2008/5. Efficiency gains and mergers. Giuseppe De Feo

Transcription

1 2008/5 Effiieny gains and mergers Giuseppe De Feo

2 CORE DISCUSSION PAPER 2008/5 Effiieny gains and mergers Giuseppe DE FEO 1 January 2008 Abstrat In the theoretial literature, strong arguments have been provided in support of the effiieny defense in antitrust merger poliy. One of the most often ited results is due to Williamson (1968) that shows how relatively small redution in ost ould offset the deadweight loss of a large prie inrease. Furthermore, Salant et al. (1983) demonstrate that (not for monopoly) mergers are unprofitable absent effiieny gains. The general result, drawn in a Cournot framework by Farrell and Shapiro (1990), is that (not too large) mergers that are profitable are always welfare improving. In the present work we hallenge the onlusions of this literature in two aspets. First, we show that Williamson's results underestimate the welfare loss due to a prie inrease and overestimate the effet of effiieny gains. Then, we prove that the onditions for welfare improving mergers defined by Farrell and Shapiro (1990) hold true only when onsumers are adversely affeted. This seems an argument to disregard their poliy presriptions when antitrust authorities are more "onsumers-oriented". In this respet, we provide a neessary and suffiient ondition for a onsumer surplus improving merger: in a two firm merger, effiieny gains must be larger than the pre-merger average markup. Keywords: mergers, effiieny gains, Cournot oligopoly. JEL Classifiation: D43, L11, L22 1 CORE, Université atholique de Louvain, Belgium; Department of Eonomis, University of Salerno, Italy and Department of Eonomis, University of Strathlyde, U.K. Giuseppe.defeo@strath.a.u.k I wish to thank Rabah Amir, Paul Belleflamme, Alfredo Del Monte and seminar audiene in Naples for helpful omments and suggestions. This paper presents researh results of the Belgian Program on Interuniversity Poles of Attration initiated by the Belgian State, Prime Minister's Offie, Siene Poliy Programming. The sientifi responsibility is assumed by the author.

3 1 Introdution The purpose of the present work is to analyze the role of effiieny gains in horizontal mergers, to evaluate the effet on welfare and to ritially review the main ontribution in the literature on this topi. Mergers represent a major eonomi issue in the eonomi literature as well as for antitrust poliy. To provide an idea of the numbers and resoure involved, over the period , there were nearly 70,000 merger announements worldwide, with eah deal worth at least 1 million U.S. dollars, of whih nearly 45,000 were atually implemented. The average deal was valued at 220 million U.S. dollars (base year 1995). Of these, 42% were horizontal mergers, defined as those involving two ompanies with sales in the same 4-digit industry, 54% were onglomerate mergers, and 4% were vertial mergers. 1 Mergers have been, also, an important soure of inrease in market onentration, partiularly outside the U.S., as desribed by Shmalensee (1989). The relationship between market onentration and welfare dominated the eonomi debate sine the first wave of mergers at the end of the nineteenth entury; the ommon wisdom was that the higher the onentration, the lower the welfare. This wisdom was hallenged some deades ago when new strands in eonomi literature questioned this relationship. Some works, suh as Salant and Shaffer (1999), showed how higher market onentration an imply higher welfare in a Cournot oligopoly. 2 In other works strong arguments were provided in support of the effiieny defense in antitrust merger poliy. One of the results most often ited was provided by Williamson (1968) that showed how relatively small redution in ost, even in a merger for monopoly, ould offset the deadweight loss of a large prie inrease. The theoretial literature on horizontal mergers largely relies on the standard Cournot model. A entral postulate is that the pre-merger and the post-merger situations are represented as Cournot equilibrium points involving different market strutures, with the merged entity being treated as a single player in the post-merger situation. An important result in this framework was provided by Salant et al. (1983). Using the simple linear demand and ost setting, they showed that mergers without ost savings are unprofitable unless they involve more than 80% of the market. Even if Davidson and Denekere (1984) showed in the Bertrand setting with differentiated produts, that every merger is profitable, the result in Cournot ompetition had some impliations. 3 One of these impliations was that soially ineffiient 1 For this data see Gugler et al. (2003). 2 At a first glane this argument seems ounterintuitive but it learly relies on an important property of Cournot equilibrium: the more effiient firms produe more. Indeed, assume linear osts and an initial equilibrium with n symmetri firms. Suppose that ost are redistributed with a mean preserving inrease in the variane: total quantity, that depends only on the average marginal ost, remains unhanged. Now more effiient firms produe more than less effiient firms. Total output is unhanged and so is soial gross surplus, but total ost are redued. Welfare has inreased together with onentration. 3 This differene in the results is due to omplementarity and substitutability of the hoie variables and to the fat that in Bertrand ompetition the merger entity ontinues to produe all the differentiated produt of the pre-merger firms. In the model of Salant et al. (1983), on the ontrary, merger resolves in a lok-up of 1

