Multi-irm Mergers with eaders and ollowers Gamal Atallah 1 University of Ottawa Deember 2011 Department of Eonomis, University of Ottawa, P.O. Box 450, STN. A, Ottawa, Ontario, Canada, 1 gatllah@uottawa.a, Tel.: 1-613-562-5800(1695), ax: 1-613-562-5999.
Multi-irm Mergers with eaders and ollowers Gamal Atallah University of Ottawa Abstrat This paper analyzes mergers involving leaders and followers in Stakelberg markets. Adding a follower to a merger inreases its profitability or redues its losses. A merger between one leader and any number of followers is profitable. When a merger involves two leaders, it requires a suffiiently high proportion of followers to partiipate in it. A merger is less likely to be profitable when the number of partiipating leaders is intermediate and the number of partiipating followers is low. All mergers involving leaders and followers are welfare reduing. Overall, Stakelberg leadership partly alleviates the merger paradox. Keywords: Mergers, Merger profitability, Merger paradox, Stakelberg JE odes: D43, 13 1. Introdution The study of mergers has revealed, starting with Salant et al. (1983) but also going bak to Stigler (1950), that mergers in a Cournot market may be unprofitable unless they involve a substantial proportion (80%) of firms in the industry. Beause the merger entails an output expansion by outsiders, if the number of the latter is large relative to the number of insiders, this expansion an be large enough to make the merger unprofitable to the merging firms. Yet, the empirial evidene does not point to a systemati unprofitability of mergers. Bruner (2002) reviews 130 empirial studies on merger profitability between 1971 and 2001, and onludes that the shareholders of target firms gain, while the returns to the shareholders of the buying firms are lose to zero. Sherer (1980) finds that on average the private gains from mergers are either negative or almost nil. Nothing suggests that the bulk of these mergers satisfy the high threshold (required for profitability) imposed by the stati Cournot model. In fat, bilateral mergers are observed in most industries (Offie of air Trading, 1999). The bulk of the empirial evidene on mergers points in the same diretion: mergers are profitable more often than what is suggested by the theoretial models. Therefore, there must be mehanisms that allow firms to benefit from suh mergers. Different mehanisms have been proposed through whih firms an evade the merger paradox: aess to sare apital (Perry and Porter, 1985); produt differentiation with Bertrand ompetition (Denekere and Davidson, 1985); non-cournot behavior (Kwoka, 1989); apaity
onstraints (Baik, 1995); properties of the demand funtion (auli-oller, 1997; Hennessy, 2000; Cheung, 1992); the short run vs. the long run (Polasky and Mason, 1998); hoie of produt range (ommerud and Sörgard, 1997); dynami Cournot ompetition (Dokner and Gaunersdorfer, 2001); open-loop vs. losed-loop strategies (Benhekroun, 2003); improved information flows inside the merged entity (Huk et al. (2004); intra-firm oordination (Higl and Welzel, 2005); and setting ompetition between the internal divisions of the merged firm (Creane and Davidson, 2004). 2 Of partiular relevane to the present paper is the interation between market leadership and the profitability of mergers. All the papers mentioned above analyzed mergers in a Cournot setting. The analysis an be different in a Stakelberg market, where some firms at as leaders and others as followers. A small number of papers have looked at the interation between Stakelberg leadership and merger profitability. Huk et al. (2001) and Kabiraj and Mukherjee (2003) find that two leaders (followers) have an inentive to merge iff there are no other leaders (followers). In addition, a merger between one leader and one follower is always profitable. Daughety (1990) onsiders the possibility that two followers merge to form a leader, and finds that suh a merger is always profitable. evin (1990) analyzes mergers starting from a Cournot setting, but allowing the merged firms to have a more general onjetural variation (for instane, beome a Stakelberg leader); he finds that suh a merger, if it indues a redution in output of the merged entity, is profitable iff it involves firms with at least a 50% pre-merger market share. Gelves (2008) and Takarada and Hamada (2006) analyze mergers involving leaders and followers, but all fous on mergers involving only two firms. One ommon assumption of the studies mentioned above analyzing Stakelberg markets is that only bilateral mergers are onsidered, whether the merger involves two leaders, two followers, or a follower and a leader. Yet, just as in a Cournot market, a merger may involve more than two firms. A omplete analysis of the profitability of mergers in Stakelberg markets requires the onsideration of mergers between arbitrary numbers of firms, involving any number of leaders and followers. To address this issue, the urrent paper sets up a Stakelberg market with leaders and 2 See Huk et al. (2005) for a survey. 2
