Stable Mergers and Cartels Involving AsymmetricFirms

Margarida Catalão Lopes¤

April 1999

Abstract

The endogenous formation of coalitions involving asymmetric …rms andtheir stability are analyzed as a function of di¤erences in e¢ciency and ofthe …xed cost of production. Results are derived for cartels as well as formergers. Players have constant but di¤erent marginal costs of productionand no rule of pro…t sharing is …xed. The analysis is illustrated for a speci…cpath of collusion. Finally welfare e¤ects are studied and some conclusionsare drawn for antitrust policy.

¤mcatalao@bportugal.pt. This paper was included as the …rst chapter of my PhD dissertationat UNL and was written while I was visiting Universitat Autònoma de Barcelona. I thank myadvisor, Luís Cabral, and also Pedro Pita Barros, Francis Bloch, José García Díaz, Kai-UweKühn, Inés Macho, Xavier Martinez-Giralt, Xavier Vives and participants at the 5th Workshopon Inter…rm Relationships, the XXI Simposio de Análisis Económico, the 2a Conferência daSPiE, the 24th Annual EARIE Conference, and the XIII Jornadas de Economia Industrial forhelpful conversations, comments and suggestions. Any remaining errors are my own. Financialsupport from the Sub-Programa Ciência e Tecnologia do 2 o Quadro Comunitário de Apoio isgratefully acknowledged.

1. Introduction

The subject of coalition formation has received in recent years a growing attentionin economic literature, accompanying the increasing importance of merging asa competitive strategy in oligopolistic industries. After a period in which theevolution of the industry has been analyzed mostly in terms of entry and exit,following major developments in noncooperative game theory, attention seemsnow to be turning to mergers and acquisitions as another source of industryevolution, giving rise to a new interest in cooperative game theory.1 Still, asthis new literature places great emphasis on the “noncooperative foundations forcooperative behavior”, links between both …elds of Game Theory are now betterunderstood than before.

A signi…cant part of merger operations occurs for …nancial reasons (speculativemotives); however, strategic motives such as increasing e¢ciency or enhancingmonopoly power are also at the origin of many mergers.

Collusive behavior may take several forms: …rst it may be explicit or implicit.The implicit form is generally known as tacit collusion and requires a dynamiccontext to be implementable. The explicit form of collusive behavior - mergers,cartels and joint ventures - can be more or less complete. Mergers are the mostcomplete type of collusive agreement: the separate entities of the constituent…rms disappear and start acting as a single unit thereafter. At the other extremelie joint ventures, which are designed for …rms to bene…t from scale economieswhile remaining separate entities.

In this paper we deal with horizontal mergers (i.e., mergers among …rms inthe same product market)2 and cartels, and analyze the role played by e¢ciencydi¤erences in their motivation and stability. This issue is still far from beingcompletely and satisfactorily explored in the literature, due to the di¢culty inhandling asymmetry. In fact, symmetry between players is usually assumed andwhen heterogeneity is allowed, a …xed rule of payo¤ division is postulated (e.g.Farrell and Scotchmer(1988) and Bloch(1996)). We do not …x any rule of pro…tsharing, rather leaving it for the bargaining process between joining …rms, whichis assumed to be costless.

We consider that an agreement is reached whenever it is pro…table for the …rms1The problem of the choice between internal growth (entry by building) and external growth

(entry by buying) has been addressed for example by Gilbert and Newbery(1992).2Mergers are often categorized as horizontal, vertical, or conglomerate. Horizontal mergers

take place between …rms in the same line of business; in a vertical merger the buyer expandsbackward or forward; conglomerate mergers involve companies in unrelated lines of business. Seefor instance Scherer and Ross(1990) and Brealey and Myers(1988) for some historical perspectiveon these various types of mergers and for some examples.

2

involved. By imposing the condition that joint pro…ts should be greater than orequal to the sum of the ex ante pro…ts of the …rms involved, we guarantee that no…rm loses when it decides to join with another; we assure that, in the absence ofbargaining costs, there can be some division of pro…ts between them which makesit worthwhile for both to join, without …xing an a priori rule of sharing.3 We aretherefore allowing for side payments, otherwise the heterogeneity of …rms wouldmake it more di¢cult to …nd admissible agreements (i.e., individually rationalagreements) and to sustain them.4

In turn, for some arrangement to be stable this kind of incentive to mergefurther (broaden the coalition) must not exist (external stability). These twoconditions - pro…tability and external stability - together with a division of thecoalitional pro…t are su¢cient to determine equilibrium mergers. For equilibriumcartels a condition for internal stability (that no …rm bene…ts from leaving theagreement) must also be set. Our stability analysis proceeds stepwise, from agiven partition to an “adjacent” partition (that is, with one player less or oneplayer more); hence the aggregation movements we consider involve just two…rms or groups of …rms and the disaggregation movements give rise to just oneadditional player.

The treatments which are closest to ours, in the sense of also dealing withheterogeneity of the agents, are Farrell and Shapiro(1990a), who consider quitegeneral cost functions but do not model the merging process, Faulí-Oller(1995),who restricts asymmetry to the existence of two di¤erent types of players, thee¢cient and the ine¢cient, and Ray and Vohra(1996), who study the endogenousformation of coalitions in the symmetric case as well as in the general case ofasymmetry. We model asymmetry by introducing a parameter ® which standsfor the (constant) marginal cost di¤erence between …rms. The analysis on theincentives to form coalitions and their stability is then performed as a functionof ® and of a parameter f representing the …xed cost of operating a …rm.

Following the positive analysis we study the consequences of the …rms’ agree-ments on social welfare and derive some conclusions about the social desirabilityof di¤erent concentration movements. It is shown that for small e¢ciency di¤er-ences no merger is good in terms of total welfare in the absence of signi…cant costsavings, whereas for large ® the optimal merger is welfare-enhancing even if thereare no savings in …xed cost (f = 0), and, as expected, involves the absorptionof a very ine¢cient …rm. The relationship between changes in market concentra-

3Many ways of modeling the process of the formation of coalitions have been proposed in theliterature. Ours is the simplest: the coalition emerges whenever it is pro…table.

4The side payments which we allow for are within coalitions. Transfers between coalitionsare not allowed (so a …rst best cannot be achieved).

3

tion and changes in economic welfare is also analyzed and conclusions are drawnfor approval rules. An antitrust policy based on concentration measures may bemisleading, but the best movement in terms of social welfare and the one whichproduces the smallest increase in concentration are the same in most of the cases.

The analysis of this paper can be extrapolated to other social situations inwhich individuals di¤ering by an observable characteristic (and having a …xedcost of performing some activity) may want to form coalitions. Indeed, thesubject of coalition formation has already been applied to di¤erent situationssuch as trading structures in bilateral oligopolies (Bloch and Ghosal(1997)), theformation of associations including R&D joint ventures or the adoption of acommon standard (Bloch(1995)), the formation of syndicates (Guesnerie(1977)and Greenberg(1979)), the possibility of mergers between domestic and foreign…rms (Horn and Persson(1997b)), trade agreements (Macho et al(1994)), customsunions (Yi(1996)), international agreements for the protection of the environment(Carraro and Siniscalco(1993)) and voting games (Peleg(1984)), among others.

The paper is organized as follows. Section 2 presents the game-theoreticsolution concepts usually employed for analyzing the formation and stability ofcoalitions and also provides a revision of the main topics usually addressed in theIndustrial Organization literature on mergers and cartels. Section 3 presents theanalytical framework and an illustration for the case of three …rms. Section 4generalizes for the case of n …rms and establishes the main results of the paper.Section 5 contains the welfare analysis and section 6 concludes. Formal proofsand details are given in the Appendix.

2. The GT tools and the IO literature

In this section we brie‡y present the game-theoretic tools usually employed tostudy the subject of coalition formation and go through the main literature onmergers and cartels, from an Industrial Organization perspective.

2.1. The GT tools

The analysis of coalition formation within the framework of cooperative GameTheory makes use of cooperative solution concepts such as the core (for example,Aumann(1967) and an application to trade agreements in Macho et al(1994)),the von Neumann and Morgenstern stable sets (Espinosa and Iñarra(1995)) orthe Shapley value (Hart and Kurz(1983) use a variant of the Shapley value, the“coalition structure value”).

This cooperative framework, however, ignores externalities among coalitionsand is not suitable to describe the formation of coalitions as a noncooperative

4

process. It seems reasonable to believe that there are spillovers (externalities,either positive or negative) to the other players when some decide to join. Hence,in recent years many authors have resorted to noncooperative solution conceptsto determine the equilibrium number, size and composition of coalitions,5 es-pecially considering games in extensive form. Examples of this approach areBloch(1996), Kamien and Zang(1990, 1991 and 1993), Nilssen and Sorgard(1998),Yi and Shin(1995) and Faulí-Oller(1995).

Extensive-form games capture situations in which players announce sequen-tially their desire to take part in a coalition (sequential games of coalition for-mation) and are especially appropriate to capture forward-looking behavior byagents. These models with “farsighted” players, in which decisions are based onthe …nal outcome, that is, considering all the chain reactions to the present moveuntil the end, are one of the …elds that currently deserve more interest (see forinstance Ray and Vohra(1996) and Espinosa(1996), this latter as an applicationof the largest consistent set notion of Chwe(1994)).6

Within the framework of strategic-form games, the most important solutionconcepts are extensions of the Nash equilibrium; these are appropriate to gamesin which all players simultaneously announce their decision to cooperate (simul-taneous games of coalition formation).

In many situations in real life it is not possible to sign binding agreementsand therefore agreements must rely on their self-enforceability. The Nash solutionis a necessary condition for self-enforceability, but is not su¢cient: in fact, it isusually possible for players to make some arrangements (form coalitions) whichare mutually bene…cial, provided that all the others do not change their strate-gies (take the actions of the complement as …xed). The simple Nash solutionexcludes this situation, but the concepts of strong Nash equilibrium (SNE) andcoalition-proof Nash equilibrium (CPNE) contemplate it. These re…nements tothe Nash solution limit the set of equilibria (usually very large) and thus allowmore accurate prediction. Since they are of a cooperative nature we can saythat simultaneous games are actually on the border line between cooperative andnoncooperative game theory.7

5This distinction between cooperative and noncooperative approaches corresponds to thedistinction between the two types of representation of games, in coalitional (or characteristic)function form and in partition function form.

6The forward-looking behavior induces a higher degree of cooperation because …rms tend toresist incentives to cheat, anticipating the consequences.

7Yi and Shin(1995) separate simultaneous games of coalition formation into two categories:open membership games, where players are free to join or leave any coalition, and exclusivemembership games, where the members of the coalition are allowed to deny membership tooutsider …rms.

5

According to the notion of SNE (Aumann(1959)) an equilibrium agreementmust be immune to deviations by every conceivable coalition, while according tothe notion of CPNE (Bernheim et al(1987) and Bernheim and Whinston(1987))an equilibrium agreement must be immune to deviations by every conceivablecoalition which are also immune to further deviations by members of the deviat-ing group (consistency requirement; notice that it embeds some forward-lookingbehavior, though players see only one step ahead).8 That is, some deviationswhich would invalidate a SNE do not invalidate a CPNE, because they are notvalid deviations and therefore a CPNE has more chance to “survive” deviatingmovements.9 In this sense the SNE is a stronger concept and more di¢cult to…nd.

More recently there have appeared cooperative tools that take externalitiesinto account. The equilibrium binding agreements of Ray and Vohra(1997) arethe “parallel” of CPNE, since they rule out coalitional deviations that are notthemselves immune to further deviations by subcoalitions.

2.2. The IO literature

According to the assumptions made and to the solution concept employed, resultsin the literature have ranged from the impossibility of merging up to monopoly(v.g., Kamien and Zang(1990 and 1991), provided that the industry has su¢-ciently numerous …rms) to models which predict the possible complete monop-olization of the industry in the absence of regulatory devices, as for example inSalant et al(1983) (henceforth sometimes referred to simply as SSR).

In SSR’s paper it is shown that in a Cournot setting exogenous mergers may beunpro…table for the …rms involved, therefore creating few incentives to join: thereason is that the new …rm tends to reduce quantity (because competition is madeless aggressive by the merger), causing an increase in the quantity produced by

8The fact that only deviations by members of the deviating group are considered in theconcept of CPNE is still a limitation. The study of abstract stable sets in which the dominancerelation allows for the possibility of a subset T of a deviating coalition S attracting a coalitionQ ½ N jS (where N is the universe of players) in order to jointly further deviate has given rise tosome equilibrium concepts, depending on the information structure. If previous agreements arecommon knowledge and under the assumption that all other players will stick to the announcedtuple of strategies, the appropriate solution is the one resulting from “coalitional contingentthreats” (Greenberg(1990), pgs.102-6): if some coalition declares that its members will adopt adi¤erent strategy pro…le any other coalition may respond in the same way to the revised proposaland the process continues until an equilibrium is found.

9The concept of CPNE can also be applied to extensive form games, giving rise to perfectcoalition-proof Nash equilibria (e.g., Matutes and Padilla(1994) for an application of perfectCPNE in pure strategies to the formation of shared ATM networks).

6

outsiders. In the absence of …xed costs and under the assumptions of symmetryand homogeneity of the product this is the sole e¤ect; the joint pro…t is thussmaller than the sum of the pre-merger pro…ts of the …rms involved and so there isno incentive to merge. Mergers must then be motivated by monopoly power aloneand the authors show that with linear demand it is necessary that at least 80%of the …rms in the industry merge to pro…tably exploit market power (minimalpro…table coalition size). Some regulatory device that prevents more than 4

5of the industry from colluding therefore makes full monopolization impossible.Cheung(1992) reduces this minimal market share to 50% by allowing for anydemand satisfying second-order conditions.

The result on the private unpro…tability of mergers can be reversed for exam-ple if the savings in …xed cost implied by the operation are su¢ciently important.When …rms are not equally e¢cient, as in our paper, there is an additional e¤ectthat may cause this merger to be pro…table even in the absence of …xed costs,the switch of production from the ine¢cient …rm to the e¢cient one. A similaridea is present in the criticism of Perry and Porter(1985) to SSR: the new costfunction must be di¤erent from the initial one, since as a result of merging, the…rm now has access to a technology which may be strictly more e¢cient than theone before; the price increase can be su¢cient to compensate for the decline inproduction, therefore making the merger pro…table.

With Bertrand competition mergers are always bene…cial (because the reac-tion of outsiders reinforces the price increase due to merger) and the more …rmsinvolved, the more pro…table they are (Deneckere and Davidson(1985)). Otherattempts to reverse the results by SSR on the private unpro…tability of merginginclude relaxing the assumption of homogeneity of the product. The existenceof product di¤erentiation reinforces the results obtained under price competitionand allows mergers to be privately pro…table even under quantity competition aslong as the degree of di¤erentiation is su¢ciently high (Granero(1997)).

The abandonment of the assumption that the newly created …rm remains aCournot player, by allowing it to behave as a Stackelberg leader if its size islarge enough,10 also reverts SSR’s results (again Granero(1997) and also Daugh-ety(1990)). Faulí-Oller(1996), in turn, has considered a more general demandfunction than that employed by SSR and has studied the e¤ects of its degree ofconcavity on the pro…tability of the agreement; Gaudet and Salant(1991) havealso reached the possibility of pro…table mergers under general demand and costfunctions (though equal for all …rms). In the current paper the symmetry as-

10See Sadanant and Sadanant(1996) for the grounds of approximating “the behavior of adominant …rm with a …nite fringe (...) by Stackelberg equilibrium”. Sha¤er(1995) also considersCournot behavior within the fringe and leader behavior by the cartel.

7

sumption is relaxed and a reversion of SSR’s results is also obtained: mergers canbe privately pro…table even with no …xed cost savings and under all the otherassumptions of SSR, provided …rms are su¢ciently asymmetric.

The incentive to free-ride (let the others merge and stay out of those groups),also known as the “hold-up” problem, subsists in all these variations, as outsidersearn more than …rms participating in the merger.11 The recognition of the free-riding problem was …rst made by Stigler(1950), who stated that the promoter of amerger could receive every encouragement from the other …rms but participation.Szidarovszki and Yakowitz(1982) show that if some …rms form a cooperative grouptheir joint pro…t can fall below that of the non-cooperative situation, whereas thepro…t of the non-cooperating …rms does not decrease. The same kind of idea ispresent in the price-leadership model of d’Aspremont et al(1983), in which fringe…rms enjoy higher pro…ts than members of the dominant cartel.12 These are clearexamples of positive spillovers in games of coalition formation (coalition membersprovide some type of public good, which external …rms enjoy without supportingits cost). The incentive to free-ride is at the origin of the potential instability ofcoalitions.

Since the work of Hart and Kurz(1983), and bearing in mind the unpro…tabil-ity characteristic of some exogenous mergers, as detected by Salant et al(1983),several papers have tried to model the process of coalition formation in an en-dogenous way, trying to predict which mergers will occur in a given situation.With a purpose similar to ours but assuming homogeneity of …rms, Rajan(1989)derives the endogenous formation of coalitions as a function of some relevant mar-ket characteristics. Kamien and Zang(1990 and 1991) use an allocation schemebased on the bids each …rm makes for all the others and on the asking price each…rm announces for itself to determine which acquisitions will take place. Rayand Vohra(1996) suggest a bargaining process in which players are farsighted andtherefore care about the end results of their moves to select which proposals willbe made and accepted. This is actually the line of research that is currentlyreceiving more attention.

The desirability of mergers in terms of welfare is ambiguous, due to thetrade-o¤ between e¢ciency gains and reduced competition (there are gains ac-

11 In our variation (asymmetry) it is not clear whether this incentive subsists or not, becausethe payo¤ for each member of the agreement is unknown (remember that we do not …x any ruleof division) and thus cannot be compared with the payo¤ of the outside …rms.

12This model is the simplest example of an open membership game. It makes use of the notionsof internal and external stability, which we also employ in the present paper. The incentive tofree-ride is not present in d’Aspremont et al(1983)’s model if the fall in price due to increasedcompetition as one …rm defects gives rise to a fall in pro…t that o¤sets the advantage of joiningthe competitive fringe (Donsimoni et al(1986)).

8

cruing to increased productive e¢ciency, and losses due to allocative ine¢ciency- Williamson(1968)). Horizontal mergers have undesirable e¤ects on price-costmargins, both through the concentration index and through the conjectural vari-ation (refer to the Lerner index of Cowling and Waterson(1976)). However theyalso have bene…cial e¤ects on cost, as the e¢cient …rms increase their marketshares at the expense of the ine¢cient ones (unless the X-ine¢ciency of Leiben-stein(1966) dominates). So, total industry pro…t and per …rm pro…t tend toincrease.

In the seminal work of Szidarovszki and Yakowitz(1982) it is shown that,under certain assumptions for the unit price function and for the (di¤erent) costfunctions of the …rms, cooperative grouping implies a decrease in productionlevels. This is true in most models of merging and is at the origin of the negativee¤ect implied on consumers’ welfare. Concerning total welfare SSR prove thatunpro…table exogenous mergers may still be socially desirable, due to e¢ciencygains resulting from savings in …xed cost. On the other hand when mergersare modeled endogenously (Kamien and Zang(1990), v.g.) socially bene…cialmergers that are unpro…table for the …rms involved will not take place and thesocially undesirable mergers that are privately bad will also fail to occur in theequilibrium, and therefore need not be of concern.

It is generally believed that horizontal mergers, by reducing the number of…rms, facilitate tacit collusion (see for example Osborne(1976), who considersthe bene…ts of a smaller number of …rms for the internal stability of a cartel interms of making it easier to share the pro…ts and to detect and deter cheating):in this sense they would hurt welfare. This is the impact on the Lerner indexvia the conjectured response of the rivals. Davidson and Deneckere(1984), how-ever, pointed out the fact that mergers increase the pro…ts earned at the threatpoint (the equilibrium to which the game reverts if cheating is detected), therebyhindering tacit collusion and potentially raising welfare.

As regards the relationship between concentration and welfare, Farrell andShapiro(1990a) show that an increase in the Her…ndahl index may not necessar-ily be welfare reducing and provide su¢cient conditions for pro…table mergersto raise welfare (analyzing the external e¤ects of the merger on the rival …rmsand on consumers); these conditions, however, may no longer be su¢cient whenmergers are interdependent, in the sense that the realization of some merger to-day may in‡uence the occurrence of mergers tomorrow, as shown by Nilssen andSorgard(1998). A positive relationship between welfare e¤ects and level of concen-tration of the non-participating …rms was found by McAfee and Williams(1992),who also proved that mergers which increase the dimension of the largest …rm orcreate a new largest …rm reduce welfare.

9

A common feature of most coalition formation models developed up to nowis the lack of heterogeneity of …rms. When di¤erences among …rms are allowedwhen modeling the merging process, a …xed rule of payo¤ division is assumed(Farrell and Scotchmer(1988) and Bloch(1996)), which highly conditions the re-sults obtained. In this paper we tr s o is o s )r peoI hssuh

allocation of outputs across facilities). Every …rm observes the merging activityamong the others and knows exactly how e¢cient its rivals are.

For the sake of simplicity we consider that there is no entry in nor exit fromthis industry, so the number of players is …xed apart from changes due to joiningor separating movements. In an industry with n …rms there are 2n ¡ 1 possiblecoalitions. The number of partitions is obviously smaller and the number of k-…rm oligopolies even smaller (1 · k · n), though all of them are of course verylarge. The formulas for the total number of partitions and for the total number ofoligopolies with k …rms can be found in the Appendix. Notice that in a symmetricindustry all k-…rm partitions yield the same level of concentration, in other words,it is indi¤erent which …rms join; however if …rms are not equally e¢cient thereare di¤erent levels of concentration associated with the various k-…rm oligopolies.

3.1.2. Payo¤s

The payo¤s are the pro…ts each …rm receives in every given market structure,after competing in quantities. The good is homogeneous, demand is linear andthere is no uncertainty about demand conditions. Firms that merge or form acartel remain Cournot players after the operation. For …rms belonging to a group,payo¤s are not clearly de…ned, but depend on the way the members agree to sharethe common pro…t, which we do not …x.

3.1.3. Strategies and equilibrium

Given that the rest of the market is unchanged, coalitions C1 6= ; and C2 6= ;will join and form C = C1 [ C2 if and only if

¼C ¡ f ¸ ¼C1 ¡ f + ¼C2 ¡ f

where, abusing notation, ¼C denotes the variable pro…t of coalition C given theexisting partition of the rest of the market (pro…tability condition).14

The new coalition C 6= ; is externally stable if and only if there exists nopro…table way of broadening it (given that the rest of the market is unchanged):for all C 0 ½ C; C0 6= ;

¼C[C0 ¡ f · ¼C ¡ f + ¼C0 ¡ f

gain to the low-cost …rm from the merger is the elimination of the high-cost …rm as a Cournotrival”.

14All notation employed shall be understood in this context: given the existing partition ofthe rest of the players.

