On mergers in a Stackelberg market with asymmetric convex costs

On mergers in a Stackelberg market with asymmetric convex costs Marc Escrihuela-Villar Universitat de les Illes Balears April, 2013 (PRELIMINARY VERSION) Abstract This paper analyzes profitability, the incentives to free ride and the welfare effects ofmergersinastackelbergmarketwithanefficient leader and a group of inefficient followers when costs are convex. It is shown that a leader-follower merger is always profitable. On the contrary, a merger of two followers is only profitable when they are inefficient enough since in this case the free-riding behavior of the non-merging followers is more than compensated by the efficient reallocation of outputs derived from the merger. Finally, we demonstrate that the presence of mergers generally leads to a price increase except when the leader and a relatively efficient follower mergebecausethemergedfirms use their leader position to increase output. JEL Classification: L13; L41 Keywords: Mergers; Stackelberg; Convex costs I am grateful to Ramon Faulí-Oller for his advice and encouragement. The author owes thanks to Wieland Müller for helpful suggestions. Financial support by the Ministerio de Educación y Ciencia through its project Políticas de salud y bienestar: incentivos y regulación (Ref: ECO2008-04321/ECON) is gratefully acknowledged. The usual disclaimer applies. Mailing address: Departamento Economía Aplicada. Edificio Jovellanos Ctra. Valldemossa km. 7.5. 07122 Palma de Mallorca, Baleares - Spain. Tel. +34 971173242. Fax: +34 971172389. Email: marc.escrihuela@uib.es. 1

1 Introduction In a symmetric linear Cournot oligopoly setting with homogenous goods, Salant, Switzer and Reynolds (1983) showed that mergers are generally not profitable since the minimum profitable merger involves at least 80 percent of the firms in the industry. Unprofitability comes from the fact that non-merging firms react to the merger by increasing their output. Mergers, therefore, create an incentive to free ride as outsiders often benefit from the merger more than participants. 1 Many papers have subsequently tried to solve this paradox by changing some of the original assumptions. For instance, in the Stackelberg model with linear demand and symmetric cost functions, Daughety (1990) showed that the merger of two followers resulting in a leader firm is potentially profitable, and Huck, Konrad and Müller (2001) showed that mergers between a leader and a follower are unambiguously profitable. Other examples are Heywood and McGinty (2008) that with convex costs obtain profitable mergers between leaders and between followers or, in an asymmetric and linear cost function setting, Escrihuela-Villar and Faulí-Oller (2008) show that a merger between a leader and several followers is always profitable regardless of the degree of cost asymmetry. In some cases, thus, the leadership assumption has been sufficient to solve the merger paradox. To the best of our knowledge, we combine for the first time the assumption of Stackelberg leadership with that of convex costs from Perry and Porter (1985) but in an asymmetric way. As in other models of merger and Stackelberg (Huck, Konrad, and Müller 2001), we take the leadership as given. We note that a substantial literature has examined the conditions under which leadership can emerge endogenously in oligopolies (see among others Saloner 1987 and Hamilton and Slutsky 1990). We consider a situation where the leader enjoys a cost advantage. Consequently, such an advantage may be seen as one of the main reasons for leadership in the first place even though we recognize that leadership in our model is an assumption. In the present paper we show that in the Stackelberg model the profitability of horizontal mergers crucially depends on cost asymmetries. We extend the analysis by Heywood 1 This well known result has been sometimes called the merger paradox. In fact, the profitability of horizontal merger depends on the degree of concavity of cost and demand functions (see for instance Perry and Porter (1985)). 2