followers. Three types of mergers are onsidered: mergers involving leaders only, followers only, and leaders and followers. or the latter type of merger, it is assumed that the merged entity forms a leader firm. or mergers between firms of the same type, it is shown that the same ritial threshold found in Cournot markets is appliable. Hene, a merger between leaders (followers) is profitable iff it involves 80% of leaders (followers). The analysis is different, however, when a merger involves both leaders and followers. Adding a follower to a merger always inreases its profitability or redues its losses. A merger between one leader and any number of followers is always profitable. When a merger involves two leaders, it requires a suffiiently high proportion of followers to partiipate in it, and this proportion dereases with the number of followers and inreases with the number of leaders present in the market. In partiular, a merger an be profitable even when the number of leaders partiipating is smaller than the standard threshold, provided the number of followers partiipating in the merger is high enough (and provided there are enough followers around to make this possible). A merger is less likely to be profitable when the number of partiipating leaders is intermediate, and the number of partiipating followers is low. This is beause with this onfiguration, the merged entity does not have enough market power (there are still many leaders around) to ompensate for the lost profits of the partiipating leaders, and the low number of followers makes their ontribution to the profitability of the merger insuffiient to ompensate for that disadvantage. When all followers partiipate to a merger, this merger is more likely to be profitable when the number of followers in the market is large. While mergers with followers are often appealing to leaders, the effets on onsumers and welfare are less enouraging. Any merger involving at least one leader and one follower, with the merged entity beoming a leader, redues onsumer surplus and welfare, and inreases total profits and the profits of eah outsider. Overall, the paper shows that Stakelberg leadership partly alleviates the merger paradox. Hene, this paper ontributes to the literature on merger profitability by analyzing the profitability of mergers involving leaders and followers. It ontributes to the literature on mergers between asymmetri firms by expliitly onsidering the asymmetri leader/follower roles in 3
Stakelberg markets. It extends the analysis of papers having onsidered mergers in Stakelberg markets by onsidering not only bilateral mergers, but also mergers involving any number of leaders and followers. The paper is organized as follows. The next setion presents the basi model and analyzes the profitability of mergers involving leaders only or followers only. Setion 3, whih onstitutes the ore of the paper, analyzes mergers involving both leaders and followers. It is divided into three parts. The first part derives some general results regarding mergers involving leaders and followers; the seond part looks at some speifi market strutures whih help illustrates the general results derived in the first part; the third part analyzes the welfare impliations of suh mergers. The last setion onludes the paper. 2.The basi model and mergers of leaders only or followers only There are leaders and followers ompeting in a Stakelberg market. Market demand is given by p=p(y)=1-y, where Y=y +y denotes total output, and where y denotes the output of an 3 individual firm. or simpliity, there are no prodution osts. The leaders determine their outputs simultaneously, and, after observing the leaders outputs, the followers hoose their outputs. The profit of a leader is given by ð =P(Y)y, while the profit of a follower is ð =P(Y)y. The outputs in the pre-merger symmetri equilibrium are given by (1) This results in per firm profits as a funtion of the number of leaders and followers: (2) Consider now how a merger an upset this equilibrium. We start with a merger involving only leaders. et Nå[2,] leaders merge and form one firm, so that the number of leaders beomes -N +1, while the number of followers is unhanged at. Substituting these values into (2) yields 3 It is well known that the profitability of mergers when firms are symmetri with onstant marginal osts and linear demand does not depend on the inverse demand interept/slope, or on the marginal ost. Hene there is no loss of generality from this assumption. 4
the per firm post-merger profits: (3) This merger is profitable iff the profits made by the merged firm after the merger exeed the profits they made before the merger, that is, iff Substituting (2) and (3) into (4) and solving for N shows that (4) is satisfied iff (4) (5) where N denotes the ritial threshold for profitability. Note that N does not depend on. Moreover, this is the same ritial threshold for the profitability of a merger in a Cournot market. Therefore, the presene of followers by itself does not alleviate the merger paradox: a substantial proportion (at least 80%) of leaders still need to merge to ensure profitability. Beause the merger entails a ontration of the output of the outsider leaders, the number of merging leaders must be substantial enough for this expansion not to hurt the merged entity too muh. Moreover, even though followers also expand their output following the merger by leaders (this is lear from (1)), this does not hange N relative to a Cournot market. We next analyze the profitability of a merger involving only followers. et Nå[2,] be the number of merging followers. There are now leaders and -N +1 followers after the merger. Note that ontrary to Daughety (1990), we do not assume that the merged entity beomes a leader. The profits following suh a merger are given by (6) This merger is profitable iff Using (2) and (3), this ondition is satisfied iff (7) (8) 5
