = (0; 0) only if V buyxx ; V buyxy ·Wx (3)(pxx; pxy) = (1; 0) only if V buyxx ¸ V buyxy ;Wx (3)(pxx; pxy) = (0; 1) only if V buyxy ¸ V buyxx ;Wx (3)(pxx; pxy) 2

©(0; 1)2 : pxx + pxy = 1

ªonly if V buyxx = V buyxy ¸Wx (3)

pxx 2 (0; 1) , pxy = 0 only if V buyxx = Wx (3) ¸ V buyxypxx = 0, pxy 2 (0; 1) only if V buyxy = Wx (3) ¸ V buyxx(pxx; pxy) 2

©(0; 1)2 : pxx + pxy 2 (0; 1)

ªonly if V buyxx = V buyxy = Wx (3) :

(10)

Combining …rm y’s three types of equilibrium conditions in (9) with the x-

…rms’ seven types of equilibrium conditions in (10) yields potentially 21 types

of symmetric Markov perfect equilibria. These di¤erent types of equilibria

are partitioned into three di¤erent categories: no-merger equilibria (NME),

immediate-merger equilibria (IME) and delayed-merger equilibria (DME).

In a NME, no …rm submits bids, that is pyx = 0 and (pxx; pxy) = (0; 0).

In an IME, at least one …rm submits a bid with certainty. For example,

pyx = 12 and (pxx; pxy) = (0; 1) constitutes an IME. In total, there are 13

types of IME.7 In a DME, no …rm bids with certainty and at least one

…rm bids with strictly positive probability. For example, pyx 2¡0; 12

¢and

(pxx; pxy) = (0; 0) constitutes a DME. In total, there are 7 types of DME.

Let the internal e¤ects of the xy- and the xx-merger, that is the prof-7In some IME, a merger occurs after a few periods. For example, consider the IME

where pyx = 0 and (pxx; pxy) = (1; 0). If …rm y is selected as bidder in the …rst period,then the xx-merger is delayed until …rm x1 or x2 is selected as bidder. However, thexx-merger occurs almost immediately as the length of the periods become very short and,as ¢ ! 0, the delay tends to 0.

10

itability of these mergers, be denoted

Ixy ´ 1r[¼xy (2) ¡ ¼x (3) ¡ ¼y (3)] ; (11a)

Ixx ´ 1r[¼xx (2) ¡ 2¼x (3)] : (11b)

Also, let the the external e¤ects of the xy- and the xx-merger, that is the

net gain compared to remaining in the triopoly of becoming an outsider to

these mergers, be denoted

Exy ´ 1r[¼x (2) ¡ ¼x (3)] ; (12a)

Exx ´ 1r[¼y (2) ¡ ¼y (3)] : (12b)

Lemma 1 in the Appendix characterizes the conditions under which the

di¤erent types of equilibria exist as ¢ ! 0. In particular, there exists at

least one type of equilibrium for all pro…t con…gurations. Henceforth, I re-

strict attention to equilibria that exist under generic pro…t con…gurations.8

Moreover, if an equilibrium is said to exist, it is meant to exist generically.

The following types of equilibria exist.

NME: pyx = 0 and (pxx; pxy) = (0; 0) ;

IMExx: pyx = 0 and (pxx; pxy) = (1; 0) ;

IMExy;yx: pyx = 12 and (pxx; pxy) = (0; 1) ;

IMExx;yx: pyx = 12 and (pxx; pxy) = (1; 0) ;

IMExx;xy;yx: pyx = 12 and (pxx; pxy) 2

©(0; 1)2 : pxx + pxy = 1

ª;

DMExy;yx: pyx 2¡0; 12

¢and pxx = 0, pxy 2 (0; 1) ;

DMExx;xy;yx: pyx 2¡0; 12

¢and (pxx; pxy) 2

©(0; 1)2 : pxx + pxy 2 (0; 1)

ª:

In the IMExx, the xx-merger occurs with certainty since …rm y does not bid

while the x-…rms bid on each other with certainty. Similarly, in the IMExy;yx,8Non-generic pro…t con…gurations are such that Ixx = 0, Ixx = Exy and so on.

11

a xy-merger occurs with certainty while in contrast both types of mergers

occur with positive probabilities in the IMExx;yx and the IMExx;xy;yx. In

the DMExy;yx, the xy-merger, even though it is delayed, occurs with cer-

tainty as in the IMExy;yx. Similarly, one may relate the DMExx;xy;yx to the

IMExx;xy;yx.9

In the remainder of the paper, I discuss the properties of the above equi-

librium structure. In particular, I am concerned with the impact of external

e¤ects on the type of merger that the …rms select in equilibrium. For instance,

do the …rms always select the most pro…table merger (with the highest in-

ternal e¤ect) or do the external e¤ects of mergers matter as well? In fact,

the conditions under which a speci…c merger may occur depend in a subtle

way on the signs and magnitudes of both internal (Ixx and Iyx) and external

(Exx and Eyx) e¤ects. To focus on the role of external e¤ects, Section 3

considers cases where they are larger than internal e¤ects. In turn, Section 4

discusses to which extent the insights found in Section 3 carry over to cases

where internal e¤ects are larger than external e¤ects. Section 5 considers a

case where internal e¤ects clearly favor one type of merger, namely when one

type of merger is pro…table while the other is unpro…table.

3 Large External E¤ects

In this section, my focus is on markets where external e¤ects are important

for the process of merger formation. To be more precise, I consider pro…t

con…gurations such that external e¤ects are large in absolute terms relative

to internal e¤ects, that is jExxj ; jExyj > max fIxx; Ixyg where Ixx; Ixy > 0.10

9Actually, other equilibria exist as well (see Lemma 1 in the Appendix). However,disregarding these equilibria (which is done in order to simplify the exposition below)equilibria, does not a¤ect any result.

10The assumption that Ixy; Ixx > 0 imply that the …rms have incentives to merge.

12

This assumption has not only the advantage to highlight the role of external

e¤ects on the endogenous formation of mergers. It has also the advantage to

put into focus the impact of mergers on consumers’ welfare. Indeed, based

on …ndings in the theoretical literature on mergers, one may use the signs of

mergers’ external e¤ects to identify whether they are anti- or pro-competitive,

that is whether they harm or bene…t the consumers.11 Throughout the paper,

I assume the following.

Assumption 1 A merger is anti- [pro-] competitive if its external e¤ect is

positive [negative].

Assumption 1 holds in many oligopoly models. For example, if goods are

perfect substitutes and …rms compete in quantities, then Assumption 1 holds

under standard assumptions about the demand and cost functions (Farrell

and Shapiro, 1990). Intuitively, the merging …rms in an anti-competitive

merger restrict their output relative to their combined pre-merger output

in order to increase the equilibrium price. As a result, the external e¤ect

of the merger is positive, since the outsider bene…ts from the higher price

without bearing the cost of reducing its own output. Similarly, Assumption

1 is usually true if goods are di¤erentiated and the …rms compete in prices.12

3.1 Anti- versus Pro-competitive Mergers

This subsection focuses attention on pro…t con…gurations such that one type

of merger has a positive external e¤ect while the other type has a negative11A leading example of an anti- (pro-) competitive merger is that it increases (decreases)

the prices of …nal goods. Other examples include mergers that reduce (increase) the varietyor the quality of …nal goods.

12To see this, consider an anti-competitive merger. Following such a merger, the merging…rms increase their prices. As a result, the outsider is better o¤, since he responds byincreasing his own prices and still gains market shares.

13

Figure 1: Ixx = Ixy = " > 0 where "! 0.

one. Note that such a con…guration of external e¤ects can be consistent with

both types of mergers being pro…table. While the type of merger with a

positive external e¤ect may be pro…table due to …xed cost savings, the other

type of merger may also be pro…table and have a negative external e¤ect due

to marginal cost savings.

Given that Assumption 1 holds, a natural question is whether the …rms

pursue anti- rather than pro-competitive mergers. While in general no def-

inite answer can be given to this question, the following analysis highlights

several mechanisms by which …rms tend to do so.

