Mergin Firms’ Strategies and the Merger Paradox

Taking a model of horizontal mergers as a reference, the purpose of this paper is to qualify the merger paradox by proving that a multidivisional fi rm formed by a merger could be sustainable even though the merger does not involve most of the fi rms in the market. Specifi cally, it is proved that the minimum number of merging fi rms to have a profi table merger, assuming both simultaneous and sequential games, is lower in our model than in the traditional literature. Furthermore, it is proved that, if the multidivisional fi rm sets the number of divisions optimally, the merger is sustainable and less harmful to welfare than in the traditional model.


Introduction
The theoretical literature on horizontal mergers states that, in quantitysetting games assuming linear demand and costs and symmetric fi rms, a merger is not typically profi table for the merging fi rms unless it involves the vast majority of industry participants (Salant et al., 1983); this result could represent the fi rst part of the merger paradox. Furthermore, a merger usually reduces the aggregate welfare, because it reduces the number of fi rms in the market and therefore decreases the existing market competition.
These theoretical results defy the empirical evidence (the wave of merger activity has increased throughout this century), and, at the same time, they are awkward for many economists (most mergers ought to be profi table), because, apart from the possible effi ciency gains derived from a merger (Perry and Porter, 1985), the merged entity gains both market powerwhich it should be able to exploit in a benefi cial manner -and informational benefi ts (Creane and Davidson, 2004;Daughety, 1990;Huck et al., 2004).
Many authors have tried to solve the merger paradox using symmetric information models, by changing some of the assumptions of the original Salant et al. (1983) model: Fauli-Oller (1997 and (2000); González-Maestre and López-Cuñat (2001); and Perry and Porter (1985). Other authors have developed their analysis by assuming asymmetric information between fi rms: Daughety (1990); Escrihuela-Villar and Fauli-Oller (2008); and Huck et al. (2004). Creane and Davidson (2004) and Huck et al. (2004) assumed symmetric information between merging and non-merging fi rms but asymmetric information between the merging fi rms in the merged entity. Méndez-Naya (2014) considered asymmetric information between merging and non-merging fi rms.
Taking Creane and Davidson (2004) and Huck et al. (2004) as a reference, it is assumed that fi rms do not just become bigger through mergers but that they also become more complex organizations. Specifi cally, it is supposed that the multidivisional fi rm formed by a merger can exploit certain strategic possibilities that are inherent from having distinct divisions.
The purpose of this paper is to complement the existing literature on the subject by providing a model of horizontal mergers in which mergers are profi table for the merging fi rms even if they do not include most of the fi rms in the industry. Furthermore, it is justifi ed that a merger is less harmful from a welfare point of view than in traditional models.
To perform the analysis, basing on the basic Salant et al. (1983) model, let us set an oligopoly model in which there are n fi rms. It is assumed that k fi rms, being k lower than n, decide to merge and, following Creane and Davidson (2004) and Huck et al. (2004), that the merged fi rm has a multidivisional structure and its output decisions are delegated to the manager of each division.
Specifi cally, it is supposed that a subset of merging fi rms, s, decides to form a single division; that is, they become a bigger fi rm and therefore cooperate with one another to maximize their joint profi ts. On the other hand, the remaining merging fi rms, k-s, behave as individual divisions in the multidivisional fi rm that results from the merger; that is, their managers choose the output to maximize each division's individual profi ts. Therefore, the merged fi rm has k-s + 1 divisions that play an internal game in which each division competes with the others, and this fact gives the merged fi rm certain strategic advantages over non-merging fi rms.
In summary, it is assumed that the merged entity plays a mixed strategy in which some divisions maximize their joint profi ts, as in the traditional models, and other divisions maximize their individual profi ts, following Creane and Davidson (2004) and Huck et al. (2004). Concretely, it is proved that this internal organization of production in the merged entity is optimal for the merged entity. Therefore, the paper contributes to the existing literature a new way of restructuring internal production that is optimal for the merged entity.
Our hypothesis about the specifi c way in which the merged entity organizes its internal production is very hard to observe and measure and therefore diffi cult to test. However, in many cases, merging fi rms stay as independent decision-making units governed by a joint headquarters, as remarked by Huck et al. (2004). Specifi cally, as tested by Hubbard and Palia (1999), fi rms acquiring other fi rms retain certain target management, and usually, as justifi ed empirically by Prechel et al. (1999) and Zey and Swenson (1999), merging fi rms become affi liates in a holding company, with some affi liates having the discretion to make independent decisions. Concrete examples of the above behaviour are the bank merger in Spain between the Bank of Santander and the Central-Hispano Bank (see Barcena-Ruíz and Garzón, 2000) and many mergers between car producers, for example those between Volvo and Ford and between Daimler and Chrysler (see Huck et al., 2004).
The purpose of this paper is to investigate how the above assumption affects some traditional results of horizontal mergers in quantity-setting games. Concretely, assuming that the multidivisional fi rm formed by a merger sets its number of divisions optimally, it is proved that the number of merging fi rms necessary for a profi table merger is lower in our model, supposing both simultaneous and sequential competition, than in the traditional Salant et al. (1983) model; furthermore, it is shown that the optimal number of cooperating divisions of the multidivisional fi rm formed by the merger is lower assuming simultaneous decisions between merging fi rms than in the sequential game. Finally, it is proved that the welfare level after the merger is always higher than in the traditional Salant et al. (1983) model.
The merger lowers the market competition and therefore increases individual fi rms' profi ts. By preserving some merged fi rms as independent divisions, the merged entity captures their profi ts, and as a consequence the merger is more profi table than in the traditional Cournot model. Therefore, the number of merging fi rms that is needed to have a profi table merger is lower in our model than in the traditional ones. This internal organization of production in the merged entity, which, as remarked, is optimal for the merged entity, renders the merger more profi table than in the traditional Cournot model and contributes to a better understanding of merger activity.
The paper is organized as follows. Section II presents the basic model. In section III, Salant et al.'s (1983) results are presented and compared with those obtained in our model, assuming that merged fi rms take decisions both simultaneously and sequentially. Finally, section IV presents the main conclusions of the paper.