Note that N =N. Therefore there is a perfet symmetry between the profitability (or not) of a merger involving only leaders and a merger involving only followers: the same proportion of firms is required in eah ase. rom (1) we know that the output of leaders is independent of, hene the only effet of the merger by followers is to indue an expansion of the output of outsiders. The result of this subsetion an be summarized as follows. Proposition 1. In a Stakelberg market, a merger between N leaders is profitable iff N >N. A merger between N followers is profitable iff N >N. Moreover, N =N =N, where N is the ritial threshold for profitability of a merger in a Cournot market, and is given by (9) 4 where N is the number of firms in the Cournot market. These results indiate that if followers (leaders) do not partiipate to a merger, they have no effet on the profitability of a merger involving only leaders (followers). They onstitute a generalization of the results of Huk et al. (2001) and Kabiraj and Mukherjee (2003) who show that two leaders (followers) have an inentive to merge iff there are no other leaders (followers). The results are different, however, when mergers between leaders and followers are allowed, as we will now see. 3. Mergers involving both leaders and followers We now onsider the more general ase where a merger involves both leaders and followers, whih onstitutes the fous of this paper. It is assumed that at least one leader and one follower partiipate in those mergers. Moreover, the merged entity beomes a leader. Hene, with Nå[1,] leaders and Nå[1,] partiipating to the merger, the number of leaders after the merger is -N +1+1, while the number of followers is -N. Suh a merger redues the number of followers more than it redues the number of leaders, beause the followers disappear. The profits after suh a merger 4 See Salant et al. (1983), footnote 6. 6
are given by (10) This merger is profitable iff (11) There is a lous of ritial values (N,N ) whih determine whether the merger is profitable or not. Muh of the analysis that follows aims at haraterizing this lous. Consider first how the partiipation of a follower to a merger affets its profitability. Proposition 2. If, for a given N, a merger is profitable when N followers partiipate in it, then it is also profitable when N +1 followers partiipate. If, for a given N, a merger is unprofitable when N followers partiipate in it, then it is also unprofitable when N -1 followers partiipate. Proof. (12) The gain to the merging entity of adding one follower exeeds the profit that was made by that follower prior to the merger. In a sense, followers benefit the merging leaders by allowing themselves to be absorbed by the merged entity. rom (1) we know that the output of a leader does not depend on the number of followers. By bringing one more follower into the merger, the output expansion of followers is redued (beause there are less followers to expand their output following the merger) and the output of eah remaining follower is redued (beause there are less followers), while the output expansion of eah leader outsider is unhanged. Hene total expansion of output by outsiders is redued, the prie is higher, and the profits of the post-merger entity are higher. And beause the profit that was made by the follower was initially small, the gain that his partiipation provides is always suffiient to ompensate for that lost profit. We now onsider the role of leaders. We start by looking at mergers involving one leader and any number of followers. This type of merger is ommon, sine it is easier for a leader to aquire a 7
small follower than to aquire a leader of equal size. Huk et al. (2001) and Kabiraj and Mukherjee (2003) find that a merger between a leader and a follower is always profitable. Proposition 2 generalizes this result, by showing that a merger between one leader and any number of followers is profitable. Proposition 3. A merger between one leader and any number of followers is always profitable. Proof. Evaluating (11) at N =1 yields Hene, beause more followers inrease the profitability of a merger, a leader and any number of followers have an advantage in merging together. This result hanges the reeived wisdom about merger profitability. With more than one leader present in the market, the merger involving one leader and any number of followers does not satisfy the standard threshold for profitability: the partiipating leader represents less than 80% of leaders. And yet, suh a merger is profitable. This is due to the partiipation of the follower(s) to the merger. We now turn to mergers involving two leaders and a number of followers. Suh mergers are also important, both theoretially and empirially. Theoretially, beause one of the often ited results of the merger paradox literature is that bilateral mergers are unprofitable whenever there are outsiders. And empirially, beause bilateral mergers between large firms are ommon, and these firms may want to further monopolize the market by predating on smaller firms. Proposition 4. or a merger involving two leaders to be profitable, it must involve at least N 2 followers, with (14) 2 0.5 where k (5-8-4). 2 N inreases linearly with, and inreases at a dereasing rate with. The proportion of followers 8