The discussion below makes frequent use of Figure 1 that illustrates the

conditions under which the di¤erent equilibria exist in the case where Ixx =

Ixy = " > 0 and " is close to 0 (to focus on the external e¤ects). The

horizontal and vertical axes in Figure 1 indicate the external e¤ects of the xy-

and the xx-merger respectively. An IMExx exists in the areas marked with

IMExx and so on. Note that there are pro…t con…gurations with multiple

equilibria.

In the north-west and south-east quadrants of Figure 1, one type of merger

is anti-competitive while the other is pro-competitive. These quadrants are

14

characterized by di¤erent types of equilibria and are therefore treated se-

quentially, starting with the north-west quadrant in Figure 1.

Proposition 1 Consider pro…t con…gurations such that the xx-merger is

anti-competitive [Exx > 0] while the xy-merger is pro-competitive [Exy <

0] and assume that mergers are pro…table [Ixx; Ixy > 0]. If external ef-

fects are large in absolute terms relative to internal e¤ects [jExxj ; jExyj >max fIxx; Ixyg], the IMExx is unique or exists simultaneously with the IMExx;yx

so that the anti-competitive xx-merger occurs with a lower bound probability

of 2/3.

All proofs are relegated to the Appendix.

Provided that the IMExx is selected when the IMExx;yx also exists, the

anti-competitive xx-merger occurs with certainty in the whole north-west

quadrant of Figure 1.13 Since the xx-merger is pro…table, the x-…rms have

incentives to merge. In turn, the positive external e¤ect Exx ensures that …rm

y has no incentive to block the xx-merger. Hence, …rm y refrains from pur-

suing the pro-competitive xy-merger in order to induce the anti-competitive

xx-merger that bene…ts him even more. This lack of incentive for merging

constitutes an example of the inducement mechanism. Note also that …rm

y’s pre-merger value is high re‡ecting that …rm y, in the IMExx, becomes an

outsider with certainty. As a result, the x-…rms bid on each other rather than

on …rm y, since each x-…rm is cheaper to buy. This incentive for buying other

…rms than a potential outsider to an anti-competitive merger constitutes an

example of the valuation mechanism.13If instead the IMExx;yx is selected when it exists, the pro-competitive xy-merger

occurs with positive probability. Nevertheless, the anti-competitive xx-merger occurs inthis case with higher probability, namely with the lower bound probability of 2/3. Ofcourse, this lower bound is precisely 2/3, since the …rms are exogenously selected as bidderswith equal probabilities.

15

Unlike the positive external e¤ect Exx, the negative one Exy plays no

role in sustaining the IMExx. Nevertheless, it has an important role in

ruling out equilibria in which the pro-competitive xy-merger occurs with

certainty. For example, suppose the IMExy;yx is an equilibrium so that the

xy-merger occurs with certainty. In such an equilibrium, each x-…rm becomes

an outsider with positive probability. In turn, each x-…rm’s pre-merger value

is low (since Exy < 0), re‡ecting the risk of becoming an outsider. As a

result, each x-…rm is better o¤ buying the other x-…rm rather than …rm y

which contradicts that the IMExy;yx constitutes an equilibrium. This out

of equilibrium incentive for buying a potential outsider to a pro-competitive

merger constitutes a second example of the valuation mechanism.

Finally, note that the above discussion illustrates the crucial role of an

endogenous split of the surplus for the process of merger formation, and more

generally for the process of forming coalitions. In particular, it contrasts

with Bloch’s (1995, 1996) model of coalition formation where the split of

the surplus within a coalition is assumed to be exogenous. Proposition 1

shows that such an assumption may be troublesome. Indeed, one could

easily construct an exogenous split of the surplus in each merger such that

there exists an equilibrium in which the pro-competitive xy-merger occurs

with certainty.

Next, consider the south-east quadrant in Figure 1.

Proposition 2 Consider pro…t con…gurations such that the xy-merger is

anti-competitive [Exy > 0] while the xx-merger is pro-competitive [Exx < 0]

and assume that Ixy > Ixx2 > 0. If external e¤ects are large in absolute terms

relative to internal e¤ects [jExxj ; jExyj > max fIxx; Ixyg], the DMExy;yx is

unique so that the anti-competitive xy-merger occurs with certainty in the

long run.

16

Once more, the negative external e¤ect, in this case Exx, rules out equilibria

such as the IMExx where the pro-competitive xx-merger occurs with cer-

tainty. In such an equilibrium, …rm y becomes an outsider with certainty

which hurts him, since Exx < 0. In turn, …rm y has an incentive to block the

xx-merger by buying one of the x-…rms, which contradicts the assumption

that the IMExx constitutes an equilibrium. Hence, …rm y pursues the anti-

competitive xy-merger in order to preempt the pro-competitive xx-merger

that would hurt him. This out of equilibrium incentive constitutes an exam-

ple of the preemption mechanism. Note also that the valuation mechanism

plays a role in ruling out the IMExx. Indeed, …rm y’s pre-merger value is

low in such an equilibrium, since it becomes an outsider with certainty. As

a result, each x-…rm is better o¤ buying …rm y rather than the other x-

…rm which again contradicts the assumption that the IMExx constitutes an

equilibrium.

At this point, the distinction between the preemption and the valuation

mechanisms should be clari…ed. In the above out of equilibrium example,

…rm y’s motive for merging is to preempt the pro-competitive xx-merger,

that is …rm y’s decision is driven by the preemption mechanism. However,

to preempt this merger, …rm y cannot choose between di¤erent types of

mergers, since …rm y only can participate in the anti-competitive xy-merger.

In contrast, the x-…rms can choose between the two di¤erent types of mergers.

Moreover, these …rms pursue the anti-competitive merger precisely because

…rm y is cheaper to buy, that is their decision is driven by the valuation

mechanism.

While the preemption and valuation mechanism hinder the pro-competitive

merger to occur, the inducement mechanism delays rather than favors the

anti-competitive xy-merger. Indeed, the DMExy;yx is unique in the south-

17

east quadrant of Figure 1. In such an equilibrium, the x-…rms gain from

merging, since Ixy > 0. However, these …rms are even better o¤ as outsiders,

since Exy > Ixy. As a result, the x-…rms delay their merger proposals, and

consequently forego valuable pro…ts, since they hope other …rms will merge

instead - much like a war of attrition. Moreover, the larger the positive exter-

nal e¤ect Exy, the larger the incentives to become an outsider. As a result,

the expected delay (until the anti-competitive xy-merger occurs) increases

with the positive external e¤ect Exy. Hence, the inducement mechanism cre-

ates a holdup problem for the …rms in the sense that a pro…table merger does

not occur immediately (see also Fridolfsson and Stennek, 2000a).14

Finally, it remains to discuss the condition Ixy > Ixx2 > 0. In cases

where Ixx2 > Ixy > 0 is ful…lled (and external e¤ects are large in absolute

terms relative to internal e¤ects), the DMExx;xy;yx is unique if Exy > 0 and

Exx < 0. In such an equilibrium the xy-merger occurs with strictly positive

probability in the long run. However, it also turns out to be impossible to

establish a lower bound probability for the anti-competitive xy-merger to

occur that is strictly larger than 0 for all pairs of external e¤ects. That is the

reason why Proposition 2 is stated in terms of the condition Ixy > Ixx2 > 0

only.14The reason for the ambiguous impact of positive external e¤ects is simple. In the

north-west quadrant of Figure 1 (Proposition 1), only …rm y gains by becoming an outsiderwhile the x-…rms loose as outsiders. As a result, there is no con‡ict of interests betweenthe …rms regarding which merger should form. In contrast, in the south-east quadrant ofFigure 1 (Proposition 2), both x-…rms are better o¤ as outsiders than as insiders due tothe positive external e¤ect Exy. As a result, a con‡ict of interests appears between …rmsx1 and x2 regarding which xy-merger should form. In turn, each x-…rm, by delaying itsmerger proposal, tries to induce the merger in which it does not participate.

18

3.2 Least versus Most Pro-competitive Mergers

Up to this point, I have only considered cases where one type of merger is anti-

competitive while the other is pro-competitive. The reason is that twofold.