and McGinty (2007) to the case where cost are asymmetric. We develop a model where a leader chooses output before a group of firms (followers) that may be less efficient than the leader. We show that a leader-follower merger is always profitable regardless of the cost asymmetry since the merged firm can take advantage of the leadership and thefactthatthemergerpermitsthefirms to combine their capital as well as reallocate outputs giving rise to scale economies. Furthermore, we show that the free riding issue may reemerge under cost convexity, in situations where it would be absent with linear costs. As followers become more inefficient theincentivetofreeridemayappearsince outsider followers profit more from the merger than the participating followers. This is true because in the efficient reallocation of outputs after the merger the inefficient merged follower is forced to drastically cut its production. We also obtain that mergers between followers become profitable only when the followers are inefficient enough compared to the leader. The intuition is as follows. One the one hand, non-merging followers react to the merger expanding their production. On the other hand, however, we have two additional effects derived form the merger, (i) merging followers may also efficiently divide the new s entity capital among its plants (formerly, firms), and (ii) the leader reduces their output when followers merge and this reduction increases as followers become less efficient. Consequently, when the followers are inefficient enough these two last effects dominate the first one rendering mergers profitable. Our analysis proves useful because it allows us to obtain that mergers of symmetric firms (followers) may be profitable in a setting where firms choose output. For instance, a merger of two followers may be profitable regardless of the number of firms in the industry. Here, the incentive to free ride reappears since even though merging firms profits increase with the merger, firms would prefer to wait for their rivals to merge, thus taking advantage from the higher prices without having to cut production. We also study the welfare impacts of mergers in the present model. It is shown that mergers between followers always increase price. However, interestingly enough, when the leader absorbs an inefficient follower market price only increases if the follower is sufficiently inefficient. The intuition behind is as follows. There are two different forces at work. First, as long as when a leader merges with a follower the new firm will stay a leader, the production of the follower is decided in a leader plan and, all else being 3

equal, should be larger than before the merger. Second, since the follower is less efficient than the leader, the former is forced to cut production due to the efficient reallocation of outputs after the merger. Obviously, as the degree of inefficiency of the follower increases the second effect dominates de the first and price increases. The rest of the paper is structured as follows. In section 2, we present the model. In section 3, we analyze firms incentives to merge and the incentives to free ride on the merger. Section 3 studies the welfare impact of mergers. We conclude in section 4. All proofs are grouped together in the appendix. 2 The Model We consider a market for a homogenous product with n firms, one leader and n 1 followers. Inverse demand is given by P (Q) =1 Q where the industry output Q = Xn 1 q L + q i isthesumoftheleader sandthen 1 followers output. Competition occurs i=1 in two stages. In the first stage, the leader chooses its output. In the second stage, the remaining n 1 followers, knowing the output chosen by the leader in the first stage, choose simultaneously their level of production. Apart from the strategic advantage, the leader is assumed to be more efficient than followers. 2 Thecostofproductionoftheleader is given by c(q L )= 1 2 q2 L, whereas the cost function of followers is given by c(q i )= c 2 q2 i for i =1,...,n 1 with c 1. When c =1we are back to the standard symmetric Stackelberg model. We take n to be exogenous. The results then are determined by the pre- and post-merger profit comparisons of the leader, the followers included in the merger, and the excluded followers. The pre-merger equilibrium quantities, profits, and price are given respectively by: 1+c q L = (1) 2+n+3c 1+n+2c q i = (n+c)(2+n+3c) p = (1+c)(1+n+2c) (n+c)(2+n+3c). (2) 2 This is assumed following the reasoning of the folk theorem that relatively large firms are committed leaders and small firms are followers. Sadanand and Sadanand (1996 ) obtain a formal result for sufficiently small amounts of uncertainty. 4

We note that contrary to the linear symmetric case, the output level of the leader depends onthenumberoffollowers. Wenotethatq L increases with the number of followers and q i decreases with the number of followers. The expressions from (1) lead to the following equilibrium profits obtained respectively by the leader and the followers: Π L = (1+c) 2 2(c+n)(2+n+3c) (3) Π i = (2+c)(1+n+2c)2 2(c+n) 2 (2+n+3c) 2. We consider two different types of mergers: (a) the merger between a leader and a follower and (b) a merger between two followers. Following the reasoning by Perry and Porter (1985), firms have the ability to allocate output across its plants. The slope of the merged firm s composite marginal cost curve is cut in half because the merged firm has two different plants. Furthermore, for the merger of type (a), the merged firm enjoys also the standard Stackelberg advantage. 3 We are assuming that the gain from merging is divided among participants according to their post-merger output as independent plants that belong to a merged entity. 4 It a standard exercise to verify that outputs, profits after merger for the leader, for (each plant of) the merged and non-merged followers and market price after the merger are respectively: Merger of type (a) ql a = c(1+c) q a 1+c 2+c(3+3c+n) m = q a 2+c(3+3c+n) nm = Π a c(1+c) L = 2 Π a 2(c+n 1)(2+c(3+3c+n)) m = p a = (1+c)(1+c(1+2c+n)) (c+n 1)(2+c(3+3c+n)). 1+c(1+2c+n) (c+n 1)(2+c(3+3c+n)) (1+c) 2 2(c+n 1)(2+c(3+3c+n)) Π a nm = (2+c)(1+c(1+2c+n))2 2(c+n 1) 2 (2+c(3+3c+n)) 2 3 We assume that if a leader merges with a follower the new firm will stay a leader mainly for two reasons: (i) the merged firm can still use the old commitment technology of the former leader firm and (ii) it can be verified that the merged firm would always rather be leader than follower. 4 An alternative approach (specially for case (a)) could consist of considering that one party acquired the other by paying it the pre-merger profit. Hence, one firm would capture all the additional profits. However, we believe that the assumption that one firm has a larger bargaining advantage than the other wouldbeanadhocassumption. (4) 5