2 required for suh a merger to be profitable, N /, dereases with and inreases with. Proof. Substituting N =2 into (11), setting it equal to zero, and solving for N yields (14). (15) 2 This last derivative is independent of, implying that N inreases linearly with. (16) (17) (18) Given that N is inreasing in, the proportion N / is also inreasing in. 2 2 If =2, the ondition is automatially satisfied sine in this ase N 2 <0, and the merger is profitable. However, even if >2, the merger will be profitable if enough followers join in. The partiipation of followers, as explained above, does not hange the output of leaders but inreases the market prie, making the merger more profitable. The ritial number of followers required varies with the market struture. It inreases with, beause the more there are followers, the more there are outsiders who will expand their output after the merger, and the more of those outsiders must be absorbed by the merger for it to be profitable. But the proportion of followers required, N 2 /, dereases with, indiating that when the number of followers is very large, a lower proportion of them is required for profitability. N 2 also inreases with, beause the larger the number of leaders outside the merger, the larger is their output expansion, and the more followers are needed to partiipate to ompensate for that effet. The 2 proportion of followers inreases with beause N inreases with. Hene, while more followers in the market inrease the sope of profitable mergers, they also imply that a larger number of them (but a smaller proportion) need to join to make the merger profitable. ollowers partiipating in the merger are a plus for the merging firms, but followers 9
remaining outside the merger redue the profitability of the merger. The alaimed result that in Cournot markets bilateral mergers are unprofitable, and that even in a Stakelberg market, two leaders (followers) have an inentive to merge iff there are no other leaders (followers) (Huk et al., 2001; Kabiraj and Mukherjee, 2003) relies heavily on the assumption that no follower is part of that merger. Before haraterizing how in general the range of profitable mergers looks, it is neessary to establish how a merger that would be profitable in a Cournot market (or, equivalently, that would be profitable in a Stakelberg market, but in whih no followers partiipate) is affeted by the partiipation of followers. The following orollary, whih follows from the results already derived, answers this question. Corollary 1. A merger involving N N leaders is profitable irrespetive of the number of followers partiipating in the merger. Proof. rom proposition 1 we know that a merger involving leaders only is profitable iff N N. And from proposition 2 we know that adding a follower to a merger inreases its profitability. Hene, if the number of leaders satisfies the unidimensional threshold N, then adding any number of followers to this merger maintains (atually, enhanes) its profitability. The results derived until now indiate that followers have a diret bearing on the profitability of mergers involving leaders and followers. ollowers are a desirable target for any solo leader: they make it more likely that a merger involving two leaders is profitable, and they enhane the profitability of mergers that were already profitable. We are now ready to haraterize the profitability of mergers with an arbitrary number of leaders, in partiular with N å[2,n ]. - + - + Proposition 5. Unprofitable mergers are in the range Nå[N,N ], with N >1 and N <N. That is, the mergers are unprofitable for intermediate values of N. Moreover, N is nondereasing and N is noninreasing in N. + - 10
Proof. The first laim of this proposition follows from propositions 1 and 3. Proposition 3 - + establishes that N >1, while proposition 1 establishes that N <N. - + The effet of N on N and N is hard to establish without fixing and. or now we fix and and illustrate the result. In the next setion it will beome lear that this result is valid for more general market strutures. et ==100. or a given value of N, solving (11) for the ritial value(s) of N suh that the merger is just profitable yields two roots. The first of these roots is what we all - + - + N. The seond root is N. igure 1 plots both urves. N is nondereasing and N is noninreasing + in N. On this figure is plotted also N, whih is an upper bound to N. [igure 1 here] igure 1 illustrates how the range of profitable mergers varies with N and N. Consider a - + given value of N. When N <N or N >N, that is, when the number of leaders involved in the - + merger is either low or large, the merger is profitable. When, however, Nå[N,N ], so that the number of leaders partiipating is intermediate, the merger is unprofitable. The intuition for this result is that the profits of leaders are initially high, therefore a substantial degree of market power of the post-merger firm is required to over their lost profits. When the number of leaders involved is intermediate and the number of followers is low, the market power of the post-merger firm is insuffiient to ompensate for the lost profits, and the merger is not profitable. - + The result that N is nondereasing and N is noninreasing in N (see the last laim of