11

where C denotes the complement of C in the universe of players.It is internally stable if and only if there exists no subcoalition of it which

receives a higher payo¤ by acting alone than by staying in (given that the rest ofthe market is unchanged): for all C00 ½ C; C00 6= ;

¼C00 jCnC00 ¡ f ·X

i2C00(¼(i)jC ¡ f)

CnC00 denotes the complement of C 00 in C and the use of ¼C00 jCnC00 is an abuse ofnotation that reinforces that coalition C 00 is competing against CnC00. Indeed, itis already clear that every coalitional pro…t is de…ned given the partition of therest of the market: we are stressing that coalition C 00 was taken out of coalitionC. This abuse of notation is employed in some other expressions throughoutthe paper. In the right hand side of the inequality the pro…t of every i 2 C 00 isdenoted by ¼(i)jC , meaning that coalition C is still “complete”, and where the useof parenthesis is intended to capture the notion that …rm i’s pro…t is unknownas long as the …rm belongs to a coalition, since we do not …x any rule of payo¤division.

It is assumed that joining and separating are costless movements.15 For thepurpose of our analysis we will consider broadening movements to include justone player more and also separating movements of just one player (C0 and C00 aresingletons). This corresponds to most of the joining and separating movementsactually observed.

Trivially the grand coalition is externally stable, whereas the totally dispersedmarket structure is internally stable.

We consider two di¤erent types of agreements: cartels, in which a number ofindependent …rms join to make price or output decisions (in our case, output)and that su¤ers from each participant having an incentive to break away and jointhe competitive fringe, and mergers, the ultimate form of collusion in which …rmslose their separate entities. We assume that the main di¤erence between cartelsand mergers is that in the …rst case …rms are free to exit the agreement, whereasin the second case they are committed to cooperation (mergers are irreversible,once an agreement is reached it cannot be broken).16 For the stability analysis

15 In particular, we consider that the buyer does not pay any premium for the selling …rm overits value as a separate entity (represented either by its market value, or by its market value plusthe valuation expected by investors following the acquisition).

16Both problems become equivalent if we allow for divestitures or for the divisions of the …rmthat came out of the merger to spin o¤ into several …rms and become independent entities at anytime (for examples of spino¤s and explanations of motives see, for instance, Habib et al (1997)),

12

of pro…table cartels we thus have to check both internal and external stability,but for the stability analysis of pro…table mergers we have to check only externalstability.

Our stability analysis is stepwise: we restrict attention to one concentrationmovement at a time (two …rms or groups of …rms joining) and we also analyzeonly one movement of exit from a group at a time. This means, for example, thatthe simultaneous exit of two …rms, each on its own, must be analyzed …rst as theexit of one and then, in the resulting market structure, as the exit of the other one,otherwise we would be analyzing deviations by more than one coalition of playersat the same time. As a consequence, we can only proceed from an oligopoly withk operating …rms to an “adjacent” oligopoly, with k ¡ 1 or k + 1 operating …rms(corresponding to external instability and internal instability of the coalition(s)that constitute the k-…rm oligopoly, respectively).

The move from an oligopoly with k operating …rms to another oligopoly withk operating …rms cannot be studied within our framework. As stated before (seesection 2.1), a solution concept based on “coalitional contingent threats” wouldbe required.

An equilibrium is de…ned by a stable partition, according to the de…nitionsgiven above, and by the corresponding payo¤ division within its coalition(s).

By setting f = 0 the results obtained can be identi…ed with an industry withno …xed cost of production, in which case the main parameter driving coalitions(apart from the search for market power) is ®; on the other hand, if ® is set tozero, then the savings in …xed cost stand as the main determinant of the joiningprocess (idem), as in the work of Espinosa and Iñarra(1995). When ® is strictlygreater than zero our analysis is an extension of the previous literature on thepro…tability and stability of coalitions to the case in which …rms are not equallye¢cient.

3.2. The case of n = 3

Consider an industry with 3 …rms. Firm 1 (the most e¢cient) has a constantmarginal cost of production of c¡ ®, …rm 2 of c and …rm 3 (the least e¢cient) ofc + ®. There is a …xed cost equal to f . Demand is linear: P = a ¡ Q where Q istotal production of the homogeneous good by the industry, a > c > ®.

With n = 3 there are three possible oligopolistic structures: monopoly (thegrand coalition, represented either by M or by {1,2,3}), duopoly and triopoly

or, alternatively, if the cartel has perfect enforcement. The occurence of divestitures is oftenmotivated by the threat of further mergers: selling the main object of the takeover bid mayinduce the proposed acquirer to drop its bid.

13

(represented either by T or by {1}{2}{3}). There are three possible duopolyconstellations: …rms 1 and 2 together against …rm 3 ({1,2}{3}), …rms 1 and 3together against …rm 2 ({1,3}{2}) or …rms 2 and 3 together competing against…rm 1 ({1}{2,3}).

All the stability analysis is extensively described in the Appendix. We re-port here only the …nal results (we consider both the cartel situation and theirreversible merger situation)

Proposition 3.1. In a market with three …rmsa) the completely disperse structure prevails for f small enough. The value of fallowed for the stability of triopoly increases with ³, the di¤erence between theexogenous demand and the marginal cost of the median …rm, and decreases withthe e¢ciency gap ®:b) the most internally stable duopoly is {1,3}{2} and the least internally stableis {1,2}{3}. Duopoly is never externally stable, so it is never stable.c1) if we allow …rms to leave the coalition when they …nd it pro…table (cartels),then the grand c Tc (a) Tj 5.16 0 TD 0.0509 Tc (v) Tj 5.4 0 TD 0.0182 Tc (e) Tj 8.76 0 TD 0.0254 Tc (t) (}) Tj 5.4 0 TD ({) Tj 5.52 0 TD (3) Tj 5.4 0 .Tj 5.52 0 TD 0.y6 Tc (h) -0.0072 Tc (6Tc (e) Tj 50.02549 Tc (y) Tj 9.6 0 TD - 492.24 Tc (v) Tj 5.4 0 TD 0.0182 Tc (e) Tj 8.:) Tj 5.4 0 TD 0.0182 Tc (e) Tj 8.:e 1 2 0 T D 0 T c ( o ) T j 5 . 4 0 T D - 0 . 0 0 7 2 T D 0 . 0 2 5 4 T c ( t ) T j 4 . 2 0 T D - 0 . 0 0 7 2 T c 6 Tj 3.12 20072 96 0.0182 Tc1.2 0 0 Tc (15 Tj 3.12 2 Tc (a) Tj 5.16 0 TD 0.0509 Tc (v) Tj 5.4 0 TD 0.0182 T 0.0509 Tc (,) Tj -380.52 -13.56 Tc (n) Tj 6.12 0 TD (d) Tj 9-0.0072 Tc (h) Tj 6.19 Tc (,) Tj 4.2 0 TD -0.0072 Tc (5.4 0 TD 0.0182 Tc (e) Tj 8.:) Tj 5.4 0 TD) Tj 9.72 0 T.52 0 TD (3) Tj 5.4 0u3s d v{ we a

,298 Tc (s) T 4.8 0 TD -0.0072 Tc (n) Tj 9.72 0 TD 90 TD2 -13.56 Tc (n) Tj 6.12 0 TD ( Tc (e) Tj 8.:) Tj 5.4 0 TD82 Tc (e12 0 TD 0.0254 Tc2 492.24 30.96 0.36 re S BT 490.08 5 0.0254 Tc (t) Tj 4.32 0 TD TD -0.0072 Tc (n) Tj 6.12 0 TD ( 0 TD -0.0298 Tc (s) Tj 8.16 0 TD 640254 Tc (t) Tj 4.32 0 TD Tj 5.16 0 TD -0.0509 Tc (w) Tj 11.76 0 TD -0.0079 Tc (l) Tj 3 0 TD 0.0182 Tc (e) 4.8 0 TD -0. 0.0437 Tc (m) Tj 9 0 TD -05 0 TD -0.0072 Tc (5.4 0 T Tc (e) Tj 8.76 0 .Tc (h) Tj 6.12 0 TD 8.76 0 Ta) Tj 5.52 0 TD 0.0509 Tc (l) Tj 3 0 TD (i) Tj 3 0 TD 0.0258 Tc (g) Tj 5.4 0 TD v TD -0. 0.0437 T0 TD ({) Tj 5.52 0 TDTj .76 0 TD -0.0079 Tc ( (3) Tj 5.4 0u) Tj 5380.52 -13.56 TD 0.0254 Tc (t) Tj 4.2 0 TD .72 0 TD 0.0182 Tc (c) Tj 4fa) Tj 5 0 0 Tc ( Tj 3.12 20072 96 0.0182 Tc1.23TD 0 Tc (15 Tj 3.12 2 0.0182 Tc (e) Tj 8. 0 TD -0.0072 Tc (5.4 0 TD 0.0182 Tj .76 0 TDq0.08 5 0.0254 T (3) Tj 5.4 0u) Tj 5.4 0 TD56 Tc (n) Tj 6.12 0 TD ( Tc (e2 Tc (e) Tj 8.76 0 TD 0 G 459.12 492.24 30.96 0.36 re S BT 4t) Tj 4.8D 0.0182 Tc (c) Tj 4fa) Tj354 Tj 4.2 0 TD .72 0 TD 0.0182 Tc (2 Tc (e) Tj 8j 3. TD - 3 0 TD -0.0298 Tc 8.76 0 TD 0.0254 Tc (t) (}) Tj 5.4 0 TD ({) Tj 5D 0.0182 Tc (c) Tj 4b) Tj -380.52 -13.56 Tc (n) Tj 6.12 0 TD ( -0.0077 Tc (r) Tj 4.30 TD (d) Tj 9.96 0 TD 0.0530 TD2 -13.56 Tc (n) Ty) Tj 4.48 Tc (r) Tj 4.30 TD (d) 3 0 TD -0.0298 T8j 3. TD -.56 Tc (n) Tj 6.12 0 TD ( Tc (e) Tj 8.:) Tj 5.4 0 TD82 Tc (e12 0 TD 0.0254 Tc 0.0182 Tc (e) Tj 8. 0 TD -0.0072 Tc (5.4 0 TD 0.0182 T 4.8 0 TD -0.0072 Tc (n) TD -0.0298 TD TD -0.0072 Tc (n) Tj 6.12 0 TD (D 0.0509 Tc 0 TD 0.0254 Tc .) Tj 9g72 0 90 TD (d) Tj (a) Tj 5.16 0 TD 0.0509 Tc (c) Tj 4.8 0 TD 0 Tc ( ( Tj 3.12 20072355 Tc (w638 0 TD4 Tj 4.2 (15 Tj 3.12 2 0 TD ({) Tj 5D 0.0182 Tc (c) Tj 40 TD 0.0254 Tc (t) Tj -3770.0298 Tc (s) (t) Tj 5.4 0 TD82 Tc (e12 0 TD 0 Tc (o) (h) Tj 6.12 0 TD Tc (e2 Tc (e) Tj 8.76 0 TD 0 G 459.12 492.24 30.96 0.T 4.8 0 TD -0.0072 Tc (n) TD -0.0298 TD TD -0.0072 Tc (n) Tj 6.12 TD 0.0509 Tc (c) Tj 4.8 0 5.4 0 TD) Tj 9g72 0 90 TD (d) Tj (a) Tj 5.16 0 0.0509 Tc (c) Tj 4.8 0 TD 0 Tc ( ( Tj 3.12 20 G44 Tc (w56 TD 0.9 Tj 4.2 (15 Tj 3.12 2 Tj 8.76 0 .Tc (h TD 03.96j 4.2 (0 8j 3.12 2 Tj6 0.011) Tj 8. 0 TD -(7) Tj -7c 8 -20.4j 4.2 (15 Tj 3.12 2 Tj072 Tc (h) Tj 6.12 0 TD 0.01822{) Tj 5.52 0 TDTj 9.96 0 T)) Tj 4.48 Tc (rTj (a) Tj 5.16 0 0.0509 Tc (c) Tj 4f0.02549 Tc (y) Tj 9.6 0 TD - 492.2455.4 0 TD82 Tc (e12 0 TD1.2 0 0 Tc36 re S BT 4t) Tj 0.0254 Tc .) Tj 9.72 0 1.26.52 0 TD 0.0509 Tc 0 TD 0.4 0 TD) Tj 9.72 0 T.52 0 TDTj 9.96 0 TD 0.054.48 Tc (rT 4.8 0 TD -0.0072 TcTj .76 0 TD -0.0079 Tc (D -0.007.0254 Tc .) Tj 9.72 0 5 Tj 6.12 0 TD 8.76 0 - 492.8 0 0 Tc36 re S BT 4(w) Tj 7.56 0 TD 0.0182 Tc (e) Tj 8.76 0 TD 0 Tc (a) Tj 5.4 090 TD (d) 3 0 TD -0.0298 T4.48 Tc (rTj t) Tj 4.32 0 TD TD -0.007) Tj 9.72 0 1.26.52 0 T 11.76 0 TD -0.0079 Tc (l) Tj 3 0 TD 0.0182 Tc (e) 4.8 0 TD -0. 0.0437 Tc (m) Tj 9 0 TD -05 0 TD -0.0072 Tc (5.4 0 1.2 0 0 TcD 0.0254 Tc (t) (}) Tj 5.4Tj 5.4 0 TD 0.0182 Tc (e) .0072 Tc (5.4 0 1.2 Tc (o) (h) Tj 6.12 0 TD 0 TD .72 0 TD 0.0182 ({) Tj 5.52 0 TDTj .76 0 TD -0.0079 Tc (5.16 0 0.0509 c (t) Tj 4.32 0 TD TD -0.0072 Tc (n) Tj 6.12 0 TD ( 0 TD .72 0 TD 0.0182 Tc (c) Tj 40 TD Tj20 0 TcD 0.0254 Tc() Tj 8.76 0 TD 0 Tc (a) Tj 5.4 090 TD (d) 0 G 459.12 492.24 30.96 0.36 re 2 Tc (e) Tj 8.76 0 TD 0) Tj 9g72 0 -0.0298 Tc (s) Tj 8.16 0 T4 30.96 0.36 re 2 Tc (e) Tj 8.76 0 TD 3 0 TD -0.0298 TET 373.76 43 0 30.9 T Tc re f 373.6 43 0 3 5 0 D4 re S BT 410.4j430.6TD -D Tj 9.96 0 T)) Tj 8. 0 TD -0.0 Tc (n) T, 0 TD 640254 Tc (t) Tj 4.32 0 TD 3) Tj 5.4Tj 5.4 0 TD 0.0182 Tc (e) .0072 Tc (5.4 0 0.0254 Tc 0.018c) Tj 40 TD Tj20 0 TcD 0.0254 Tc (t) (}) Tj 5.4Tj 5.4 0 TD 0.0182 0 TD (d) 0 G 459.12 492.21.2 0 0 TcD) Tj 9g72 0 -0.0298 T36 re 2 Tc (e) Tj 8.76 0 TD 0) Tj 9({) Tj 5D 0.0182 Tc (c) Tj 40 TD 0.0254 Tc (t) Tj -3770.0298 Tc (s) Tj072 Tc (h) Tj 6.12 0 TD 0.0182.72 0 T.52 0 TD0 TD -0.0072 TcTj .76 0 TD -0.0079 Tc (D.16 0 0.0509 c (t) Tj 4.32 0 TD TD -0.0072 Tc (n) Tj 6.12 0 TD ( 0 TD .72 0 TD 0.0182 Tc (c) Tj 40 TD 0TD 0.0182 Tj .76 0 TD Tj 4.30 TD (d) 3 0 TD -0.0298 T90 TD (d) 0298 Tc 8.76 0 TD 0.0254 Tc (t) (}) Tj 5.4 0 TD ({) Tj 55) Tj 5.4Tj 5.4 0 TD b) Tj 5.4 0 TD56 Tc (n) T -0.0079 Tc (l) Tj 3 0 TD 0.01TD 0 Tc ( T 4.8 0 TD -0.0072 Tc (n) TD -0.0298 T9 Tc 4 0 TD56 Tc (n) T -0.0079 Tc (l0 TD .72 0 TD 0.0182 Tc (c) Tj 40 TD 0.0254 Tc .) Tj 9g72 0 0T352 0 TD0 TD -0.0072 Tc 3 0 TD -0.0298 T90 TD (d) ( Tj 3.12 20 G44 Tc (w56 TD Tj88TD (d) (15 Tj 3.12 2 0 TD ({) Tj 55) Tj 5.4Tj 5.4 0 TD 0 TD 0.4 0 T(t) Tj 0TD 0.0182 ( Tj 3.12 20072096 0.01f TD Tj4TD (d) (15 Tj 3.12 2 0 TD ({) Tj 5.0298 T36 re 2 Tc (e) Tj 8.76 0 TD 0) Tj 3 0 TD 0.01TD 0 Tc ( Tj 5.4 0 TD 0.0182 Tc (e) .0 Tc (n) Tj 6.12 0 TD ( 0 TD g{) Tj 55) Tj 5.4Tj 5.4 0 TD h TD 0TD 0.0182 Tj s) Tj 8.16 0 T4 30.96 0.36 re c) Tj 40 TD 0.0254 Tc .) Tj 9.72 0 TD 0.0182 Tc (c) Tj 4u TD 0.0254 Tc .) Tj 9g72 0 TD 0.0182 Tc (c) Tj 4h TD 0TD 0.0182 (n) TD -0.0298 TD TD -0.007) Tj 9.72 0 0T352 0 TDD 0.0254 Tc (t) (}) Tj 5.4Tj 5.4 0 TD 0.0182 Tc (e) 0 TD ({) Tj 55) Tj 5.4Tj 9.96 0 TD 0.054. TD (d) 0 (t) (}) Tj 5.4Tj 5.42 Tc (e) Tj 8.76 0 TD 0) Tc (n) Tj 6.12 0 TD ( 0 TD .72 0 TD 0.0182 Tc (c) Tj 4p TD 0.30.4 0 TD) Tj 9.72 0 T.52 0 TDTj .76 0 TD -0.0079 Tc (Dy72 0 0T55.4 0 TD Tj 4.30 TD (d) 3 0 TD -0.0298 T90 TD (d) D 0.0509 Tc 0 TD 0.4 0 TD) Tj 9.72 0 T.52 0 TDTj 9.96 0 TD 0.05-379.76 98 440182 Tj s) Tj 8.16 0 T4 30.96 0.Tj .76 0 TDx72 0 T762 0 TDD 0.0254 Tc (t) (}) Tj 5.4 .0072 Tc (5.4 0 0.0254 Tc 0.0182 Tc (e) Tj 8. TD (d) D 0.0509 Tc 0 TD 0.0254 Tc .) Tj 9 TD -0.0072 TcTj .76 0 TD -0.007. TD (d) 0 -0.0079 Tc (Dy72 0 9.30.4 0 T (n) TD -0.0298 TD TD -0.007 0.0254 Tc (t) (}) Tj 5.4 0 TD ({) Tj 55) Tj 5.4Tj 5.4 0 TD b) Tj 5.4 0 TD56 Tc (n) T -0.0079 Tc (l) Tj 3 0 TD 0.014TD 0.0182 Tj .76 0 TD. 0.05-61TD 0-24.4 0 ) (0 Tj 3.12 20072.76 0 TDT 0.057 30.96 0.36 re c) Tj 40.0182 0 TD (d) 0 G 459.12 492.270.0254 Tc 0.018c) Tj 4f-0.007.30.4 0 TD) Tj 9.72 0 T0072 TcTj .76 0 TD -0.007. TD (d) 0 -0.0079 Tc (D) Tj 9.72 0 TTj 6.12 0 TD 8.76 0 - 492.7 30.96 0.Tj .76 0 TD Tj 4.30 TD (d) 3 00509 Tc 0 TD 0.4 0 TD) Tj 9g 0 TD 5) Tj 5.4Tj 5.4 0 TD (w) Tj 7.5.0254 Tc .) Tj 9g72 0 TD 0.0182 Tc (c) Tj 4u TD 0.0254 Tc 0.0182 Tc (e) Tj 8. TD (d) .0072 Tc (5.4 0 0.0254 Tc 0.0 TD -0.0298 T7.76 0 TD 0) Tc (n) Tj 6.12 0 TD (D -0.007. TD (d) 0 -0.0079 Tc ( Tc (c) Tj 4u TD 0.4 0 T (n) TD -0.0298 TD TD -0.007 0.0254 Tc (t) (}3254 Tc 0.0182 Tc (e) Tj 8. TD (d) 0 TD ({) Tj 55) Tj 5.4Tj 9.96 0 TD 0.058. TD (d) .0072 Tc (5.4 0 7TD 0.0182 Tj .0254 Tc (t) (}) Tj 5.4Tj 5.4 0 TD 0.0182 0 TD (d) 0 G 459.12 492.270.0254 Tc 0.018c) Tj 4pTj 7.5.0254 Tc .) Tj 9({) Tj 5.0298 T36 re 2 Tc (e) Tj 8.76 0 TD 0) 9.96 0 TD 0.058. TD (d) .0 Tc (n) Tj 6.12 0 TD ( (t) Tj 4.32 0 TD TD -0.0072 Tc (n) Tj 6.12 0 TD ( 0 TD .72 0 TD 0.0182 Tc (c) Tj 40 TD 9.0254 Tc .) Tj 9.72 0 T5) Tj 5.4Tj 5.4 0 TD f TD 0.30.4 0 TD (t) Tj 4.32 0 TD 35.4 0 7TD 0.0182 (7 Tj 3.12 20 9.96 0 T( 0 TD 3) Tj 5.4 ( Tj 3.12 20 G44 Tc (w56 TD 6.95.4 0 TD56 Tc (n) T; 0 T4 30.96 0.36 re96 0.01f TD 6.60.0182 (7 Tj 3.12 20 9.96 0 T)98 T7.76 0 TD (0 Tj 3.12 20072 TD -0.0298 TD TD -0.00 0.018c) Tj 4pTj 7.5TD (d) 0 TD ({) Tj 55) Tj 5.4Tj 072 Tc (h) Tj 6.12 0 TD2 492.270.0254 Tc 0.018c) Tj 4f-0.007.30.4 0 TD) Tj 9.72 0 T0072 Tc36 re 2 Tc (e) Tj 7 440Tj 5.4Tj 072 Tc (h) Tj 6.12 0 TD.ed Tc TD -0.0298 T7.440Tj 5.4Ttem T D 9 . 0 2 5 4 T c . j s e5.4 0 0.0254 Tc 0.0182 Tc (e) Tj 8. TD (d) D 0. TD -0.0298 TD TD -0.007 0.76 0 TD,98 T7.56254 Tc 0.0182 Tc (e) Tj 8. TD (d) .0072 Tc (5.4 0 0.0254 Tc 0.0 TD -0.0298 T(}) Tj 5.4Tj 5.4 0 TD p TD 0.30.4 0 TD.0072 Tc (5.4 0 0.0254 Tc(h) Tj 6.12 0 TD. (t) Tj 4.32 0 TD TD -0.0072 Tc (n) Tj 6.12 0 TD ((v72 0 TD 0.0182 .0072 Tc (5.4 0 0.0254 TcTj .76 0 TD -0.0079 Tc (Dy72 0 0T012 0 TD. (t) Tj 4.3( 0 TD TD (d) 0 (t) (}) Tj 5.4Tj 5.4 0 TD 0.0182 0 TD (d) 0 G 459.12 492.29.0254 Tc .j .76 0 TD -0.0079 Tc (Dj 6.12 0 TD (D 0.0509 Tc 0 TD 0.0254 Tc . G 459.12 492.29.0254 Tc ( Tj 3.12 236 re96 0.01f TD 6.412 0 TD. (t69 Tc (m TD 8 96 0 TD (0 Tj 3.12 272 Tc (n) Tj 6.12 0 TD (D 0.0509 Tc 0 TD 0T352 0 TD (w) Tj 7.5.0254 Tc .) Tj 9g72 0 TD 0.0182 Tc (c) Tj 4u TD 0.0254 Tc 0.0182 Tc (e) Tj 8. TD (d) .0072 Tc (5.4 0 9.0254 Tc .) Tj 93{) Tj 55) Tj 5.4Tj 432 Tc (--0.007.0.4 0 TD) Tj 91.4 0 9.7254 Tc 0.0182 Tc (e) Tj 8.3254 Tc .j s) Tj 8.16 0 T4 30.96 0.36 re c) Tj 4pTj 7.5.0254 Tc 0.0182 Tc (e) Tj 8. TD (d) .0072 Tc (5.4 0 0.0254 Tc 0.0 TD -0.0298 T(}3254 Tc .j s) Tj 8.16 0 T4 30.96 0.36 re c) Tj 4n{) Tj 5762 0 TDD 0.0254 Tc (t) (}) Tj 5.4 0.0 TD -0.0298 T8.642 0 TDD 0.0254 Tc (t) (}) Tj 5.4 0.0.4 0 TD 0.0182 0 TD (d) 0 G 459.12 492.29.0254 Tc (h) Tj 6.12 0 TD.) Tj 9.72 0 T5) Tj 5.4Tj 5.4 0 TD 0 TD 0.4 0 T(t) Tj 6.0254 Tc .j .76 0 TDj 6.12 0 TD ( (t) Tj 4.32 0 TD 254 Tc .j .76 0 TDj 6.12 .0254 Tc .) Tj 9.72 0 T.0298 T36 re 09 Tc 0 TD 0T352 0 TD 0.018c) Tj 4f-0.007.30.4 0 TD) Tj 9.72 0 T0072 Tc36 re 2 Tc (e) Tj -379.) 98 564 TcTj .0254 Tc (t) (}) Tj 5.4Tj 5.4 0 TD 0.0182 0 TD (d) 0 G 459.12 492.27030.96 0.Tj .76 0 TD Tj 4.30.96 0.36 re c) Tj 4n{) Tj 5762 0 TDD 0.0254 Tc (t) (}) Tj 5.4 .0072 Tc (5.4 0 0.0254 Tc 0.0182 Tc (e) Tj 8. TD (d) D 0.0509 Tc 0 TD 0.0254 Tc . Tj 9({) Tj 5.0298 T.j .76 0 TD -0.000.4 0 T (n) TD -0.0298 TD TD -0.007 0.0254 Tc (t) (}3254 Tc T) Tj 9({) Tj 5.0298 T36 re 09 Tc b) Tj 6.0254 Tc .j .76 0 TDj 6.12 0 TD (D -0.0079 Tc (Dj 6.12 0 TD (7 0.0254 Tc (t) 3.95.4 0 TD56 Tc (n) Ty98 T8.70.4 0 TD) Tj 9.72 0 T0072 Tc36 re 0 TD f TD 0.30.4 0 TD (432 Tc (m TD 9.0254 Tc .) Tj 9.72 0 T.0298 T36 re 09 Tc 0 TD 0.0254 Tc . Tj 9.72 0 T.0298 T36 re 09 Tc p TD 0.30.4 0 TD) Tj 9.72 0 T5) Tj 5.4Tj .76 0 TD -0.0079 Tc (Dy72 0 5762 0 TDD 0.0254 Tc)98 TD 254 Tc .j .76 0 TD. 6.12 .0253.95. TD (0 8j 3.12 272 6) Tj 91.4 0 D TD (d) 08 TD 9.30.-3.95. TD (0 Tj 3.12 27) Tj 9N98 T8.10.4 0 T(.72 0 T.0298 T. (t) Tj 4.32 0 TD TD -0.0072 G 459.12 492.27030.96 0.Tj .0254 Tc (t) (}) Tj 5.4Tj 5.4 0 TD 0.0182 0 TD (d) 0 Tj 9({) Tj 5.0298 T.j .0254 Tc (t) 7T352 0 TD 0.0.76 0 TDw492.27030.96 0.Tj 5.4 0 TD 0.0182 0 TD (d) 0 G 459.12 492.24 30.96 0.36 re c) Tj 4n{) Tj9.0254 Tc .) Tj 9a98 T8.40Tj 5.4Tj 072 Tc (h) Tj 6.12 0 TD.ee m T D 9 . 0 2 5 4 T c . j s t5.4 0 0.0254 Tc 0.0182 Tc (e) Tj 8. TD (d) 72 G 459.12 492.270.0254 Tc(h) Tj 6.12 0 TD.5 . 4 0 8 5 5 e5.4 0 8.642 0 TDD. 7 6 0 T D v 7 2 0 T 1 6 5 4 T c T 0 T D 3 2 5 4 T c ( T j 3 . 1 2 2 . j s 4 T j 4 . 3 ® T D 0 . 9 5 . 4 0 T D 5 6 T c ( n ) T ; 0 T 4 3 0 . 9 6 0 . 3 6 r e 9 6 0 . 0 1 f T D 6 . 6 0 2 9 8 T / F 7 T j 3 . 1 2 2 7 2 t ) T j 4 . 3 ) 4 9 2 . 2 7 0 . 0 2 5 4 T c ( 0 T j 3 . 1 2 2 0 . 0 1 8 c ) T j 4 f - 0 . 0 0 7 . 3 0 . 4 0 T D e ( 0 7 2 T c ( h ) T j 6 . 5 6 T j 5 . 4 T j 5 . 4 0 T D 0 . 0 1 8 2 9 . 8 . 0 2 9 8 T 0 T D 0 . 4 0 T D ) T j 9 . 7 2 0 9 . 0 2 5 4 T c ( ( { ) T j 5 5 2 5 4 T c 9 g 7 2 0 T D 0 . 0 1 8 2 T c ( 2 T c ( e ) T j 8 . 3 2 5 4 T c . j s ) T j 8 . 1 6 0 T 4 3 0 . 9 6 0 . ( 5 . 4 0 0 . 0 2 5 4 T c 7 2 4 3 2 T c ( m T D 9 T j 5 . 4 . 0 0 7 2 T c ( 5 . 4 0 0 . 0 2 5 4 T c 0 . 0 1 8 c ) T j 4 n { ) T j 5 7 6 2 0 T D D 0 . 0 2 5 4 T c ( t ) 7 0 . 0 2 5 4 T c 7 2 . 7 6 0 T D T j 4 . 3 0 . 9 6 0 . 3 6 r T D - 0 . 0 2 9 8 T 8 . 0 . 0 2 9 8 T 0 2 9 8 T D T D - 0 . 0 0 7 0 . 0 2 5 4 T c ( t ) ( } 2 5 4 T c T ) T j 9 ( { ) T j 5 5 2 5 4 T c 0 . 0 1 8 c ) T j 4 b . 0 1 8 2 T j 5 . 4 . 0 . 7 6 0 T D - 0 . 0 0 7 . 0 2 5 4 T c 7 2 G 4 5 9 . 1 2 4 9 2 . 2 4 0 3 0 . 9 6 0 . T j . 7 6 0 T D , T D 6 . 7 5 2 0 T D 0 . 0 . 7 6 0 T D w 4 9 2 . 2 7 0 9 ) T j 5 . 4 T j 5 . 4 0 T D 0 . 0 1 8 2 T j 5 . 4 . 0 0 7 2 T c ( 5 . 4 0 0 . 0 2 5 4 T c 0 . 0 1 8 2 T c ( e ) T j 8 . T D ( d ) 7 2 G 4 5 9 . 1 2 4 9 2 . 2 0 . 0 2 5 4 T c 7 ) T j 9 ( { ) T j 5 . 0 2 9 8 T 3 6 r T D - 0 . 0 2 9 8 T 8 . 0 . 0 2 9 8 T T j . 7 6 0 T D T j 4 . 3 0 . 9 6 0 . 3 6 r e c tm T D 9 . 0 2 5 4 T c . j s e5.4 0 0.0254 Tc 0.0182 Tc (e) Tj 8. TD (d) D 0. TD -0.0298 T8.0.0298 TTj .0254 Tc (t) (}) Tj 5.4Tj 5.4 0 TD 0.0182 Tj 5.4 .0072 Tc (5.4 0 0.0254 Tc 0.0182 Tc (e) Tj 8. TD (d) 72 G 459.12 492.28T5) Tj 5.4Tj .76 0 TD Tj 4.30.96 0.36 r TD -0.0298 T7..0254 Tc7 0.0.76 0 TDw492.2705654 Tc T td 2 T c ( e ) T j 8 . 3 2 5 4 T c . j . 0 2 5 4 T c ( t ) ( } 2 5 4 T c T . 0 . 7 6 0 T D j 6 . 1 2 0 T D ( ( t ) T j 4 . 3 2 0 T D T D - 0 . 0 0 7 2 . 7 6 0 T D j 6 . 1 2 0 T D ( ) T j 9 . 7 2 0 T 0 0 7 2 T c 3 6 r e 0 T D n { ) T j 9 . 7 0 2 5 4 T c 7 ) T j 9 ( { ) T j 5 . 0 2 9 8 T 3 6 r T D - 0 . 0 2 9 8 T D T D - 0 . 0 0 0 2 9 8 T D T D - 0 . 0 0 7 c.4 0 0.0254 Tc72 .76 0 TDj 6.12 0 TD ( ) Tj 9({) Tj 5.0298 T.j .0254 Tc (t) 8.3254 Tc .j stee 7 T j 3 . 1 2 2 7 2 t 0 T D 2 5 4 T c ( T j 3 . 1 2 2 . j s 4 T j 4 . 3 ® T D 7 T 0 1 2 0 T D . ( . 7 6 0 T D ; 0 T 4 3 0 . 9 6 0 . 3 6 r e 9 6 0 . 0 1 f T D 6 . 4 8 2 5 4 T c ( 7 T j 3 . 1 2 2 7 2 t ) T j 4 . 3 ) 0 T D 3 2 5 4 T c ( T j 3 . 1 2 2 . j . 7 6 0 T D : 0 T E T 1 8 8 8 3 2 1 . 0 2 5 1 5 3 . 2 . 0 2 . 2 . 0 r e f 1 8 7 3 2 1 . 0 2 5 1 5 3 . 3 0 . 4 . 3 0 . r e S B T 1 2 2 . 6 4 2 3 1 4 5 5 ) T c ( 0 6 . 1 2 2 . 0 T D 1 - 0 . 0 0 7 . 0 2 5 4 T c ( 7 0 T D 2 5 - 3 . 8 . 0 T c ( 0 9 1 2 2 0 . 0 1 T D - 0 . 0 T T D 6 . 6 0 2 9 8 T D 5 6 4 5 0 T D 0 . 0 1 8 2 5 1 5 . 4 0 T D 5 6 1 T D - 0 . 0 T j 4 . 2 5 5 ) T j 5 . 4 T j . 4 D - 0 . 0 2 9 8 T 7 . 5 5 . 4 0 T D 5 6 1 T D - 0 . 0 l T j 4 . 2 5 5 ) T j 5 . 4 T j 6 ) T j 9 ( { ) T j 4 . 6 8 T j 5 . 4 T j . 4 D - 0 . 0 2 9 8 T 7 . 0 2 5 4 T c 3 6 r 1 5 4 T c ( t ) 7 . 4 . 0 2 9 8 T 3 6 r 4 8 T c ( e ) T j 7 . 0 2 5 4 T c 3 6 r 3 6 T j 8 . 1 6 0 T 4 0 8 T j 5 . 4 T j . 4 D - 0 . 0 2 9 8 T 7 . 0 2 5 4 T c D 5 6 4 5 0 T D u . 0 1 8 2 5 1 5 . 4 0 T D 5 6 1 T D - 0 . 0 l T j 4 . 2 5 5 ) T j 5 . 4 3 6 r 1 5 4 T c ( t ) 7 . 0 2 5 4 T c D 5 6 1 T D - 0 . 0 , T D 6 . 7 5 2 0 T D 0 . 0 3 6 T j 8 . 1 c 0 T 4 0 8 T j 5 . 4 T j 6 ) T j 9 . 7 2 0 4 . 5 5 . 4 0 T D 5 6 4 5 0 T D n . 4 0 0 . 0 2 5 4 T c 0 . 0 1 5 4 T c ( t ) 7 . 0 2 5 4 T c 3 6 r 4 8 T c ( e ) T j 7 . 0 2 5 4 T c T j 6 e y 4 9 2 . 2 8 T 7 0 2 5 4 T c 3 6 r 1 5 4 T c ( t ) 7 . 0 2 5 4 T c T j 6 v 7 2 0 4 . 6 8 T j 5 . 4 3 6 r 3 6 T j 8 . 1 6 0 T 4 0 8 T j 5 . 4 T j 4 5 0 T D n . 4 0 0 8 2 5 4 T c 3 6 r 1 5 4 T c ( t ) 7 . 0 2 5 4 T c D 5 6 1 T D - 0 . 0 T j 4 . 2 5 6 4 T j 5 . 4 T j 6