4 mergers are unprofitable as long as they are not merger for monopoly. Some subsequent works, suh as Perry and Porter (1985), hallenged this point of view suggesting that antiompetitive profitable mergers are possible when fixed asset ombination is taken into aount and when mergers our in industries with inreasing marginal ost in the short run. 4 But then Farrell and Shapiro (1990) provided an important result: they defined onditions under whih profitable mergers are welfare improving in a general Cournot setting. Considering synergies, learning and asset ombination they are able to show that the realloation of prodution to the more effiient firms an lead to welfare improvement. Their benhmark work is based on the haraterization of the external effet of a merger, that is the sum of the effets on onsumers and outsiders inumbent firms not involved in the merger. The empirial literature on merger profitability exhibits some degree of ontroversy, mostly of a quantitative sort. Gugler et al. (2003), in the largest ross-national study to date, report that nearly 60% of all the horizontal mergers were profitable. On the other hand, two other broad-base studies onluded that the profitability of aquired firms delined after the merger for U.K. firms (Meeks, 1977) and for U.S. firms (Ravensraft and Sherer, 1988), while in Mueller (1980) divergent results studying OECD ountries are reported. Trying to explain this rather mixed piture, Amir et al. (2003) modeled the effet of a merger when non-merging firms are unertain about the effiieny gains reahed by the merging firms. They show how the result depends not only on the magnitude of ost redution but also on the belief of the outsiders; a merger an be profitable even if effiieny gains do not atually materialize, provided that non-merging firms suffiiently believe that the merger will generate large enough ost redution. The ontribution of this work is twofold. First of all we hallenge the ommon wisdom that small ost redutions more than offset the deadweight loss due to a prie-inreasing merger. Indeed, even though this result is due to the pioneering work of Williamson (1968), it remains very popular in the literature. Many works suh as Bloh (1995); Sapienza (2002); Shmalensee (2004); Shapiro and Willig (1990); Willig et al. (1991), just to provide few examples refer to Williamson s result as the learest theoretial evidene of the importane of effiieny gains in merger analysis. In the simple linear Cournot framework, we show that the ost redution needed to offset the alloative effiieny ost of a prie-inreasing merger are muh larger than Williamson s. To provide an example of the magnitude of this differene, Williamson reports that, with a demand elastiity of 1, a redution of just 0.5% of the average ost an offset the negative effet on welfare of a 10% prie inrease. In our setting the effiieny gains must exeed 11%. There are two main reasons for this huge differene. First of all, in the Williamson s analysis effiieny gains are omputed on the whole prodution of the industry. This is of ourse adequate only when a merger for monopoly is onsidered. Seond, he assumes a ompetitive prie in the pre-merger situation, so that the deadweight loss is just a loss in the onsumer surplus and not in profits. all the merging firm but one. Perry and Porter (1985) for this argument. 4 On merger profitability see also MAfee and Williams (1992). Faulí-Oller (1997) shows that profitability of mergers inreases with the onavity of demand funtion. 2