First, when both types of mergers are anti-competitive (the positive quadrant

in Figure 1) there are multiple equilibria. In fact, both the xx- and the xy-

merger may occur with certainty depending on which equilibrium is selected.

Intuitively, there is a con‡ict of interests between all …rms regarding which

merger should form and this con‡ict materializes into multiple equilibria.

Unfortunately, I am not aware of any method of equilibrium selection that

can be applied straightforwardly to the present problem. Therefore, I abstain

from making any prediction in this region. Second, it is not straightforward

to identify which type of merger is the most anti- or pro-competitive when

external e¤ects have the same sign. However, the following assumption is

motivated in many contexts.

Assumption 2 If both types of mergers are anti- [pro-] competitive, then the

type of merger with the largest positive [negative] external e¤ect is the most

anti- [pro-] competitive.

Assumption 2 has weaker theoretical support than Assumption 1. Neverthe-

less, it is easy to construct examples by means of simple oligopoly models

that validate this assumption. For example, consider an homogeneous good

Cournot oligopoly where …rms have constant marginal costs, where demand

is linear and where both types of mergers are pro-competitive due to large

marginal cost savings. Then the merger inducing the largest marginal cost

savings is the most pro-competitive, that is bene…ts the consumers the most.

Moreover, this merger has the largest negative external e¤ect, as long as the

19

outsider is not driven out of the market.15

Proposition 3 Consider pro-competitive mergers [Exx; Exy < 0] and as-

sume that mergers are pro…table [Ixx; Ixy > 0]. If Assumption 2 holds and if

external e¤ects are large in absolute terms relative to internal e¤ects [jExxj ; jExyj >max fIxx; Ixyg], the IMExx;yx or the IMExx;xy;yx is unique and the least pro-

competitive merger occurs with a lower bound probability approximately equal

to 0,16.

Proposition 3 focuses on the negative quadrant in Figure 1. In this area,

the preemption mechanism provides all the …rms with incentives to pursue a

merger in order to avoid becoming an outsider. It is not mainly this mech-

anism, however, that induces the …rms to pursue the least pro-competitive

merger. Indeed, …rm y has an incentive to pursue the xy-merger irrespec-

tive of whether it is the most or the least pro-competitive one. Rather it is

the valuation mechanism. To see this, consider the behavior of the x-…rms

who can choose between the two di¤erent types of mergers. Since …rms’ pre-

merger values incorporate the risk of becoming outsiders, these …rms tend to

buy the …rm that is potential outsider to the most pro-competitive merger

with the largest negative external e¤ect. Thereby they preempt this merger

and induce the least pro-competitive merger.

To be more precise, consider the simplest case, namely the area in Figure

1 where the IMExx;yx is unique. In this area, the x-…rms lose more as

outsiders than …rm y does (since Exy < Exx < 0) and therefore their pre-

merger value is low. In turn, the x-…rms bid on each other with certainty and15One may construct examples where Assumption 2 does not hold if the negative external

e¤ect of one type of merger drives the outsider out of the market. To see this, note thatthe triopoly pro…ts of the …rm becoming an outsider constitutes an upper bound on thenegative external e¤ect. Hence, if these pro…ts are small, then the negative external e¤ectmust also be small, even though the merger induced marginal cost savings may be large.

20

therefore the xx-merger, that is the least pro-competitive merger, occurs with

high probability, namely 2/3. In the area where the IMExx;xy;yx is unique,

each x-…rm bids on both other …rms with positive probabilities. However,

they bid with highest probability on the …rm that is a potential outsider

to the most pro-competitive merger (if Ixx = Ixy). In fact, the x-…rms bid

on …rm y almost with certainty as Exy ! 0. As a result, the preemption

and the valuation mechanisms complement each other so that the least pro-

competitive merger, in this case the xy-merger, occurs almost with certainty.

4 Large Internal E¤ects

The present section discusses brie‡y cases where internal e¤ects may be large

relative to external e¤ects, keeping the assumption that mergers are prof-

itable. In connection to the previous analysis, a natural question is whether

pro…tability considerations reinforce the tendency for …rms to pursue anti-

rather than pro-competitive mergers. The answer to this question is am-

biguous. This is not surprising, however; for assume that one merger is

pro-competitive while the other is not, that the former merger is su¢ciently

pro…table; then the …rms tend to pursue the pro-competitive merger. Con-

versely, if the anti-competitive merger is su¢ciently pro…table, the …rms tend

to pursue the anti-competitive merger. While recognizing that …rms, in many

markets, pursue the most desirable merger, the paper proceeds by identifying

further instances in which the opposite is true.

To test the robustness of Propositions 1, 2 and 3 with respect to large

internal e¤ects, consider pro…t con…gurations such that Ixy > Ixx2 > 0. If

external e¤ects are equal to 0, the IMExx;yx or the IMExx;xy;yx is unique.

Hence, although some merger occurs with certainty, no speci…c type of merger

21

Figure 2: Ixy > Ixx2 > 0.

occurs with certainty.16 In this sense, the condition Ixy > Ixx2 > 0 implies that

pro…tability considerations do not favor too much one type of merger over

the other. In turn, if external e¤ects di¤er from 0 and from each other, their

impacts on …rms’ merger decisions are similar to the ones discussed in Section

3.

To be more precise, consider Figure 2 that illustrates in the (Exy; Exx)-

plane the conditions under which the di¤erent equilibria exist when Ixy >Ixx2 > 0. The solid lines represent the horizontal and vertical axes and an

equilibrium area is delimited by the dashed lines.17

First, consider the north-west and south-east quadrants of Figure 2 where

one type of merger is anti-competitive, while the other is pro-competitive.

Since the IMExx;yx or the IMExx;xy;yx is unique in these areas if the positive

external e¤ect is su¢ciently small relative to the internal e¤ect, the anti-

competitive type of merger occurs with strictly positive probability in such16Interestingly, this observation implies that the …rms may fail to pursue the most

pro…table merger even in the absence of external e¤ects.17Note that Figure 1 is obtained by letting the internal e¤ects tend to 0 in Figure 2.

22

cases. Furthermore, there are pro…t con…gurations in the south-east quadrant

of Figure 2 where the IMExy;yx is unique. Unlike the case when internal

e¤ects are small, the anti-competitive xy-merger may thus occur, not only

with certainty, but also immediately.18 In these cases, large internal e¤ects

thus strengthen Propositions 1 and 2.

Next, consider the negative quadrant in Figure 2 where both types of

mergers are pro-competitive. Since the IMExx;yx or the IMExx;xy;yx is unique

in the negative quadrant of Figure 2 also when external e¤ects are small in

absolute terms, both types of mergers occur with strictly positive probability

in this quadrant. In this sense, also Proposition 3 is robust to large internal

e¤ects. Finally, note that equilibria are unique in the positive quadrant

of Figure 2 if at least one external e¤ect is su¢ciently large relative to the

internal e¤ects. Therefore, one may conclude that also more anti-competitive

mergers in some cases preempt less anti-competitive ones (in particular, if

Assumption 2 holds).

The conclusion of this discussion is thus that external e¤ects being large

relative to internal e¤ects is a su¢cient, but not a necessary condition, for

the results in the previous section to hold.

5 The Preemptive Merger Hypothesis

The present section relaxes the assumption that all mergers are pro…table. In

particular I will focus attention on cases where the xy-merger is unpro…table

(Ixy < 0) while the xx-merger is pro…table (Ixx > 0). Clearly, this assumption

favors the xx-merger. Nevertheless, I will show that the signs and magnitudes

of the external e¤ects may be crucial in order to determine which merger will18Also, the expected delay associated with the DMExy;yx decreases as Ixy increases.

23

Figure 3: Ixx > 0 and Ixx2 > Ixy.

occur.

To be more precise, consider Figure 3 that illustrates in the (Exy; Exx)-

plane the conditions under which the di¤erent equilibria exist when Ixx > 0

and Ixx2 > Ixy (actually, Figure 3 is drawn such that Ixx > 0 and Ixy < 0).