m T D T j 5 e

e e y 4 9 2 . 2 0 . 0 2 5 4 T c 0 . 0 5 2 T c ( m T D 7 . 6 8 T j 5 . 4 ( m T D 7 . 6 8 T j 5 . 4 3 6 r 3 6 T j 8 . 1 6 0 T 4 0 8 T j 5 . 4 3 6 r 1 5 4 T c ( t ) 7 . 0 2 5 4 T c 3 6 r 4 8 T c ( e ) T j 7 . 0 2 5 4 T c T j 4 8 T c ( y 4 9 2 . 2 8 T . 0 2 9 8 T D 5 6 4 5 0 T D b { ) T j 5 . 0 2 9 8 T 3 6 r 3 6 T j 8 . 1 6 0 T 4 0 8 T j 5 . 4 3 6 r 1 5 4 T c ( t ) 7 . 2 . 0 2 9 8 T 3 6 r 1 T D - 0 . 0 w T D 6 . 4 8 T j 5 . 4 3 6 r 3 6 T j 8 . 1 6 0 T 4 0 8 T j 5 . 4 1 6 0 T 4 0 8 T j 5 . 4 T j 4 5 0 T D n . 4 0 8 5 6 4 T j 5 . 4 3 … . 0 1 8 2 5 1 5 . 4 0 T 3 6 r 4 8 T c ( e ) T j 7 . 0 2 5 4 T c 3 6 r 5 2 T c ( m T D 7 . 6 8 T j 5 . 4 T j . 4 D - 0 . 0 2 9 8 T 7 . 0 8 T j 5 . 4 3 6 r 1 5 4 T c ( t ) 7 . 0 2 5 4 T c T j 4 5 0 T D 0 . 0 1 8 2 5 1 5 . 4 0 T 3 6 r 3 6 T j 8 . 1 6 0 T 7 0 5 0 2 5 4 T c 3 6 r 5 2 T c ( m T D 7 . 6 8 T j 5 . 4 T j 6 ) T j 9 . 7 2 0 4 . 6 8 T j 5 . 4 3 6 r 4 8 T c ( e ) T j 7 . 0 2 5 4 T c 3 6 r 3 6 T j 8 . 1 6 0 T 7 . 5 5 . 4 0 T D 5 6 4 5 0 T D d . 4 0 5 1 5 . 4 0 T D 5 6 1 T D - 0 . 0 T j 4 . 2 5 5 ) T j 5 . 4 3 6 r 5 2 T c ( ¢ T D 7 . 6 8 T j 5 . 4 3 6 r 3 6 T j 8 . 1 c 0 T 4 0 8 T j 5 . 4 T j 4 5 0 T D u . 0 1 8 2 5 1 5 . 4 0 T D 5 6 1 T D - 0 . 0 l T j 4 . 2 5 5 ) T j 5 . 4 3 6 r 1 5 4 T c ( t ) 7 . 2 . 4 0 T D 5 6 1 T D - 0 . 0 T j 4 . 2 5 5 ) T j 5 . 4 3 6 r 1 5 4 T c ( t ) 7 . 0 8 T j 5 . 4 T j 1 T D - 0 . 0 T j 4 . 2 5 6 4 T j 5 . 4 T j . 4 D - 0 . 0 2 9 8 T 7 . 0 8 T j 5 . 4 3 6 r 1 5 4 T c ( t ) 7 . 0 2 5 4 T c T j 6 ) T j 9 o 4 9 2 . 2 8 T 1 0 2 5 4 T c D 5 6 . 4 D - 0 . 0 2 9 8 T 7 . 0 2 5 4 T c D 5 6 4 5 0 T D u . 0 1 8 2 5 1 5 . 4 0 T D 5 6 . 4 D - 0 . 0 2 9 8 T 7 . 0 2 5 4 T c 3 6 r 1 5 4 T c ( t ) 7 . 0 2 5 4 T c T j 6 m T D 7 . 6 8 T j 5 . 4 D 5 6 1 T D - 0 . 0 , T D 6 . 2 . 0 2 9 8 T 0 T j 4 . 2 5 5e y 4 9 2 . 2 9 . 4 8 T j 5 . 4 3 6 r 1 5 4 T c ( t ) 7 . 0 2 5 4 T c T j 6 y 4 9 2 . 2 0 . 0 2 5 4 T c 0 . 0 5 2 T c ( m T D 7 . 6 8 T j 5 . 4 3 6 r 3 6 T j 8 . 1 6 0 T 4 0 8 T j 5 . 4 T j 4 5 0 T D n . 4 0 0 8 T j 5 . 4 3 6 r 1 5 4 T c ( t ) 7 . 0 2 5 4 T c T j . 4 D - 0 . 0 2 9 8 T 7 . 7 2 . 4 0 T D 5 6 1 T D - 0 . 0 . T D 1 1 5 1 5 . 4 0 T 3 6 r 3 D - 0 . 0 A T D 6 . 8 4 . 4 0 T 3 6 r 3 6 T j 8 . 1 c 0 T 4 0 8 T j 5 . 4 0 . 0 1 5 4 T c ( t ) 7 . 0 2 5 4 T c D 5 6 4 5 0 T D u . 0 1 8 2 5 1 5 . 4 0 T D 5 6 6 ) T j 9 ( { ) T j 4 . 5 0 2 5 4 T c D 5 6 1 T D - 0 . 0 l T j 4 . 2 5 6 4 T j 5 . 4 3 l T j 4 . 2 5 5 y 4 9 2 . 2 4 0 8 T j 5 . 4 D 5 6 1 T D - 0 . 0 , 0 T 7 0 5 0 2 5 4 T c 3 6 r 1 T D - 0 . 0 w T D 6 . 0 2 5 4 T c T j 4 5 0 T D 0 . 0 1 8 2 5 1 5 . 4 0 T 3 6 r 3 6 T j 8 . 1 6 0 T 4 0 8 T j 5 . 4 T j 4 5 0 T D n . 4 0 9 . 7 5 2 0 T D 0 . 0 1 5 4 T c ( t ) 7 . 0 2 5 4 T c T j 4 5 0 T D 0 . 0 1 8 2 5 0 4 . 4 0 T 3 6 r 3 6 T j 8 . 1 6 0 T 8 5 7 5 . 4 0 T 1 6 0 T 4 0 8 T j 5 . 4 3 6 r 5 2 T c ( ¢ T D 7 . 6 8 T j 5 . 4 3 6 r 3 6 T j 8 . 1 c 0 T 4 0 8 T j 5 . 4 T j 1 T D - 0 . 0 T j 4 . 2 5 5 ) T j 5 . 4 3 6 r 3 6 T j 8 . 1 6 0 T 4 e m T D 1 2 . 2 . 0 2 9 8 T 3 6 r 3 6 T j 8 . 1 c 0 T 4 0 8 T j 5 . 4 T j 6

m T D 7 . 6 8 T j 5 . 4 D 5 6 4 5 0 T D p . 0 1 8 2 5 4 . 4 0 T 3 6 r 3 6 T j 8 . 1 6 0 T 4 0 8 T j 5 . 4 T j 4 5 0 T D n . 4 0 5 1 5 . 4 0 T D 5 6 . 4 D - 0 . 0 2 9 8 T 7 . 0 2 5 4 T c T j 6 t ¢ T D 7 . 6 8 T j 5 . 4 3 6 r 3 6 T j 8 . 1 c 0 T 4 2 . 4 0 T D 5 6 1 T D - 0 . 0 T j 4 . 2 5 5e e

e v { e

t e e y 4 9 2 . 2 4 0 8 T j 5 . 4 D 5 6 1 T D - 0 . 0 , 0 T 6 . 7 5 2 0 T D 0 T j 4 . 2 5 6 4 T j 5 . 4 T j . 4 D - 0 . 0 2 9 8 T - 3 7 9 . . 0 2 - 1 0 . . 0 2 8 T D 5 6 1 5 0 T D f T j 4 . 2 5 7 5 . 4 0 T D 5 6 6 e t m T D 7 . 6 8 T j 5 . 4 3 6 r 3 6 T j 8 . 1 6 0 T 6 . 0 2 5 4 T c 1 6 0 T 4 0 8 T j 5 . 4 T j 6 e e

e

ev {

e v { e k 4 9 2 . 2 7 . 3e m T D 7 . 6 8 T j 5 . 4 3 6 r 3 6 T j 8 . 1 6 0 T 4 0 8 T j 5 . 4 T j 4 5 0 T D n . 4 0 0 8 T j 5 . 4 0 . 0 1 5 4 T c ( t ) 6 . 1 e t

eeev {q { e m T D 7 . 6 8 T j 5 . 4 3 6 r 3 6 T j 8 . 1 6 0 T 4 0 8 T j 5 . 4 T j 4 5 0 T D n . 4 0 0 8 T j 5 . 4 0 . 0 1 5 4 T c ( t ) 7 . 0 2 5 4 T c T j . 4 D - 0 . 0 2 9 8 T 6 . 1 5 2 0 T D D 5 6 1 T D - 0 . 0 T j 4 . 2 5 6 4 T j 5 . 4 T j 4 5 0 T D n . 4 0 0 8 T j 5 . 4 T j 4 8 T c ( v { ) T j 4 . 6 8 T j 5 . 4 D 5 6 6v { e

e

m TD 7.68 Tj 5.4Tj 6

e m TD 7.68 Tj 5.4Tj .4D -0.0298 T6.152 0 TD-Tj 1TD -0.0w98 T6.7

e t em TD 7.68 Tj 5.4Tj .4D -0.0298 T5.88 Tj 5.4D566¢ TD 7.68 Tj 5.436 r36Tj 8.1c 0 T4 08 Tj 5.4Tj 1TD -0.0 Tj 4.255 ee S . 4 0 5 0 4 . 4 0 T 3 6 r 3 6 T j 8 . 1 c 0 T 4 0 8 T j 5 . 4 T j 6 m T D 7 . 6 8 T j 5 . 4 3 6 r 3 6 T j 8 . 1 6 0 T 4 e) e¢ TD 7.68 Tj 5.436 r36Tj 8.1c 0 T4 08 Tj 5.4Tj 1TD -0.0 Tj 4.2564.4 0 T36 r36Tj 8.16 0 T4 08 Tj 5.4Tj 450 TD n.4 0 504.4 0 T36 r36Tj 8.1c 0 T4 08 Tj 5.4Tj 4TD -0.0y.4 0 7 ee eee e¢ TD 7.68 Tj 5.436 r36Tj 8.1c 0 T4 08 Tj 5.4Tj 1TD -0.0 Tj 4.255e m T D 1 0 0 8 T j 5 . 4 T j 1 T D - 0 . 0 T j 4 . 2 5 5

m T D 7 . 6 8 T j 5 . 4 T j 4 5 0 T D p . 0 1 8 2 5 4 . 4 0 T 3 6 r 3 6 T j 8 . 1 6 0 T 4 0 8 T j 5 . 4 T j 4 5 0 T D n . 4 0 5 1 5 . 4 0 T D 5 6 . 4 D - 0 . 0 2 9 8 T 7 . 0 2 5 4 T c D 5 6 6

e

e e

e m T D 7 . 6 8 T j 5 . 4 3 6 r 3 6 T j 8 . 1 6 0 T 4 0 8 T j 5 . 4 T j 1 T D - 0 . 0 . 0 T 9 . 0 2 5 4 T c 3 6 r 1 T D - 0 . 0 T 9 8 T 6 . 7 ee g { e m TD 7.68 Tj 5.436 r36Tj 8.16 0 T4 08 Tj 5.4Tj 450 TD n.4 0 08 Tj 5.436 r 154 Tc (t) 7.68 Tj 5.4Tj 1TD -0.0 Tj 4.255e e m T D 7 . 6 8 T j 5 . 4 D 5 6 . 4 D - 0 . 0 2 9 8 T 6 . 7 e

m T D 7 . 6 8 T j 5 . 4 3 6 r 3 6 T j 8 . 1 6 0 T 4 0 8 T j 5 . 4 T j 4 5 0 T D n . 4 0 0 9 8 0 T 4 2 - 3 . 8 4 5 . 4 / F 0 9 T j f 3 6 r 3 D - 0 . 0 N 9 8 T 6 . 8 4 T j 5 . 4 T j 6

e e e m T D 1 1 5 7 0 2 5 4 T c T j 6

e e m T D 1 0 . 6 8 T j 5 . 4 3 6 r 1 5 4 T c t e

4. The general case of n …rms

In this section we present the main results that can be obtained concerning thepro…tability and stability of coalitions under a general framework; these resultsare then particularized for a speci…c collusion path.