Merger of type (b) q b L = q b m = q b nm = (1+c)(2+c) 2(1+n)+c(10+3c+n) (5) (1+c)(2n+c(5+2c+n)) 2(2(n 1)+c(2+c+n))(2(1+n)+c(8+3c+n)) (2+c)(2n+c(5+2c+n)) (2(n 1)+c(2+c+n))(2(1+n)+c(8+3c+n)) Π b L = (1+c)2 (2+c)(8+c(72+c(340+3c(98+3c(10+c))))+2(2+c)(6+c(41+c(23+3c)))n+(2+c) 2 (4+c)n 2 ) 2(2(n 1)+c(2+c+n))(2(1+n)+c(8+3c+n))(2(1+n)+c(10+3c+n)) 2 Π b m = Π b nm = p b = (1+c) 2 (4+c)(2n+c(5+2c+n)) 2 2(2(n 1)+c(2+c+n)) 2 (2(1+n)+c(8+3c+n)) 2 (2+c) 3 (2n+c(5+2c+n)) 2 2(2(n 1)+c(2+c+n)) 2 (2(1+n)+c(8+3c+n)) 2 (1+c)(2+c)(2n+c(5+2c+n)) (2(n 1)+c(2+c+n))(2(1+n)+c(8+3c+n)). 3 The mergers A merger is considered to be profitable if the profits of merging firmsincreaseaftermerger. In case (a) it implies that: Π a L + Π a m > Π L + Π i. (6) Foramergeroftype(b)itimpliesthat: Π a m > Π i. (7) Regarding case (a) the results of the symmetric case analyzed by Huck, Konrad and Müller (2001) (with linear costs) and Heywood and McGinty (2008) (with convex costs) extend to the asymmetric case with convex costs: the merger between a leader and a followers is always profitable. Proposition 1 For all n and c, a merger between the leader and a follower is always profitable. As mentioned above, Proposition 1 parallels for the asymmetric case previous results in Stackelberg merger models that assume symmetric costs. Note that in the Cournot model with convex costs merger is profitable only for sufficiently large level of cost convexity (Perry and Porter (1985)) since in this case the reduction in output by the merged firm brings sufficient costsavingsandprofit increases. In this sense, the introduction of 6

leadership makes these cost savings inherently larger. Furthermore, the inefficiency of the follower (compared to the leader) exacerbates the cost savings derived from the merger sinceinthiscasewehavetheadditional positiveeffect of allowing also some cost savings by transferring output from a high cost firm to a low cost firm. Forcase(b),however,mergerprofitability cannot rely on the leadership effect. In this case, thus, we can observe that two different forces are at stake. On the one hand, the larger the inefficiency of the followers, the larger are also the cost savings originated by the merger through the optimal reallocation of output between the merging plants. In this case, however, the cut reduction effort of the participants creates an incentive to free-ride, as non-participants might benefit from the merger more than the participants. On the other hand, as c increases, the leader takes less into account the rivalry of the followers. When after the merger the production of (some) inefficient followers is reduced, the leader uses less the strategic power to anticipate a large output, and consequently leaders production increase becomes smaller as c increases. Proposition 2 For all n, a merger between two followers is profitableonlyifc is high enough. In other words, the merger is only profitable when the followers are inefficient enough becauseinthiscasetheefficient reallocation of output, in addition to the (smaller) increase of output by the leader, compensates the free rider problem that convex costs do not eliminate. We analyze now how the incentive to free ride is affected by the presence of cost asymmetries for both types of mergers. The incentive to free ride is given by Π a nm Π a m and Π b nm Π b m for case (a) and (b) respectively that capture whether non-participant followers profit more from the merger than the participating firms. This effect is important since the existence of free-riding may stop profitable mergers from taking place. At first sight, one could expect that the presence of cost saving mergers (due to convex costs), would help solving the free rider problem. However, Heywood and McGinty (2007) show that in the case of Cournot competition with convex costs there is always an incentive to free ride and that this incentive decreases with the degree of cost convexity. Our interest, however, is in the effect of the cost asymmetry in the incentives to free ride. For the type (a) of mergers, we obtain 7