The basic model
Let us take as a reference an oligopoly model of n fi rms that produce a homogeneous good. Quantity competition is assumed among fi rms, and, for simplicity, it is supposed that every fi rm has the same technology with no fi xed costs and constant and equal zero marginal costs.
Given the above assumptions, each fi rm's profi t is given by its overall income, that is, P is the market price and x i the output of the ith fi rm.
Consumers are assumed to have quadratic utility functions, which are additively separable and linear in a competitive numeraire good. In this case, consumers' utility function is given by: where , then, the inverse demand function is given by Therefore, consumers' surplus is given by Social welfare is measured as the sum of consumers' surplus and fi rms' profi ts and is given by: Let us take, as an anchor case, the premerger scenario. In this case, we use the standard Cournot model and the assumed output competition results in the following equilibrium values: where the superscript B denotes the scenario before any merger between fi rms.

Merging fi rms' strategies and merger effects
The existing literature on horizontal merger profi tability in Cournot oligopoly models states that a merger could be profi table for the merging , ., , 0 fi rms under certain hypotheses. Particularly, Salant et al. (1983) stated that at least 80% of fi rms in the industry have to merge to make the merger profi table and that, even in this case, the merger would be harmful from a welfare point of view.
The purpose of this section is to qualify the above results by proving that the number of fi rms needed to have a profi table merger is lower than that stated by Salant et al. (1983). Furthermore, this section verifi es that a merger is less welfare harmful than in the traditional model. To prove these results, similarly to Huck et al. (2004) and Creane and Davidson (2004), it is assumed that a merger is not strictly a fusion; that is, it is not just a bigger fi rm but a more complex organization made up of several divisions. Therefore, merging fi rms, instead of cooperating with each other and choosing the output to maximize their joint profi ts, could exploit the strategic possibilities that result from having several divisions or subsidiaries.
Specifi cally, it is assumed that the merged entity allows some divisions to act individually. In this case, the merged entity has two types of divisions: those divisions that set their output to maximize their joint profi ts -that is, they become a single division in the multidivisional fi rm formed by the merger -and those divisions that set their output to maximize their individual profi ts -that is, they behave as individual divisions in the merged entity.
To develop the analysis, the section is organized as follows. Firstly, as an anchor case, the traditional Salant et al. (1983) model is considered. Secondly, it is assumed that the merged entity allows some divisions to act individually. Finally, similarly to Creane and Davidson (2004) and Huck et al. (2004), it is considered that the strictly merged divisions have an informational advantage over the individual optimizer divisions.