The model is as described in subsection 3.1. The game is two-stage: …rms form(or not) coalitions and then the resulting groups compete in quantities. For all n…rms to be in the market we have to set some prior conditions on the values of ®and f , in order to assure positive quantities: these conditions derive directly fromimposing qnn > 0 (where the subscript refers to the n-th …rm and the superscriptsignals an n-…rm oligopoly) and ¼nn ¡ f > 0 respectively (¼ denotes the variablepro…t). The …rst condition gives rise to the restriction on ®: ® <® (³; n) and thesecond condition gives rise to the restriction on f : f <f (³; ®; n) = ¼nn. (See theAppendix for the formal derivations).

4.1. Main results

Having a completely dispersed market, in which all n …rms operate separately,we will analyze its stability in terms of the parameters ® and f and study therelative pro…tability of the various two-…rm coalitions that transform the marketin an (n¡1)-…rm oligopoly. We analyze then the stability of these agreements (byadding to the condition for internal stability the condition for external stability)and try to generalize to broader coalitions, and thus more concentrated marketstructures.

4.1.1. Two-…rm agreements

The n-…rm oligopoly is unstable whenever it is pro…table for any …rms i and j tomerge,19 that is, whenever

¼n¡1ij ¡ f ¸ ¼ni ¡ f + ¼nj ¡ f

This condition says that in the assumed absence of bargaining costs there isat least one way of sharing the joint pro…t of i and j such that each …rm is bettero¤ by joining than by staying alone (the merger represents a Pareto improvementfor i and j). It is thus clear that the condition for external instability of theeht T c ( t ) T 9 2 0 ( d ) T j T 0 2 1 8 2 T c ( e ) T j D 0 0 T D - 0 . 0 0 7 7 T c ( r ) nairy faew to s t bn vq tu Tj 4.32 0 TD -D 0 Tc (o) Tj 8.52 0 (o) Tj 5.1.32 0 TD -0.0072 Tc (n) Tj 6 0 TD 0 Tc (a) Tj 4.92 0 TD 0.0509 Tc (l) Tj32 0 TD 0.0182 Tc (e) Tj 4.8 0 TD -0.0072 Tc (n) Tj 11.04 Tf (s) Tj 4.32 0 TD 0.0254ot eioc (h) Tj 6.12b4 0.0182 Tc (e) Tj 4.8 0 TD 0.0254 Tc (t) Tj 4.2 0 TD 0 Tc (a) Tj 5.4 0 TD -0.0072 Tc (b) Tj 6.12 0 TD 0.0509 Tc (i) Tj 3 TD (l) Tj 3 0 TD (i) Tj 3.9.32 0 TD -D 0 Tc (y) Tj 5.4 0 TD -00509 Tc (y) Tj 10.68 0 TD 0 Tc (o) Tj 7j56 0 TD 0.0072 Tc (f) Tj 8.16 0 TD 0.0254 Tc (t) Tj 4.2 0 TD -0.0072 Tc (h) Tj 9 0 TD 0.0254 Tc (t) Tj 4.2 7 Tc (r) Tj 4.2 00.0437 Tc (e) Tj 4.92 0 TD -0.0509 Tc (v) Tj 5.4 0 TD (r) Tj 4.32 0 TD 0.0.0509 Tc (i) Tj 3 0 TD TD 0.0509 Tc (l) Tj32 0 TD 0.0182 Tc (e) Tj 4.8 0 TD -0.0072 Tc (n) Tj 11.04 Tf 0.0254 Tc (t) Tj 9.12 0 TD 0182 Tc (e) Tj 4.8 0 TD -0.0D 0 Tc (o) Tj 5.16 0 TD .0077 Tc (r) nai vl i

…rms i and j) coincides with the condition for the internal stability of coalitionfi; jg. Hence pro…tability and internal stability are equivalent for two-membercoalitions and so it is indi¤erent whether we are focusing on cartels or on mergers(because internal stability is always met for pro…table agreements).

We assume that …rm i is more e¢cient than …rm j, so its marginal cost prevailsafter merging has taken place. This will imply in our notation j > i. Solving theabove expression for f we get

f ¸ fij = ¼ni + ¼nj ¡ ¼n¡1ij

For values of f above fij the n-…rm oligopoly is not stable. This boundary isa function of the usual parameters (³, ® and n) and also, now, of i and j (thatare discrete variables, just as n) (see the Appendix for the complete expression).The minimum of fij over i and j gives the upper bound for the stability of then-…rm oligopoly. This minimum is found to be f1n, giving rise to the followingLemma.

Lemma 4.1. An oligopolistic structure with every …rm operating separately isstable if and only if f < f1n(³; ®; n); for all ³; ® and n, where f1n(³; ®; n) denotesthe values of the …xed cost above which the most e¢cient and the least e¢cient…rms want to join. This is the two-…rm agreement that generates the highestsurplus as compared with the situation in which …rms operate separately.Proof. See Appendix.

An immediate corollary of this Lemma is that when n=3, triopoly is stable ifand only if f < f13(³; ®), as we have already shown in subsection 3.2.

The economic intuition for this …rst result is the following: the coalitionthat minimizes fij = ¼ni + ¼nj ¡ ¼n¡1ij is the one which equivalently maximizes¼n¡1ij ¡ ¼ni ¡ ¼nj and so it is the most attractive for the merging parties since itgenerates the highest surplus to be shared; this movement is therefore the onewhich most “threatens” the stability of the n-…rm oligopoly. The fact is that, dueto the constant marginal cost assumption, the high-cost …rm is shutdown and itsproduction is transferred to the other …rm: the gains are maximized when thistransfer occurs from the least e¢cient to the most e¢cient …rm in the market.

There is various evidence that this type of collusive agreement between verye¢cient and very ine¢cient …rms may actually happen. See for example Bar-ros(1998) for an application to a sample of concentration operations in the Por-tuguese industry. Brealey and Myers(1988) argue that most gains from combiningcomplementary resources occur when small …rms are acquired by large ones. Theyalso refer to evg

unyfer tasr gayfgaoet a l …rm

for acquisition (see Palepu(1986)), which reinforces the importance of eliminatingine¢ciencies as a motive for merger.

This result may seem counterintuitive in that it contradicts the conventionalwisdom according to which …rms would prefer good partners. Actually, accord-ing to our …ndings, in a constant marginal cost context …rms always prefer thebad partners, since they have to pay less for them and can thus keep most ofthe surplus generated by the agreement. The rivalry e¤ect that would induce…rms to choose closer rivals to merge with, and which is implicit in the Cournottype of behavior, is more than compensated for by this incentive to “buy” cheappartners. This result still holds if we consider non-constannd ciedeid iehe dedp TD -0.0072 Tc (u) Tj 4.32 0 TD 0.0254 Tc (t) Tj 8.4 0 TD -0.0072 T 0 TD 0.0182 Tc j 8.4 A0 TD -1.0072 Tc (h) Tj 6.12p0 TD -0.0298 T6 0 TD 0.0182 Tc (e) Tj 4.92 0 TD -.0072 Tc (d) Tj 9.6 TD 0.0-0.0298 T60 TD 0509 Tc (i) Tj 3.12 0 TD 0298 T6x0.0072 Tc (n) Tj 6 0 (e) T)0 TD 0.0509 Tc (.) Tj 8.04 0 TD11-1.0072 Tc (h 0 (e) TI0 TD -9.0072 Tc (h) Tj 6.12 TD 0.10 Tc (a) Tj 5.52 0 TD10 .0072 Tc (n) Tj 6 0 TD 0.0437 Tc j 5.52 0 TD -0.0072 Tc (n) Tj 6 0 TD 0.0437 Tc (-) Tj 3.72 0 TD 0.0182 Tc (c) Tj 4.8 0 TD 0 Tc (o) Tj 5.4 0 TD -0.0072 Tc (n) Tj 6.12 0 TD -0.0298 Tc (s) Tj 4.32 0 TD 0.0254 Tc (t) Tj 4.2 0 TD 0 Tc (a) Tj 5.TD -0.0072 Tc (n) Tj 6 0 T TD -0.0072 Tc (n) Tj 6 0 (e) Tj 4.8 0.0182 Tc (c) Tj 9 0 TD -0.0 Tc (o) Tj 5.4 0 TD -0.0077 Tc (r) Tj 4.2 0 TD 0.0254 Tc j 4.2 g0 TD -0.0077 T (i) Tj 3 0 TD -0.0072 Tc (n) Tj 6.12 0 TD Tc (o) Tj 5.4 0 TD -0.0077 T (l) Tj 3 0 TD -00.0509 Tc (i) Tj 4.8 0 TD 0 Tc (o) Tj 5.4 0 TD -0.0072 Tc (n) Tj 4.32 0 TD 0.0254 Tc (t) Tj 4.2 0 TD 9509 Tc (i) Tj 4.8 0 TD 0 Tc (o) Tj 5.4 0 TD -0 Tc (n) Tj 6 0 T TD -0.0072 Tc (n) Tj 6 0 (e) Tj 4.8 0.0182 Tc (e) Tj 4.92 0 TD -0.0298 T (l) Tj 3 0 x0.0072 Tc (n) Tj 6 0 (e) Tj 4.8 0.0182 Tc (e) Tj 3 0 ,0 TD -0.0072 Tc (n) Tj 6 0 hTD 0.0437 Tc j 5.52 0 TD -1.0072 Tc (h) Tj 7.56 0 TD 060509 Tc (i) Tj 4.8 j 9.6 0 TD -0.0072) Tj 7.560 TD 0.0182 Tc (e) Tj 8.4 0 TD 4-0.0072 Tc (h) Tj 4.2 0 TD 0.0182 Tc (e) Tj 3 0 ,0 TD -0.0072 T2 0 TD -0.0298 T072 0 (e) Tj 4.8 0.0182 Tc (c) Tj 9 0 TD -0.0 Tc (o) Tj 5.4 0 TD -10509 Tc (i) Tj 3.12y0 TD10 0.0072 Tc (h) Tj 6.12b0 TD 0.0182 Tc (e) Tj 4.92 0 TD3.480.01 (p) Tj 6 0c (r) Tj 4.2 0 TD 0.0182 Tc (e) Tj 4.92 0 TD -0.0298 T072ehut desidi 1.0072 Tc (h) Tj 6.12b0 TD 0.0182 Tc j 5.4 0 TD -0.0072 T (t) Tj 4.32 0 TD 0.0509 Tcc (n) Tj 6 0 hTD 0.90.0182 Tc (e 0 (e) Tj 4.8 0.0182 Tc (e) Tj 4.92 0 TD -0.0298 T 0 TD 0 4.0072 Tc (n) Tj 6 0 hTD 0.0-0.0298 T6 TD 0.0437 Tc j 5.52 0 TD -0.0072 T (l) Tj 3 0 TD -0.0072 T j 5.52 0 TD -0.0072 T g0 TD -0509 Tc (i) Tj 3 0 TD -0.0072 T (e) Tj 8.4 0 TD 4-0.0072 Tc (h) Tj 4.32 0 TD 0.0182 Tc2 0 TD 0.0254 Tc (t) Tj 4.2 0 TD 0 Tc (a) T(i) Tj 3.12 0 TD 0298 T6l0 TD -0.0298 T6l0 TD6.2.0072 Tc (n) Tj 6 0 p0 TD -0.0298 Tc (r) Tj 4.2 0 TD 0.0254 Tc j 4.2 0 TD -7.0072 Tc (h) Tj 6.12 0 TD 0.0182 Tc(uTD 0.0182 Tc (e) Tj 4.8 0 TD 0.0509 Tc 0 TD (1.0072 TTj 5.4 0 TD -.0072 Tc (n) Tj 6 0 f0 TD -.0182 Tc (e 0 (e) Tj 4.8 0.0182 Tc (e) Tj 4.92 0 TD -0.0298 Tc (r) Tj 4.2 0 TD 0.0182 Tc (c) Tj 9 0 TD -0.182 Tc (e) Tj 8.4 0 TD 4-0.0072 Tc (h) Tj 4.2 0 TD 0.0182 Tc 4.2 g0 TD -0.0077 T (i) Tj 3 0 TD -0.0072 Tc (n) Tj 6.12 0 TD 0 Tc (o) Tj 5.4 g TD (7.0072 Tc (h) Tj 6.12hTD 0.0437 Tc j 5.52 0 TD -0.0077 Tc (r) Tj 4.32 0 TD 0.0182 Tc (t) Tj 4.2 0 TD 0 Tc (a) Tj 5.TD -0.0072 Tc (n) (i) Tj 3 0 k TD 0.0182 Tc (e) Tj 8.4 0 TD 4-.0072 Tc (d) Tj 9.6 TD 0.90.0182 Tc p0 TD -0.0298 T (l) Tj 3 0 TD -0.0072 T j 5.52 -0.0072 Tc (n) (i) Tj 4.8 0 TD 0 Tc (o) 2 0 TD0 TD -0.0072) Tj 7.56;0 TD -48182 Tc (t) Tj 4.2 0 TD 0 Tc (a) c (n) Tj 6 0 hTD 0.0-0.0298 T (i) Tj 3 0 TD -0.0072 Tc (n) Tj 4.32 0 TD3.44 TD (p) Tj 6 0j 6) Tj 9 0 TD -0.182 Tc j 5.TD -0.0072 Tc (n) (i) Tj 3 0 k TD 0.0182 Tc (e) Tj 8.4 0 TD 4-0.0072 Tc (h) Tj 4.32 0 TD8.60509 Tc (i) Tj 3 0 TD -0.-0.0298 T072 0 (e) Tj 4.8 060509 Tc (i) Tj 9 0 TD -0.182 Tc j 5.TD0 TD -0 Tc (n) Tj 6 0Tj 4.2 0 TD 0..0254 Tc (t) Tj 8.4 0 TD 9.2.0072 TTj 5.4 0 TD -.0072 T (t) Tj 4.32 0 TD 0.0509 Tc 0 TD 0 Tc (a) c (n) Tj 4.2 0 TD 0.0182 Tc 4.2 -0.0072 Tc (n) (i) Tj 4.8 0 TD 0 Tc (o) (t) Tj 4.32 0 TD 0.0509 TcT(i) Tj 3.12 0 TD 0298 T60 TD 0.0182 Tc (e) Tj 8.4 0 TD 90.0182 Tcc (s) Tj 6 0 f0 TD -2.0072 TTj 5.4 0 TD -0 Tc (n) Tj 6 0Tj 4.2 0 TD 060509 Tcc (s) Tj 6 0 …0 TD -0.0298 Tc (r) Tj 4.2 0 TD 0.0254 Tc (i) Tj 9 0 TD -0(p) Tj072 TTj 5.4 1 TD 908.0072 T (t) Tj 4.32 0 TD 00254 Tc j 4.2 0 TD9.90182 Tcc (s) Tj 6 0 j0 TD -2.0072 TTj 5.4 0 TD -0 Tc (n) (i) Tj 3 0 TD -0.0072 Tc (n) Tj 6.12 0 TD10 4.0072 TTj 5.4 0 TD9.90182 Tc g0 TD -0509 Tc 0 TD -7.0072 T 0 TD -7.0072 Tc (h) Tj 6.12 0 TD10 4.0072 T p0 TD -0.0298 T j 5.4 0 TD -.0072 TTj 6 0Tj 4.2 0 TD 0..0254 Tc (t) Tj 4.2 0 TD 0 Tc (a) c (n) Tj 6 0 0 TD -0.0298 T (i) Tj 4.8 0 TD 4-.0072 Tc (d) Tj 4.2 0 TD 0..0254 Tc (t) Tj 3 0 ,0 TD 0.0182 Tc (t) Tj 4.92 0 TD -0.0298 T072) Tj 7.560 TD 0.0182 Tc (e) Tj 8.4 0 TD 4-.0072 Tc (d) Tj 9.6 TD 0.10 0.0072 T (t) Tj 4.2 0 TD 0 Tc (a) c (n) Tj 6 0 hTD 0.0-0.0298 T j 5.4 0 TD -00509 Tcc (s) Tj 6 0 u0 TD 0 Tc (o) Tj 5.4 g TD -00509 Tcc (s) Tj 6 0 h0 TD10 4.0072 TT(i) Tj 3 0 TD -0.-0.0298 T072 0 (e) Tj 4.8 060509 Tc-T(i) Tj 3 0 w0 TD 0 TD -0.0072) Tj 7.56 TD -0.-0.0298 T TD -0.0072 T TD -0-3800 T (p) Tj 6 0c (r) Tj 6.12hTD 0.0437 Tc j 5.52 0 TD -10509 Tc (i) Tj 3.120 TD 0. Tc (n) (i) Tj 4.8 0 TD (.0072 T (t) Tj 4.32 0 TD 00254 Tc j 4.2 0 TD9.1 Tc (a) c (n) Tj 6 0 pTD 0.0437 Tc j 5.52 0 TD -10509 Tc (i) Tj 3.12yTD 0.90.0182 Tc (e) Tj 9 0 TD -0.-0.0298 T j 5.4 0 TD -00509 Tcc (s) Tj 4.2 0 TD 0..0254 Tc (t) Tj 8.4 0 TD 8-.0072 Tc (n) Tj 6 0 f0 TD -.0182 Tc j 5.4 0 TD -00509 Tcc (s) Tj 4.2 0 TD 7-0.0298 T072) Tj 7.56 TD -0.0072 T (e) Tj 4.2 0 TD 0 Tc (a) T(i) Tj 3.12. TD 7-0.0298 T- (e) Tj 4.2 I0 TD -90182 Tcc (s) Tj 6 0 TD 0.0437 Tc2 0 TD 0.0182 Tc (e) Tj 8.4 0 TD 4-.0072 T 0 TD 4-.0072 Tc (d) Tj 9.6 dTD -0.-7 Tc (n) (i) Tj 3 0 TD -0.0072 Tc (n) Tj 6.12 0 TD9.0437 Tc j 5.52 0 TD9.1 Tc (a) c (n) Tj 4.32 0 TD 0.0509 TcT(i) Tj 3.12 0 TD 0298 T (e) Tj 9 0 TD -0.-0.0298 Tc (n) Tj 6 0 pTD 0.0437 Tc (l) Tj 3 0 TD -0.0072 T (t) Tj 8.4 0 TD 8-50254 Tc (t) Tj 4.2 0 TD 0 Tc (a) c (n) Tj 6 0 hTD 0.0-0.0298 Tc (r) Tj 4.2 0 TD 0.0254 Tc (i) Tj 4.92 0 TD -0.0298 T2 0 TD0 TD -0.0072) Tj 9 0 -0 TD -0182 Tcc (s) Tj 6 0 …0 TD -0.0298 Tc (r) Tj 4.2 0 TD 0.0254 Tc (i) Tj 9 0 TD -0(2-7 Tc (n) c (n) Tj 6 0 f0 TD -.0182 Tcc (n) Tj 4.2 0 TD 0.0182 Tc 4.2 -0.0072 Tc (n) (i) Tj 9 0 TD -0.-0.0298 T (e) Tj 8.4 0 TD 4-.0072 Tc (d) Tj 3 0 w0 TD 0 Tj072 TTj 5.4 0 TD -00509 Tcc (s) Tj 4.2 0 TD 0..0254 Tc (t) Tj 3 0 k TD 90.0182 Tcc (s) Tj 3 0 w0 TD 00.0298 T072) Tj 7.56 TD -0.0072 T (e) Tj 4.2 0 TD 0 Tc (a) c (n) Tj 6 0 hTD 0..-7 Tc (n) (i) Tj 3 0 q0 TD -7.0072 Tc (h) Tj 6.12uTD 0.0437 Tc j 5.52 0 TD -5 Tc (a) c (n) Tj 6 0 dTD 0.0437 Tcc (s) Tj 4.2 0 TD 0..0254 Tc j 5.52 0 TD -.0072 T (t) Tj 4.32 0 TD 00254 Tc (i) Tj 3 0 TD -0.-0.0298 T072p ntn n

Further results on the pro…tability and internal stability of two-member coali-tions can be found in the Appendix.

Based on the analysis of the shape of fij as a function of i and j (see Appendix)we can say that the potential external instability of the agreement fi; jg comesfrom the enlargement to include either the most e¢cient …rm left in the market(let us denote it by 1’ and note that 1’ is …rm 1 if this does not yet belong tothe group) or the least e¢cient …rm left in the market (n’); which one is morepro…table will depend on the relative e¢ciency of the coalition as compared withthe other …rms that are still operating, as shows the expression in the Appendix.

Let us denote fij by f intij and the minimum of ffij10 ; fijn0g by f extij and computefextij ¡ f intij . A necessary and su¢cient condition for coalition fi; jg to be stablein a range (®; f) is that fextij ¡ f intij is positive.

The expression f extij ¡f intij is a function of i; j; n; ³; ®: When ® = 0 it is positivefor all n ¸ 5, so all two-member coalitions are stable in some non-empty rangeof f , provided the market stays with at least four (groups of) operating …rms.21

These …ndings can be related to Selten(1973)’s results, according to which whenthe number of competitors is less than or equal to 4 they have a tendency tocooperate and maximize joint pro…t.