Proposition 3 If the leader and a follower merge, non-merging followers profit more from the merger than the participating follower if and only if c is large enough. As c increases, even though the merger is (always) profitable and besides the fact that the merged follower enjoys a leadership position, the inefficient merged follower is forced to further cut its production compared to the pre-merger output in order to maximize joint profits with its merged partner. Consequently, the free-riding incentives reappear if c is large enough. Regarding the type (b) of mergers Proposition 4 If two followers merge, non-merging followers always profit morefrom the merger than the participating followers. From Proposition 2 we know that, obviously, free riding cannot appear when c is small becauseinthiscasethemergerisnotprofitable. Then, the intuition of Proposition 4 is derived from two different forces. Firstly, as c increases, the participants can further efficiently reallocate outputs through the merger. Secondly, since participants also cut production compared to the pre-merger equilibrium, the leader reacts to the merger by expanding its production. As mentioned above this output expansion is smaller as c increases and this benefits the non-participating followers. Interestingly enough, the second force dominates the first one and there is always a free riding incentive. 4 Welfare To study the welfare effects of mergers we use in this subsection the total consumer surplus as a welfare measure. 5 Therefore, we just have to compare the price after the merger and before the merger. Proposition 5 The price increases after the merger between the leader and a follower only if c is large enough. Otherwise, the merger decreases price. 5 Several recent papers call for antitrust agencies to use a consumer surplus standard rather than a total welfare standard (see for instance Pittman (2007)). Often, the argument against a consumer welfare standard is that it implies a tolerance for monopsony. However, it is well known that this is more likely to occur in markets for intermediate goods and we focus here in a final good market. 8

The intuition is as follows. The merged follower has a leader role after the merger and as a consequence its production tends to increase. On the other hand, however, as long as themergedfollowerismoreinefficient than the leader, an efficient reallocation of output inside the merged entity tends to decrease the production of former follower. When c is relatively small, the first effect dominates and consequently price decreases. Regarding the merger of two followers Proposition 6 For all n and c the merger of two followers strictly increases price. In this case the merger unambiguously increases price since the reduction in the output of the followers compensates the output expanding reaction to the merger from the leader and the non-merging followers. 5 Concluding comments The purpose of this paper is to analyze mergers in a Stackelberg market with convex costs when the leader is more efficient than the followers since often a cost advantage can be seen as one of the main reasons for leadership. We have shown that the combination of convex costs and Stackelberg leadership does not always eliminate the well-known merger paradox when costs are asymmetric. In fact, the free riding incentives are restored when mergers between followers are considered. In this case, the merger of two followers is only profitable if the followers are inefficient enough because in this case an optimal reallocation of output after the merger compensates the free riding behavior of the non-participants. A welfare analysis shows that price normally increase after the merger and the output reduction effort of the merged entity offsets the optimal reallocation of output derived from the merger. The only exception to this occurs when the leader merges with a relatively efficient follower. In this case, price decreases because the merged follower can use the old commitment technology of the former leader by significantly increasing production. The present paper thus highlights that even with convex costs the free riding incentives of non-participants could prevent mergers from taking place. One question this note does not address is the extension to multilateral mergers. Following the logic of Perry and Porter s (1985), however, it seems clear thatifamergerbetweenaleaderandasin- gle follower is profitable, it would seem that a merger with multiple followers would be 9