5 In the present work we onsider the ase in whih a merger ours in an oligopoly in whih firms ompete à la Cournot. Extending the analysis to a more general Cournot setting the external effet defined by Farrell and Shapiro (1990) is fully haraterized. We first show that the effet of a merger on onsumers has always the opposite sign to the effet on outsiders; so it is possible to show that the ondition defined by Farrell and Shapiro for a welfare-improving merger holds only when onsumers are adversely affeted. The latter seems an argument to disregard their poliy presriptions when antitrust authorities are onerned about onsumer protetion. At this regard we provide a lear-ut ondition for a merger not to redue onsumer surplus under Cournot ompetition: the effiieny gains must be larger than (m 1) times the average pre-merger markup-to-ost (MC) ratio, where m is the number of merging firms. This work is organized as follows: Setion 2 is devoted to the analysis of the Williamson (1968) s result and to the omparison with the linear Cournot equilibrium analysis. Setion 3 extends the analysis and desribes, in the linear Cournot setting, the Farrell and Shapiro (1990) s approah and the underlying fores that drive the result. It is shown that the effiieny gains needed to offset the welfare loss due to the inreased market power of the merger is a proportion of the pre-merger MC ratio that is dereasing in the number of firms. On the ontrary, it is shown that the effiieny gains needed to have a profitable merger is a proportion of the pre-merger MC ratio that is inreasing in the number of firms. So a threshold in the market share of the merging firms is found as a neessary and suffiient ondition to have a profitable merger that inreases total welfare. Setion 4 extends the Farrell and Shapiro (1990) s analysis using their marginal approah. We show that the effet of a merger on onsumers is always opposite to the effet on outsiders profits, and that the ondition defined by Farrell and Shapiro for a welfare-improving merger holds only when onsumers are adversely affeted. Moreover a neessary and suffiient ondition for a merger not to redue onsumer surplus is provided. Setion 5 ontains the onlusions. Most of the omputations are gathered in the Appendix. 2 Williamson s model ompared with an equilibrium analysis In his pioneering work, Williamson (1968) analyzes the welfare tradeoff of market power and effiieny motivations of mergers, and shows that relatively small ost redutions more than offset the alloative effiieny ost of a prie inrease. Assuming a market in whih two or more firms produe a homogeneous good, he analyzes the overall welfare effet of a merger that inreases the prie and ahieves effiieny gains. In Figure 1 his welfare analysis is depited. If the demand is approximate to a linear funtion, the deadweight loss due to the prie inrease is given by 1 2 P Q, while the benefit from ost savings is equal to ACQ. After some transformation, Williamson defines the ondition under whih the total effet is positive; that is, AC AC k ( ) P 2 2 η 0 (1) P 3

6 Figure 1: The total welfare effet of a merger, (Williamson, 1968, p.21). where AC is the average ost of prodution, η is the elastiity of the demand in the initial equilibrium and k is a measure (P/AC) of the pre-merger market power. Assuming that the initial market power is negligible, so k = 1, expression (1) gives rise to the results in Table 1. 5 ( P ) P η %.25%.12%.06% 10% 1.00%.50%.25% 20% 4.00% 2.00% 1.00% 30% 9.00% 4.50% 2.25% Table 1: Perentage ost redutions needed to offset some given prie inreases for seleted values of η. Aording to his hypothesis, a merger for monopoly, that ours when η = 1 and that rises the prie of 20%, an be welfare improving if it ahieves a redution in ost of just 2%. Unfortunately, his formulation is erroneous, in the sense that it overestimates the effet of ost 5 This table is taken from Williamson (1968, p. 23). 4

7 redution. 6 After several attempts, Jakson (1970) provided the (orret) following expression: AC AC and numerial examples shown in Table 2. 7 ( P ) P ) 2 k 2 η ( P P 1 η P P η %.27%.13%.06% 10% 1.24%.55%.26% 20% 6.66% 2.49% 1.11% 30% 22.49% 6.42% 2.64% 0 (2) Table 2: Perentage ost redutions needed to offset some given prie inreases for seleted values of η; Jakson (1970) s version. Results are not very different, but formulation (2) is not general, sine it works only when the prie is ompetitive at the pre-merger equilibrium; that is, when k = 1. Indeed, the welfare loss aounts only for the onsumer surplus loss, and not for the loss in profits on the quantity no longer produed. 8 Moreover, and most importantly, the ost redution in Williamson (1968) and Jakson (1970) is omputed on the whole prodution of the industry and not only of the merging firms. Then, his analysis is orret just in ase of a merger for monopoly when the starting prie is ompetitive; a situation that is at least unommon. Notwithstanding these limitations, Williamson s results are still onsidered as the learest evidene of the importane of effiieny gains in merger analysis and his work is one of the most ited paper on this topi. 9 In what follows Williamson s results are ompared with an equilibrium analysis based on the standard Cournot model of oligopolisti ompetition. 2.1 Equilibrium analysis: the linear Cournot model Consider an industry desribed by a linear inverse demand P = a bq and, in the pre-merger situation, n symmetri firms with onstant marginal ost that produe a homogeneous good and ompete à la Cournot. Assume a > > 0 and b > 0. The equilibrium in the pre-merger situation is desribed by the following values of total prodution and prie, Q = n (a ) b (n + 1) P = a + n n Indeed, the welfare effet of ost redution is omputed on the pre-merger level of output, while it should be omputed at the post-merger level, that is lower. 7 Jakson (1970, p. 441). 8 See DePrano and Nugent (1969) and Jakson (1970) on this point. 9 See, for example, Bloh (1995); Sapienza (2002); Shmalensee (2004); Shapiro and Willig (1990); Willig et al. (1991). 5

