Sequential mergers with differing differentiation levels March 31, 2008 Discussion Paper No.08-03 Takeshi Ebina and Daisuke Shimizu
Sequential mergers with differing differentiation levels Takeshi Ebina and Daisuke Shimizu March 31, 2008 Abstract Merger incentives under product differentiation are found to be stronger for two firms producing closely related goods than more differentiated goods. Also, after one merger, other firms are willing to follow with their own merger, resulting in a sequential merger. JEL classification numbers: L13, L41 Keywords: Sequential merger, Product Differentiation. 1 Introduction Mergers have been rampant in many industries throughout the world. A reality or perception of increasing competition has recently led to more mergers in order to gain ground or even survive. One example of this is the video game software industry in Japan. A massive shock occurred in 2003, as Enix, known for Dragon Quest series, the role playing game (RPG) series that is arguably the most popular series in Japan, and Square, known for its Department of Social Engineering, Graduate School of Decision Science and Technology, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro, Tokyo, Japan, 152-8522 E-mail: ebina-t@soc.titech.ac.jp Faculty of Economics, Gakushuin University, 1-5-1, Mejiro, Toshima-ku, Tokyo 171-8588, Japan. Phone: (81)-3-5992-3633. E-mail: daisuke.shimizu@gakushuin.ac.jp 1
also very popular RPG Final Fantasy series, merged. Dragon Quest series have sold over 41 million copies in Japan and Final Fantasy series 75 million copies throughout the world. Since these two firms were seen as giants, the merger foresaw how severe these firms felt the game markets will be thereafter. Following their footstep in 2005, Namco and Bandai, the former known for games with innovative ideas and smooth graphics and the latter known for character-oriented games from anime or comic series, set up a holding company. Several other firm combinations formed thereafter, including purchase of Sega by Sammy and Konami making Hudson its subsidiary. Thus, the initial shock of Square-Enix merger led to a sequence of mergers and other business combinations. A key in this series of M&A s is that in many cases firms with similar type of strong suits, thus goods with low degree of differentiation and high degree of substitution, merged. One reason may be that firms can enjoy scale merit due to consolidation of assets. Our focus is rather on the post-merger effect where merging with firms producing similar products would make the later competition softer. Nilssen and Sørgard (1998) offer analysis where two groups of firms in an industry can sequentially merge. They allow cost savings from mergers and show that the profitability and actual pattern of mergers depend on the size of cost savings. We abstract from cost reduction and introduce product differentiation in the demand function. 1 We find the result in the video game software industry can be explained by our model. If we in addition consider cost reduction, merger with similar degree of differentiation may be even more attractive, since they are more likely to operate using similar facilities and capitals, possibly enabling a reduction or an elimination of now unnecessary fixed cost. Throughout the paper, we adopt a four-firm setting because it is the simplest case with a possible sequential merger. The main results are not affected by an introduction of outsider firms producing differentiated goods 1 There have been several earlier works that deal with mergers and differentiation using the spatial competition context. See Levy and Reitzes (1992) and Matsushima (2001). 2