The solid lines represent the horizontal and vertical axes and an equilibrium

area is delimited by the dashed lines.

Not surprisingly, the xx-merger occurs with certainty for many pro…t

con…gurations. More interestingly, however, note that the xx-merger does

not occur with certainty if Exx < Ixy ¡ Ixx2 . Hence:

Proposition 4 Assume that the xx-merger is pro…table and pro-competitive

[Ixx > 0 and Exx < 0] while the xy-merger is unpro…table and anti-competitive

[Ixy < 0 and Exy > 0]. Then the unpro…table and anti-competitive xy-merger

24

occurs with strictly positive probability if Exx < Ixy ¡ Ixx2 .19

The condition Exx < Ixy ¡ Ixx2 guarantees that equilibria such as the IMExx

where the pro…table xx-merger occurs with certainty, do not exist. In such

an equilibrium, the x-…rms split the surplus equally. In turn, if …rm y buys

one of the x-…rms (which contradicts that the IMExx is an equilibrium), it

must compensate the selling x-…rm for its foregone share of the surplus in the

xx-merger, that is Ixx2 . Thereby, …rm y loses Ixy ¡ Ixx2 relative to the status

quo. But such a behavior constitutes a best reply if …rm y is even worse o¤ as

an outsider, that is Exx < Ixy¡ Ixx2 . Hence, …rm y’s decision is driven by the

preemption mechanism discussed previously. Note also that the valuation

mechanism plays a role in ruling out the IMExx, since …rm y becomes an

outsider with certainty in such an equilibrium. As a result, the x-…rms are

better o¤ pursuing the unpro…table xy-merger rather than the pro…table xx-

merger, since …rm y’s pre-merger value is very low. Finally, note that …rm y

bears more than the whole cost associated with the xy-merger. Otherwise,

it could not be a best-reply for the x-…rms to bid on …rm y with positive

probability, as they do in the IMExx;xy;yx and the DMExx;xy;yx.20

The preemptive motive for unpro…table mergers has already been studied

by Fridolfsson and Stennek (1999). They show in a setting with three sym-

metric …rms that unpro…table mergers may occur in equilibrium, if being an

outsider is even more disadvantageous. The value added of the present paper

is thus to extend their analysis to asymmetric …rms. Thereby, Proposition 419It can be shown that the unpro…table xy-merger may be very likely. For instance,

the probability with which the unpro…table xy-merger occurs, tends to 1 as Exy ! Ixx2

(given that Exx < Ixy ¡ Ixx2 ). Note also that there are pro…t con…gurations such that an

unpro…table and anti-competitive xx-merger occurs with strictly positive probability eventhough the xy-merger is pro…table and pro-competitive.

20Hence, assuming an exogenous split of the surplus could once more be troublesome.In particular, if both merging …rms in the unpro…table merger were exogneously assignedto bear a share of the cost associated with the merger, then the unpro…table merger wouldnot occur.

25

shows that the preemptive motive may be so strong that unpro…table merg-

ers occur even though other mergers are pro…table. In addition, the present

analysis strengthens their results by showing that some equilibria entailing

unpro…table mergers are unique. In Fridolfsson and Stennek (1999), unprof-

itable mergers only occur when all mergers are unpro…table (due to symme-

try). In that case a NME exist as well. In contrast a NME does not exist

in Figure 3, since the xx-merger is pro…table.21

Proposition 4 deserves a few more remarks. First, restricting attention

to symmetric Markov perfect equilibria is not crucial for that result. While

asymmetric equilibria may exist for the pro…t con…gurations indicated in

Proposition 4, they must entail that the unpro…table merger occurs with

positive probability. Indeed, the condition Exx < Ixy ¡ Ixx2 re‡ects that

equilibria such that the pro…table xx-merger occurs with certainty, do not

exist even if the analysis is extended to asymmetric equilibria.

Second, one can easily generate examples by means of simple oligopoly

models such that one type of merger is anti-competitive and unpro…table

while the other is pro-competitive and pro…table. For instance, it is well

known that anti-competitive mergers often are unpro…table (Salant, Switzer

and Reynolds, 1983; Perry and Porter, 1985). Moreover, substantial average

cost savings are necessary for a merger to reduce the equilibrium price (Farrell

and Shapiro, 1990). Thereby, pro-competitive mergers tend to be pro…table.

Third, preemptive mergers may be relevant for vertical mergers which

aim at raising the rivals’ costs. A downstream …rm may buy a supplier to

foreclose other downstream …rms’ access to the input market (Ordover, Sa-

loner and Salop, 1990). Note that the reason for such a merger is closely21Note also in Figure 3 that one merger being anti-competitive while the other is pro-

competitive, is not a necessary condition for an unpro…table merger to occur with positiveprobability. Indeed, the IMExx;yx and the IMExx;xy;yx also exist (and are unique) ifExx; Exy < 0 (that is, both mergers are pro-competitive), Exx < Ixy ¡ Ixx

2 and Ixx > 0.

26

related to its negative externality on the competitor. The present analysis

suggests that downstream …rms may buy the supplier even if vertical in-

tegration is ine¢cient in itself, and even if the gains from foreclosure are

dominated by reduced internal e¢ciency. The reason is that the relevant

alternative is that a rival integrates with the supplier. Hence, by allowing for

bidding-competition, this work extends and strengthens the previous analysis

of foreclosure.

Fourth, the welfare e¤ects of mergers may be quite perverse. To see this,

note that the change in the producers’ surplus relative to the initial market

structure is given by ¢PSxx ´ Ixx+Exx and ¢PSxy ´ Ixy+Exy for the xx-

and the xy-merger respectively. The two dotted lines in Figure 3 separate

the areas where ¢PSxx > 0 (¢PSxy > 0) and ¢PSxx < 0 (¢PSxy < 0).

In particular, consider the area to the left of the vertical dotted line (so

that ¢PSxy < 0) and above the horizontal dotted line (so that ¢PSxx >

0). In this area there are pro…t con…gurations such that the xx-merger is

pro…table (Ixx > 0) and pro-competitive (Exx < 0) while the xy-merger is

unpro…table (Ixy < 0) and anti-competitive (Exy > 0). Hence, in this area,

the pro…table xx-merger increases both the producers’ and the consumers’

surpluses relative to the initial market structure. In contrast, the unpro…table

xy-merger reduces both these surpluses. Nevertheless, the xy-merger occurs

with strictly positive probability if Exx < Ixy ¡ Ixx2 .22

22In such cases the pro…table and pro-competitive xx-merger occurs with highest prob-ability. Nevertheless, the unpro…table and anti-competitive xy-merger occurs with a lowerbound probability of 1

3 .

27

6 Concluding Remarks

In a framework where mergers are mutually excluding, I …nd that …rms pur-

sue an anti-competitive merger when alternative mergers are pro-competitive.

This result is driven by three distinct mechanisms related to the signs and

magnitudes of mergers’ external e¤ects.

Some indirect evidence, such as the challenged Volvo-Scania merger dis-

cussed in the Introduction, indicate that the issue addressed in this paper, is

not merely a theoretical concern.23 Unfortunately, it is di¢cult to …nd direct

evidence, since preempted mergers, by de…nition, are not observed. However,

some further empirical investigations could be pursued. Most obviously, fur-

ther indirect evidence could be collected by investigating more systematically

whether the prohibition of horizontal mergers have triggered other mergers.

Less obviously, the event-study methodology could potentially be used to

identify whether preemption is an important motive behind mergers. If the

stock market is e¢cient, in the sense that share prices re‡ect …rms’ true

values, then share prices should re‡ect, not only the possibility of becoming

an insider, but also the risk of becoming an outsider. Expressed di¤erently,

share prices may, prior to the merger, incorporate information on di¤erent

mergers, including information related to mergers that never occur. This is

precisely the case in IMExx;yx and IMExx;xy;yx where all …rms may become

insiders and outsiders. In such equilibria, the testable prediction is that the

combined stock market value of the merging …rms increases when a merger

is announced, while the share prices of the outsider decrease at that time.23Fridolfsson and Stennek (1999) discuss further indirect evidence indicating that pre-

emption is an important motive behind many mergers. In particular, they show thatpreemption may be an explanation for the empirical puzzle that unpro…table mergersoccur, and that the combined stock market value of the merging …rms nevertheless isincreased.