8 of eah firm s prodution and profit q i = a b(n + 1) π i = (a )2 b (n + 1) 2 and of onsumer surplus and total welfare CS = 1 n 2 (a ) 2 2 b (n + 1) 2 W n (n + 2) (a )2 = 2b (n + 1) 2 Assume now that two firms merge experiening not only an inreased market power, but also effiieny gains so that the merged entity has marginal ost m = with 0. At the new equilibrium an asymmetry between the merged entity and the outsiders arises. As a general result in Cournot oligopoly, the most effiient firm produes more than the others and the post-merger equilibrium is desribed by the following values of total prodution and prie, Q m (n 1) (a ) + = nb of the merged firm s prodution and profit, P m = a + (n 1) n q m m = a + (n 1) nb of the outsiders prodution and profit, 10 π m m = [(a ) + (n 1) ]2 bn 2 q m o = a nb π m o = [(a ) ]2 bn 2 and of onsumer surplus and total welfare CS m = 1 [(n 1) (a ) + ] 2 ( 2 bn 2 n 2 1 ) (a ) (n 1) (a ) + ( 2n 2 2n 1 ) 2 W m = Comparing the pre-merger and post-merger equilibrium welfare, the following proposition holds: Proposition 1 In a Cournot industry with linear demand, linear ost and n symmetri firms, a two-firm merger is welfare improving if and only if it ahieves a redution in ost suh that 2bn 2 γ (n) P where γ (n) is a dereasing funtion that depends only on n with 0 < γ (n) < 1 and P is the markup-to-ost (MC) ratio at the pre-merger equilibrium. Then, the effiieny gains needed to offset the negative alloative effet of a prie inrease is an inreasing funtion of the initial MC ratio. 10 As ommon in the literature we denote as outsiders the firms that are not involved in the merger. (3) 6

9 Proof. See Appendix. This result an be interpreted as a support to the ommon wisdom that the larger the market power of the firms, the greater the alloative effiieny ost of a merger. Then, the effiieny gains needed to offset this negative alloative effet should inrease with the premerger market power. In order to ompare this equilibrium analysis with Williamson s results, we an rearrange ondition (3), and express the effiieny gains needed to keep the welfare onstant as a funtion of the prie inrease and of the elastiity at the pre-merger equilibrium. The results are summarized in Table ( P ) P η % 4.64% 5.33% 6.37% 10% a 11.86% 15.76% 20% a a 53,24% 30% a a b Table 3: Perentage ost redution needed to offset some given prie inreases for seleted values of η; equilibrium analysis in a linear Cournot model. a It is impossible to have this prie inrease with this value of elastiity at the pre-merger equilibrium. b To offset this prie inrease the ost redution should be larger than 100%. The differene with Williamson s results is striking and has two main explanations previously highlighted. On the one hand his analysis does not take into aount the presene of a markup in the industry, so that the alloative effiieny ost of a prie-inreasing merger is a loss in onsumer s surplus, only. On the other hand the ost redution in Williamson (1968) is omputed on the whole prodution of the industry. 3 Comparative statis in a linear Cournot model The results in the previous Setion highlight how large the effiieny gains should be in order to offset the negative welfare effet of a prie-inreasing merger, and how optimisti Williamson s analysis is. Treating n as a disrete variable, it is possible to ompute the effiieny gains needed to keep the welfare onstant and the resulting equilibrium hange in prie when two firms merge. Table 4 shows this results for seleted values of the elastiity and of the number of symmetri firms in the pre-merger equilibrium. Some omments an be made. First of all, the ost redution needed to keep the welfare onstant is inversely related to the elastiity, as in Table 2.1. This is due to the fat that, in 11 This analysis is arried out treating n as a ontinuous variable in order to hoose given values of the elastiity of the demand and given prie inreases due to the merger. See Appendix for omputational details. 7