but do not engage in mergers. In addition, the sequential merger result of Proposition 2 remains with more than two sets of merger pairs. We restricted ourselves to the current setting because of simplicity in presentation. 2 Model Let there be four firms producing differentiated goods. Consumers maximize U(q 1, q 2, q 3, q 4 ) 4 i=1 p iq i, where q i is firm i s output and p i its price. We let gross utility U take the form U(q 1, q 2, q 3, q 4 ) = q 0 + a 4 q i 1 { 4 qi 2 + 2β 1 (q 1 q 2 + q 3 q 4 ) 2 i=1 } + 2β 2 (q 1 q 3 + q 1 q 4 + q 2 q 3 + q 2 q 4 ), i=1 where a > 0 and q 0 is the quantity of the numeraire good. Since this is a quasi-linear utility, we can proceed with the partial equilibrium analysis. β i [0, 1], i {1, 2} is a parameter describing the degree of differentiation among the four goods produced by the four firms. β i = 1 implies perfect substitutes, whereas β i = 0 implies independent goods. Here, we assume that each pair of firms 1 and 2 and firms 3 and 4 produces goods with relatively low differentiation, denoted by β 1. On the other hand, the degree of differentiation between goods produced by firm 1 (and 2) and firm 3 (and 4) is high and is denoted by β 2. Thus we have β 1 β 2. Next we describe the game structure. We compare two different regimes, depending on whether the firms producing goods with high or low degree of differentiation attempts to merge. In the low case, in stage 1, firms 1 and 2 jointly decide whether to merge. Then firms 3 and 4 jointly do the same in stage 2 after looking at the decision in stage 1. In the high case, firms 1 and 3 jointly make the merger decision in stage 1, and firms 2 and 4 do the same in stage 2. In both regimes, all firms engage in Cournot competition in stage 3. If the firms merge, the merged firms would maximize the joint profit of the firms. Thus depending on the merger regime, there can be two to four competing firms in the third stage. We assume mergers with three or more 3
firms does not occur. 2 Finally in this setting, we derive the inverse demand function. From the consumer utility maximization problem, the system of inverse demand for good 1 is given by p 1 = a q 1 β 1 q 2 β 2 q 3 β 2 q 4 and other prices are given similarly. 3 Equilibrium In solving for the subgame perfect equilibrium, we solve the game using backward induction. In the second stage, a firm pair decides whether to merge after observing the merger decision in stage 1. Given two regimes of differentiation, we need to compare Cournot equilibrium profit levels for the case with (1) no merger, (2) only firms 1 and 2 merge, (3) only firms 1 and 3 merge, (4) firms 1 and 2 merge and firms 3 and 4 merge, and (5) firms 1 and 3 merge and firms 2 and 4 merge. We solve for Cournot equilibrium output, price, and profit levels for each of the regimes, and obtain the following. The parameter after Π i indicates how the merger partitioned the firms. ( ) 2 Π a 1(1, 2, 3, 4) =, 2 + β 1 + 2β 2 [ ] 2 Π a(1 + β 1 β 2 ) 3(12, 3, 4) =, 2 + 3β 1 + β1 2 2β2 2 Π 12(12, 3, 4) = 1 + β [ ] 2 1 a(2 + β1 2β 2 ), 2 2 + 3β 1 + β1 2 2β2 2 [ ] 2 Π a(2 β 1 + β 2 ) 2(13, 2, 4) =, 4 β1 2 + 6β 2 2β 1 β 2 + β2 2 [ ] 2 Π a(2 β 1 ) 13(13, 2, 4) = 2(1 + β 2 ), 4 β1 2 + 6β 2 2β 1 β 2 + β2 2 Π 12(12, 34) = 1 + β ( ) 2 1 a, 2 1 + β 1 + β 2 ( ) 2 Π a 13(13, 24) = 2(1 + β 2 ). 2 + β 1 + 3β 2 2 One reason for this may be antitrust issues. Scherer and Ross (1990) observe that most mergers in the US since World War II have been two-firm combinations. 4
We use these results and analyze how the firms choose to merge. First we have the following result on the myopic merger incentives in the low differentiation case. Lemma 1 The range of (β 1, β 2 ) where two firms with low degree of differentiation prefer to merge rather than to operate individually when the other two firms independently compete is given by { β 1 if 0 β 1 β 2 β (2 + β1 )(1 + β 1 1 + β 1 ) / 2 if β (1) β 1 1, where β 0.307 is a root to 9β1 3 19β1 2 8β 1 + 4 = 0. Similarly, (β 1, β 2 ) where two such firms merge when the others have merged is given by { β 1 if 0 β 1 β 2 ˆβ (1 + β1 )(1 + β 1 1 + β 1 )/2 if ˆβ (2) β 1 1, where ˆβ 0.369 is a root to β 3 1 5β 2 1 β 1 + 1 = 0. Proof. Without loss of generality, let us consider whether firm 1 decides to merge (with firm 2). Firm 1 would merge with firm 2 if the half of the joint profit is higher than the profit when there is no merger. We are looking for parameters (β 1, β 2 ) for this to occur. For this we need 1 2 Π 12(12, 3, 4) Π 1(1, 2, 3, 4) = a 2 [ (1 + β1 )(2 + β 1 2β 2 ) 2 4(2 + 3β 1 + β 2 1 2β 2 2) 2 1 (2 + β 1 + 2β 2 ) 2 ] 0. (3) Among β 2 that satisfies (3) with equality, only β 2 = (2 + β 1 )(1 + β 1 1 + β 1 ) / 2 (4) fills the requirement of 0 β 2 1. 3 In addition, since β 1 β 2 and the right hand side (RHS) of (4) is increasing in β 1, the solution to β 1 = 3 When (3) is satisfied with equality, there are four roots. Of them, two are negative and one is, when 0 β 1 < 1 holds, 1 < β 2 1.61. Thus the remaining solution is the unique one between 0 and 1, given by (4) 5
