## 1 Introduction

In a symmetric linear Cournot oligopoly with homogenous goods, Salant, Switzer, and Reynolds (1983) (henceforth, SSR) showed that horizontal mergers are generally not profitable and that the minimum profitable merger involves at least 80 percent of the firms. Mergers typically are not profitable for insiders but are profitable for non-merging firms. Unprofitability comes from the fact that non-merging firms react to the merger by increasing their output. This is often known as the merger paradox and makes it hard to explain how merger activity gets started, since a firm would always prefer to remain an outsider.
^{1} This approach, however, leaves out the competitive intensity and assumes Cournot behaviour. In fact, most of the literature considers the absence of collusion as the most plausible scenario in which mergers take place and only very few papers address related aspects of mergers in a (possibly) collusive environment. Among the exceptions, Escrihuela-Villar (2008) analyses the price effects of mergers in a dynamic collusive environment. Also Kwoka (1989) and Rodrigues (2001), that using the conjectural variations approach and linear demand functions, illustrate the determinants of merger occurrence showing that the equilibrium market concentration is increasing in the expected competitive intensity. The main purpose of this note is to fill this gap in the literature by studying the effect of pre-merger competitive intensity on merger profitability in a quantity-setting market under fairly general conditions.

Following a recent trend in the literature, we model the intensity of competition assuming that firms care about their profits plus a weighted average of the profits of the other firms.
^{2} Using a payoff function based on relative performance, the intensity of competition can be parameterised. Intermediate values of this parameter then, might represent imperfect collusion and may be justified by reference to some implicit dynamic zmodel of collusion, a reduced-form representation of which being the quantity competition subgame of this model. This formulation is closely related to the “coefficient of cooperation”, defined by Cyert and DeGroot (1973) and has also been justified by other arguments. For instance, it captures the relative performance approach that is evolutionary stable (Vega-Redondo 1997). Furthermore, it can also be assumed that sufficiently patient firms always achieve the highest level of collusion that is sustainable as equilibrium of an infinitely repeated game given a number of parameters taken as exogenous at the competition stage. Under this interpretation, a fall in the importance given to rivals’ profits might correspond to a lower critical discount factor in an infinitely repeated game.

Our main result shows that the incentive to merge is increasing in the intensity of competition for a wide class of demand functions. Intuitively, in industries where firms are already cooperating, mergers add less to the high profits that firms are otherwise achieving. A simple illustration is provided using a linear demand function to show that the result of SSR mentioned above is only valid in Cournot environments and fails to hold in a quantity-setting model with a different competitive intensity since the well-known output expanding response of non-merging firms is smaller in a more competitive market. Consequently, the minimum required proportion of participants in the merger decreases with the intensity of competition.
^{3} A possible implication for policy makers is that, since mergers may be seen as an alternative to cooperation, it seems plausible to expect greater merger movements in more competitive markets.

Despite its theoretical viewpoint, this note is also motivated by real cases. Empirical evidence is copious suggesting that merger incentives are related to the pre-merger degree of competition, showing that in industries characterised by cartelist activities firms resort to mergers when cartels become less viable. Bittlingmayer (1985) shows that during the Great Merger Wave (1898–1902), mergers in several (cartelised) industries in the US (cotton oil, sugar, cast iron pipe, oil, meat packaging or steel and railroading), appear to have been the result of antitrust actions taken against cartels after the Sherman Act was passed. Equivalently, the Restrictive Trade Practices Act (1956), which outlawed cartels, triggered a wave of mergers in the UK (see Symeonidis 2002). In the same line, Evenett, Levenstein, and Suslow (2001) examine international cartels during the 1990s and observe that mergers are among the different measures adopted by firms for survival in collusive industries where cartel formation is restricted. A similar trend regarding the substitutability between merging and colluding has also been noticed in Germany where Neumann (2001) argues that German industries like cement, food processing machine building, etc. adopted cartelist activities in order to attain monopolistic power only when mergers were not possible. One could argue thus that a stricter legal enforcement against a cartel increases the relative cost of collusion as compared to merger and consequently firms resort to mergers.

## 2 Model and results

Assume an *n*-firm homogenous product oligopoly where 2 ≤ *n*. The inverse demand function is given by *P(Q*) where *q _{i}* is the output of the

*ith*firm. The demand function is twice continuously differentiable, downward sloping, and has a downward sloping associated marginal revenue function. Production costs are normalised to 0. Demand at price 0 is positive but finite and satisfies 2

*P′*(

*Q*) +

*P*″

*(Q)Q <*0 so that the second-order conditions always hold implying that

*β*(

*Q*) > –2, where

*β*(

*Q*) ≡

*i*, to be maximised by choice of

*q*, is:

_{i}with *Q*_{–i} = *Q* – *q _{i}* representing the quantity supplied by the (

*n*–1) competitors of firm

*i*.