10 n ab η (4.59 ) (15.62 ) 21.28(10.30 ) 3.51(2.75 ) (17.68 ) 16.71(10.41 ) 9.46(6.87 ) 2.05(1.84 ) (27.53 ) 20.44(13.26 ) 10.03(7.81 ) 6.08(5.15 ) 1.44(1.38 ) (18.35 ) 10.22(8.84 ) 5.57(5.21 ) 3.55(3.43 ) 0.91(0.92 ) (12.23 ) 5.84(5.89 ) 3.34(3.47 ) 2.18(2.29 ) 0.58(0.61 ) (9.18 ) 4.09(4.42 ) 2.39(2.60 ) 1.58(1.72 ) 0.43(0.46 ) γ (n) d Table 4: Perentage ost redution needed for a 2 firm merger to keep welfare onstant for different values of η and n. a number of firms at the pre-merger equilibrium b in brakets the orresponding prie inrease, in perentage the elastiity of demand at the pre-merger equilibrium, in absolute value d effiieny gains, as a proportion of the initial MC ratio, needed to keep the welfare onstant the linear Cournot setting, the elastiity of the demand is inreasing in the equilibrium prie; and the latter, in turn, is inreasing in the marginal ost. But the MC ratio at equilibrium is dereasing in ost; so, a larger elastiity at equilibrium is assoiated with a lower ratio. Given that the effiieny gains needed to offset the alloative effiieny ost of the prie inrease are diretly related to the MC ratio, they are inversely related to the demand elastiity at the pre-merger equilibrium. A similar argument an explain the inverse relation between effiieny gains and the number of firms. In fat, the more ompetitive the market, the lower the alloative effiieny ost of a merger, the smaller the effiieny gains needed to keep the welfare onstant. Finally, it should be emphasized that the effiieny gains must always be larger than the prie inrease (exept for very high values of the elastiity) in order to avoid a redution in welfare. This is the most evident ontrast between our results and Williamson (1968) s. As highlighted in the Introdution, Farrell and Shapiro (1990) provide onditions under whih a profitable merger is welfare enhaning when firms ompete à la Cournot. These onditions essentially affirm that a merger, in order to be profitable, should entail redutions in ost. When the market share of the firms involved in the merger is below some threshold, these effiieny gains imply that the merger inreases the welfare. To illustrate the fores underlying this result we provide an illustration in the linear framework. The first step is to analyze merger profitability. Considering a two firm merger, it is profitable if πm m 2πi. Salant et al. (1983) show that a merger in a linear Cournot setting with onstant marginal ost and no fixed ost is not profitable if it does not ahieve any ost redution. In the following Proposition the magnitude of these effiieny gains is defined. 8

11 Proposition 2 In a Cournot industry with linear demand, linear ost and n symmetri firms, a two firm merger is profitable if and only if it ahieves a redution in ost so that φ (n) P where φ (n) is an inreasing funtion that depends only on n. Comparing the two thresholds for profitable and welfare improving merger Proof. See Appendix. φ (n) γ (n) n 1 2 (4) ( ) (5) Farrell and Shapiro (1990) find a suffiient ondition for a profitable merger to be welfare improving: the sum of the market share of the merging firms must be at most half of the market. 12 In the following Corollary, using the result (5) in Proposition 2, we define a neessary and suffiient ondition for the simple linear symmetrial Cournot oligopoly. Corollary 1 In a Cournot industry with linear demand, linear ost and n symmetri firms, a neessary and suffiient ondition for a profitable merger to be always welfare improving is that the sum of the market shares of the merging firms is lower than An important feature of Corollary 1 is that it holds even when the merger redues prie, while Farrell and Shapiro suffiient ondition holds only when the prie inreases. This is due to the fat that their results are based on the analysis of the external effet of the merger, that is the sum of the effets on onsumer surplus and outsiders profits. Using some properties of Cournot ompetition they are able to sign this external effet under general onditions; but only when the prie inreases they are able to define the overall effet on welfare. In setion 4 these properties of Cournot ompetition are presented and disussed in a more general setting. The threshold for a merger to be onsumer surplus inreasing an be omputed. Comparing the pries (or, alternatively, the onsumer surplus) defined in Setion 2.1 before and after the merger, the following Proposition holds: Proposition 3 In a linear Cournot model, a two firm merger is onsumer surplus improving if it ahieves a ost saving P that is, effiieny gains should be at least equal to the pre-merger MC ratio. Proof. See Appendix In this ase of linear demand and onstant marginal ost. 13 Note that this result holds in the general Cournot setting, as shown in Proposition 6. 9

12 n a η b welfare π CS welfare π CS welfare π CS welfare π CS 0,5 d d d 40,88 (17,68 ) 24,26 e 9,46 (6,87 ) 17,85 66,67 2,05 (1,84 ) 8,73 25,00 1,00 26,60 (18,35 ) -17,16 100,00 10,22 (8,84 ) 6,07 50,00 3,55 (3,43 ) 6,69 25,00 0,91 (0,92 ) 3,88 11,11 1,50 13,30 (12,23 ) -8,58 50,00 5,84 (5,89 ) 3,47 28,57 2,18 (2,29 ) 4,12 15,38 0,58 (0,61 ) 2,49 7,14 2,00 8,87 (9,18 ) -5,72 33,33 4,09 (4,42 ) 2,43 20,00 1,58 (1,72 ) 2,98 11,11 0,43 (0,46 ) 1,84 5,26 γ (n) f 26, 60 20, 44 14, 19 8, 18 φ (n) f 17, 16 12, 13 26, 77 34, 91 Table 5: Perentage ost redution needed for a 2 firm merger to keep welfare, insiders profit and onsumer surplus onstant for different values of η and n. a number of firms at the pre-merger equilibrium b the elastiity of demand at the pre-merger equilibrium in brakets the orresponding prie inrease d there is no pre-merger equilibrium at this level of elastiity e ost redution needed is higher than 100% f as perentage of the initial MC ratio 10