proposition 5 and figure 1) means that the partiipation of more followers to the merger redues the number of leaders required for profitability. Moreover, the partiipation of followers redues the number of leaders required for profitability ompared with the ase where only leaders merge. Similarly, the partiipation of leaders redues the number of followers required for profitability ompared with the ase where only followers merge. This result reverses one of the main results of the merger paradox literature, that if a merger by a speified number of firms auses losses (respetively, gains), a merger by a smaller (larger) number of firms will ause losses (gains). (Salant et al., 1983:193), so that the solution to merger unprofitability was to enlarge the pool of partiipating firms. In a Stakelberg market, a merger may 11
beome unprofitable by adding more leaders, and may beome profitable by removing leaders. Until now we onentrated on the number of leaders required for profitability, noting that followers failitate and enhane profitability. We now take a loser look at the market side of followers. In fat, for a ertain number of followers to join a merger and make it profitable, there must be enough followers in the market in the first plae. At the same time, as explained in proposition 4, the number of followers required to make a merger involving two leaders (this will also be true for any N ) profitable inreases with. The following two propositions larify the role of in merger profitability. Consider the profitability of mergers involving all followers in the market. Is this suffiient to ensure profitability? The following proposition gives a negative answer to this question. Proposition 6. There exists a ritial number of followers that must be present in the market prior to the merger,, suh that, for a merger involving N leaders and all followers (N =), the merger is profitable iff >. Moreover, inreases with, and inreases and then dereases with N. Proof. Setting N = in (11), we an solve for : (19) This funtion depends in a omplex way on and N. It is illustrated on figure 2. It inreases steadily with, and inreases and then dereases with N. [igure 2 here] Proposition 6 takes the extreme ase of all followers joining the merger and asks the question: is this suffiient to ensure profitability? The answer to that question depends on and N. The ritial threshold indiates the minimal number of followers in the market suh that, if all followers joint the merger, it will beome profitable. This threshold inreases with : as the number of leaders beomes large relative to the number of followers, the likelihood inreases that even if all followers join, the merger will not be profitable (for a given N ). Hene, a small number of followers redues the sope of benefitting from them to ensure merger profitability. Also, inreases and then dereases with N. As explained above, mergers are more likely to be profitable when a small or a 12
large number of leaders join in. When N is low, the merger is more likely to be profitable, and less followers are needed (assuming they all partiipate in the merger) to make the merger profitable. As N inreases, however, the merger beomes unprofitable, and more followers are needed to join the merger, hene the total number of followers available for the merger must inrease. Beyond a ertain point, however, the joining leaders have less and less market power prior to the merger, and they require a lesser inrease in profits by the merged entity, hene the need for followers delines, and delines. The result that inreases and then dereases with N (from proposition 6) is another faet of the result that when mergers are unprofitable, they are so for intermediate numbers of leaders partiipating in the merger (from proposition 5). reahes its maximum for intermediate values of N, whih means that it is for those values of N that the most followers are needed to make the merger profitable. Whereas, is lower at low and high values of N, meaning that for those values of N only a few followers are needed to make the merger profitable. As for mergers involving only a portion of followers, figure 1 also gives, indiretly, the - + number of followers required for a merger to be profitable, for a given N. N and N define a funtion, N suh that a merger is profitable iff N >N. This funtion has an inverted-u shape. or N very low, the merger is profitable with any stritly positive number of followers. or N very high, in partiular for N >N, the merger is profitable even without any followers. In between, N first inreases then dereases with N. It is easy to verify that the number of follower required for profitability, N, follows the same pattern as : it inreases linearly with, and inreases and then dereases with N. The following orollary takes this argument one step further, and asks the question: suppose all followers join the merger; when are enough of them around for a merger to beome profitable, irrespetive of the number of leaders joining in (but at least one, to indue leadership)? Corollary 2. If > =max, then a merger involving all followers is profitable for any number of leaders partiipating in the merger. Moreover, is inreasing in. Proof. This follows from proposition 6. And is inreasing in beause is inreasing in. 13