In this baseline framework, a particular way of modelling the intensity of competition is considered. Following the approach of the coefficient of cooperation defined in Cyert and DeGroot (1973), we obtain a model in which firms are concerned about relative profits (the average profits of the other firms) as well as their own profits.
^{4}

The payoff function that firm *i* maximises is given by *i* = 1, 2,..., *n* and *λ* ∈ [1–*n*, 1] that we assume to be symmetric across firms. The parameter *λ* thus can be interpreted as representing the degree of reciprocal preference. A positive *λ* is equivalent to firms envying their rivals’ success, whereas a negative *λ* occurs when firms have reciprocal (namely, altruistic) payoff functions.
^{5} The present model encompasses the Cournot case if *λ* = 0, the joint profit maximizing allocation if *λ* = 1–*n* and the perfectly competitive outcome for *λ* = 1.
^{6} Hence *λ* may represent the severity of competition and enables us to treat the severity of competition as a continuous variable. Additionally, we consider the possibility that a group of *m* +1 firms merge without cost synergies associated. Once *m* +1 firms have merged there will be *n* – *m* firms that behave like we described above. Admittedly, since the range of values of *λ* depends on *n* it could also be argued that the pre-merger degree of competition is modified after the merger.
^{7} This, however, is not our approach. Instead, we consider how the expected pre-merger competitive intensity affects the incentives to merge. A merger is considered to be profitable if the profits of merging firms increase after merger. Since the function in eq. [1] depends on *n* and *λ*, we can denote the incentives to merge for *m* +1 firms by:

We assume *β* (*Q*) ≥ *β* *>* –2 for any *Q*. In other words, *β* is the lower bound of the degree of concavity and a set of demands satisfying *β* (*Q*) ≥ *β* is considered. We obtain:

*The incentive to merge defined by eq. [2] is increasing in λ*.

See Appendix. The intuition can be easily illustrated applying our model to the linear demand case as in SSR (*β* (*Q*) = 0).
^{8} Assume that *P(Q) = a*–*Q*. Then, firm’s output in equilibrium is given by:

Firms’ profits are given by
^{9}:

We can easily obtain:

*If β* (*Q*) = 0, *the merger of m*+1 *firms is profitable if**Hence, the minimum proportion of participants in order for the merger to be profitable**decreases with λ*.

See Appendix. In a Cournot setting even though mergers increase price, they are (generally) not profitable because outsiders react to the merger by expanding their output. This is still true in our model because in eq. [3] *λ* ∈[1–*n*, 1]. In words, the reaction of non-merging firms is to expand production but in a more competitive market this expansion is smaller rendering mergers more profitable. In fact for instance if *P* = *a – Q*, the reaction function of a firm *i* can be expressed as follows: *λ* ∈[1–*n*, 1]. Then, as *λ* increases, the reaction function of a firm becomes gentler and the free rider effect is weaker.

The result of SSR thus is only valid in a Cournot competition environment. If *λ* > 0, a higher degree of competition compared to Cournot competition increases firms’ incentives to merge. *λ* < 0 firms are already sustaining a collusive agreement and mergers lose attractiveness as an anti-competitive device.
^{10}

Admittedly, we treat *λ* as an exogenous parameter and although we build a direct link to the degree of competition, the reader might feel more comfortable when such link is made explicit and based on a direct behavioural assumption like the output produced. It can be easily shown that Proposition 2 also holds in a model where firms’ strategy set is ^{11} Therefore, the profitability of horizontal mergers crucially depends on the pre-merger output (namely, the intensity of competition).

## 3 Conclusions

A theoretical framework has been developed to study merger profitability in an imperfectly competitive environment. We found that mergers are more likely profitable in rivalrous environments. Consequently, the results in SSR are sensitive to the assumption of pre-merger Cournot competition. The well-known merger paradox regarding merger profitability thus might be alleviated or exacerbated if the pre-merger degree of competition is considered. It is to be expected thus more intense merger movements where collusion is absent.
^{12} The results presented are also robust to other approaches to the intensity of competition like conjectural variations and seem consistent with existing empirical evidence regarding the substitutability between merging and colluding.