13 Table 5 ompares the different ost redution thresholds needed to keep the welfare onstant, to have a just profitable merger and to keep the onsumer surplus at the pre-merger equilibrium. Some omments an be made. First of all, a merger for monopoly, when n = 2, is profitable even if it does not ahieves effiieny gains. Negative values means that even worsening effiieny the merger is profitable. Seond, as shown in Corollary 1, merger profitability implies welfare improvement for n > 3. In fat, looking at the values of γ (n) and φ (n), this result is due to the fat the effiieny gains needed for a profitable merger inreases as n inreases, while the redution in ost needed to have a welfare improvement is dereasing in n. Third, the ondition for profitability is worth disussing. Salant et al. (1983) pointed out that, as the number of firms inreases, it beomes more diffiult for a merger to be profitable. This result does not hold if we define this diffiulty in terms of the neessary ost savings. In fat, realling ondition (4), depends on φ (n) and on the MC ratio. While φ (n) is inreasing in n, the MC ratio is dereasing in the same variable; so, the effiieny gains needed to have a profitable merger is the result of these two opposite fores. Table 5 shows that it is first inreasing than dereasing when n > 5. Forth, the effiieny gains needed to keep the onsumer surplus onstant is always equal to the pre-merger MC ratio. Then, profitability never implies a redution in prie. Finally this table allows us to understand the underlying fores that drive the Farrell and Shapiro (1990) s result. For n > 3 (see for example the values for n = 5) welfare an inrease even when the effiieny gains are not large enough to have profitability of the merger and onsumer surplus improvement. This means that the (positive) effet on outsiders profit is larger than the (negative) effet on onsumer surplus. This in turn means that, in the Cournot setting, when the sum of the merging firms market shares is less than the half of the market (n > 4), the onsumer surplus effet is dominated by the outsiders profit effet. In the next Setion we will see that this relation is true in the general Cournot setting and is reversed when the market share of the merging firms is above the threshold. 4 The external effet of a merger in a general Cournot setting Assume that an industry, in whih firms produe a homogeneous good and ompete hoosing quantity levels, an be desribed by an inverse demand funtion P (Q) with P (Q) < 0, and eah firm s ost struture is desribed by the funtion i (q i ). We define the reation funtions of the firms in the standard way r i (q i ) arg max q i 0 π i (q i, q i ) i = 1,..., n where π i and q i are profit and quantity of firm i, and q i is the quantity produed by all the other firms. To have the usual downward sloping reation funtions it is suffiient to assume 11

14 that: 14 2 π i (q i, q i ) q i q i < 0 Expressing this ondition in terms of the primitives: P (Q) + q i P (Q) < 0 i = 1,..., n. This ondition for all the firms is respeted if we assume that the demand funtion is suh that P (x) + xp (x) < 0. (6) Moreover, to have uniqueness of the equilibrium, assume that P (Q) i (q i ) < 0 i = 1,..., n. (7) Assumptions (6) and (7) imply that the reation funtions are dereasing ontrations; that is, their slopes r i P (q i + q i ) + q i P (q i + q i ) (q i ) = 2P (q i + q i ) + q i P (q i + q i ) i (q (8) i) belongs to the interval ( 1, 0). This is a suffiient ondition to have uniqueness and stability of the equilibrium. In the equilibrium eah firm s maximization program implies that: P (Q) + q i P (Q) i (q i ) = 0 q i > 0 and i = 1,..., n. (9) where qi is the best response to the quantity produed in equilibrium by the others. In what follows we will use a marginal approah to analyze the effet of a disrete exogenous variation of the rivals output. This approah, used also by Farrell and Shapiro (1990), is originally due to Gaudet and Salant (1988). Starting from the equilibrium, suppose that the quantity produed by some firms has an exogenous variation. This variation an our beause some firms at differently with respet to a Cournot player as well as some firms experiene effiieny gains that redue their prodution osts: a merger with effiieny gains ould be one of the auses of this exogenous variation of the output. The equilibrium responses of the other firms and the overall effet on market performane are analyzed. Note that the resulting situation is a Cournot equilibrium for the outsiders and the one (those) that auses the hange an either be in a Cournot equilibrium or not. So, the following analysis applies to Cournot market, but also to ases in whih some firms at differently This is the ondition to have a submodular game, that is a game with strategi substitutes. For more on submodularity of Cournot games see Amir (1996). 15 For example it an be used to analyze mixed oligopoly where there is a publi firm ompeting with private firms and the effet of a privatization. 12