28

The …nding that the relevant alternative to a merger may be another

merger rather than the original market structure, have some policy implica-

tions. Current policies mainly evaluate the impact of mergers relative to the

original market structure. Propositions 1, 2 and 4 imply that such a policy

may underestimate the bene…ts of blocking anti-competitive mergers. Propo-

sition 3 implies that even blocking pro-competitive mergers may bene…t the

consumers.

An immediate implication of these …ndings is that competition authori-

ties should try to assess the relevant alternative to a proposed merger. Un-

fortunately, the implementation of such an ambitious policy is likely to be

problematic. Indeed, the authority would have to assess, not only the con-

sequences of the proposed merger, but also the impact of mergers that have

not been proposed, both regarding their pro…tability and their impact on

competitors’ as well as on consumers’ welfare. Clearly, such a policy requires

that antitrust authorities have access to a substantial amount of information.

In particular, implementing such a policy requires more information than the

implementation of current policies.

One may even argue that assessing the consequences of proposed mergers

is less di¢cult than assessing the consequences of potential ones. For ex-

ample, the participating …rms in potential mergers (that have not been pro-

posed) may be reluctant to reveal relevant information. Fridolfsson (2000)

suggests that, in such cases, it may be optimal to delegate a welfare standard

to competition authorities which assigns a larger weight to the consumers’

than to the producers’ surplus.

References

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ticipants, Economic Letters; 60(1), July 1998, 113-19.

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No 4064/89 Merger Procedure, 2000a.

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A Appendix

A.1 Equilibrium Structure

Lemma 1 Consider the set of symmetric Markov perfect equilibria as ¢ !0. Such an equilibrium exists for all pro…t con…gurations. The following

equilibria exist.

1. A NME exists if, and only if, Ixx · 0 and Ixy · 0.

2. An IMExx exists if, and only if, Ixx ¸ 0 and Exx ¸ Ixy ¡ Ixx2 .

3. An IMExy;yx exists if, and only if, Exy · Ixy and Exy ¸ Ixx2 .

4. An IMExx;yx exists if, and only if, Exx · Ixy¡ Ixx2 + 1

5

¡Ixx2 ¡Exy

¢and

Exx ¸ Ixy ¡ Ixx2 ¡ 2

5

¡Ixx2 ¡Exy

¢.

5. An IMExx;xy;yx exists if, and only if, Exy < Ixx2 and Exx < Ixy ¡ Ixx2 ¡

25

¡ Ixx2 ¡ Exy

¢.

6. An IME such that pyx 2¡0; 12

¢and (pxx; pxy) = (1; 0) exists if, and

only if, Exx < Ixy ¡ Ixx2 + 1

5

¡Ixx2 ¡Exy

¢and Exx > Ixy ¡ Ixx

2 .

32

7. ADMExy;yx as well as DME such that (i) pyx 2¡0; 12

¢and (pxx; pxy) =

(0; 0), and (ii) pyx = 0, pxx = 0 and pxy 2 (0; 1) exist if, and only if,

Ixy ¸ Ixx2 and ªx ´ Ixy

Exy¡Ixy > 0.

8. A DMExx;xy;yx as well as DME such that (i) pyx 2¡0; 12

¢, pxx 2 (0; 1)

and pxy = 0 and (ii) pyx = 0 and (pxx; pxy) 2©(0; 1)2 : pxx + pxy 2 (0; 1)

ª

exist if, and only if, £y ´ Ixy¡ Ixx2Exx¡(Ixy¡ Ixx2 ) > 0 and £x ´

Ixx2

Exy¡ Ixx2> 0.

All other types of equilibria exist only for non-generic pro…t con…gurations.24

Proof: The following proof restricts attention to the type of equilibria that

exists generically. To analyze the equilibria that only exist for non-generic

pro…t con…gurations is time consuming, but not di¢cult.

The proof starts by rewriting the de…nitions of V buyyx , V buyxx , V buyxy , Wy (3)

and Wx (3). By equations (2) and (8), we have:

V buyyx = Wxy (2) ¡Wx (3) (13a)

V buyxx = Wxx (2) ¡Wx (3) (13b)

V buyxy = Wxy (2) ¡Wy (3) (13c)

Let ± = e¡r¢ and rearrange (5) in the following way.

(1 ¡ ±)³Wy (3) ¡ ¼y(3)

r

´= 2±

3

£pyx

¡V buyyx ¡Wy (3)

¢+ pxy

¡V sellxy ¡Wy (3)

¢

+pxx¡V outy ¡Wy (3)

¢¤:

By equations (3) and (8), we have V sellxy = Wy (3). Eliminate V sellxy . Use

equations (4) and (13a) to eliminate V outy and V buyyx .

(1 ¡ ±)³Wy (3) ¡ ¼y(3)

r

´= 2±

3 [pyx (Wxy (2) ¡Wx (3) ¡Wy (3))+pxx (Wy (2) ¡Wy (3))] :

(14)

24Non-generic parameter con…gurations are such that Ixx2 = Exy, Ixy = 0 and so on.

33

Rearrange (6) in a similar way. Use equations (3) and (8) to eliminate V sellxx

and V sellyx . Use equations (4), (13b) and (13c) to eliminate V outx , V buyxx and

V buyxy .

(1 ¡ ±)³Wx (3) ¡ ¼x(3)

r

´= ±

3 [pxy (Wxy (2) ¡Wy (3) ¡Wx (3))+ pxx (Wxx (2) ¡ 2Wx (3))

+ (pxy + pyx) (Wx (2) ¡Wx (3))] :

(15)

Next, I derive the conditions under which each type of equilibrium exists. The

proof ends by showing that an equilibrium exists for all pro…t con…gurations.

Proof of point 1: A NME is characterized by pyx = 0 and (pxx; pxy) =

(0; 0). By equation (14), we have Wy (3) = ¼y (3) =r. By equation (15), we

have Wx (3) = ¼x (3) =r.

First, consider …rm y’s equilibrium condition in (9). By equation (13a), we

have Wxy (2) ·Wy (3) +Wx (3). Eliminate Wy (3) and Wx (3) by using their

equilibrium values. Use (1) to eliminate Wxy (2). Rearrange the inequality

so as to use de…nition (11a). Then it simpli…es to Ixy · 0.

Second, consider the x-…rms’ equilibrium conditions in (10). By equations

(13b) and (13c), we have Wxx (2) · 2Wx (3) and Wxy (2) ·Wy (3) +Wx (3).Eliminate Wy (3) and Wx (3) by using their equilibrium values. Use (1) to

eliminateWxx (2) andWxy (2). Rearrange the inequalities so as to use de…ni-

tions (11b) and (11a). Then they simplify to Ixx · 0 and Ixy · 0 respectively.

Proof of point 2: An IMExx; is characterized by pyx = 0 and (pxx; pxy) =

(1; 0). Use these values to simplify equations (14) and (15). Solve for Wy (3)

in (14) and for Wx (3) in (15). Rearrange the solutions in the following way.

Wy (3) =¼y(3)r + 2±

3¡±

hWy (2) ¡ ¼y(3)

r

i= ¼y(3)

r + 2±3¡±Exx;

Wx (3) = ¼x(3)r + 2±

3¡±12

hWxx (2) ¡ 2¼x(3)r

i= ¼x(3)

r + 2±3¡±

Ixx2 :

The second equality in the solution for Wy (3) [for Wx (3)] follows from the

de…nitions in (1) and (12b) [in (1) and (11b)].

34

First, consider …rm y’s equilibrium condition in (9). By equation (13a),

we have Wxy (2) · Wy (3) +Wx (3). Eliminate Wy (3) and Wx (3) by using

their equilibrium values as ± ! 1 (¢ ! 0). Use (1) to eliminate Wxy (2).

Rearrange the inequality so as to use de…nition (11a). Then it simpli…es to

Exx ¸ Ixy ¡ Ixx2 .