For ® strictly positive the di¤erence fextij ¡ f intij is concave in j and increasingmost of the time (except when f extij = fij1 and ® is too small - see Appendix). Thereason is that when j is small i is small too, and the agreement does not enjoyeither large external or internal stability; however as j increases internal stabilityis reinforced and that is the main reason for the stability of the coalition to beimproved (that means, its range of stability is enlarged). As to i the evolution offextij ¡ f intij depends a lot on whether fextij = fij1 or f extij = fijn, as described inthe Appendix. Besides fij1 and fijn, note that a third possible expression for fextijis fin(n¡1), which is the relevant one when the agreement already includes …rmn, and i is such that the most pro…table enlargement is to include …rm (n ¡ 1).The function fextij ¡ f intij in this case can be shown to increase with i, so thecoalition with the largest range of stability will have an intermediate i, such thatfextin = fin(n¡1).

There are three di¤erent candidates to be the most stable agreement with…rms i and j: fi = 1; j = n ¡ 1g, fi = i1; j = n ¡ 1g, where i1 is such thatfijn = fij1, and fi = i2; j = ng, where i2 is such that fin(n¡1) = fin1. Themost stable agreement thus involves a very ine¢cient …rm (j = n or j = n ¡ 1)and either a very e¢cient …rm (i = 1) or a …rm with an intermediate level ofe¢ciency. The solution will depend on the parameters of the problem (®; ³ and

21Notice that when ® = 0 fij10 = fijn0 :

19

n). The analysis carried out at the extremes of ® (® = 0 and ® = ®) shows thatfor su¢ciently low e¢ciency di¤erences an intermediate i is optimal, whereas fora level of asymmetry large enough i = 1 is optimal.

Proposition 4.2. i) When …rms are symmetric and n ¸ 5 all two-member coali-tions are stable in a well de…ned interval for the values of the …xed cost. Forn = 4 or n = 3 no two-…rm agreement can be stable.ii) For low enough asymmetry levels the most stable two-…rm agreement involvesa …rm i with an intermediate level of e¢ciency (i such that the coalition is indif-ferent between broadening to include the most e¢cient or the least e¢cient …rmleft in the market), and a …rm j very ine¢cient (j = n or j = n ¡ 1). Whenthe asymmetry level is su¢ciently large, then i = 1 and again j very ine¢cientmaximizes the stability of the agreement.Proof. In the Appendix.

4.1.2. Broader agreements

In this section we discuss some of the results that can be obtained for a genericcoalition with more than two elements (r > 2, where r denotes the cardinality ofthe coalition). At this level of generality results are not very conclusive.

When r ¸ 3 the pro…tability condition and the condition for internal stabil-ity no longer coincide. As we said before, we restrict ourselves to the internalstability condition based on avoiding every …rm from separating alone. Actually,under asymmetry it is not clear that members prefer to exit alone, it may bemore attractive to leave in group (a simple example that illustrates the problemis included in the Appendix); however, for tractability reasons we con…ne ouranalysis to the simpler case, which may actually be the most relevant in a realworld economy.22

22 It is clear that in the symmetric case …rms always prefer to deviate alone, since they thendo not have to share the deviating pro…t with anyone else; in the asymmetric case, however,there is an additional e¤ect, which works in the opposite direction, thus making the outcomeuncertain, namely that it may be interesting to leave with the most e¢cient …rm left in thecartel so that this new rival stays with a weaker technology. What is sure is that if the moste¢cient …rm in the coalition wants to leave in the company of another …rm it will choose thesecond most e¢cient in the coalition, if it wants to leave with two more …rms it will choose thesecond and the third, and so on. The preference for one or another solution will depend on thesize and composition of the coalition, as well as on the value of ®. For n = 3 we are able to showthat …rms always prefer to deviate from the grand coalition alone rather than in the companyof another …rm (section 3.2 and the corresponding part of the Appendix): this result, however,is not generalizable for higher n and higher cardinality of the coalition.

20

Consider the coalition C = fe1; e2; :::; erg; with its members ranked in de-scending order of e¢ciency and #C = r. Denote by f intC the condition for itsinternal stability:

¼C ¡ f ¸X

i2C(¼ijCnfig ¡ f) , f ¸ f intC =

Pi2C

¼ijCnfig ¡ ¼C

r ¡ 1

External stability is de…ned either by

f · f ext1C = ¼C + ¼1jC ¡ ¼C[f1g (e1 > 1)

or byf · f extnC = ¼C + ¼njC ¡ ¼C[fng (er < n)

The pro…tability condition (above which the coalition forms) is given by

f ¸ fprofC = ¼Cnfefg + ¼ef jCnfefg ¡ ¼C

where ef denotes the last …rm that joined the agreement.If coalition C is a cartel, then a necessary and su¢cient condition for it to

have a range of stability is that

fextC ¡ maxffprofC ; f intC g > 0

where fextC = minffext1C ; fextnC g.23 Instead, if we consider an irreversible merger,this condition is simply

f extC ¡ fprofC > 0

which is clearly at least as easy to verify as the condition for the stability of thecartel. Thus, whenever a cartel is stable the corresponding merger is also stable,as is patent in the …gures below (note that if fextC was not the outer line, as canhappen, there would be no stability area).

If ® = 0 the two conditions for external stability are the same and the con-dition for internal stability is f ¸ f intC = r:¼ijCnfig¡¼C

r¡1 . It can be shown thatfextC ¡ fprofC > 0 for all r · n ¡ 3, so we conclude that symmetric mergers are

23We can have fprofC ><f intC : the fact that some coalition has formed may not be su¢cient to

ensure that it will not be dissolved.

21

fcext fcext

fcprof

fcint

fcint

fcprof

α α

f f

cartel

merger

both

Figure 4.1: Stability areas

stable in [fprofC ; fextC ], provided they do not lead to triopoly, nor to duopoly.24

Monopoly is the stable market structure for all values of the …xed cost such thatf > fextC (r = n ¡ 3; :) = fprofC (r = n ¡ 2; :).

As to cartels, f intC > fprofC for all r > 2 (for two-member agreements weknow that f intC = fprofC ), so a necessary and su¢cient condition for the cartelwith r symmetric …rms to have a range of stability is that fextC ¡ f intC > 0. Thiscondition is veri…ed for n ¸ 5 + 4(r ¡ 2), or, equivalently, for r · n+3

4 , so thehigher the cardinality of the cartel, the higher has to be n. In other words,cartels which imply a too high degree of concentration cannot be stable: morethan 75% of the initial number of …rms must remain operating. The reason hasto do with internal stability, since external stability is stronger the higher r is (aswill be illustrated in the next subsection). Actually higher concentration makes itmore attractive to deviate and enjoy the pro…ts of a highly concentrated market,therefore weakening internal stability. This e¤ect is su¢ciently strong to outweighthe e¤ect on external stability. For a given n the stability of symmetric cartels isthus more di¢cult the more members they enclose.

Note that the condition for the stability of symmetric cartels (r · n+34 ) is

much more stringent than the condition for the stability of symmetric mergers(r · n ¡ 3). Mergers can thus enclose a much higher proportion of the total

24 In the work of Espinosa and Iñarra(1995) duopolies and triopolies have a range of stability,because they consider the x-oligopoly to be externally stable whenever the (x-1)-oligopoly is notinternally stable (for example, duopoly is externally stable for values of the …xed cost belowwhich monopoly is internally unstable).

22

If stabm(r; :) < 0 then C is obviously unstable, but if stabm(r; :) > 0 it has arange of stability.

Note that in this speci…c partition fprofC (r + 1; :) = fextC (r; :). For ease ofexposition fprofC (r; :) will be denoted by f1n(n¡1):::(n¡r+2) and its arguments willbe suppressed.25

It can be shown (see Appendix) that stabm(r; :) > 0 for all r · n¡5 (at leastsix (groups of) operating …rms); for all n; ³; and ® 2 [0; ®(:)); f 2 [0; f(:)), i.e.,that f1n < f1n(n¡1) < f1n(n¡1)(n¡2) < ::: < f1n(n¡1)(n¡2):::7 < f1n(n¡1)(n¡2):::76.This result says that as the number of coalition members increases it becomesmore di¢cult to form pro…table agreements with the remaining …rms, since thesurplus generated by these agreements is declining.26 It can also be proventhat the di¤erence between f1n:::(n¡r+2)(n¡r+1) and f1n:::(n¡r+2) is increasing inr: the di¢culty in broadening the coalition increases as it becomes broader or,in other words, the more …rms have merged the more stable is the merger. Itcan be shown that f1n(n¡1)(n¡2):::76 < f1n(n¡1)(n¡2):::765 < f1n(n¡1)(n¡2):::7654, butf1n(n¡1)(n¡2):::76543 < f1n(n¡1)(n¡2):::7654, so, as expected, mergers which implytriopoly or duopoly are never stable. Monopoly is stable once it forms (that is,once there is an incentive to join to triopoly, i.e., for f > f1n(n¡1):::4(:)).

The following …gures illustrate for n = 8 the analysis for mergers and cartelswith this speci…c collusion path.

f187654

f18765

f1876f18 f187

M

α

f

r=2

r=3

r=4

r=5

Figure 4.2: Stable mergers (n=8)25The notation f1n(n¡1), for example, means that …rms 1 and n joined …rst and then …rm

(n-1) was aggregated. The order is not irrelevant, in fact f1n(n¡1) 6= f1(n¡1)n.26Outside …rms are the more e¢cient and therefore require a higher payment. In can also

be argued that the decline in the number of …rms as mergers occur increases the pro…ts of thestill stand-alone …rms and hence the value that the acquiring …rm has to pay for them (so akind of race for being the last to be acquired would take place). If the acquired …rm was paidaccording to the Shapley value, i.e., in accordance with its marginal contribution to the pro…t ofthe coalition, then this payment would increase in the degree of market concentration, as well.

24

f187654

f1876f18

f187

M

α

f

r=2 r=3r=4

r=5

f18765

fcint|r=4

fcint|r=5

fcint|r=3

fintM

Figure 4.3: Stable cartels (n=8)

According to our previous …ndings, when ® = 0 the only stable cartel hasjust two elements. However, for ® > 0 cartels with r ¸ 3 can be stable (providedthey lead to neither triopoly, nor duopoly), since as ® increases internal stabilityis more reinforced than external stability is weakened.

When we are dealing with cartels there can be some combinations f®; fg forwhich no agreement is stable, contrary to what occurs for mergers. Monopoly isnot stable for all f ¸ f1n(n¡1):::4(:), but only for higher f . Mergers with r > 2members clearly have a larger stability range than cartels with the same numberof elements. Given all the parameters of the problem either a unique stableoligopoly is determined or no stable oligopoly exists.

As is clear, the stability of an oligopolistic structure depends crucially onmany parameters: the di¤erences in e¢ciency between …rms (®), the value ofthe …xed cost each …rm has to pay in order to produce (f), the dimension of themarket and the common marginal cost component (³ = a ¡ c), and the numberof operating …rms (related to the initial number n).27

27Without complicating too much the analysis we could have considered a slightly more generaldemand function in which the sensitivity of quantity to price was not forced to -1. By assumingthe demand P = a ¡ bQ an additional parameter would enter the analysis. With b > 1 (resp.b < 1) the quantity produced would be smaller (resp. larger) than with b = 1 and the pricewould be higher (resp. lower). Coalitions’ external stability appears to be decreasing in b, which

25

Notice that departing from an industry totally dispersed or with a high levelof dispersion we can get complete monopolization by sequential acquisition, thatis, by allowing for successive rounds of mergers. This result is in line with thepapers by Kamien and Zang(1990, 1991 and 1993) in which the authors obtainthe impossibility of merging up to monopoly in just one round if the industry hassu¢ciently numerous …rms (…rst two papers), but show that sequential acquisitionmakes it easier to monopolize (the 1993 paper).

In this fourth section we have been dealing with the positive aspects of theformation of coalitions between asymmetric …rms. The following section presentssome results on welfare in the context of our model.

5. Welfare analysis

So far we have been dealing with the positive aspects of the formation of coalitionsbetween asymmetric …rms. A normative analysis on the welfare consequences ofthese operations and its implications for antitrust policy is the subject of thepresent section. We shall treat the case of two-member coalitions, so it is in-di¤erent if they are cartels or mergers, though we mainly use the denomination“mergers”. Full proofs can be found in the Appendix.

5.1. Social welfare

In analyzing the impact of a merger on social welfare we will …rst go brie‡ythrough the e¤ects on the various sectors (consumers, outside …rms, and partici-pants) and then concentrate on the overall economic impact.

When …rms are symmetric the e¤ect of a merger on consumer surplus is nega-tive and invariant with respect to the …rms that join (it only varies with n and ³);in an asymmetric industry, too, it is negative, but we can make it softer or harderdepending on the producer which is “eliminated”. As expected, consumers preferthat the high-cost …rms disappear from the market, so that production is moree¢cient and thus quantity is less reduced and price less increased in the sequenceof the merger. The deeper the asymmetries the more important this is.

As to the surplus of the …rms that do not participate in the merger, it can beshown that every outside …rm derives a positive bene…t from the merger of tworivals, independently of the value of ®; its production and its market share bothincrease.28 E¢cient …rms are those which bene…t most and the more e¢cient is

means that the lower the sensitivity of demand to price (and the price elasticity of demand), themore attractive it is for …rms to collude and exploit the resulting market power.

28The no-entry assumption is important here, for the increased pro…tability of the marketafter the merger might trigger the entry of further …rms and thereby erode the bene…ts of the

26

the …rm whose production ceases following the merger, the better it is for non-participants. This is a curious result for, somehow counterintuitively, it showsthat mergers between “giants” do not hurt competitors - on the contrary. Andthe competitors who bene…t most from these arrangements are exactly the closestrivals of the …rms that merge. This is a consequence of our constant marginalcost assumption.

Similarly to what happens under symmetry, there are thus positive external-ities for the rivals from two …rms merging.29 Notice however that it is not clearwhether or not the incentive to free-ride subsists in the asymmetric context, be-cause this will depend upon which …rms merge and on how they divide the jointpro…t (which, remember, we do not state), as well as on the speci…c outside …rmwe are looking at. Since the global gain of non-participating …rms is positive,30

the conditional Her…ndahl (the Her…ndahl of non-participating …rms) increaseswith the merger, i.e., production by non-participants becomes more concentrated(see the Appendix for the proof).

It is thus apparent that consumers and outside …rms are in con‡ict as to thetype of merger which is preferable in terms of welfare: since their interests arestrictly opposed, the merger will never yield a Pareto improvement. If the sumof the variations in consumer surplus and outside …rms’ surplus is taken as asu¢cient condition for the merger to be welfare-improving (in the spirit of Farrelland Shapiro(1990a)),31 then n ¸ 4 is found to be su¢cient for the merger to besocially desirable in the symmetric case,32 whereas under asymmetry a su¢cientcondition is n ¸ 6. The condition for the symmetric case is equivalent to amarket share of the participating …rms lower than 50 per cent. With asymmetrythis value is also su¢cient but there are some socially desirable mergers whichinvolve …rms producing ex ante up to 60% of the total quantity placed in themarket; however when the sum of the market shares exceeds 60% the merger

move.29See Boyer(1992) for an analysis of mergers that harm competitors. In his model, this happens

because mergers lead to a decrease in the level of marginal costs of the merged …rm.30Thus, according to the classi…cation introduced in section 2, our game is one with positive

spillovers. Negative spillovers arise in oligopolistic games where the formation of a coalition leadsto a reduction in the production cost of member …rms, making them behave more aggressively(for example, the case of research joint ventures), or, more generally, in games where the forma-tion of agreements in the …rst stage improves the position of cooperating …rms for the secondstage of the game.

31Which is equivalent to assuming the private pro…tability of the operation, on the beliefthat privately unpro…table mergers will not be proposed. This condition has the considerableadvantage of not requiring any information provided by the joining partners.

32See the more general results derived by Levin(1990) in studying “the 50-percent benchmark”and also Salant et al(1983) and Cheung(1992).

27

surely has negative aggregate e¤ects. As long as the market share of the joiningpartners satis…es these conditions it does not matter which type of agreementwe have, that is, whether it involves two …rms of approximate size or two veryasymmetric producers.

As we have seen above (see section 4.1) i = 1 and j = n is the mergerwhich maximizes internal gains, so consumers and member …rms have the samepreferences as to the technology of production that is going to be “abandoned”.The production of the new …rm is lower than the sum of the production of itscomponents before merger and so is its market share (although higher than themarket share of its largest component). The production increase of the outside…rms is not su¢cient, then, to compensate for the decline in participants’ output,since total quantity falls. The welfare e¤ects thus depend a lot on this reallocationof production, in the sense that the better the response of the rival …rms, the lessconsumers su¤er.

Let us consider total welfare changes and denote by

¢SW = ¢¼in + ¢¼out + ¢CS

the (variable) change in the level of aggregate social welfare in the sequence ofthe merger of …rms i and j > i; composed of the changes in participating …rms’pro…t, in outsiders’ aggregate pro…t and in consumer surplus (note that thereis also a …xed gain corresponding to the saving in …xed cost, which will not beconsidered here, but that reinforces the desirability of the operation). When…rms are identical a merger between two of them always hurts social welfareunless there are signi…cant …xed cost savings.33 The e¢ciency e¤ect associatedwith the operation when …rms are unequally e¢cient may render some mergerssocially desirable.

Recalling that the optimal j for consumers and participating …rms is j = nand that the optimal j for non-participating …rms is j = 2, the maximization ofour social welfare function, which gives the same weight to all agents, will yieldsome j¤ between 2 and n. If the weights were changed, j¤ would approach thelimits of this interval, accordingly. Choosing j = 2 never has a positive e¤ect onsocial welfare. On the contrary, j = n may have a positive e¤ect if ® is not toosmall.

There is a range of ® where the change in welfare is surely negative (unlessthe savings in …xed cost are su¢ciently high), no matter what technology “dis-

33This is not in contradiction with the former result of the external desirability of the mergerwhenever n ¸ 4. In fact that result is based on the presumption that participants bene…tfrom the operation, which is true only if there are signi…cant (…xed) cost savings (see Salant etal(1983)).

28

appears”, but as n increases this interval tends to shrink, so the higher n is, themore likely it is that the merger is socially bene…cial even with no …xed costsavings. The interval corresponds (approximately) to the 28% lowest values of ®when n = 3, 22% when n = 4, 9.5% when n = 10 (1% when n = 100): mergers inthis range should not be allowed unless savings in …xed cost are high enough tocompensate for this loss. Since we are not taking into account the …xed bene…t ofthe merger (f), mergers which are socially good in accordance with our analysisshall be authorized with accrued reason and mergers which are “almost there”may also be permitted if f is known to be signi…cantly di¤erent from zero.34

Due to the assumptions of linear demand and constant marginal costs, thesocial welfare variation can be written, by simple manipulation, as

¢SW = (Qn¡1jfi;jg)2 [Hn¡1jfi;jg + 12 ] ¡ (Qn)2 [Hn + 1

2 ]

where the terms with 12 refer to the e¤ect on consumer surplus and the terms

with the Her…ndahl index (H) refer to the e¤ect on all …rms in the industry(participants and outside …rms). Since total quantity declines (Qn¡1jfi;jg < Qn),a necessary condition for the merger to improve social welfare is Hn¡1jfi;jg > Hn.The increase in concentration is thus necessary (though it may not be su¢cient)for the overall economic e¤ect to be positive (since consumers lose, producers mustgain),35 which shows that refusing mergers primarily on the basis of the variationinduced in concentration or on the basis of established critical values for H maybe seriously misleading.36 It is good to let large …rms (which is equivalent toe¢cient, in our model) grow, especially at the expense of small (ine¢cient) ones.

34This is clearly a partial analysis. In a general equilibrium framework the e¤ects on all othersectors should also be taken into account.

35Notice that if we were increasing the number of operating …rms in the industry (insteadof decreasing it by merging) then the rise in H would no longer be necessary for the positivesocial e¤ect, since quantity would increase (this is clearly true in the symmetric case, where anincrease in n causes H to fall and SW to rise - see next footnote).

36This is exactly what Farrell and Shapiro(1990a) argued: when …rms are symmetric, in aconstant marginal cost Cournot oligopoly SW and H evolve in opposite directions; however if…rms are unequally e¢cient there is no reason to suppose that this relationship will still hold - adecrease in the number of …rms by merger may be welfare-improving (such as shifting productionfrom high-cost to low-cost producers) and yet it causes a rise in concentration. Consequentlythe concentration criterion shall not be used in isolation. Daughety(1990) also claims thatdecreasing concentration does not imply increasing welfare and vice-versa, in a model where…rms are identical but asymmetric in behavior (merged groups act as Stackelberg leaders, playingCournot against each other, whereas the fringe plays Cournot). A similar result is obtained forthem, and also for us, if a concentration ratio measure (for example C4) is used instead of H .For a wide discussion of the relationship between changes in H and changes in SW , see Farrelland Shapiro(1990b).

29

It can be proven that j¤ does not fall below 3n=4, so the optimal merger neverinvolves the absorption of a low-cost …rm.37 This kind of merger induces a smalldeterioration in the concentration index, so although welfare and concentrationboth evolve in the same direction, their variations seem to follow the (correct)inverse pattern.38 In the next subsection this relationship is discussed in moredetail. For the moment let us summarize the social welfare results in the following

Proposition 5.1. When …rms are not su¢ciently asymmetric (® < ®0(n), with®0 decreasing in n and ®0(3) = 0:28®, where ® is the upper limit for ® suchthat all …rms are active) the change in social welfare due to the merger of two…rms is negative unless there are signi…cant …xed cost savings associated with theoperation. When the asymmetry level is large enough (® > ®0(n)) the mergermay be socially desirable even without …xed cost savings. The welfare maximizingoperation, fi; j¤g, involves the absorption of an ine¢cient …rm (34n < j¤ · n).Proof. In the Appendix.

5.2. Concentration and welfare

Let Hn(®; ³) stand for the pre-merger Her…ndahl concentration index in our in-dustry. The level of concentration after the merger of …rms i and j > i is denotedby Hn¡1jfi;jg(®; ³).39 Its minimum occurs for j = n.

The change in the level of concentration is given by ¢H(®; ³; n; j), which isequal to Hn¡1jfi;jg(:)¡Hn(:). As we have seen in the previous subsection ¢H > 0is a necessary condition for social welfare to increase. The minimum increase inthe level of concentration is achieved, by the above argument concerning Hn¡1,when the least e¢cient …rm is “absorbed”, which is very intuitive, because this…rm is the smallest in the market. Thus, although there is an inevitable increasein concentration due to merger, the operation that maximizes social bene…ts(34n < j¤ · n) does not imply a too serious increase in H (which is the importantmeasure to look at), and actually the optimal mergers in both cases coincide for

37From a social point of view it is good to close down ine¢cient …rms, on the one hand, and,on the other hand, to let large …rms compete. This result is in accordance with the observationmade by Farrell and Shapiro(1990a): “(...) it often enhances economic welfare - de…ned in theusual way - to close down small or ine¢cient …rms, or, failing that, to encourage them to mergeso that they produce less output. This observation may call for some rethinking of our views onpolicy toward competition, including horizontal merger policy” (pgs.122-3).