15 With a little abuse of notation, we an define the optimal response of firm i to an infinitesimal exogenous hange of the quantity produed by the other firms in the following way: dq i = r i (q i ) dq i Adding on both sides r i (q i) dq i, we an express the variation of the quantity of the firm i in terms of total output variation: ( 1 + r i (q i ) ) dq i = r i (q i ) (dq i + dq i ) dq i = r i (q i) 1 + r i (q dq (10) i) Equation (10) identifies the equilibrium response of firm i to a total output hange dq aused by a hange in prodution of some firm. This analysis is learly a marginal analysis but, under some regularity onditions, it an be extended to disrete variations, as we will see soon. By assumptions (6) and (7) the variation of eah firm i s output is inversely related to the total output variation. Indeed, defining R i (q i ) r i (q i) 1 + r i (q i) (11) it has negative sign beause R i (q i ) = P (Q) + q i P (Q) P (Q) i (q i) < 0 This result is important to show some properties of the Cournot ompetition. Proposition 4 The effet of an exogenous variation of the output of some firms on the equilibrium profit of the others is opposite to the effet on onsumers. In fat, if some firms inrease the output exogenously, total output inreases, and prie and the other firms profit dereases. If these firms redue the output, prie inreases and the other firms profit inreases, too. Proof. The fat that an exogenous variation of the output of some firms hanges the equilibrium output in the same diretion is due to reation funtions being ontrations. Indeed, assuming that the output of some firms j hanges for exogenous reasons (for example a hange in her ost funtion or a hange in the objetive), summing up the variation of all the firms but j dqi = dq j = R i (q i ) dq i j i j 13

16 Adding dq j on both sides dq = i j R i (q i ) dq + dq j 1 R i (q i ) dq = dq j i j dq = 1 1 i j R i (q i) dq j (12) Sine R i is always negative for every firm, dq has always the same sign of dq j even if it is smaller; so, if firm j inreases the output, at the new equilibrium Q is larger and prie is lower. The onsumers are then better off. To see the effet on the other firms equilibrium profit of a hange in total output due to a hange of the output of some firms j πi = P (Q) qi i (qi ) dπi dq = P (Q) qi + P (Q) dq i dq (qi ) dq i dq = P (Q) q i + [ P (Q) (q i ) ] dq i dq From the first order ondition (9), P (Q) (qi ) = P (Q) qi. Substituting it in the previous expression and by equation (10) and definition (11), we have that: dπ i = P (Q) q i ( 1 R i (q i ) ) dq (13) By equation (12) dq = 1 1 P i j R i (q i) dq j and so: dπ i dq j = 1 R i (q i) 1 i j R i (q i) P (Q) q i < 0 Moreover this ondition always holds for any hange in Q and we an extend the result to disrete variation of quantity, whih ompletes the proof. This result an easily be applied to mergers with or without effiieny gains. Consider now j to be the subset of firms that merge and assume that the merged entity ats as Cournot player: if it ahieves effiieny gains suh that it produes more than the sum of pre-merger output of firms j, then total output inreases. 16 In this ase onsumers are better off and outsiders are worse off. On the other hand, if there is no ost saving, or it is not large enough, the merged firm produes less than pre-merger output, then total output dereases. The onsumers are worse off while the outsiders are better off. These results are summarized in the following Corollary. 16 In the Cournot setting if a firm has lower ost produes more. See an interesting appliation of this result in Salant and Shaffer (1999). 14