Second, consider the x-…rms’ equilibrium conditions in (10). By equations

(13b) and (13c), we haveWxx (2) ¸ 2Wx (3) andWxx (2)¡Wx (3) ¸Wxy (2)¡Wy (3). EliminateWx (3) in the …rst inequality by using its equilibrium value

(where ± < 1). Use (1) to eliminate Wxx (2). Rearrange the inequality so

as to use de…nition (11b). Then it simpli…es to Ixx ¸ 0. Next, eliminate

Wy (3) and Wx (3) in the second inequality by using their equilibrium values

as ± ! 1 (¢ ! 0). Use (1) to eliminate Wxx (2) and Wxy (2). Rearrange

the inequality so as to use de…nitions (11a) and (11b). Then it simpli…es to

Exx ¸ Ixy ¡ Ixx2 .

Proof of point 3: An IMExy;yx is characterized by pyx = 12 and (pxx; pxy) =

(0; 1). Use these values to simplify equations (14) and (15). Then one gets a

system of two equations with two unknowns: Wy (3) and Wx (3). Solve this

system. Use (1) to eliminate Wxy (2) and Wx (2) in the resulting solutions.

Rearrange so as to use de…nitions (11a) and (12a).

Wy (3) =¼y(3)r + ±

6¡5± [(2 ¡ ±) Ixy ¡ ±Exy] ;Wx (3) =

¼x(3)r + ±

6¡5± [2 (1 ¡ ±) Ixy + (3 ¡ 2±)Exy] :

First, consider …rm y’s equilibrium condition in (9). By equation (13a), we

have Wxy (2) ¸Wy (3) +Wx (3). Eliminate Wy (3) and Wx (3) by using their

equilibrium values (where ± < 1). Use (1) to eliminate Wxy (2). Rearrange

the inequality so as to use de…nition (11a). Then it simpli…es to Exy ·(2 ¡ ±) Ixy. Let ± ! 1 (¢ ! 0) to get Exy · Ixy.

Second, consider the x-…rms’ equilibrium conditions in (10). By equations

35

(13b) and (13c), we have Wxy (2) ¸ Wy (3) +Wx (3) and Wxy (2)¡Wy (3) ¸Wxx (2) ¡ Wx (3). We already know that the …rst inequality simpli…es to

Exy · Ixy as ± ! 1. Eliminate Wy (3) and Wx (3) in the second inequality

by using their equilibrium values as ± ! 1 (¢ ! 0). Use (1) to eliminate

Wxy (2) and Wxx (2). Rearrange the inequality so as to use de…nitions (11a)

and (11b). Then it simpli…es to Exy ¸ Ixx2 .

Proof of point 4: An IMExx;yx is characterized by pyx = 12 and (pxx; pxy) =

(1; 0). Use these values to simplify (14) and (15). Then one gets a system of

two equations with two unknowns: Wy (3) and Wx (3). Let ± ! 1 (¢ ! 0)

in both equations. The LHS in both equations then equals to 0, sinceWy (3)

and Wx (3) are bounded. Solve the resulting system of equations. Use (1)

to eliminate Wxy (2), Wxx (2), Wx (2) and Wy (2). Rearrange so as to use

de…nitions, (11a), (11b), (12a) and (12b).

Wy (3) =¼y(3)r + 1

3

£Ixy ¡

¡45Ixx2 + 1

5Exy¢¤

+ 23Exx;

Wx (3) = ¼x(3)r + 4

5Ixx2 + 1

5Exy:

First, consider …rm y’s equilibrium condition in (9). By equation (13a), we

have Wxy (2) ¸Wy (3) +Wx (3). Eliminate Wy (3) and Wx (3) by using their

equilibrium values. Use (1) to eliminateWxy (2). Rearrange the inequality so

as to use de…nition (11a). Then it simpli…es to Exx · Ixy¡ Ixx2 + 15

¡ Ixx2 ¡ Exy

¢.

Second, consider the x-…rms’ equilibrium conditions in (10). By equations

(13b) and (13c), we haveWxx (2) ¸ 2Wx (3) andWxx (2)¡Wx (3) ¸Wxy (2)¡Wy (3). EliminateWx (3) in the …rst inequality by using its equilibrium value.

Use (1) to eliminateWxx (2). Rearrange the inequality so as to use de…nition

(11b). Then it simpli…es to Exy · Ixx2 . Next, eliminate Wy (3) and Wx (3) in

the second inequality by using their equilibrium values. Use (1) to eliminate

Wxy (2) and Wxx (2). Rearrange the inequality so as to use de…nition (11a)

and (11b). Then it simpli…es to Exx ¸ Ixy ¡ Ixx2 ¡ 2

5

¡Ixx2 ¡ Exy

¢.

36

Finally, note that Exy · Ixx2 is ful…lled if the two other conditions are

ful…lled.

Proof of point 5: An IMExx;xy;yx is characterized by pyx = 12 and (pxx; pxy) 2

©(0; 1)2 : pxx + pxy = 1

ª. Eliminate pyx in equations (14) and (15) as well

as pxy, using the fact that pxy = 1 ¡ pxx. Then one gets a system of two

equations with two unknowns: Wy (3) and Wx (3). Let ± ! 1 (¢ ! 0) in

both equations. The LHS in both equations then equals to 0, since Wy (3)

and Wx (3) are bounded. Solve the resulting system of equations. Use (1)

to eliminate Wxy (2), Wxx (2), Wx (2) and Wy (2). Rearrange so as to use

de…nitions, (11a), (11b), (12a) and (12b).

Wy (3) =¼y(3)r

+ 11+4pxx

£Ixy ¡Exy + 4pxxExx + 2

3pxx (Ixy ¡ Ixx + Exy ¡ Exx)¤;

Wx (3) =¼x(3)r + Ixx

2

¡ 11+4pxx

£Ixx2 ¡ Exy ¡ 4

3pxx (1 ¡ pxx) (Ixy ¡ Ixx + Exy ¡ Exx)¤:

First, consider the equality V buyxx = V buyxy , that is one of the x-…rms’

equilibrium conditions in (10). By equations (13b) and (13c), we have

Wxx (2) ¡Wx (3) =Wxy (2) ¡Wy (3). Eliminate Wx (3) and Wy (3) by using

their equilibrium values. Use (1) to eliminateWxx (2) andWxy (2). Rearrange

the equality so as to use de…nitions (11a) and (11b). Finally, rearrange in

the following way.

f (pxx) ´ 3 (1 ¡ 2pxx)pxx (7 ¡ 2pxx)

=Ixy ¡ Ixx + Exy ¡ Exx

Ixx2 ¡Exy

: (16)

Equation (16) de…nes pxx implicitly as a function of the exogenous variables

in the RHS of (16), since f 0 (pxx) < 0 8pxx 2 (0; 1).

Second, consider the inequality V buyxy ¸ Wx (3), that is the x-…rms’ other

equilibrium condition in (10). By equation (13c), we haveWxy (2) ¸Wx (3)+Wy (3). Eliminate Wx (3) and Wy (3) by using their equilibrium values. Use

37

(1) to eliminate Wxy (2). Rearrange so as to use de…nition (11a). The in-

equality then simpli…es to

¡6µIxx2

¡ Exy¶

· (3 + 2pxx) (Ixy ¡ Ixx + Exy ¡ Exx) :

Assume that Ixx2 > Exy and rearrange the inequality in the following way.

¡ 63 + 2pxx

· Ixy ¡ Ixx + Exy ¡ ExxIxx2 ¡ Exy

By equation (16), the RHS equals f (pxx). Simplify to get 3+10pxx¡8p2xx ¸0, which is true 8pxx 2 (0; 1). Conversely, the inequality V buyxy ¸ Wx (3)simpli…es to 3 + 10pxx ¡ 8p2xx · 0 if Ixx2 < Exy, which is not true for any

pxx 2 (0; 1). Hence, if V buyxx = V buyxy , then V buyxy ¸ Wx (3) if, and only if,

Exy < Ixx2 .

Third, note by (13a), that …rm y’s equilibrium condition in (9) is equiv-

alent to Wxy (2) ¸Wx (3) +Wy (3), that is the inequality treated above.