This orollary says that if is high enough relative to, then a merger where all followers join will be profitable, irrespetive of N. The ritial value of required for this to our is, for a given, the maximum number of followers that was required by a speifi number of leading firms. If is large enough for the merger to be profitable with this value of N, then it will also be large enough for the merger to be profitable for any other value of N. This is why is obtained by maximizing over N : if the merger is profitable under harsh onditions (N is suh that a very large number of followers are needed), then it will be profitable under easier onditions (N requires less followers to join the merger to make it profitable). ollowers provide a double benefit from the point of view of merger profitability for leaders. On the one hand, the standard threshold is unhanged from a Cournot market, hene even if they don t join a merger they do not redue the likelihood of its profitability. On the other hand, they an inrease the likelihood of a merger beoming profitable by partiipating in it, and ating as substitutes for other leaders who haven t joined. igures 3 and 4 illustrate the regions of profitable and unprofitable mergers, using the different thresholds derived above, in two ases: the ase where >, so that a merger involving all followers is profitable for any number of leaders partiipating in the merger (figure 3), and the ase <, so that even partiipation by all followers does not guarantee profitability (figure 4). These two figures inorporate also mergers involving only leaders or only followers. [igures 3 and 4 here] 3.4 Speifi market strutures Now that we have analyzed some general properties of mergers in Stakelberg markets, we turn to some speifi market strutures to obtain more speifi results. Three symmetri and two asymmetri market strutures are onsidered: (=2,=2), (=3,=3), (=10,=10), (=10,=3) and (=3,=10). These hoies represent suffiient variety in terms of degree of onentration and asymmetry between leaders and followers to illustrate the results derived above. Consider first the market strutures (=2,=2) and (=3,=3). In the (=2,=2) market, there are four possible mergers (with at least one leader and one follower): (N =1,N =1), (N =1,N =2), (N =2,N =1), and (N =2,N =2). In the (=3,=3) market, there are 9 possible mergers. 14
Proposition 7. With (=2,=2) or (=3,=3), all mergers involving at least one leader and one follower, with the merged entity ating as a leader, are profitable. Proof. Substituting =2 and =2 into (11) yields (20) This expression is positive for all ombinations of N å{1,2} and N å{1,2}. Substituting =3 and =3 into (11) yields (21) This expression is positive for all ombinations of N å{1,2,3} and N å{1,2,3}. In the (=2,=2) market, all possible mergers are profitable. Exept for the merger (N =2,N =2), all other 3 mergers entail that only 50% of leaders and/or 50% of followers are engaged in the merger. This is well below the 80% threshold found when only leaders or only followers merge together (see proposition 1). Yet, these merges are profitable. They are profitable beause they entail the partiipation of firms with different market positions. Take the merger (N =2,N =1). This merger is profitable even though it involves only one follower; by inorporating the two leaders into the merger, the total output expansion by outsiders is redued, and the remaining follower annot hurt the merging entity. The other market struture onsidered in proposition 7 is (=3,=3). There are 9 possible mergers, all of them profitable. Take for instane the merger (N =2,N =1). This merger is between firms whih have a market share well below what is required when only leaders are involved. Yet, it is profitable beause one follower joined in. The joining of the follower does not hange the output of the merging entity (see (1)), but inreases the prie suffiiently (beause that follower disappears) to make the merger profitable, in spite of the output expansion by outsiders. In a sense, the follower ats as a substitute for the third leader who is not part of this merger. Consider now the market struture (=10,=10). Based on (11), a merger of N and N firms in this market is profitable iff 15
(22) There are 100 possible mergers in this market. igure 5 illustrates the possible mergers and their profitability, based on (11). When N 6, a merger is profitable iff N is very low or very high; when N >6, all mergers involving at least one leader are profitable. This is onsistent with proposition 5, whih establishes that mergers an be unprofitable for intermediate numbers of partiipating leaders. [igure 5 here] To make more preise the profit omparisons behind figure 5, onsider the row of figure 5 with N =6. This row indiates that a merger involving 6 followers is profitable iff Nå{1,2,6,7,8,9,10}. With ==10 and N =6, a merger involving N leaders is profitable iff (23) This expression has two roots: N =2.3, and N =5.2. Hene, the merger is not profitable for Nå[3,5] and is profitable otherwise. To better understand this result, figure 6 illustrates the pre-merger and post-merger profits as a funtion of N with N =6. When N =1, the merger is highly profitable. When N =2, the merger is still profitable, but only marginally relative to the no-merger equilibrium. When the seond leader joins in, his ontribution to the profits of the merged firms is less than the profits sarified when that firm was not merged (this is refleted by the slope of the urve, whih is lower than the slope of the straight line at that point). However, beause the merger profits were so high at N =1, the resulting merger profits at N =2 are still higher than with no merger. However, as the third and subsequent leaders join, the merger is not profitable anymore. Only when N =6 does the merger beome profitable again. Hene, there is a disontinuity in the relationship between the number of leaders joining the merger and its profitability. [igure 6 here] We now want to inquire how the results hange when the number of leaders and followers in the pre- merger equilibrium differ. Consider the asymmetri market struture (=10,=3). igure 16