(2 + β1 )(1 + β 1 1 + β 1 ) / 2 around β 0.307 becomes the threshold between the two ranges. 4 9β 3 1 19β 2 1 8β 1 + 4 = 0 when β 1 0. Note that this equation can be rearranged to be For the second merger part, let firms 1 and 2 merge. For firms 3 and 4 also to merge, 1 2 Π 34(12, 34) Π 3(12, 3, 4) = a 2 β 1 [ β1 (1 + β 1 ) 3 4β 2 2(1 + β 1 β 2 )(1 + β 1 + β 2 ) 4(1 + β 1 + β 2 ) 2 (2 + 3β 1 + β 2 1 2β 2 2) 2 ] 0 (5) must hold. Among β 2 satisfying (5) with equality, only β 2 = (1 + β 1 )(1 + β 1 1 + β 1 )/2 (6) is within [0, 1]. result. Using the same method as in the first part, we have the (Q.E.D.) Now let us consider the case with high degree of differentiation. As before, we first derive the range of (β 1, β 2 ) in which one such merger occurs when four firms originally are operating individually. Lemma 2 The range of (β 1, β 2 ) where two firms with high degree of differentiation prefer to merge rather than to operate individually when the other two firms independently compete is given by β 1 β 2 + (2 + β 2 )(1 + β 2 1 + β 2 ). (7) Similarly, (β 1, β 2 ) where two such firms merge when the others have merged is given by β 1 β 2 + 2(1 + β 2 )(1 + β 2 1 + β 2 ). (8) 4 We have (3) positive when (β 1, β 2 ) = (0.307, 0.307), but negative when (β 1, β 2 ) = (0.308, 0.308). 6
The proof is similar to that of Lemma 1 and is omitted. available from the authors upon request. Full proofs are We now summarize our result so far. In Figure 1 there are six regions in the lower right (the relevant range is β 1 β 2 ). The lower dashed curve is equation (6) and two firms in the low differentiation case have an incentive to merge in the region below this curve (A, B, and D). The upper dashed curve is equation (4) and in the region below this curve (A to E), the second pair has an incentive to merge given the first pair has merged. The dotted curve to the left is equation (7) with equality and two firms in the high differentiation case have an incentive to merge in the region to the left of this curve (A only). The dotted curve to the right is equation (8) with equality and the second pair has an incentive to merge given the first pair has merged (regions A to C). When the degrees of substitution are very high, i.e. β 1 and β 2 are very close to 1, no firm would want to merge, as seen from the analysis by Salant, et al. (1983). The figure confirms this result. Given that one merger has occurred, would the other pair also merge, resulting in a merger wave? The following proposition offers an affirmative answer to this. Proposition 1 Irrespective of the differentiation regime, if one pair of firms has the short-term incentive to merge, the remaining pair will also have an incentive to merge. Proof. Since the RHS of (1) is larger than that of (2) for all β 1 in [0, β], [ β, ˆβ], and [ ˆβ, 1], the remaining firms merge if the initial firms have a myopic incentive to merge in the low differentiation case. The results for the high differentiation case is proved in a similar fashion by comparing the RHSs of (7) and (8). (Q.E.D.) This leads to our main result. Proposition 2 In the subgame perfect equilibrium, either sequential mergers or no mergers occur. 7