38 In other words, the …rst derivative has the same sign for both, but second derivatives haveopposite signs.

39 It shall be calculated on the basis of the new market shares and not taking (erroneously, aswe saw before) the share of the resulting …rm as equal to the sum of the pre-merger shares ofthe participants.

30

most values of ®. However, for ® large the merger which maximizes the increasein social welfare (34n < j < n) induces an increase in concentration above theminimum (reached for j = n); in turn the merger which minimizes the variationin the level of concentration is not optimal with respect to social welfare bene…ts.

Proposition 5.2. The merger which minimizes the increase in concentration(j = n) and the merger which maximizes welfare gains (34n < j · n) coincide for® < ®00(n) and tend to diverge for high values of ® (®00(n) is decreasing in n andconverges to 1

2®, where ® is the upper limit for ® such that all …rms are active).Proof. In the Appendix.

The above results highlight the risks of an antitrust policy based simply onthe analysis of concentration indexes. The higher the di¤erences among …rms inthe industry, the more the welfare maximizing merger will be distinct from theone which minimizes the e¤ects on concentration.

6. Conclusion

We have analyzed the formation and stability of coalitions in an industry withasymmetric …rms and constant marginal cost where no rule of pro…t sharing is…xed and side transfers are allowed. Cartels and mergers are considered sepa-rately. They form whenever they are pro…table and for them to be stable, bothinternal and external stability are required for cartels, while for mergers (whichare assumed to be irreversible) external stability alone is necessary. As in Ra-jan(1989) the emerging oligopolistic structures are characterized as a function ofsome market parameters: the …xed cost (as in Espinosa and Iñarra(1995)) andthe e¢ciency asymmetry (to our knowledge, addressed for the …rst time). Thewelfare consequences of these agreements have also been studied.

The existing literature has been dealing in its majority with the formation ofcoalitions among identical …rms and when heterogeneity is allowed a …xed rulefor dividing the joint pro…t is assumed. Our paper is an attempt to overcomeboth limitations, admitting asymmetry and …xing no rule of pro…t sharing.

In our model, when two …rms join, aside from the savings in …xed costs there isan e¢ciency e¤ect associated with the transfer of production from the ine¢cientto the e¢cient …rm which also works in favor of the operation. It was shownthat this e¤ect dominates the rivalry e¤ect based upon which …rms would preferto join good partners and so, contrary to what occurs when side payments arenot allowed, the larger the asymmetry between merging parties, the higher thepro…tability of the agreement.

31

e¢ciency e¤ect associated with the spreading of a better technology by mergermay be su¢cient to guarantee the pro…tability of the move, even in the absence ofthe other factors that have been pointed to in the literature as possible solutionsto this problem, such as savings in …xed cost, di¤erentiated product, non-lineardemand and modi…cations in the behavior of the participants.

The subject of the endogenous determination of mergers and cartels withasymmetric …rms has many limitations arising from the di¢culty of the task. Ouranalytical framework is still simpli…ed in that costs are linear and so is demand.However it has allowed us to identify the main problems related to this subjectand to specify a mechanism for accurately determining the stable oligopoliesunder given realizations of the relevant parameters without need to …x the rule ofdivision of the joint pro…t. This type of approach can be useful in analyzing themerging phenomenon in many industries. It may facilitate the understanding ofsome acquisitions and even allow some prediction. We believe that we have alsoprovided some relevant insights for antitrust policy in asymmetric industries.

33

APPENDIXA. Number of partitions

The number of k partitions (partitions in k blocks) of a set with n elements iscalled stirling number of the second kind and denoted S (n; k). We have S (n; k) >0 for all 1 · k · n and S(n; k) = 0 if k > n ¸ 1. S(n; k) can be computedaccording to

S(n; k) = 1k!

P1·i·k

(¡1)k¡i¡ki

¢in

For example S (n; 1) = 1; S (n; 2) = 2n¡1¡1; S (n; 3) = (3n¡1+1)2 ¡2n¡1. This

says that the number of duopolies when n = 3 is 3 and when n = 4 is 7; thenumber of triopolies for n = 3 is 1 and for n = 4 is 6.

There exist “triangular”, “vertical” and “horizontal” recurrence relations thathelp in the computation of S(n; k) and which can be found in any advancedcombinatorics text book (see for example Comtet(1974), pgs 208-10).

For values of n and k up to considerably large limits there are double-entrybuilt tables that immediately tell the desired number. The increase with n ismore than exponential (for n = 15, for instance, there are 420,693,273 6-…rmoligopolies).

The number of all partitions of a set with n elements is called Bell number orexponential number and is denoted by $(n). The following de…nition is clear:

$(n) =P

1·k·nS(n; k); n ¸ 1

The recurrence relation $(n + 1) =P

0·h·n

¡nh

¢$(h); n ¸ 0 simpli…es the com-

putation of $(n) when it is not available in the previously built tables.

B. Stability analysis when n=3

With n = 3 there are three possible oligopolistic structures: monopoly (the grandcoalition, represented either by M or by {1,2,3}), duopoly and triopoly (repre-sented either by T or by {1}{2}{3}). There are three possible duopoly constel-lations: …rms 1 and 2 together against …rm 3 ({1,2}{3}), …rms 1 and 3 togetheragainst …rm 2 ({1,3}{2}) or …rms 2 and 3 together competing against …rm 1({1}{2,3}).

Variable pro…ts in each case are

34

¼M =¡a¡c+®

2

¢2

¼f1;2gf3g12 =³(a¡c)+3®

3

´2; ¼f1;2gf3g3 =

³(a¡c)¡3®

3

´2

¼f1;3gf2g13 =³(a¡c)+2®

3

´2; ¼f1;3gf2g2 =

³(a¡c)¡®

3

´2

¼f1gf2;3g23 =³(a¡c)¡®

3

´2; ¼f1gf2;3g1 =

³(a¡c)+2®

3

´2

¼T1 =³(a¡c)+4®

4

´2; ¼T2 =

³(a¡c)4

´2; ¼T3 =

³(a¡c)¡4®

4

´2

Denote the di¤erence (a ¡ c) by ³. Notice that for all …rms (in whateveroligopolistic structure) to produce positive quantities (that is, to be in the market)the parameter ® cannot exceed ³4 (since ® > ³

4 ) qT3 = 0) and the parameter f

cannot exceed³³¡4®4

´2(since f >

³³¡4®4

´2) ¼T3 ¡ f < 0, so qT3 will be equal

to 0). Denote these upper bounds by ® and f , respectively, and notice that ®is a function of ³ and f is a function of both ³ and ®. Let us now examine indetail the stability of the various oligopolistic structures. Recall that the grandcoalition is externally stable and that triopoly is internally stable. Recall alsothat our stability analysis is stepwise, so the only possible movements are: fromtriopoly to duopoly, from duopoly either to triopoly or to monopoly, and frommonopoly to duopoly.

To begin with, suppose that all …rms are operating separately. Then,

² Stability of triopoly

We only have to analyze one deviation:

¤ two …rms agree to join (external instability)

In this case the market will become a duopoly. To guarantee that this willnot happen we have to assure that the following three conditions are veri…ed:

¼T1 ¡ f + ¼T2 ¡ f ¸ ¼f1;2gf3g12 ¡ f¼T1 ¡ f + ¼T3 ¡ f ¸ ¼f1;3gf2g13 ¡ f¼T2 ¡ f + ¼T3 ¡ f ¸ ¼f1gf2;3g23 ¡ f

Solving for f we get

35

f · ftf1;2gf3g(³; ®) = ¡12®³+³272

f · ftf1;3gf2g(³; ®) = 112®2¡32®³+³272

f · ftf1gf2;3g(³; ®) = 64®2¡20®³+³272

where ftf1;2gf3g(³; ®), for example, is the superior limit for the stability oftriopoly as regards deviations to the duopoly f1; 2gf3g. Putting these three con-ditions together in a graph it appears that the binding one is ftf1;3gf2g(³; ®), andso triopoly is stable for values of ® and f low enough so that f · ftf1;3gf2g(³; ®).

fd1{1,2}{3}

fd1{1,3}{2}

fd1{1}{2,3}

f

α

Figure B.1: Triopoly stability

² Stability of the duopolies

We have to analyze two types of deviations from duopoly:

¤ one of the …rms which are together deviates and stays alone (inter-nal instability, relevant only if we are considering that …rms formcartels and not irreversible mergers)

In this case we will revert to triopoly. In order to avoid this we have to assurethat

¼f1;2gf3g(1) ¸ ¼T1 ¡ f

¼f1;2gf3g(2) ¸ ¼T2 ¡ f

s:t: ¼f1;2gf3g(1) + ¼f1;2gf3g(2) = ¼f1;2gf3g12

36

for the duopoly f1; 2gf3g, where ¼(i) denotes the unknown pro…t of …rm i(recall that it is unknown because …rms are di¤erent and we do not …x any ruleof payo¤ division). These conditions are equivalent to

¼f1;2gf3g12 ¡ f ¸ ¼T1 + ¼T2 ¡ 2f

Similarly we get the inequality

¼f1;3gf2g13 ¡ f ¸ ¼T1 + ¼T3 ¡ 2f

for the duopoly f1; 3gf2g and the inequality

¼f1gf2;3g23 ¡ f ¸ ¼T2 + ¼T3 ¡ 2f

for the duopoly f1gf2; 3g.Solving for f we get the three following conditions, which are exactly the re-

verse of the inequalities for the stability of triopoly vis à vis the various duopolies:40

f ¸ fd1f1;2gf3g(³; ®) = ftf1;2gf3g(³; ®) = ¡12®³+³272

f ¸ fd1f1;3gf2g(³; ®) = ftf1;3gf2g(³; ®) = 112®2¡32®³+³272

f ¸ fd1f1gf2;3g(³; ®) = ftf1gf2;3g(³; ®) = 64®2¡20®³+³272

By looking again at …gure B-1, it appears that the most stable duopoly re-garding moves to triopoly is the one formed by …rms 1 and 3 together against…rm 2 (because it is stable for the largest range of ® and k) and the least stableis the one formed by …rms 1 and 2 together against …rm 3. If the latter is stable,then all the others are, as well, but there are values of f and ® for which onlyf1; 3gf2g is stable and values of f and ® for which f1; 3gf2g and f1gf2; 3g arestable, but f1; 2gf3g is not. For f ¸ ³2=72 there is no risk of instability of anyduopoly versus triopoly and the same for ® ¸ ³=12 (since above this value allcurves lie below the horizontal axis). For ® < ³=28 all duopolies are unstableunless f is high enough, for ³=28<®<³=16 the duopoly f1; 3gf2g is internallystable for whatever f but the others only for f high enough, for ³=16<®<³=12duopolies f1; 3gf2g and f1gf2; 3g are internally stable for all f but the duopolyf1; 2gf3g is internally stable only for high f .

The intuition for these results is the following: since these conditions forstability have to do with no …rm of the block having an incentive to deviate,

40This is due to the fact that the coalitions whose internal stability we are analyzing are madeup of just two elements. For coalitions with three members or more this is no longer so.

37

the most internally stable duopoly is the one which entails the highest gain ascompared with the deviating situation. When …rms 1 and 3 join this gain ismaximized, because due to our constant marginal cost assumption, all productionis transferred from 3 to 1, meaning, from the least e¢cient to the most e¢cient…rm in the market, and thus the e¢ciency e¤ect is maximum; on the contrarywhen …rms 1 and 2 join the gain is minimized.

¤ the three …rms agree to move to monopoly (external instability)

This movement is always in the …rms’ interest, because monopoly pro…t ishigher than the sum of duopoly pro…ts (¼M > ¼f1gf2;3g1 + ¼f1gf2;3g23 = ¼f1;3gf2g13 +¼f1;3gf2g2 > ¼f1;2gf3g12 + ¼f1;2gf3g3 ), so duopoly is never externally stable and thus itis never stable.

² Stability of monopoly

Since we are considering separating movements by single …rms, there is onlyone possible deviation from monopoly, which is relevant only if we are consideringthat …rms form cartels (and not irreversible mergers, in which case monopoly isstable once it forms):

¤ one …rm deviates alone, leaving the other two together (internalinstability)

In this case we will revert to a situation of duopoly. In order to avoid this wehave to assure that each …rm receives in monopoly at least as much as it wouldreceive in the deviating situation. We thus have the following conditions whichguarantee that the deviation will not occur:

¼M(1) ¸ ¼f1gf2;3g1 ¡ f

¼M(2) ¸ ¼f1;3gf2g2 ¡ f

¼M(3) ¸ ¼f1;2gf3g3 ¡ fs:t: ¼M(1) + ¼M(2) + ¼M(3) = ¼M

Adding the three inequalities and subtracting the …xed cost from the variablemonopoly pro…t we get

¼M ¡ f ¸ ¼f1gf2;3g1 + ¼f1;3gf2g2 + ¼f1;2gf3g3 ¡ 3f

Solving for f we get the condition of internal stability:

38

f ¸ fm(³; ®) = 47®2¡34®³+3³272

If the e¢ciency di¤erence happens to be larger than 0.103³ or if the …xed costis larger than ³2=24, the grand coalition is stable. Higher values of ³ require highervalues of f for the stability of this cartel. On the contrary, higher values of ®facilitate its stability as f is decreasing in ®. The economic intuition behind theseresults is clear: the better the demand conditions (higher ³) the more attractiveit is for a …rm to separate from the others and try to make pro…t for itself; themore acute the di¤erences in e¢ciency, the more important is the e¢ciency e¤ectassociated with the coalition and the easier it is to compensate the “acquired”…rms (the ine¢cient ones).41

C. Results from section 4.

P = a ¡ Qn …rms, n oddMC1 = c ¡

¡n¡12

¢®; MC2 = c ¡

¡n¡32

¢®; ...; MCn = c +

¡n¡12

¢®

…xed cost f

qnn =a¡n(c+(n¡12 )®)+

n¡1Pi=1MCi

n+1 = ³¡(n2¡12 )®n+1 > 0 , ® < 2³

n2¡1 =® (³; n)

f < ¼nn = (qnn)2 , f < [ ³¡(n2¡12 )®n+1 ]2 =f (³; ®; n)

® (³; n) is a vertical line in the (®; f) space and f (³; ®; n) is a convex curve inthe same space.

MCi = c ¡¡n+1¡2i

2

¢®

fij(³; ®; n) = 14n2(n+1)2 [2³+®(n2+2n+1)¡2®j(n+1)][2³(n2¡2n¡1)¡2®j(n3+

n2 ¡ n ¡ 1) + 4®in(n + 1) + ®(n4 ¡ 4n2 ¡ 4n ¡ 1)]

C.1. Two-…rm agreements

Proof of Lemma 1:In this proof, such as in others below, we will abuse notation and take deriv-

atives in order of i and j, though they are discrete variables.fij is linearly increasing in i, so the lower i, the lower fij :

41For n = 3 it is easy to see that exiting in groups of two is less attractive than exiting alone(the stability condition in the …rst case lies below the stability condition in the second andactually below the horizontal axis for all ®). In general this is not clear under asymmetry.

39

@fij@i = ®(®+2³¡2®j+2®n¡2®jn+®n2)

n(n+1) > 0 , j < 2³+®(n2+2n+1)2®(n+1) ¸ n

fij is convex in j, the minimum being reached for n ¡ 1 < j < n :@2fij@j2 = 2®2(n2¡1)

n2 > 0@fij@j = 0 , j = j¤ = 2³(n2¡n¡1)+®(n4+n3¡2n2¡3n¡1)+2®ni(n+1)

2®(n¡1)(n+1)2

j¤ is increasing in i and decreasing in ®; which implies n ¡ 1 < j¤ < n:Therefore the minimum of fij occurs for i = 1; j = n ¡ 1 or i = 1; j = n. Let

us compare:

f1n ¡ f1(n¡1) = ®(®n4¡2³n2¡3®n2+2³n+2³+2®)n2(n+1) < 0 , ® < 2³(n2¡n¡2)

n4¡3n2+2 <® (³; n)

But for ® > 2³(n2¡n¡2)n4¡3n2+2 f1n < 0 and also f1(n¡1) < 0:

f1(n-1)

f1n

f

α_α

Figure C.1: Two-…rm agreements

Hence f1n < f1(n¡1) in the entire range where f1n and f1(n¡1) are both positiveand ® <® :

We still have to prove that f1n < f . Actually f ¡ f1n is concave in ® and > 08 0 · ® < ®, so f > f1n 8n; ³; ® 2 [0; ®):¥

Non-constant e¢ciency di¤erences:

Suppose n = 3, MC1 = c ¡ ®, MC2 = c, MC3 = c + ¯®. So far we have had¯ = 1; let us now consider ¯ > 1 and 0 < ¯ < 1 and see what happens to theprevious ordering f13 < f23 < f12. The relevance of this analysis can be based,

40

for example, on the following observation: if …rm 3 is much more ine¢cient than…rm 2 (and not just the same as …rm 2 is di¤erent from …rm 1), then does themost e¢cient …rm still prefer to join with 3? We show that if the e¢ciency e¤ectdominates the rivalry e¤ect when …rms di¤er equally, then this result is reinforcedwhen the least e¢cient …rm becomes clearly even more ine¢cient than the others.

The …gures below illustrate the three situations. As expected, for ¯ > 1 theordering is preserved, with f12 growing more and more apart from the other linesas ¯ increases (the previous dominance of the e¢ciency e¤ect over the rivalrye¤ect is reinforced), whereas f13 and f23 tend to become closer (1 and 2 areincreasingly more similar in terms of “acquiring” 3 as this …rm becomes moreand more ine¢cient relative to them). For 0 < ¯ < 1 the ordering can be eitherf13 < f12 < f23 or f13 < f23 < f12, depending on ¯ being < 0:75 or > 0:75,respectively; the lines f13 and f12 tend to converge as ¯ decreases, as …rms 2 and3 become closer.

f12

f23

f13

α

f

Figure C.2: Constant e¢ciency di¤erences: ¯ = 1

f23f12

f13

α

f

Figure C.3: Non-constant e¢ciency di¤erences: ¯ = 4

41

f23

f12

f13

α

f

Figure C.4: Non-constant e¢ciency di¤erences: ¯ = 0:1

In any case the minimum of fij when n is equal to 3 and the e¢ciency di¤er-ences between …rms are not constant is always f13. This result can be generalizedfor any n by induction and so we can conclude that the e¢ciency e¤ect alwaysprevails over the rivalry e¤ect when marginal cost is constant, independently ofthe e¢ciency di¤erence between “adjacent” …rms being constant or not.

More results on the pro…tability and internal stability of two-member coali-tions:

By maximizing fij = ¼ni + ¼nj ¡ ¼n¡1ij in i and j we obtain the least pro…tabletwo-…rm agreement, since it is the one which requires the highest values of ® andf to be pro…table. By performing the appropriate calculations f12 arises as themaximum of fij over i and j :

Given the shape of fij as a function of i and j (and given that j > i) the twonatural candidates for the highest fij appear to be f12 and f(n¡1)n:

f(n¡1)n ¡ f12 > 0 , ® > 2³(n2¡2n¡1)n3+n2¡n¡1 >® (³; n) 8n ¸ 4, which shows that

f12 > f(n¡1)n over the relevant range.However we still have to verify that f12 > f23; f24; :::; f2n (j is larger, but

i is also larger), that f12 > f34; f35; :::; f3n, ..., that f12 > f(n¡2)(n¡1) (we havealready seen that f12 > f(n¡1)n). But f23 > f24 > ::: > f2n, f34 > f35 > ::: >f3n, ..., so it su¢ces to show that f12 > f23, f12 > f34; ..., f12 > f(n¡2)(n¡1): Ifwe can prove that fi(i+1) > f(i+1)(i+2), then f12 > f23 appears as su¢cient. Butf12 > f23 is itself a condition of the type fi(i+1) > f(i+1)(i+2), so we have only toprove this last inequality:fi(i+1)¡ f(i+1)(i+2) = ®(2®¡2³+2®i+3®n¡4³n+6®in¡3®n2+2³n2+2®in2¡3®n3¡2®in3+®n4)

n2(n+1) ,

42

f12

f23f34

f13f24f14

α

f

Figure C.5: Internal stability of two-…rm coalitions (n=4)

fij1 ¡ fijn = ®n(n¡1)2 (¡® ¡ 2³ + 2®j + 4³n + 4®in + 4³in ¡ 4®jn ¡ 4®ijn ¡

6³n2 ¡ ®in2 ¡ 2³in2¡2®i2n2+6®jn2+2®ijn2¡3®n3+2³n3+2®in3+®i2n3¡2®jn3+®n4¡®in4)The di¤erence fij1 ¡ fijn is decreasing in j 8®; ³; n; i · n ¡ 2. Denote by j1

the value of j such that fij1 = fijn. Then

j1 = ®+2³¡4³n¡4®in¡4³in+6³n2+®in2+2³in2+2®i2n2+3®n3¡2³n3¡2®in3¡®i2n3¡®n4+®in42®(1¡2n¡2in+3n2+in2¡n3)

For j > j1 fij1 < fijn, that is, the most attractive enlargement is to include…rm 1, but if j < j1 then fijn < fij1 (the most attractive enlargement is to include…rm n). The choice of fij1 or fijn thus depends on the relative e¢ciency of thecoalition (i) as compared with the other …rms that are still operating (all but j).

fij1¡fijn is convex in i and has a single root in the relevant range (the secondroot is > n¡2 8j; n; ³; ®). Let us denote it by i1(®; ³; n; j). For i < i1 fijn < fij1and for i > i1 fij1 < fijn.

Proof of Proposition 2:

i) ® = 0 ) f extij ¡ f intij = ³2(2n3¡9n2+3)n2(n2¡1)2 > 0 8n ¸ 5

ii)@2(fij1¡f intij )

@j2 =@2(fijn¡f intij )

@j2 = 2®2(3¡6n+n2+2n3¡n4)n2(n¡1)2 < 0 8n ¸ 2.

It can be shown that fijn ¡ f intij is always increasing in j, because the valueof j for which fijn ¡ f intij is maximum is larger than (n ¡ 1); fij1 ¡ f intij is alsoincreasing if ® < ®0(³; n; i; j) (because the maximum occurs for j > n) and for

44

® > ®0 it can decrease for j high enough (j > j0(®; ³; n; i), which can be provento be > n ¡ 2).

@2(fij1¡f intij )@i2 = 2®2n(n¡2)

(n¡1)2 > 0 8n ¸ 2, so fij1 ¡ f intij is convex in i. It isdecreasing 8i, since we already know that f intij increases with i 8i (see proof ofLemma 4.1) and since the minimum of fij1 takes place for i > (n ¡ 1), so fij1decreases in all the relevant range. The optimal i is thus i1 such that fij1 = fijn.

@2(fijn¡f intij )@i2 = 0, so fijn ¡ f intij is linear in i.

@(fijn¡f intij )@i = 1

n(n2¡1) [2®(® +2³ ¡ 2®j + 2®n ¡ ®jn + ®jn2 ¡ ®n3)], which can be proven to be > 0 when® < ®00(³; n; j) and < 0 when ® > ®00. So, the optimal i can be i = i1 or i = 1.