7 illustrates the results. The range of profitable mergers is qualitatively similar to the range obtained for the previous market struture (=10,=10). With (=10,=3), if all followers partiipate, then the merger is profitable for any number of partiipating leaders; if at least one follower does not partiipate, then the merger is profitable iff N is very low (N =1) or relatively high, again with a substitution between leaders and followers: the more followers partiipate, the less leaders are required to do so. [igure 7 here] The last market struture onsidered, (=3,=10), yields a similar result. As figure 8 shows, the only unprofitable mergers are those with (N =2,N 2). Again, an intermediate number of leaders and a small number of followers yield an unprofitable merger. Given the small number of leaders in this market, there is no sope for substitution between leaders and followers here. [igure 8 here] The omparison between figures 7 and 8 reveals that the range of unprofitable mergers is larger when is large relative to than when is large relative to. This is beause a large makes the range of intermediate values of N, for whih mergers are more likely to be unprofitable, larger. The study of these speifi market strutures illustrates the results derived above for more general market strutures. 3.3 Welfare What is the impat of mergers involving at least one leader and one follower on welfare? Suh mergers inrease the degree of onentration and, not surprisingly, are welfare reduing. The next proposition formalizes the impat of these mergers on onsumers, outsiders, and total welfare. Define onsumer surplus as (24) and define welfare as a funtion of the number of leaders and followers as (25) Proposition 8. Any merger involving at least one leader and one follower, with the merged entity 17
beoming a leader, redues onsumer surplus and welfare, and inreases total profits and the profits of eah outsider. Proof. The hange in onsumer surplus is given by This expression is negative for all,>0, N å[1,],n å[1,]. The hange in the profit of any outsider leader (if any) is given by (26) (27) This expression is positive for all,>0, N å[1,-1],n å[1,]. The hange in the profits of a follower are (28) This expression is positive for all,>0, N å[1,],n å[1,-1]. The hange in total profits is given by (29) This expression is positive for all,>0, N å[1,],n å[1,]. The hange in welfare is given by (30) 18
Substituting onsumer surplus and profits, it is easy to verify that (30) is negative for all N å[1,],n å[1,]. Beause the mergers entail an inrease in onentration and firms are symmetri, even though they may or may not benefit the merging firms, they always benefit outsiders, in addition to reduing 5 onsumer surplus (i.e. inreasing the prie) and welfare. Hene, the vigilane of antitrust authorities need not be relaxed beause a merger involves a mixture of leader and follower firms. 4. Conlusions This paper has extended the analysis of the profitability of mergers to Stakelberg markets. ollowers alleviate the merger paradox, in that they inrease the range of profitable mergers for leaders. Unprofitable mergers are typially those involving an intermediate number of leaders and a small number of followers. All mergers involving at least one leader and one follower are welfare reduing. Note that it is the partiipation of followers to the merger, rather than their presene per se, whih makes mergers profitable more often. Hene, it is not really Stakelberg leadership, but the absorption of followers, whih enhanes mergers profitability. ollowers are typially smaller in size than leaders. When merger between a number of leaders is unprofitable, it may be easier to bring in one or more followers, than to bring in additional leaders, beause followers are smaller in size. Yet, in spite of their small size, they may be suffiient to make the merger profitable. The model an also be interpreted as saying that small firms, in spite of their small size, an ontribute to making mergers between larger firms profitable. One extension of this model whih has diret impliations for antitrust analysis of mergers is to onsider the possibility that follower/smaller firms have higher prodution osts. In this ase, the merger may be soially benefiial, beause it eliminates produers with higher prodution osts. It would be straightforward to inorporate this extension into the model. 5 Experimental evidene suggests that mergers in Stakelberg markets benefit outsiders and redue onsumer surplus, as predited by theory. Buy they do not hange the profits of the merging firms, ontrary to theoretial preditions. see Huk (2005). 19
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