From eq. [1] we obtain the first-order condition (FOC) for firm *i*;

*Q(n)*. We abuse notation and suppress the functions in the derivatives. Differentiating eq. [5] with respect to

*n*we obtain

*n*, and simplifying notation we obtain

*n*is given by:

*m*+l firms is

*m*+l firms is profitable if

Since *m* +l firms is profitable if

I am grateful to Ramon Faulí-Oller for his advice and encouragement. Financial support by the “Ministerio de Educación, Cultura y Deporte” through its project “Análisis de Economia Pública y Bienestar” (Ref: ECO2012-36480) is gratefully acknowledged. The usual disclaimer applies.

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## Footnotes

^{1}

This paradox though is weakened for instance by convex demand (Fauh-Oller, 1997) or cost functions (Perry and Porter 1985). Furthermore, this result is valid only in Cournot environments and generally fails to hold in differentiated Bertrand models. Deneckere and Davidson (1985) show that when reaction functions are upward sloping, mergers of any size are beneficial.

^{2}

See for instance Symeonidis (2008) for a duopoly or Matsumura, Matsushima, and Cato (2013) for an oligopoly. Equivalently, d’Aspremont and Ferreira (2009) add some positive weight also to the total surplus.

^{3}

We also note that this result resembles a well-known result in the literature on managerial compensation where the increased profit from any reduction in the number of firms is higher under delegation (see for instance Ziss 2001 or Straume 2006). In these papers thus, the profitability of merger in oligopoly is significantly enhanced if firms delegate output decisions to a manager.

^{4}

Escrihuela-Villar (2015) shows that using conjectural variations and the coefficient of cooperation leads to equivalent closed-form solutions. However, the rationale behind in order to model the degree of competition in the market is fairly different. Additionally, several authors have already shown the preference of the present approach instead of the conjectural variations approach. As an example, Symeonidis (2000) pp. 58 footnote 4 states that this formulation “is free of some of the theoretical problems encountered in other approaches to modelling the intensity of short-run competition by way of a reduced-form parameter, such as the conjectural variations approach”.

^{5}

Admittedly *λ* > 0 might seem counterintuitive at first sight. We follow here however the justifying reasoning of Matsumura and Matsushima (2012): (i) objective functions are often based on relative performance since, for instance, evaluations of management activities are also based on their relative performances, (ii) experimental works already pointed out reciprocal (but not necessarily altruistic) behaviour (see for instance Cason, Saijo, and Yamato 2002) that is closely related to firms’ objective functions based on relative performance, (iii) following Fershtman and Judd (1987) where managers are paid in proportion to a combination of profits and sales, we can replace sales with (negative) profit of the rival in their model and the firms would in fact adopt a positive *λ*, (iv) it can be shown that an increase in *λ* makes collusion less stable in a repeated game also when *λ* is positive. This suggests thus that a larger *λ* indicates more intensive competition.

^{6}

Note that the first-order condition for a firm i when *λ* = 1 becomes *q _{j} = q_{i}* and therefore

^{7}

In this line, the literature on horizontal merger assessment is huge and has often focused on coordinated effects (if a merger induces rivals to alter their strategies, resulting in some form of price-increasing coordination). As an example Vives (2001) obtains that reducing the number of competitors facilitates collusion.

^{8}

Using the present formulation, it can also be easily checked that the result in Faulí-Oller (1997) where the degree of concavity of the demand function reduces merger profitability in a Cournot setting can also be generalized to a model with several competition intensities.

^{9}

Note that the outcome gradually approaches the competitive outcome as increases. So, in the view of behaviourism (or the revealed preference approach), λ can be regarded as the parameter of competitiveness. Additionally, we can also justify λ as a measure of the intensity of competition since the equilibrium price is monotonically decreasing with λ, while an increase in competitiveness increases consumer surplus and decreases producer surplus.

^{10}

In fact, in the linear case if λ is low enough mergers are never profitable since the required proportion of firms involved in the merger is larger than 1. More precisely, if

^{11}

Further details are available upon request.

^{12}

We note also that the mergers considered here are all welfare reducing as they involve a reduction in output and no efficiency gains. However, following Farrell and Shapiro (1990), we could still examine the effects of merger on the nonparticipant firms and consumers, whom they call “outsiders”. It can be easily checked that in a very competitive market, outsiders can also benefit from the merger since in a very competitive market the increase in price could be more than compensated by the increase in firms’ profits due to the merger.