Sequential mergers under general symmetric product differentiation with four firms

Abstract

This paper studies the types of sequential mergers that emerge and how they emerge when goods produced by firms in an industry are differentiated. In particular, we employ the most general differentiation setting with four firms by introducing a new parameter describing the highest degree of differentiation. We analyze how the degree of product differentiation affects the emergence of sequential mergers and find that when the value of this new parameter decreases, sequential mergers are more likely to emerge. Furthermore, we also provide welfare and numerical analyses to discuss the product differentiation ranges that would or would not lead to sequential mergers or no mergers in equilibrium. Consequently, policymakers can use this new parameter as a rule-of-thumb parameter in predicting the future merger structure.

Keywords:

• Sequential mergers
• merger waves
• product differentiation
• three differentiation levels
• merger policy

Acknowledgements

We thank Hong Hwang, Toshihiro Matsumura, and all other participants at the University of Tokyo and National Taiwan University Joint Conference on Industrial Organization in July 2015 for many helpful comments. We also thank two anonymous referees for their insightful comments.

Notes

No potential conflict of interest was reported by the authors.

1 One may consider the four-firm, four-market model to be somewhat specific. The model in our paper can be extended to more than four (even) firms, but the differentiation levels cannot be as general as in the current paper without making the analysis intractable. Four-firm, four-market model is also considered in the spatial connotation in Matsumura, Ohkawa, and Shimizu (2005).

2 Xing and Zhao (2008) examined a model with three differentiation parameters with three firms. That is, all three pairwise differentiation levels can be different. We thank an anonymous referee for this information.

3 Brander and Eaton (1984) considered a problem of a multi-product firm producing four substitute goods sequentially. Here, commodity pairs (1, 2) and (3, 4) are close substitutes, while pairs (1, 3), (1, 4), (2, 3), and (2, 4) are more distant substitutes. Thus the substitution pattern is similar to that in Ebina and Shimizu (2009). This is about the simplest structure in which the question of whether a firm should produce close or distant substitutes can be addressed. Although there is one major difference between their paper and ours since they considered a problem of monopoly, these two papers can be considered to be broadly in the same line of the literature, focusing on the nature of product differentiation and on four products. We thank one of the referees for noting this point.

4 Over the past few decades since Salant, Switzer, and Reynolds (1983), a considerable number of studies have been conducted on mergers under Cournot and Bertrand competition (e.g. Deneckere and Davidson 1985). Recently, merger problems are discussed in free entry markets. See Davidson and Mukherjee (2007) and Cato and Matsumura (2013).

5 For related analyses, see Levin (1990) and McAfee and Williams (1992).

6 Throughout this paper, we adopt a four-firm setting because it is the simplest with possible sequential mergers. The main results are not affected by an introduction of outsider firms producing differentiated goods but do not engage in mergers. Namely, the range in which sequential mergers arise in equilibrium shrinks; however, the lemmas and propositions still hold. In addition, the sequential merger result of Proposition 1 remains with more than two sets of merger pairs. We restricted ourselves to the current setting because of the simplicity in presentation.

7 Note that this inequality is newly assumed in our setting, whereas the previous studies without explicitly considering did not need to assume this explicitly. This is because the inequality is always satisfied when as in Dobson and Waterson (1996) and as in Ebina and Shimizu (2009). See Appendix 1 for more on this. We thank one of the referees for noting this point. .

8 One reason for this may be antitrust issues. Scherer and Ross (1990) observed that most mergers in the United States since World War II have been two-firm combinations. This assumption has been incorporated in many of the merger literature, including Ebina and Shimizu (2009) and Nocke and Whinston (2010).

9 The relevant parameter range in this paper is also limited by the assumption . Lemmas 1 and 2 are not affected by this. When looking at Figure 3, note that this assumption needs to be taken into account. See Section 5 for the cases with selected values of .

10 Because the firm pairs may not be able to decipher the merger decision of the rival pair, also considering the case in which the firms pairs choose merger decisions simultaneously may be natural. The equilibrium outcome depends on the parameter values of the s. In addition to the range in which only sequential or no mergers occur, a range in which multiple equilibria of sequential and no mergers may occur exists. In this range, Equation (1) is not satisfied but Equation (2) is satisfied. We thank Hong Hwang for noting this point.

11 One can consider a similar result for Regime 2 and , although the condition for is restrictive: let lie at the midpoint between and , i.e. . Then, mergers in Regime 2 occur if and only if . The proof structure is basically the same as the proofs of Propositions 2 and 3.

12 When discussing synergy effects, cost reduction can be considered as a part of activities in research and development (R&D). For papers on R&D presented in this special issue, see Chang, Lin, and Tsai (2016), Chang, Hwang, and Peng (2016), and Wang, Wang, and Liang (2016).

13 When (A2(A2) ) is satisfied with equality, four roots exist. Of these roots, two are negative and one is when holds. Thus, the remaining solution is the unique one between 0 and 1, given by (1(1) ).

14 Regarding the assumption , the area in which sequential merger occurs is not affected by the assumption in Regime 3, as holds. In the cases with other two regimes and the case with no mergers in Regime 3, the relevant parameter range is limited by this assumption.

Additional information

Funding

The authors gratefully acknowledge financial support from a Grant-in-Aid for Young Scientists [grant number 15K17047]; Basic Research [grant number 15H03349], [grant number 24530264]; from the MEXT and the Japan Society for the Promotion of Science. Needless to say, we are responsible for any remaining errors.