@(fijn¡f intij )@® j®=0 = ³

n2(n3¡n2¡n+1) [3¡6j+3n¡4in+6jn¡11n2+4in2+10jn2¡3n3¡8jn3+6n4+2jn4¡2n5], which is increasing in i, so for values of ® su…cientlylow i = i1 induces higher stability than i = 1.

@(fijn¡f intij )@® j®=® = 2³

(n¡1)3(n2+n3) [¡3j + 6j2 + 3n ¡ 2in ¡ 6jn + 8ijn ¡ 12j2n +2n2 ¡ 4in2 + 18jn2 ¡ 12ijn2 +2j2n2 ¡ 10n3 + 10in3 ¡ jn3 + 4ijn3 +4j2n3 +n4 ¡4in4 ¡ 7jn4 ¡ 2j2n4 + 3n5 + 3jn5 ¡ n6], which is decreasing in i, so for values of® su…ciently close to ® i = 1 induces higher stability than i = i1.

@(fin(n¡1)¡f intin )@i = 2®(®+2³+®n)

n(n2¡1) > 0 8n ¸ 2: Denote by i2 the value of i suchthat fin(n¡1) = fin1. The optimal i in this case is thus i2.

Hence, the possible candidates to be the most stable two-…rm agreement are:fi = i2; j = ng, fi = i1; j = n ¡ 1g, and fi = 1; j = n¡ 1g. We have seen that forlow ® an intermediate i is optimal, whereas for ® close to its upper limit i = 1 isoptimal.¥

C.2. Broader agreements

Internal stability condition for coalitions with three members or more:Consider the three-member cartel fi; j; lg, with i < j < l. The condition that

implies that …rms do not want to exit from this cartel alone is ¼fi;j;lg¡f ¸ ¼ijfj;lg¡f +¼jjfi;lg¡f +¼ljfi;jg¡ f , f ¸ f1(i; j; l; n; ³; ®) = ¼ijfj;lg+¼j jfi;lg+¼ljfi;jg¡¼fi;j;lg

2On the other hand, the condition that implies that …rms do not want to exit

in groups of two is 2¼fi;j;lg ¡ 2f ¸ ¼fi;jgjflg ¡ f + ¼fi;lgjfjg ¡ f + ¼fj;lgjfig ¡ f ,f ¸ f2(i; j; l; n; ³; ®) = ¼fi;jgjflg + ¼fi;lgjfjg + ¼fj;lgjfig ¡ 2¼fi;j;lg.43

43This condition is derived from the following system:¼(i)jfi;j;lg + ¼(j)jfi;j;lg ¸ ¼fi;jgjflg ¡ f¼(i)jfi;j;lg + ¼(l)jfi;j;lg ¸ ¼fi;lgjfjg ¡ f¼(j)jfi;j;lg + ¼(l)jfi;j;lg ¸ ¼fj;lgjfig ¡ fs:t: ¼(i)jfi;j;lg + ¼(j)jfi;j;lg + ¼(l)jfi;j;lg = ¼fi;j;lg

45

So, it is not clear whether f1 > f2 or f2 > f1: the sum of the three positiveterms is higher in f2 than in f1 and also f2 is not being divided by 2, but thenegative term is much higher in f2 than in f1.

® = 0:fext1C = fextnC = ¼n¡r+1

c + ¼n¡r+1i jC ¡ ¼n¡rC[fig = 2

³³

n¡r+2

´2¡

³³

n¡r+1

´2

fprofC = ¼n¡r+2cnfig + ¼n¡r+2

i jCnfig ¡ ¼n¡r+1C = 2

³³

n¡r+3

´2¡

³³

n¡r+2

´2

fextC ¡ fprofC = ³2(¡17¡12n+3n2+2n3+12r¡6nr¡6n2r+3r2+6nr2¡2r3)(1+n¡r)2(2+n¡r)2(3+n¡r)2 > 0 8r · n ¡ 3,

since the numerator decreases in r 8r · n ¡ 2 and is positive for r = n ¡ 3.When considering mergers, monopoly is stable 8f > f extC (r = n ¡ 3; :) =

fprofC (r = n ¡ 2; :) = 7³2400

f intC = r(¼n¡r+2i jCnfig)¡¼n¡r+1

Cr¡1 =

r( ³n¡r+3)

2¡( ³n¡r+2)

2

r¡1f intC ¡ fprofC = ³2(r¡2)(5+2n¡2r)

(2+n¡r)2(3+n¡r)2(r¡1) > 0 8r > 2 (the fact that C forms is notsu¢cient to ensure that it will not be dissolved).

fextC ¡ f intC = ³2(27+36n+15n2+2n3¡58r¡54nr¡12n2r+39r2+18nr2¡8r3)(1+n¡r)2(2+n¡r)2(3+n¡r)2(r¡1) . The denomi-

nator of this expression is positive 8r > 1.When r = 2 the numerator is equal to (apart from ³2) 2n3 ¡ 9n2 + 3, which

is > 0 8n ¸ 5:r = 3 ) numerator (apart from ³2) = 2n3 ¡ 21n2 + 36n ¡ 12 > 0 8n ¸ 9r = 4 ) numerator (apart from ³2) = 2n3 ¡ 33n2 + 108n ¡ 93 > 0 8n ¸ 13r = 5 ) numerator (apart from ³2) = 2n3 ¡ 45n2 + 216n ¡ 288 > 0 8n ¸ 17r = 6 ) numerator (apart from ³2) = 2n3 ¡ 57n2 + 360n ¡ 645 > 0 8n ¸ 21r = 7 ) numerator (apart from ³2) = 2n3 ¡ 69n2 + 540n ¡ 1212 > 0 8n ¸ 25r = 8 ) numerator (apart from ³2) = 2n3 ¡ 81n2 + 756n ¡ 2037 > 0 8n ¸ 29It seems then that as r increases by one unit, n has to increase by four units

in order to preserve stability of the cartel, the minimum n required for each rbeing 5 + 4(r ¡ 2). Actually, for n = 5 + 4(r ¡ 2) the numerator takes thevalue (apart from ³2) r(9r ¡ 4), which is always positive, while for n one unitlower (n = 5 + 4(r ¡ 2) ¡ 1) the numerator is equal to ¡9r2 + 14r ¡ 5, whichis negative 8r ¸ 2. The numerator of f extC ¡ f intC is cubic in n and has onlyone real root, which, by the above arguing, must lie between n = 4r ¡ 4 andn = 4r ¡ 3. Taking the derivative of the numerator in order of n, one can seethat it is positive 8n > 3r ¡ 3 (negative otherwise), so the numerator is growingin n 8n ¸ 5 + 4(r ¡ 2) = 4r ¡ 3. This concludes the proof that fextC ¡ f intC > 08n ¸ 5 + 4(r ¡ 2).

46

n ¸ 5+4(r ¡2) , r · n+34 . For r = n+3

4 there are n¡ r +1 = 3n4 + 1

4 playersin the market, so more than 75% of the initial number of …rms are still operating.

@fextC@r = 2³2(¡6¡6n+n3+6r¡3n2r+3nr2¡r3)

(1+n¡r)3(2+n¡r)3 > 0 8r · n¡3, because the numeratordecreases in r 8r · n¡2 and is positive for r = n¡3, so it is positive 8r · n¡3.

For monopoly f intC = ³2(4n¡9)36(n¡1) (it su¢ces to replace r for n in the general

expression for f intC ).fprofC (r = n ¡ 1; :) = ³2

72 < f intM 8n ¸ 3, so f intM is the relevant condition.@f intM@n = 5³2

36(n¡1)2 > 0, i.e., the more members it has, the more di¢cult it is tosustain internal stability of the grand coalition.

® > 0 :Consider the generic coalition C = fe1; e2; :::; erg with members ranked in

descending order of e¢ciency and #C = r. To simplify, denote by x the marginalcost of the most e¢cient …rm in C (e1), by y the marginal cost of the last …rmthat entered the agreement (ef , say), and by z the sum of the marginal costs ofthe rivals of C (z =

Pi=2C

MCi).

@2(fext1C ¡fprofC )@z2 = @2(fextnC ¡fprofC )

@z2 = 1(1+n¡r)2(2+n¡r)2(3+n¡r)2 [2(¡17¡12n+3n2+

2n3 + 12r ¡ 6nr ¡ 6n2r + 3r2 + 6nr2 ¡ 2r3)]. The denominator of this expressionis always positive. As to the numerator, it is decreasing in r for all r < n ¡ 1(duopoly excluded, as usual); for r = n ¡ 2 it is negative, but for r = n ¡ 3 it ispositive, so it is positive 8r · n ¡ 3 (triopoly also excluded). So fextC ¡ fprofC isconvex in the sum of the marginal costs of the outsiders 8r · n ¡ 3.

@2(fext1C ¡fprofC )@x2 = 1

(2+n¡r)2(3+n¡r)2 [2(7 + 18n + 20n2 + 8n3 + n4 ¡ 18r ¡ 40nr ¡24n2r¡4n3r+20r2+24nr2+6n2r2 ¡8r3¡4nr3+r4)]. It can be shown that thenumerator is decreasing in r 8r; n and positive for the highest r, so it is alwayspositive, and hence f ext1C ¡ fprofC is convex in x.

@2(fextnC ¡fprofC )@x2 = 2(7+32n+27n2+6n3¡32r¡54nr¡18n2r+27r2+18nr2¡6r3)

(1+n¡r)2(2+n¡r)2(3+n¡r)2 . The nu-merator decreases with r and is positive for the maximum r, so it is alwayspositive, and hence fextnC ¡ fprofC is convex in the marginal cost of the most e¢-cient …rm in the agreement.

@2(fext1C ¡fprofC )@y2 = @2(fextnC ¡fprofC )

@y2 = 2(¡5¡4n¡n2+4r+2nr¡r2)(3+n¡r)2 , which is < 0 8n; r.

It is proven, then, that fextC ¡ fprofC is concave in the marginal cost of the last…rm to enter C.

47

Based on this analysis we can isolate the mergers which are the candidates tobe the most stable, and that are mentioned in the text.

As to cartels, it can be shown that fextC ¡ f intC is convex in the marginal costof the most e¢cient …rm in the cartel, concave in the marginal cost of the secondmost e¢cient …rm in the cartel, concave in the sum of the marginal costs of therest of the members of the cartel, and convex (concave) in the sum of the marginalcosts of the outside …rms for r · n+3

4 (for r > n+34 ). The possible candidates

are thus: fi intermediate; all the other members very inefficientg, for all r,fi = 1; all the other members very efficientg, for r · n+3

4 , and for r > n+34

fi = 1; all the other members intermediately efficientg, fi = 1; some of theother members very efficient and some very inefficientg, or fi = 1; all theother members very inefficientg. For most cartels observed r · n+3

4 , so the…rst two solutions are the more relevant in practice.¥

C.3. Speci…c collusion path

stabm(r = n¡h; :)¡stabm(r = n¡h+1; :) = 12h2(1+h)2(2+h)2(3+h)2 (18®2+72®³+

72³2 ¡ 42®2h + 48®³h + 264³2h ¡ 203®2h2 ¡ 248®³h2 + 172³2h2 ¡ 262®2h3 ¡260®³h3 ¡ 226®2h4 ¡ 72®³h4 ¡ 12³2h4 ¡ 146®2h5 ¡ 4®³h5 ¡ 58®2h6 ¡ 12®2h7 ¡®2h8+36®2n+72®³n+24®2hn+264®³hn¡124®2h2n+172®³h2n¡130®2h3n¡36®2h4n ¡ 12®³h4n ¡ 2®2h5n + 18®2n2 + 66®2hn2 + 43®2h2n2 ¡ 3®2h4n2)

This expression is concave in ® 8h; ³; n and negative 8® when h ¸ 5, that is,when r · n¡ 5. The stability area is thus increasing in r 8r · n¡ 5 (at least six(groups of) operating …rms in the market). For the minimum r (r = 2) we havestabm(r = 2; :) = f1n(n¡1) ¡f1n = 1

4n2(n2¡1)2 (3®2 +12®³ +12³2 ¡4®2n¡8®³n¡19®2n2¡40®³n2¡36³2n2¡10®2n3¡8®³n3+8³2n3+9®2n4+12®³n4+8®2n5¡®2n6 ¡ 2®2n7)which is concave in ® 8n, and positive 8n ¸ 5 in the entire range 0 · ® < ®where f1n and f1n(n¡1) are both > 0 (see proof of Lemma 4.1).

Hence there is a stability area when r = 2 8n ¸ 5 and so there is a stabilityarea 8r · n¡5, 8n; ³; ® 2 [0; ®(:)); f 2 [0; f(:)) (note that r · n¡5 already guar-antees n ¸ 5), that is, f1n < f1n(n¡1) < f1n(n¡1)(n¡2) < ::: < f1n(n¡1)(n¡2):::7 <f1n(n¡1)(n¡2):::76.

We then have:stabm(r = n ¡ 4; :) = f1n(n¡1)(n¡2):::765 ¡ f1n(n¡1)(n¡2):::76 =

= ®2(37n2+1614n¡189523)+®(148³n+3228³)+148³258800 , which is > 0 8®; ³; n ¸ 7

stabm(r = n ¡ 3; :) = f1n(n¡1)(n¡2):::7654 ¡ f1n(n¡1)(n¡2):::765 =

= ®2(7n2+614n¡38993)+®(28³n+1228³)+28³214400 , which is also > 0 8®; ³; n ¸ 7

48

stabm(r = n ¡ 2; :) = f1n(n¡1)(n¡2):::76543 ¡ f1n(n¡1)(n¡2):::7654 =

= ®2(¡13n2+1054n¡31333)+®(¡52³n+2108³)¡52³214400 , which is < 0 8®; ³; n, and hence

f1n(n¡1)(n¡2):::76543 < f1n(n¡1)(n¡2):::7654. Triopoly is not stable.

D. Results from section 5.

D.1. Social welfare

Consumer surplusConsidering the linear demand of our model, consumer surplus is given by

CS = Q2

2 .

In the pre-merger situation each …rm produces qni (®; ³) = ³+®[ (n+1)(n+1¡2i)2 ]

n+1 ;

total quantity in the market is Qn(³) =nPi=1

qni = ³nn+1 ; independent of ®,

and price Pn(a; c) = a ¡ Qn(³) = a+ncn+1 : Consumer surplus is equal to CSn(³) =

(Qn)22 = ³2n2

2(n+1)2

Qn(:) is increasing in ³ and in n. Pn(:) grows with a and c and falls with n.CSn(:) varies with its arguments in the same direction as Qn(:).

@Qn(:)@n = ³

(n+1)2 > 0@Qn(:)@³ = n

n+1 > 0Since Pn = a¡Qn, the proof that Pn increases with c and decreases with n is

immediate. As to the parameter of exogenous demand, @Pn(:)@a = 1

n+1 > 0: Since

CSn = (Qn)22 it is clear that the signs of the derivatives of Qn(:) apply to CSn(:)

too.

If …rms i and j > i merge …rm l 6= i; j has qn¡1l jfi;jg(®; ³) = ³+®[n2+1¡2j¡2n(l¡1)

2 ]n ;

the newly created …rm produces qn¡1i jfi;jg(®; ³) = ³+®[n2+1¡2j¡2n(i¡1)

2 ]n ;

total production is Qn¡1jfi;jg(®; ³) = ³(n¡1)n ¡ ®(n+1¡2j)

2n ;Pn¡1jfi;jg(®; a; c) = a+(n¡1)c

n + ®(n+1¡2j)2n ; and

CSn¡1jfi;jg(®; ³) = [2³(n¡1)¡®(n+1¡2j)]28n2 :

¢Q(®; ³; n; j) = Qn¡1jfi;jg(:)¡Qn(:) = ¡2³+®(n2+2n+1)¡2®j(n+1)2n(n+1) : This expres-

sion is negative whenever j < 2³+®(n2+2n+1)2®(n+1) ; which is larger than n 8® < ®(:),

49

so ¢Q < 0 8j; ® < ®; ³; n and ¢P (®; ³; n; j) = Pn¡1jfi;jg(:) ¡ Pn(:) = ¡¢Q > 08j; ® < ®; ³; n:44

¢CS(®; ³; n; j) = CSn¡1jfi;jg(:)¡CSn(:) < 0 8j; ® < ®; ³; n as an immediateconsequence of ¢Q < 0 and ¢P > 0:

@¢CS(:)@j = ®

2n2 [2³(n ¡ 1) ¡ ®(n + 1 ¡ 2j)] > 0 8® < 2³(n¡1)n+1¡2j ; which is larger

than ®(:) 8j, so @¢CS(:)@j > 0 8j; ® < ®; ³; n:@2¢CS(:)@j@® = ³(n¡1)¡®(n+1¡2j)

n2 , which is also > 0 8j; ® < ®; ³; n:Hence a merger between two …rms always involves a decrease in consumer

surplus. The more ine¢cient is the …rm whose technology is abandoned in thesequence of the merger, the softer are the e¤ects on the welfare of consumers(quantity is less reduced and price less increased). This e¤ect is reinforced forhigher ®.

Surplus of non-participating …rmsThe outside …rm l earns a variable pro…t

¼nl (®; ³) = (qnl (:))2 =

µ³+®[ (n+1)(n+1¡2l)

2 ]n+1

¶2

before the merger and

¼n¡1l jfi;jg(®; ³) = (qn¡1l jfi;jg(:))2 =µ³+®[n

2+1¡2j¡2n(l¡1)2 ]

n

¶2

after the merger. It

derives a bene…t denoted by¢¼l(®; ³; n; l; j) = ¼n¡1l jfi;jg(:) ¡ ¼nl (:) == (¡®¡2³+2®j¡2®n+2®jn¡®n2)(¡®¡2³+2®j¡4®n¡4³n+2®jn+4®nl¡5®n2+4®n2l¡2®n3)

4n2(n+1)2 fromthe operation.

Expanding ¢¼l(:) it becomes clear that it is convex in j, its minimum beingattained at j = ³

® + n2+2n+1¡2nl2 , which can be proven to be > n 8® < ®(:).

So ¢¼l(:) is decreasing in j 8j · n. For j = n its value declines with l; beingpositive for the maximum l (l = n¡1), so it is positive 8l and hence it is positive8j; l; ® < ®; ³; n:

@¢¼l(:)@l = ®(¡®¡2³+2®j¡2®n+2®jn¡®n2)

n(n+1) , which is < 0 for j < n+12 + ³

®(n+1) . Butthis value is larger than n for ® < ®(:), so it is clear that ¢¼l(:) decreases with l.

44This rise in price is a trivial consequence of the observation that the merger between …rmsi and j generates no synergies, for it only accounts for a better allocation of production acrossthese two …rms (see Proposition 2 of Farrell and Shapiro(1990a)). Signi…cant economies of scaleor learning e¤ects are required for an oligopolistic merger to increase aggregate industry outputand reduce price (again Farrell and Shapiro(1990a)).

50

Since we have proven that ¢¼l > 0, it is immediate that also ¢ql > 0 8l:

sni (®; ³) = qni (:)Qn(:) = 1

n + ®(n+1)(n+1¡2i)2³n , decreasing in i:

@sni (:)@® = (n+1)(n+1¡2i)

2³n = ¡ ³®@sni (:)@³ > 0 , i < n+1

2The market share of any …rm is an increasing function of ® (decreasing func-

tion of ³) if this …rm is among the more e¢cient (in the …rst half of …rms asranked according to e¢ciency) and diminishes with ® (increases with ³) if the…rm is a high-cost …rm (in the second half as ranked according to e¢ciency).

sn¡1l jfi;jg(®; ³) = qn¡1l jfi;jgQn¡1jfi;jg = 2³+®(n+1)2¡2®(j+nl)

2³(n¡1)¡®(n+1)+2®j

¢sl(®; ³; n; l; j) = sn¡1l jfi;jg(:) ¡ snl (:) = [2³+®(n+1¡2l)][2³+®(n+1)(n+1¡2j)]2³n[2³(n¡1)¡®(n+1¡2j)] > 0

8® < ®; ³; n; j; l. This is proven by showing that the …rst term in the numeratoris positive for ® < ® and l · n, the second is positive for ® < ® and j · n, andthe denominator is positive for ® < ® and j · n; so the ratio is positive:

@¢sl(:)@l = ®[2³+®(n+1)(n+1¡2j)]

³n[¡2³(n¡1)+®(n+1¡2j)] < 0, bearing in mind the analysis just madefor ¢sl(:). This result means that the market share of the more e¢cient outside…rms is more increased than the market share of the ine¢cient.

¢¼out(®; ³; n; i; j) =Pl 6=i;j

¢¼l(®; ³; n; l; j) =

= 14n2(n+1)2 [(® + 2³ ¡ 2®j + 2®n ¡ 2®jn + ®n2)(¡2® ¡ 4³ + 4®j ¡ 7®n ¡ 6³n +

4®in + 6®jn ¡ 8®n2 + 4³n2 + 4®in2 + 2®jn2 ¡ 3®n3)]

¢¼out =nPl=1l6=i;j

(qn¡1l jfi;jg)2¡nPl=1l6=i;j

(qnl )2 =

=nPl=1l 6=i;j

(sn¡1l jfi;jg:Qn¡1jfi;jg)2¡nPl=1l 6=i;j

(snl :Qn)2 =

= (Qn¡1jfi;jg)2:nPl=1l 6=i;j

(sn¡1l jfi;jg)2 ¡ (Qn)2:nPl=1l 6=i;j

(snl )2 ,

, ¢¼out = (Qn¡1jfi;jg)2:Hn¡1c ¡ (Qn)2:Hncwhere Hnc denotes the conditional Her…ndahl index in the n-…rm oligopoly.

Bearing in mind that total quantity decreases in the sequence of the merger, so(Qn¡1jfi;jg)2 < (Qn)2, and that ¢¼out > 0, Hn¡1c > Hnc .

51

Consumers and non-participating …rms’ welfareDenote by ¢CS¼out(®; ³; n; i; j) = ¢CS(®; ³; n; j) + ¢¼out(®; ³; n; i; j) =1

8n2(n+1)2 (® + 2³ ¡ 2®j + 2®n ¡ 2®jn + ®n2)(¡3® ¡ 6³ + 6®j ¡ 12®n ¡ 12³n +8®in+10®jn¡15®n2+4³n2+8®in2+4®jn2¡6®n3) the variation in the welfareof consumers and outside …rms in the sequence of the merger of …rms i and j > i.

® = 0 ) ¢CS¼out(³; n) = ³2(2n2¡6n¡3)2n2(n+1)2 > 0 8n ¸ 4. When n = 3 the merger

is socially harmful.This is the familiar result which says that in a symmetric industry with more

than three …rms operating at constant marginal cost and facing linear demand,antitrust authorities should not prevent the occurrence of proposed mergers (in-volving two …rms). The fact that ¢CS¼out is negative for n = 3 points outthe social undesirability of merging to duopoly, as also noted by Farrell andShapiro(1990a, pg.118).