Finally, note that limpxx!0 f (pxx) = +1 and f 0 (pxx) < 0 8pxx 2 (0; 1).

Since pxx 2 (0; 1), equation (16) has a unique solution if, and only if,

Ixy ¡ Ixx + Exy ¡ ExxIxx2 ¡Exy

> f (1) = ¡35

Since Ixx2 > Exy, this inequality simpli…es to Exx < Ixy¡ Ixx2 ¡ 2

5

¡Ixx2 ¡ Exy

¢.

Proof of point 6: In this equilibrium, pyx 2¡0; 12

¢and (pxx; pxy) = (1; 0).

Eliminate pxx and pxy in equations (14) and (15). Let ± ! 1 (¢ ! 0) in both

equations. Since Wy (3) and Wx (3) are bounded, the two equations simplify

to

0 = pyx [Wxy (2) ¡Wy (3) ¡Wx (3)] +Wy (2) ¡Wy (3) (17)

0 = Wxx (2) ¡ 2Wx (3) + pyx [Wx (2) ¡Wx (3)] (18)

First, consider …rm y’s equilibrium condition in (9). By equation (13a), we

have Wxy (2) = Wy (3) +Wx (3). This equation and equation (17) constitute

38

a system of two equations with two unknowns: Wy (3) and Wx (3). Solve

this system. Use (1) to eliminate Wy (2) [Wxy (2)] in the solution for Wy (3)

[Wx (3)]. Rearrange so as to use de…nitions (11a) and (12b). The solutions

are then Wy (3) =¼y(3)r + Exx and Wx (3) =

¼x(3)r + Ixy ¡ Exx.

Second, consider the x-…rms’ equilibrium conditions in (10). By equations

(13b) and (13c), we haveWxx (2) ¸ 2Wx (3) andWxx (2)¡Wx (3) ¸Wxy (2)¡Wy (3). EliminateWx (3) in the …rst inequality by using its equilibrium value.

Use (1) to eliminate Wxx (2). Rearrange so as to use de…nition (11a). The

inequality then simpli…es to Exx ¸ Ixy¡ Ixx2 . Similarly, the second inequality

also simpli…es to Exx ¸ Ixy ¡ Ixx2 .

Finally, it is required that pyx 2¡0; 12

¢. To obtain an expression for

pyx, insert the equilibrium values of Wy (3) and Wx (3) into (18). Use (1)

to eliminate Wxx (2) and Wx (2). Rearrange so as to use de…nitions (11b)

and (12a). Solving for pyx yields that pyx = 2Exx¡(Ixy¡Ixx=2)Ixy¡(Exy+Exx) . Hence, it is

required that 12 > 2Exx¡(Ixy¡Ixx=2)Ixy¡(Exy+Exx) > 0. Since Exx ¸ Ixy ¡ Ixx

2 , these two

inequalities imply that Ixy > Exy + Exx and Exx > Ixy ¡ Ixx2 and Exx <

Ixy ¡ Ixx2 + 1

5

¡Ixx2 ¡Exy

¢. The two latter inequalities imply that the former

one is ful…lled.

Proof of point 7: I only prove the conditions under which a DMExy;yx

exists. Following the same steps as below, it is straightforward to prove that

the two other DME exist under the same conditions as the DMExy;yx.

A DMExy;yx is characterized by pyx 2¡0; 12

¢and pxx = 0, pxy 2 (0; 1).

Consider …rm y’s equilibrium condition in (9). By equation (13a), we

have Wxy (2) = Wy (3) +Wx (3). This equation and equation (14) constitute

a system of two equations with two unknowns: Wy (3) andWx (3). Solve this

system, using the fact that pxx = 0. The solutions are Wy (3) = ¼y(3)r and

Wx (3) = Wxy (2) ¡ ¼y(3)r = ¼x(3)

r + Ixy (the second equality follows from the

39

de…nitions (1) and (11a)).

Next, eliminate Wx (3) in the LHS in (15) by using its equilibrium value.

The LHS then equals Ixy. The RHS in (15) equals 13±

1¡± (pxy + pyx) (Wx (2) ¡Wx (3)),since pxx = 0 and Wxy (2) = Wy (3) +Wx (3). Note that Wx (2) ¡Wx (3) =Exy ¡ Ixy (which follows from the equilibrium value of Wx (3) and the de…n-

itions in (1) and (12a)). Hence:

pxy + pyx = 31 ¡ ±±

IxyExy ¡ Ixy

= 31 ¡ ±±

ªx: (19)

Since pxy+pyx > 0, it is necessary that ªx > 0. As ± ! 1 (¢ ! 0), the RHS

tends to 0 so that there exists probabilities pxy and pyx satisfying the above

equality (in fact there exists a continuum of such probabilities). Thus, the

condition ªx > 0 is also su¢cient in order to satisfy the above equality.

Finally, consider the x-…rms’ equilibrium conditions in (10). By equations

(13b) and (13c), we have Wxy (2) =Wy (3) +Wx (3) and Wxy (2)¡Wy (3) ¸Wxx (2) ¡Wx (3). We already know that the equality is ful…lled. Use (1) to

eliminateWxy (2) andWxx (2) in the inequality. EliminateWy (3) andWx (3)

by using their equilibrium values. Rearrange so as to use de…nitions (11a)

and (11b). The inequality then simpli…es to Ixy ¸ Ixx2 .

Proof of point 8: I only prove the conditions under which a DMExx;xy;yx

exists. Following the same steps as below, it is straightforward to prove that

the two other DME exist under the same conditions as the DMExx;xy;yx.

ADMExx;xy;yx is characterized by (pxx; pxy) 2©(0; 1)2 : pxx + pxy 2 (0; 1)

ª

and pyx 2¡0; 12

¢.

Consider the x-…rms’ equilibrium condition in (10). By equations (13b)

and (13c), we have Wxy (2) = Wy (3) + Wx (3) and Wxx (2) ¡ Wx (3) =

Wxy (2) ¡Wy (3). Use these equations to solve for Wy (3) and Wx (3). The

solutions are Wy (3) = Wxy (2) ¡ Wxx(2)2 = ¼y(3)

r + Ixy ¡ Ixx2 and Wx (3) =

40

Wxx(2)2 = ¼x(3)

r + Ixx2 (the second equality in the …rst [second] solution follows

from the de…nitions in (1), (11a) and (11b) [(1) and (11b)]).

Next, eliminate Wy (3) in the LHS in (14) by using its equilibrium value.

The LHS then equals Ixy¡ Ixx2 . The RHS in (14) equals 23±

1¡±pxx (Wy (2) ¡Wy (3)),since Wxy (2) = Wy (3) + Wx (3). Note that Wy (2) ¡ Wy (3) = Exx ¡¡Ixy ¡ Ixx

2

¢(which follows from the equilibrium value of Wy (3) and the def-

initions in (1) and (12b)). Hence:

pxx =321 ¡ ±±

Ixy ¡ Ixx2

Exx ¡¡Ixy ¡ Ixx

2

¢ = 31 ¡ ±±

£y:

Since pxx > 0, it is necessary that £y > 0. As ± ! 1 (¢ ! 0), the RHS tends

to 0 so that there exists a probability pxx satisfying the above equality. Thus,

the condition £y > 0 is also su¢cient in order to satisfy the above equality.

Similarly, by simplifying equation (15), it is straightforward to show that

there exists probabilities pxy and pyx satisfying equation (15) if, and only if,

£x =Ixx=2

Exy¡Ixx=2 > 0.

Finally, consider …rm y’s equilibrium condition in (9). By equation (13a),

we have Wxy (2) = Wy (3) +Wx (3), which we know is ful…lled.

Existence: To complete the proof, it remains to show that at least one type

of equilibrium exists for all pro…t con…gurations.

Consider the case in Figure 3, that is pro…t con…gurations such that

Ixx > 0 and Ixx2 > Ixy. Next, I show that there exists an equilibrium for all

pairs (Exx; Exy) given that the above conditions on Ixx and Ixy are ful…lled.