Proof. First, note that sequential merger always increases joint profit if β 1 and β 2 are not zero. Π 12(12, 34) 2Π 1(1, 2, 3, 4) = 2a2 β 1 (β 1 + β 2 1 + 4β 2 + 4β 2 2 + 4β 1 β 2 ) 4(1 + β 1 + β 2 ) 2 (2 + β 1 + 2β 2 ) 2 > 0, Π 13(13, 24) 2Π 1(1, 2, 3, 4) = 2a2 β 2 (2β 1 + β 2 1 + 3β 2 + 4β 2 2 + 4β 1 β 2 ) (2 + β 1 + 3β 2 ) 2 (2 + β 1 + 2β 2 ) 2 > 0. In the low differentiation case, in the second stage subgame where one merger has taken place, second merger also occurs in regions A to E in Figure 1, whereas it does not in region F. In the subgame where no merger has taken place, merger occurs in regions A, B, and D and does not in regions C, E, and F. Solving backwards, the initial merger occurs in regions A to E (from Proposition 1, there are no parameters where an initial merger does not lead to the second merger), and does not occur in region F. Thus only sequential mergers or no mergers occur. Solving the high differentiation case similarly, we have sequential mergers in regions A to C and no mergers in D to F. Thus we have the desired result. (Q.E.D.) As seen in the proof, sequential mergers are more likely to occur in the regime of merger between firms with lowly differentiated goods than in the high regime. In either regime, subgame perfect equilibrium leads to either sequential or no mergers but not just a single merger. Comparing the post-merger profit levels, we have the following proposition. Proposition 3 The post-merger profit level is higher when the merger is between firms with high degree of substitution than when it is between firms with low degree of substitution. Proof. Subtracting the profit in high degree of substitution case by the low case, we have Π 12 (12, 34) Π 13 (13, 24) = a2 (β 1 β 2 )(β 1 + β 2 1 + 3β 2 + 3β 1 β 2 + 4β 2 2) 4(1 + β 1 + β 2 ) 2 (2 + β 1 + 3β 2 ) 2. Thus this is nonnegative and the numerator is equal to zero only when β 1 = β 2 8 (Q.E.D.)
To summarize, as mentioned in the proof of Proposition 2, the area of (β 1, β 2 ) where firms with a high degree of substitution merge contains the area of the same with a low degree of substitution. From Proposition 3, if the goods produced by the firms are not independent (completely differentiated), profits of the merged firm when the merger is between firms with a highly substitutable goods are larger than when it is between firms with a lowly substitutable goods. Thus when mergers do occur, it seems more likely that firms with a high degree of substitution would merge. 4 Concluding Remarks In this paper, we have shown that in a sequential merger setting, firms with low degree of differentiation merging is more likely, in that parameter range of differentiation that allows this is larger than mergers with high degree of differentiation and the profit level is higher in the low differentiation case. Thus there exists a post-merger effect in which merging first with firms producing similar products mitigates the later competition. We have also shown that either a sequence of mergers occurs or mergers do not occur in equilibrium. This analysis can be easily, but with tedious calculation, extended to a case of an n firm industry, with n 4 outsider firms in addition to the four firms present in the current model. The main results remain the same. The model can also have 2m firms with m pairs sequentially having chance to merge. As in Proposition 1, if the first pair decides to merge, all pairs thereafter would have an incentive to merge. This is because due to mergers the number of outsider firms decreases. Mergers have an anti-competitive effect to welfare, but the welfare analysis must also take cost reducing effect into account. 9
References Levy, David T., and James D. Reitzes, 1992. Anticompetitive Effects of Mergers in Markets with Localized Competition, Journal of Law and Economics, and Organization, 8, pp. 427 40. Matsushima, Noriaki, 2001. Horizontal Mergers and Merger Waves in a Location Model Australian Economic Papers, 40(3), pp. 263-286. Nilssen, Tore, and Lars Sørgard, 1998. Sequential Horizontal Mergers European Economic Review, 42, pp. 1683-1702. Salant, S., Switzer, S., Reynolds, R., 1983. Losses from Horizontal Merger: The Effects of an Exogenous Change in Industry Structure on Cournot- Nash Equilibrium, Quarterly Journal of Economics, 158, pp. 185-99. Scherer, Frederic M., and David R. Ross, 1990. Industrial Market Structure and Economic Performance. 3rd ed, Houghton Miffin, Boston. 10
Figure 1: Six different regions of parameters (β 1, β 2 ). In regions A to E sequential merger occurs in the low differentiation case, whereas it only occurs in regions A to C in the high differentiation case. 11