For ® 6= 0 ¢CS¼out(:) is an increasing function of i : @¢CS¼out(:)@i = 1n(n+1) [®(®+

2³ ¡2®j +2®n¡2®jn+®n2)], which is > 0 for j · n and ® < ®.45 It is concavein j, since @

2¢CS¼out(:)@j2 = ¡®2(2n+3)

n2 < 0, so its minimum is reached for i = 1 andj = 2 or j = n:

¢CS¼out(i = 1; j = 2; :)¡¢CS¼out(i = 1; j = n; :) = 12n2(n+1) [®(n¡2)(¡3®¡

6³ + 7®n ¡ 8³n + 2®n2 + 2³n2 ¡ 2®n3)], which is < 0 8® < ® and n ¸ 7, beingnull for n = 6. For n = 3; 4; 5 it is negative for ®0 = 2³(n2¡4n¡3)

2n3¡2n2¡7n¡3 < ® < ® andpositive for ® < ®0.

The minimum of ¢CS¼out(:) is thus reached for i = 1 and j = 2 8n ¸ 6:with these values for i, j and n it is concave in ®, being positive for ® = 0 (asexpected) and for ® = ® (null for n = 6 and ® = ®), so, due to monotonicity, wecan conclude that ¢CS¼out(:) > 0 8i; j 8® < ®;8n ¸ 6:

For n = 5 ¢CS¼out(:) > 0 80 < ® < ® 8i ¸ 2; j > i and also when i = 1 andj ¸ 4. For i = 1; j = 3 it is positive for ® low, more precisely 80 < ® < 17³

240 , i.e.,in 85% of the range of variation of ® and < 0 817³

240 < ® < ®; for i = 1; j = 2 wehave ¢CS¼out(:) > 0 80 < ® < 17³

318 , which corresponds to approximately 64% ofthe range allowed.

45So the higher i, the more desirable is the merger. Notice that i is the …rm which is notconsidered in the expression of ¢CS¼out and which is assumed to bene…t from the merger. Thefact that i is high means that two ine¢cient …rms are excluded from this calculation (sincej > i), so the value of ¢CS¼out tends to rise with i.

52

we have j¤ = n and since 2³(2n+1)2n4+2n3¡n2¡2n¡1 < 2³(n+1)

2n3¡n¡1 ¢SW can be negative, aswe have seen above.

¢SW =nPl=1l 6=j

(qn¡1l jfi;jg)2¡nPl=1

(qnl )2 + (Qn¡1jfi;jg)2

2 ¡ (Qn)22 =

=nPl=1l 6=j

(sn¡1l jfi;jg:Qn¡1jfi;jg)2¡nPl=1

(snl :Qn)2 + (Qn¡1jfi;jg)2

2 ¡ (Qn)22 =

= (Qn¡1jfi;jg)2:nPl=1l 6=j

(sn¡1l jfi;jg)2 ¡ (Qn)2:nPl=1

(snl )2 + (Qn¡1jfi;jg)2

2 ¡ (Qn)22 ,

, ¢SW = (Qn¡1jfi;jg)2:[Hn¡1 + 12 ] ¡ (Qn)2:[Hn + 1

2 ]

Proof of Proposition 3:We have seen above that ¢SW (j = n; :) is negative for ® lower than its

…rst root (which is 2³(2n+1)2n4+2n3¡n2¡2n¡1) and that j¤ = n for ® · 2³(n+1)

2n3¡n¡1 . Since2³(2n+1)

2n4+2n3¡n2¡2n¡1 < 2³(n+1)2n3¡n¡1 8³; n, then j¤ = n for ® < 2³(2n+1)

2n4+2n3¡n2¡2n¡1 and so¢SW is surely negative in that interval of ® 8j; ³; n. For 2³(2n+1)

2n4+2n3¡n2¡2n¡1 < ® ·2³(n+1)2n3¡n¡1 j¤ is still equal to n, but then the merger with j¤ is socially desirable aswe have seen above. Finally, for ® > 2³(n+1)

2n3¡n¡1 j¤ < n, but then ¢SW (j¤; :) > 0

too, so ¢SW (j¤; :) > 0 8® > 2³(2n+1)2n4+2n3¡n2¡2n¡1 :

@j¤@® = ¡ ³(n+1)

®2(2n2+2n+1) < 0Because j¤ is decreasing in ® its minimum is attained for ® = ®, where it

takes the value n(3n2+5n+2)

4n2+4n+2 . Dividing it by n, observing that this ratio declineswith n and then taking the limit as n ! 1 we see that j¤ does not fall below3n4 :¥

D.2. Concentration and welfare

Hn(®; ³) =nPi=1

(sni )2 = 1

n + ®2(n4+2n3¡2n¡1)12³2n

® = 0 ) Hn = 1n

As expected Hn(:) is increasing in ® and decreasing in ³ (thus decreasing inthe dimension of the market a and increasing in the marginal cost component c).

@Hn(:)@® = ®(n¡1)(n+1)3

6³2n = ¡ ³®@Hn(:)@³ > 0 8® < ®

Hn¡1jfi;jg(®; ³) = 13(®+2³¡2®j+®n¡2³n)2 (¡3®2 ¡ 12®³ ¡ 12³2 + 12®2j + 24®³j ¡

55

12®2j2¡9®2n¡12®³n+12³2n+24®2jn¡12®2j2n¡12®2n2+24®2jn2¡12®2j2n2¡10®2n3 + 12®2jn3 ¡ 3®2n4 + ®2n5)

@Hn¡1jfi;jg(:)@j = 4®2n3[®(n2¡1)¡6³(n+1¡2j)]

3[®(n+1¡2j)¡2³(n¡1)] : The denominator is < 0 8j · n; ³; n;® < ®. For all 0 < ® < ® the numerator decreases with j when j · n¡1

2 andincreases with j when j ¸ n¡1

2 ; so Hn¡1jfi;jg increases with j when j is small anddecreases when j is large.

In order to …nd the minimum of Hn¡1jfi;jg(:) let us compare Hn¡1jfi;jg(j =2; :) with Hn¡1jfi;jg(j = n; :) :

Hn¡1jfi;jg(j = 2; :)¡Hn¡1jfi;jg(j = n; :) = 4®2n3(n¡2)(®2+16®³+12³2+®2n¡6®³n+2®³n2)3(n¡1)(2³+®)2(¡3®+2³+®n¡2³n)2

The long term in the numerator is convex in ® and increasing 8 0 < ® < ®; beingpositive for ® = 0, so it is positive 8®; ³; n. Therefore Hn¡1jfi;jg(j = 2; :) >Hn¡1jfi;jg(j = n; :) and so it is proven that the minimum of Hn¡1jfi;jg(:) occursfor j = n.

The sign of the derivative of ¢H(®; ³; n; j) with respect to ®; ³ and n dependson the realizations of the parameters. It can be proven that it varies in oppositedirections with ® and ³.

@¢H(:)@® = ¡ ³®

@¢H(:)@³ ; so ¢H varies in opposite directions with ® and ³:

Proof of Proposition 4:

Recall that j¤ = (n+1)[2³+®(2n2+2n+1)]2®(2n2+2n+1) .

Then, as we have seen before, j¤ · n , ® ¸ 2³(n+1)2n3¡n¡1 < ®. Therefore j¤ = n

for all 0 < ® · 2³(n+1)2n3¡n¡1 and j¤ = (n+1)[2³+®(2n2+2n+1)]

2®(2n2+2n+1) < n 8® > 2³(n+1)2n3¡n¡1 :

2³(n+1)2n3¡n¡1® = (n+1)2

2n2+2n+1 is decreasing in n and is equal to (approximately) 0.64for n = 3, 0.61 for n = 4, 0.59 for n = 5; 0.55 for n = 10, 0.51 for n = 50. In thelimit this ratio is equal to 1

2 , so j¤=n whenever the e¢ciency di¤erence is amongthe 50% lowest values allowed, independently of n.¥

56

References

[1] Aumann, R.(1959), Acceptable points in general cooperative n-person games,in Contributions to the Theory of Games IV, Princeton University Press.

[2] Aumann, R.(1967), A survey of cooperative games without side payments,in Essays in Mathematical Economics-in honor of Oskar Morgenstern, ch.1,Shubik, M. (ed.), Princeton University Press.

[3] Barros, P.(1998), Endogenous mergers and size asymmetry of merger partic-ipants, Economics Letters 60, 113-9.

[4] Bernheim, D., Peleg, B. and Whinston, M.(1987), Coalition-proof Nash equi-libria - I. Concepts, Journal of Economic Theory 42, 1-12.

[5] Bernheim, D. and Whinston, M.(1987), Coalition-proof Nash equilibria - II.Applications, Journal of Economic Theory 42, 13-29.

[6] Bloch, F.(1995), Endogenous structures of association in oligopolies, RandJournal of Economics 26, 537-56.

[7] Bloch, F.(1996), Sequential formation of coalitions in games with externali-ties and …xed payo¤ division, Games and Economic Behavior 14, 90-123.

[8] Bloch, F.(1997), Noncooperative models of coalition formation in games withspillovers, mimeo, Groupe HEC, Jouy-en-Josas.

[9] Bloch, F. and Ghosal, S.(1996), Buyers’ and sellers’ cartels on markets withindivisible goods, mimeo, Groupe HEC, Jouy-en-Josas, July.

[10] Bloch, F. and Ghosal, S.(1997), Stable trading structures in bilateraloligopolies, Journal of Economic Theory 74, 368-84.

[11] Boyer, K.(1992), Mergers that harm competitors, Review of Industrial Or-ganization 7, 191-202.

[12] Brealey, R. and Myers, S.(1988), Principles of corporate …nance, McGraw-Hill.

[13] Budd, C., Harris, C. and Vickers, J.(1993), A model of the evolution ofduopoly: does the asymmetry between …rms tend to increase or decrease?,Review of Economic Studies 60, 543-73.

[14] Carraro, C. and Siniscalco, D.(1993), Strategies for the intern

[15] Cheung, F.(1992), Two remarks on the equilibrium analysis of horizontalmerger, Economics Letters 40, 119-23.

[16] Chwe, M.(1994), Farsighted coalitional stability, Journal of Economic The-ory 63, 299-325.

[17] Comtet, L.(1974), Advanced combinatorics - the art of …nite and in…niteexpansions, D. Reidel Publishing Company.

[18] Cowling, K. and Waterson, M.(1976), Price-cost margins and market struc-ture, Economica 43, 267-74.

[19] d’Aspremont, C., Jacquemin, A., Gabszewicz, J. and Weymark, J.(1983), Onthe stability of collusive price leadership, Canadian Journal of Economics 16,17-25.

[20] Daughety, A.(1990), Bene…cial concentration, American Economic Review80, 1231-7.

[21] Davidson, C. and Deneckere. R.(1984), Horizontal mergers and collusive be-havior, International Journal of Industrial Organization 2, 117-32.

[22] Deneckere, R. and Davidson, C.(1985), Incentives to form coalitions withBertrand competition, Rand Journal of Economics 16, 473-86.

[23] Donsimoni, M.-P.(1985), Stable heterogeneous cartels, International Journalof Industrial Organization 3, 451-67.

[24] Donsimoni, M.-P., Economides, N. S. and Polemarchakis, H. M.(1986), Sta-ble cartels, International Economic Review 27, 137-27.

[25] Espinosa, M. P.(1996), Farsightedness in coalition formation, mimeo, Uni-versidad del País Vasco, February.

[26] Espinosa, M. P. and Iñarra, E.(1995), Von Neumann and Morgenstern stablesets in a merger game, SEEDS Discussion Paper #143, December.

[27] Farrell, J. and Scotchmer, S.(1988), Partnerships, Quarterly Journal of Eco-nomics 103, 279-97.

[28] Farrell, J. and Shapiro, C.(1990a), Horizontal mergers: an equilibrium analy-sis, American Economic Review 80, 107-26.

[29] Farrell, J. and Shapiro, C.(1990b), Asset ownership and market structure inoligopoly, Rand Journal of Economics 21, 275-92.

58

[30] Faulí-Oller, R.(1995), Takeover waves, PhD. Dissertation, UniversitatAutònoma de Barcelona.

[31] Faulí-Oller, R.(1996), Mergers for market power in a Cournot setting andmerger guidelines, CEPR Discussion Paper #1517.

[32] Faulí-Oller, R.(1997), On merger pro…tability in a Cournot setting, Eco-nomics Letters 54, 75-9.

[33] Fisher, F.(1987), Horizontal mergers: triage and treatment, Journal of Eco-nomic Perspectives 1, 23-40.

[34] Gaudet, G. and Salant, S.(1991), Increasing the pro…ts of a subset of …rmsin oligopoly models with strategic substitutes, American Economic Review81, 658-65.

[35] Gilbert, R. J. and Newbery, D. M.(1992), Alternative entry paths: the buildor buy decision, Journal of Economics & Management Strategy 1, 129-50.

[36] Granero, L.(1997), Horizontal mergers in large markets, mimeo, Institutd’Anàlisi Econòmica (CSIC), Barcelona, June.

[37] Greenberg, J.(1979), Existence and optimality of equilibrium in labour-managed economies, Review of Economic Studies 46, 419-33.

[38] Greenberg, J.(1990), The theory of social situations-An alternative game-theoretic approach, Cambridge University Press.

[39] Greenberg, J.(1994), Coalition structures, in Handbook of Game Theory withEconomic Applications, vol.2, ch.37, Aumann, R. and Hart, S. (eds.), North-Holland.

[40] Guesnerie, R.(1977), Monopoly, syndicate, and Shapley value: about someconjectures, Journal of Economic Theory 15, 235-51.

[41] Habib, M., Johnsen, D. and Naik, N.(1997), Spino¤s and information, Jour-nal of Financial Intermediation 6, 153-76.

[42] Hart, S. and Kurz, M.(1983), Endogenous formation of coalitions, Econo-metrica 51, 1047-64.

[43] Horn, H. and Persson, L.(1997a), Endogenous mergers in concentrated mar-kets, mimeo, Institute for International Economic Studies, Stockholm Uni-versity, May.

59

[44] Horn, H. and Persson, L.(1997b), The equilibrium ownership of an interna-tional oligopoly, mimeo, Institute for International Economic Studies, Stock-holm University, August.

[45] Jacquemin, A. and Slade, M.(1989), Cartels, collusion, and horizontalmerger, in Handbook of Industrial Organization, vol.1, ch.7, Schmalensee,R. and Willig, R. (eds.), North-Holland.

[46] Kamien, M. and Zang, I.(1990), The limits of monopolization through ac-quisition, Quarterly Journal of Economics 105, 465-99.

[47] Kamien, M. and Zang, I.(1991), Competitively cost advantageous mergersand monopolization, Games and Economic Behavior 3, 323-38.

[48] Kamien, M. and Zang, I.(1993), Monopolization by sequential acquisition,Journal of Law, Economics & Organization 9, 205-29.

[49] Kwoka, J.(1989), The private pro…tability of horizontal mergers with non-Cournot and Maverick behavior, International Journal of Industrial Organi-zation 7, 403-11.

[50] La Manna, M.(1993), Asymmetric oligopoly and technology transfers, Eco-nomic Journal 103, 436-43.

[51] Leibenstein, H.(1966), Allocative e¢ciency vs X-e¢ciency, American Eco-nomic Review 56, 392-415.

[52] Levin, D.(1990), Horizontal mergers: the 50-percent benchmark, AmericanEconomic Review 80, 1238-45.

[53] Macho-Stadler, I., Pérez-Castrillo, D. and Ponsati, C.(1994), Stable multi-lateral trade agreements, Working Paper 281.94, Departament d’Economiai d’Història Econòmica- Universitat Autònoma de Barcelona and Institutd’Anàlisi Econòmica (CSIC).

[54] Matutes, C. and Padilla, J.(1994), Shared ATM networks and banking com-petition, European Economic Review 38, 1113-38.

[55] McAfee, R. P. and Williams, M.(1992), Hor P55.ome om. an0 TD (5) Tj 5.5359 0.0.0528 Tc (P) Tj 6.48 02 0 TD0 TD 0 TD 0 Tc (5) Tj 5.4 (r) Tj 4.32 0 TD 0.0 0 TD TD 0 TD 0 Tc (530182 Tc (o) Tj 5.52 0 TD -0.0077 T (m)8 Tc (P)u TD -0.0072 Tc (m) Tj 9.12 0 TD -0.0298 Tc (sTD 0 TD 0 Tc 0.60.03 Tc (-) Tj -357.48 -13.56 TD -0.0072 Tc 50 (.) Tj 8.04 0 TD 0528 Tc (P) l0 TD Tj 3 0 TD (l) Tj 3 0 (g) Tj 8.76 0 TD 0.0182 Tc 28 Tc (P) y0 TD 0.0182 T2 0 TD 0.0326 Tc (,) Tj 6.72 0 TD3Tj 8.4 JTj 6.48 02 0 TD (n) Tj 6.24 0 TD -0.0014 Tc (o) Tj 7.32 0 TD -0.0485 Tc (u) Tj 5.88 0 TD 0502 0 TD0 ) Tj 5.52 0 TD 0.0072 Tc (m) Tj 4.44 0 TD -0.0014 Tc (t) 52c (P) l0 TD 0.0509 Tc (n) Tj 6.24 0 TD -0.0014 Tc (o) Tj 5.04 f TD -00014 Tc (o)50j 6.24 I Tc (5) Tj 5.4 (r) Tj 5.52 0 TD 0.0355 Tc (n) Tj 6.24 d TD -0.0014 Tc (o) Tj 7.32 0 TD -0.0485 Tc (u4) Tj 8.40 TD -04 0 TD 0528 470 TD 0 Tc 0.0437 Tc(u) Tj 5.88 0 TD 0502 0 TD (v) Tj 5.04 0 TD -0.0293 Tc (i) Tj 4.44 0 TD -0.0014 Tc (t) 52c (P) l0 TD 0.0509 Tc (n) /F4 11.04 Tf -0.0451 Tc (E) Tj 7.44 0 TD -0.0384 Tc (c) Tj 4.44 0 TD -0.0014 Tc (a) Tj 5.64 0 TD 0.0072 Tc (m) Tj 4.44 0 TD -0.0014 Tc (o) Tj 5.52 0 TD 92 0 TD (v) Tj 5.04 0 TD -0.0293 Tc (i) Tj 7.44 0 TD -0.0182 Tc 284) Tj 8.40 TD 9326 Tc () Tj 12.12 0 TD /F4(9) Tj 5.4 0 TD 0Tj 8.04 0 TD 0528 Tc (P) 0 TD 0.0509 Tc(() Tj 4.32 0 TD 0 Tc (3) Tj 5.4 0 TDj 4.32 0 TD 0 TD (3) Tj 5.4 0 TD 0.0437 Tc (-) T75

[ 5] Moar P 5an ao.(11, c mt vi . ov ui im.oc

vi i mimt . T j 5 . 5 2 - T c 0 . 7 0 . 0 3 T c ( - ) T j - 3 5 7 T j 5 . 5 2 0 T D 8 0 . 0 4 8 5 T c ( v ) T j 7 . 4 4 0 T D 5 0 . 0 5 0 9 T c ( . ) T j 5 . 5 2 0 T D 0 . 0 0 7 2 T c 5 2 8 4 7 0 T D 0 T c 0 . 0 4 3 7 T ( ) T j 1 2 . 1 2 0 2 8 T c ( P ) 0 T D 0 0 3 2 6 T c ( M ) T j 8 . 0 4 0 T D - 0 . 0 3 8 4 T c ( c )mP530437 Tc528 Tc (P) TD (l) Tj 3 0 (g) Tj 8.76 0 TD 08509 Tc6 0 TD-0.0182 Tc 284P530437 Tc528ea 5P

[57] Neven, D., Nuttall, R. and Seabright, P.(1993), Merger in daylight - theeconomics and politics of European merger control, CEPR.

[58] Nilssen, T. and Sorgard, L.(1998), Sequential horizontal mergers, EuropeanEconomic Review 42, 1683-702.

[59] Osborne, D. K.(1976), Cartel problems, American Economic Review 66, 835-44.

[60] Palepu, K.(1986), Predicting takeover targets, Journal of Accounting andEconomics 8, 3-35.

[61] Peleg, B.(1984), Game thoeretic analysis of voting in committees, CambridgeUniversity Press.

[62] Perry, M. and Porter, R.(1985), Oligopoly and the incentive for horizontalmerger, American Economic Review 75, 219-27.

[63] Porter, R.(1983), A study of cartel stability: the Joint Executive Committee,1880-1886, Bell Journal of Economics 14, 301-14.

[64] Rajan, R.(1989), Endogenous coalition formation in cooperative oligopolies,International Economic Review 30, 863-76.

[65] Ray, D. and Vohra, R.(1996), A theory of endogenous coalition structure,mimeo, Boston University, September.

[66] Ray, D. and Vohra, R.(1997), Equilibrium binding agreements, Journal ofEconomic Theory 73, 30-78.

[67] Sadanant, A. and Sadanant, V.(1996), Firm scale and the endogenous timingof entry: a choice between commitment and ‡exibility, Journal of EconomicTheory 70, 516-30.

[68] Salant, S., Switzer, S. and Reynolds, R.(1983), Losses from horizontalmerger: the e¤ects of an exogenous change in industry structure on Cournot-Nash equilibrium, Quarterly Journal of Economics 98, 185-99.

[69] Salop, S.(1987), Symposium on mergers and antitrust, Journal of EconomicPerspectives 1, 3-12.

[70] Scherer, F. and Ross, D.(1990), Industrial market structure and economicperformance, Houghton Mi­in.

61

[71] Schmalensee, R.(1987), Horizontal merger policy: problems and changes,Journal of Economic Perspectives 1, 41-54.

[72] Selten, R.(1973), A simple model of imperfect competition, where 4 are fewand 6 are many, International Journal of Game Theory 2, 141-201.

[73] Sha¤er, S.(1994), Structure, conduct, performance and welfare, Review ofIndustrial Organization 9, 435-50.

[74] Sha¤er, S.(1995), Stable cartels with a Cournot fringe, Southern EconomicJournal 61, 744-54.

[75] Shapiro, C.(1989), Theories of oligopoly behavior, in Handbook of Indus-trial Organization, vol.I, ch.6, Schmalensee, R. and Willig, R. (eds.), North-Holland.

[76] Stigler, G.(1950), Capitalism and monopolistic competition: I. The theory ofoligopoly-Monopoly and Oligopoly by Merger, American Economic Review,40, Papers and Proceedings, 23-34.

[77] Szidarovszky, F. and Yakowitz, S.(1982), Contributions to Cournot oligopolytheory, Journal of Economic Theory 28, 51-70.

[78] Tirole, J.(1992), Collusion and the theory of organizations, in Advances inEconomic Theory, vol.2, ch.3, La¤ont, J.-J. (ed.), Cambridge UniversityPress.

[79] von Neumann, J. and Morgenstern, O.(1944), Theory of games and economicbehavior, Princeton University Press, Princeton, New Jersey.

[80] White, L.(1987), Antitrust and merger policy: a review and critique, Journalof Economic Perspectives 1, 13-22.

[81] Williamson, O.(1968), Economies as an antitrust defense: the welfare trade-o¤s, American Economic Review 58, 18-36.

[82] Yi, S.-S.(1996), Endogenous formation of customs unions under imperfectcompetition: open regionalism is good, Journal of International Economics41, 153-77.

[83] Yi, S.-S.(1997), Stable coalition structures with externalities, Games andEconomic Behavior 20, 201-37.

[84] Yi, S.-S. and Shin, H.(1995), Endogenous formation of coalitions in oligopoly,Working Paper 95-2, Department of Economics, Dartmouth College.

62