First, assume that Exx > Ixy¡ Ixx2 . By point 2, there exists an IMExx, since

Ixx > 0. Second, assume that Exx < Ixy ¡ Ixx2 and Exy > Ixx

2 . By point 8,

there exists a DMExx;xy;yx, since Ixy ¡ Ixx2 < 0 and Ixx > 0. Third, assume

that Exx < Ixy ¡ Ixx2 and Exy < Ixx

2 . By point 5, there exists an IMExx;xy;yx

if Exx < Ixy¡ Ixx2 ¡ 2

5

¡Ixx2 ¡Exy

¢. If instead Exx > Ixy¡ Ixx

2 ¡ 25

¡ Ixx2 ¡ Exy

¢,

there exists an IMExx;yx. Indeed, this latter inequality constitutes one of the

41

two conditions for an IMExx;yx to exist (see point 4). Moreover, the second

condition for such an equilibrium to exist is Exx < Ixy ¡ Ixx2 + 1

5

¡ Ixx2 ¡ Exy

¢.

This condition is ful…lled if Exx < Ixy ¡ Ixx2 and Exy < Ixx

2 .

To check existence for all possible pro…t con…gurations, repeat similar

arguments for the following three cases: (i) Ixy > Ixx2 > 0 (that is, the case

in Figure 2), (ii) Ixy > 0 and Ixx < 0 and (iii) Ixx; Ixy < 0. QED.

A.2 Proofs of Propositions:

All proofs below build upon the equilibrium structure derived in Lemma 1.

A.2.1 Proof of Proposition 1:

By Lemma 1, Figures 2 and 3 illustrate the conditions under which each

equilibrium exists when Ixy ¸ Ixx2 ¸ 0 and Ixx

2 ¸ Ixy ¸ 0 respectively. First,

assume that Ixy ¸ Ixx2 ¸ 0. Then Exx > Ixy ¡ Ixx

2 , since Ixy; Ixx > 0, Exx > 0

and jExxj > Ixy. Moreover, Exy < 0 < Ixx2 . By Figure 2, the IMExx is then

unique or exists simultaneously with the IMExx;yx. Consequently, the xx-

merger occurs with probability 1 (in the IMExx) or 2/3 (in the IMExx;yx).

Second, assume that Ixx2 ¸ Ixy ¸ 0. Then Exx > Ixy ¡ Ixx2 , since Exx > 0 ¸

Ixy ¡ Ixx2 . The same conclusion as in the …rst case follows from Figure 3.

QED.

A.2.2 Proof of Proposition 2:

By Lemma 1, Figure 2 illustrates the conditions under which each equilibrium

exists when Ixy ¸ Ixx2 ¸ 0. Since Exx < 0 and Ixy > Ixx

2 , we have that

Exx < Ixy ¡ Ixx2 . Moreover, Exy > Ixy, since Exy > 0 and jExyj > Ixy. By

Figure 2, the DMExy;yx is then unique.

42

It remains to show that the xy-merger occurs with probability 1 in the

long run. Note that there are t=¢ time periods between time 0 and time t. In

aDMExy;yx, the triopoly remains until time t with probability¡1 ¡ 2

3 (pyx + pxy)¢t=¢ =³

1 ¡ 21¡e¡r¢e¡r¢ ªx

´t=¢where the second inequality follows from (19) and the

fact that ± = e¡r¢. Let q0 (¢) ´ 1 ¡ 21¡e¡r¢e¡r¢ ªx and de…ne the cumula-

tive distribution function indicating the probability that a merger has not

occurred before time t, as

G0 (t) = lim¢!0

[q0 (¢)]t=¢ :

Since the logarithm is continuous

lnG0 (t) = t lim¢!0

ln q0 (¢)¢

:

Note that lim¢!0 q0 (¢) = 1. Hence, lim¢!0ln q0(¢)

¢ = "00". By l’Hopital’s

rule: lim¢!0ln q0(¢)

¢ = lim¢!0q00(¢)q0(¢) = lim¢!0 q00 (¢). Hence:

lnG0 (t) = t lim¢!0q00 (¢) = ¡2rtªx:

Thus, G0 (t) = e¡2rtªx. De…ne the probability that a xy-merger has occurred

at time t as G (t) ´ 1 ¡ e¡2rtªx. limt!1G (t) = 1 for all ªx > 0. QED.

A.2.3 Proof of Proposition 3:

By Lemma 1, Figures 2 and 3 illustrate the conditions under which each

equilibrium exists when Ixy ¸ Ixx2 ¸ 0 and Ixx

2 ¸ Ixy ¸ 0 respectively. First,

assume that Ixy ¸ Ixx2 ¸ 0. By Figure 2, the IMExx;yx or the IMExx;xy;yx

is unique if Exy; Exx < 0. Second assume that Ixx2 ¸ Ixy ¸ 0. Then Exx <

Ixy ¡ Ixx2 , since Ixy; Ixx > 0, Exx < 0 and jExxj > Ixx. By Figure 3, the

IMExx;yx or the IMExx;xy;yx is then unique if Exy; Exx < 0.

It remains to prove the lower bound probability. Consider …rst the simple

case when the IMExx;yx is unique. In such an equilibrium, the xy- (xx-)

43

merger occurs with probability 1/3 (2/3). Hence, the least pro-competitive

merger must occur with a lower bound probability of 1/3.

Next, consider the more di¢cult case when the IMExx;xy;yx is unique. In

such an equilibrium, the xy-merger occurs with a lower bound probability

of 1/3 while the xx-merger may occur with an arbitrarily small probability.

It thus remains to determine the lower bound probability for the xx-merger

to occur in the cases when it is the least pro-competitive, that is when 0 >

Exx > Exy (by Assumption 2). This amounts to …nd a lower bound for

pxx which is de…ned implicitly by equation (16). Since f 0 (pxx) < 0 and

limpxx!0 f (pxx) = 0, the lower bound of pxx is found by solving the following

maximization problem.

maxfIxx;Ixy;Exx;Exyg

Ixy ¡ Ixx + Exy ¡ ExxIxx2 ¡ Exy

;

subject to Exy ¡ Exx · 0, Ixx; Ixy ¸ 0 and Exy; Exx · ¡max fIxx; Ixyg.

By noting that the existence of an IMExx;xy;yx requires that Ixx2 > Exy, it

is straightforward to solve the above maximization problem. The solutions

are given by f(Ixx; Ixy; Exx; Exy) : Ixx = 0 and Exx; Exy = ¡Ixyg. The maxi-

mized expression equals 1 at its maximum. Replacing the RHS in (16) by 1

and solving the resulting equation yields that pxx = 134 ¡ 1

4

p145. Moreover,

the xx-merger occurs with probability 23pxx. By continuity, it follows that the

xx-merger occurs with a lower bound probability of 23

¡134 ¡ 1

4

p145

¢' 0:16.

QED.

A.2.4 Proof of Proposition 4:

By Lemma 1, Figure 3 illustrates the conditions under which each type of

equilibrium exists when Ixx > 0 and Ixy < 0. First, consider pro…t con-

…gurations such that Exx < Ixy ¡ Ixx2 and Exy < Ixx

2 . By Figure 3, either

44

the IMExx;yx or the IMExx;xy;yx is then unique. Hence, the xy-merger then

occurs immediately with strictly positive probability. Second, consider pro…t

con…gurations such that Exx < Ixy ¡ Ixx2 and Exy > Ixx

2 . By Figure 3, the

DMExx;xy;yx is then unique. In the proof of Proposition 2, it was shown that

in a DMExy;yx, the probability that a xy-merger has occurred at time t is

1 ¡ e¡2rtªx. Similarly, it can be shown that in a DMExx;xy;yx, the proba-

bility that some merger has occurred at time t is 1 ¡ e¡rt(2£x+£y). Since,

limt!1 1 ¡ e¡rt(2£x+£y) = 1, it follows that some merger occurs with proba-

bility 1 in the long run. Moreover, it is easy to show that, conditional on the

event that some merger has occurred, the probability that a xy-merger has

occurred is 2£x2£x+£y

> 0. QED.

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