more profitable because the merged firm gains the cost advantage of allocating production changes across even more plants. The same reasoning applies to the merger among multiple followers. Finally, the limited context of the present model is acknowledged: to analyze real-world cases of mergers a wider range of demand functions or capacity constraints should also be considered. Appendix Proof of Proposition 1. From (4) and (6) we can obtain that this merger is always profitable if the following expression is positive 4+c(1+c)(24+c(29+c(32+c(10+3c))))+8n+2c(12+c(15+c(15+2c(2+c))))n+(c 4 2 9c 5c 3 )n 2 2(1+(c 1)c)n 3 +n 4. 2(c+n 1)(c+n) 2 (2+3c+n) 2 (2+c(3+3c+n)) With the Mathematica computer algebra system it can be easily proved that there is no root for c>1and n>3 for the last expression being equal to 0. Consequently, we just have to check that for instance when c =1, Π a L +Π a m (Π L +Π i )= 200+n(104+(n 5)n(3+n)) > 2n(1+n) 2 (5+n) 2 (8+n) 0. Proof of Proposition 2. From(5)and(7)this merger is profitable if ((1+c) 2 (4+c)(2n+c(5+2c+n)) 2 (2(n 1)+c(2+c+n)) 2 (2(1+n)+c(8+3c+n)) 2 (2+c)(1+2c+n) 2 (c+n) 2 (2+3c+n) 2 > 0. We can easily check that the last expression with equality has only one root in c for c>1. Then, when c =1, Π a m Π i = 20(7+3n) 2 (13+42n+9n 2 ) 2 3(3+n)2 (5+6n+n 2 ) 2 is negative if n is large enough. Consequently, since Π a m Π i has a unique maximum with respect to c (for 1 c), we just have to check that lim c Π a m Π i =0. Proof of Proposition 3. From (4) we have to check that Π a nm > Π a m if c is large enough. Then Π a nm Π a m = (2+c+c2 n)(2+c(4+c(5+c)+n)) > 0 c if n =3and for n>3 is positive only 2(c+n 1) 2 (2+c(3+3c+n)) 2 if c> 1( 4n 7 1). 2 Proof of Proposition 4. From (5) we just have to check that Π b nm Π b (4+3c)(2n+c(5+2c+n)) m = 2 > 0. 2(2(n 1)+c(2+c+n)) 2 (2(1+n)+c(8+3c+n)) 2 Proof of Proposition 5. We just have to compare prices before merger in (2) and after the merger in (4). We can easily obtain with the Mathematica computer algebra system that for a given n, p a p = (1+c)(2+4c3 (n 2)n+c(7+n)+c 2 (7+2n)) =0has only one root in c (c+n 1)(c+n)(2+3c+n)(2+c(3+3c+n)) that we denote by c Then, the result holds since (pa p) > 0. c c= c Proof of Proposition 6. AS in the last Proposition, we just have to compare prices before merger in (2) and after the merger in (5). Then 10

p b p = 2(1+c)(5c3 +4c 2 (4+n)+2(1+n+n 2 )+c(10+n(10+n))) (c+n)(2+3c+n)(2( 1+n)+c(2+c+n))(2(1+n)+c(8+3c+n)) > 0. References -Daughety, A. F. (1990), Beneficial concentration. American Economic Review 80, 1231-1237. -Escrihuela-Villar, M. and Faulí-Oller, R. (2008). Mergers in asymmetric Stackelberg markets. Spanish Economic Review 10, 279-288. -Hamilton, Jonathan H., and Steven M. Slutsky. (1990). Endogenous timing in duopoly: Stackelberg of Cournot equilibria. Games and Economic Behavior 2, 29-46. -Heywood, John S. and Matthew McGinty (2007). Mergers among leaders and mergers among followers. Economics Bulletin 12, 1-7. -Heywood, John S. and Matthew McGinty (2008). Leading and Merging: Convex Costs, Stackelberg, and the Merger Paradox. Southern Economic Journal, 74 (3), 879-893. -Huck, S. Konrad, K. and Müller, W. (2001). Big fish eat small fish: on merger in Stackelberg markets. Economic Letters 73, 213-217. -Perry, M.K., Porter, R.H. (1985). Oligopoly and the incentive for horizontal merger. American Economic Review 75, 219-227. -Pittman, Russell W. (2007). "Consumer Surplus as the Appropriate Standard for Antitrust Enforcement". Competition Policy International, 3(2), 204-224. -Sadanand, A. and V. Sadanand, (1996). Firm Scale and the Endoneous Timing of Entry: a Choice between Commitment and Flexibility. Journal of Economic Theory 70, 516-530. -Salant, S. Switzer, S. and Reynolds, R. (1983). The Effects of an Exogenous Change in Industry Structure on Cournot-Nash Equilibrium. Quarterly Journal of Economics 98(2), 185-199. -Saloner, Garth. (1987). Cournot duopoly with two production periods. Journal of Economic Theory 42, 183-187. 11