Traditional model
Taking as a reference the basic oligopoly model set in section II, it is assumed that k fi rms decide to merge. In this case, after the merger, Cournot competition leads to the following equilibrium values: where the subscript A represents the situation after the merger.
The merger will be profi table if it is verifi ed that.
That is, mergers are not benefi cial for the merging fi rms unless they include the vast majority of industry participants.

Some divisions act individually: The simultaneous game
In this part, as in the above subsection, it is assumed that k fi rms decide to merge; furthermore, in this case, it is assumed that the merged entity allows some divisions to set their output individually. Specifi cally, it is assumed that s fi rms merge by forming a single division and the other fi rms in the merged entity, that is, k-s divisions, set their output to maximize their individual profi ts. Therefore, after the merger, there are n-s + 1 fi rms in the market: n-k non-merging fi rms, k-s individual profi t optimizer divisions and one division that results from the fusion of s fi rms. In this case, following Baye et al. (1996) and Schwartz and Thompson (1986), it is assumed that the manager of each division in the merged entity chooses the output to maximize his or her division's profi ts.
In this case, the market equilibrium, derived from Cournot competition between both merging and non-merging fi rms, is given by the following values: where the subscript AIC represents the situation after the merger, assuming both that some divisions act individually and that Cournot competition exists. The above expressions show that the market equilibrium depends on the existing relationship between the number of fi rms in the market, n, and the number of cooperating divisions in the merging entity, s. Let us assume that the merging entity decides its number of cooperating divisions optimally. In this case, the following results can be stated:

If
, it will be optimal for the merged entity to have two kinds of divisions: those that maximize joint profi ts and those that maximize individual profi ts.

Proof
In this case, let us assume that the merged entity sets s to maximize the merged entity's joint profi ts, that is, ; , by solving, the optimal s is derived and it is given by s AIC = 2k − n. Therefore, it is verifi ed that , which proves the result.

Result II
The number of merging fi rms that is needed to have a profi table merger is lower in our model than in the traditional Salant et al. (1983) model.

Proof
In this case, the merger would be profi This reduction in the number of merging fi rms necessary for a merger to be profi table is due to the fact that the fusion of some divisions (s) is profi table for the remaining (k-s) divisions in the multidivisional fi rm resulting from the merger and that the global profi ts of the merged entity therefore increase. As a result, the number of merging fi rms needed to guarantee a profi table merger is lower in our model than in the traditional Salant et al. (1983) model. The above result is related to the existing literature on the subject. Specifi cally, some papers have tried to identify suffi cient conditions by which certain mergers should be approved; see Cheung (1992), Farrell and Shapiro (1990) and Levin (1990), among others. These papers show that, in the absence of cost synergies, what matters for a profi table merger is not the number of merging fi rms but the merger's market share.

Result III
Assuming that s is set optimally, that is, s AIC = 2k − n the merger will be harmful to consumers and profi table for non-merging fi rms. Furthermore, it will be more profi table from a welfare point of view than in the traditional Salant et al. (1983) model.

Proof
The above result can be obtained directly by comparing the market equilibria values before and after the merger. Specifi cally, the following relationships are derived: Π AIC − Π B > 0; CS AIC − CS B < 0 and W AIC − W B > 0, which prove the result.
It is well known that mergers increase market concentration and therefore fi rms' profi ts increase and consumers' surplus decreases. However, in our model, as some divisions behave as individual profi t optimizers, the merger increases both their output and their profi ts. Furthermore, as a con-sequence, the negative effect on consumers is lower than in the traditional literature, and, as a result, the merger is less harmful from a welfare point of view than in the traditional Salant et al. (1983) model.

Some divisions act individually: The sequential game
Once more, it is assumed that the manager of each division sets the output to maximize his or her division's profi t. However, similarly to Creane and Davidson (2004) and Huck et al. (2004), it is assumed that some managers have an informational advantage with respect to other managers. Consequently, although the multidivisional fi rm has no informational advantage with respect to non-merging fi rms, the merger gives some divisions a fi rstmover advantage with respect to other divisions.
Specifi cally, it is assumed that the single division that results from the fusion of s individual fi rms is a 'partial leader' of the game; that is, it acts as a Stackelberg leader against the other k-s divisions, which are 'partial followers'. However, as indicated, the multidivisional fi rm that results from the merger and the non-merging fi rms are Cournot competitors.
The above game can be solved by backwards induction as follows. First, the k-s Stackelberg followers set the output to maximize their individual profi ts. In this case, their reaction function can be derived as follows: where the superscripts PF , NM and PL refer to partial followers, non-merging fi rms and the partial leader, respectively.
The partial leader, knowing the partial followers' reaction functions, sets its output to maximize its profi ts, which enables us to derive its reaction function as follows: The non-merging fi rms' reaction function can be derived similarly and is given by: Solving the above reaction functions, the market equilibrium is derived and is characterized as: where the subscript AIS represents the situation after the merger, assuming both that some divisions act individually and that Stackelberg competition exists. Once more, the market equilibrium depends on the existing relationship between n, k and s. Assuming that the number of cooperating divisions, s, is set optimally, the following results can be stated:

Result IV
Assuming that merged fi rms take their decisions sequentially, if it is verifi ed that , it will be optimal for the merged entity to adopt two kinds of divisions: those that maximize joint profi ts and those that maximize individual profi ts. Furthermore, the optimal number of cooperating divisions that maximize joint profi ts after the merger is greater in the sequential than in the simultaneous game. Proof Once more, the multidivisional fi rm formed by the merger sets s to maximize the merged entity's joint profi ts, that is, , and, solving, the optimal s is derived and is given by ,; it is also verifi ed that. , Furthermore, , which proves the result. The optimal number of cooperating divisions increases assuming sequential competition, because, although neither partial followers' divisions nor non-merging fi rms change their output, the partial leader increases both its output and its profi ts.

Result V
Assuming that s is set optimally, that is, the number of merging fi rms needed to have a profi table merger assuming sequential competition is the same as in the simultaneous game and therefore lower than in the traditional Salant et al. (1983) model. lowers (k-s) decreases, which increases the number of merging fi rms needed to have a profi table merger. The two effects offset each other; as a result, the number of merging fi rms (k) is the same in the two considered contexts. Taking into account the above two results, the following corollary can be stated:

Corollary
The number of divisions of the multidivisional fi rm formed by the merger is lower in the sequential game than in the simultaneous one.

Result VI
If s is set optimally, it is verifi ed that partial followers' profi ts, non-merging fi rms' profi ts, consumers' surplus and welfare are the same under both a simultaneous and a sequential game. However, the partial leader's profi ts are greater in the sequential game than in the simultaneous one.

Proof
The result is proved directly by comparing the above-mentioned values in the two considered scenarios, and the following results are obtained: which prove the result.
As indicated, in the sequential game, the partial leader's output increases and both partial followers and non-merging fi rms do not change their output, s increases and the number of partial followers decreases. However, neither the overall market output nor the equilibrium price varies, which justifi es the result.

Concluding remarks
This paper analyses the effects on the market equilibrium derived from a multilateral horizontal merger assuming both that a multidivisional fi rm is formed by the merger and that this multidivisional fi rm delegates output decisions to the manager of each division. These hypotheses give the merged fi rm strategic advantages over outsiders, and this fact has important implications for the traditional literature on horizontal mergers in quantity-setting games.
Specifi cally, it is proved that, from the merged fi rm's point of view, it could be optimal to organize its production according to two types of divisions: those that maximize their joint profi ts and those that maximize their individual profi ts. To the best of our knowledge, this is an original contribution to the existing literature on the subject. Particularly, it is proved that the optimal number of divisions in the multilateral fi rm resulting from the merger is lower when assuming simultaneous than when assuming sequential competition between merging fi rms.
In this context, it is proved that the number of merging fi rms needed for a merger to become profi table, assuming both simultaneous and sequential competition, is lower in our model than in the seminal paper by Salant et al. (1983). Furthermore, supposing that the merged fi rm sets the number of divisions optimally, it is proved that the merger is both sustainable and more benefi cial, from a welfare point of view, than in the traditional Salant et al. (1983) model. In summary, the paper complements the existing literature on mergers by proving that, in the stated model, a merger has different effects on relevant variables from those stated in the traditional literature, which could have important policy-making implications.