17 Corollary 2 Under assumptions (6) and (7), if a merger is harmful for onsumers, it inreases outsiders profits; if a merger redues the prie, it redues the outsiders profits. Farrell and Shapiro (1990) use the sum of the effet on onsumer surplus and outsiders profits to define the external effet of an exogenous hange in the prodution of some firms. They find onditions under whih this effet is positive when the total output shrinks. In what follows, we haraterize this external effet for a general variation of the output, both positive and negative. This allows us to extend and fully haraterize the result, that is summarized in the following Proposition. Proposition 5 Assume that P, P, i (.) are nonnegative and i (.) is nonpositive in the relevant range. Suppose that an exogenous variation of the output of some firms j ours. Then, the external effet has always opposite sign to the effet on onsumer surplus when the output of the insiders is smaller than a ertain threshold q j. When the market share of the insiders is larger, the external effet has the same sign as the onsumer surplus effet. In the first ase the effet on outsiders profits dominates the effet on onsumer surplus. The reverse is true in the latter ase. Proof. Total welfare is defined as the sum of onsumer surplus and profits, that is W = = Q 0 Q 0 [P (z) P (Q)] dz + π j + i j P (z) dz QP (Q) + π j + i j π i π i where, as usual, j represent the insiders firms that experiene an exogenous variation of the output and i j the outsiders. Now we want to ompute the marginal effet on welfare of a marginal equilibrium hange in Q due to the exogenous hange in q j ; that is, the effet on welfare when the outsiders optimally respond to the variation in the output of the subset of firms j. dw dq = P (Q) P (Q) + QP (Q) + dπ j dq + i j = dπ j dq + QP (Q) + i j Sine it is not possible to determine dπ j dq dπ i dq dπ i dq without knowing the reasons of the hange in the output, we an define the external effet as the differene between the effet on welfare and the effet on insider profits. From equation (13), we an substitute dπ i dq : dw dq dπ j dq = QP (Q) + P (Q) q ( i 1 R i (q i ) ) i j dw dq dπ j dq = QP (Q) + P (Q) qi P (Q) R i (q i ) qi i j i j 15

18 Sine i j q i = Q q j dw dq dπ j dq = q jp (Q) P (Q) i j R i (q i ) q i dw dπ j = P (Q) q j + R i (q i ) q i dq (14) i j So, the external effet has opposite sign with respet to the equilibrium variation of Q when q j + R i (q i ) qi < 0 i j that is, being R i (q i) always negative, q j < q j i j R i (q i ) q i (15) The external effet has the same sign when the opposite is true. By Proposition 4, the sign of dq is the same as the onsumer surplus, and outsider s profits move always in the opposite diretion. When ondition (15) is true, the effet on outsiders profits dominates the effet on onsumers, while onsumer surplus effet dominates if ondition (15) does not hold. To extend this result to disrete hanges of the insiders output, onditions on seond and third derivatives of inverse demand and ost funtions are needed. 17 The most important assumption of the previous Proposition is that ost funtions should be onvex for all the firms. However, this is not a great limitation sine onave osts give usually rise to natural monopolies. 18 To better understand the meaning of this threshold, some examples may be useful. Under linear ost and demand, R i (q i) = 1 so q j = i j q i. This means that half of the market is the threshold. In ase of linear demand and quadrati osts it an be shown that R i (q i) = q i η where η is the elastiity of the demand at the pre-merger equilibrium. 19 So the threshold is q j = 1 η i j q 2 i. In terms of antitrust poliy presriptions, Farrell and Shapiro affirm that mergers between firms whose market share is lower than the threshold should be allowed beause the external effet is positive. Looking at the external effet of a merger is then their guideline for antitrust srutiny. However, their result holds only when onsumers end up paying higher prie. Indeed: 17 See Farrell and Shapiro (1990, p. 116). 18 However this is not always true. Even when osts are onave (but not too muh) it an exists a unique equilibrium in Cournot oligopoly with all firms produing. This is atually a different definition of natural monopoly and it turns out to depend on the demand funtion, too. The ondition under whih this equilibrium exists is ondition (7) for whih an be negative but bigger than P. See on this Amir (2005). 19 See Farrell and Shapiro (1990, p. 118) for this result. 16

19 Corollary 3 Under the onditions of Proposition 5, if a merger between some firms whose market shares q j < q j redues the prie, the external effet of the merger is negative as long as the output inrease is not too large. To see this result,note that by (12) we an define the external effet in terms of q j. From equation (14) dw dπ j = P (Q) q j + i j R i (q i) q i 1 i j R i (q i) So W π j, that an be defined as the external surplus, is dereasing when q j < q j and inreasing when q j > q j. Under the onditions of Proposition 5, we graph in Figure 2 this external surplus as a funtion of the level of the output of insiders. We an see that when q j < q j and the merger determines an inrease in the output, so that total output inreases and onsumers are better off, the external effet is undetermined sine it an end up with a quantity higher than q j. But if q j is small enough, so that the post-merger output annot be larger than the threshold, we have the paradoxial effet that the external effet is negative even when welfare inreases. To get the intuition, suppose that the merged firm produes just the same quantity as the merging firms in the pre-merger situation. It is possible only if it ahieves effiieny gains. So, soial gross benefit is unhanged, total prodution osts are lower and total welfare inreases. The larger the effiieny gains, the larger the inrease in prodution, the larger the inrease in welfare. This result of ourse undermines the validity of the Farrell and Shapiro (1990) s poliy presription for whih the external effet is the key element. dq j Figure 2: External surplus as a funtion of insiders output. 17

20 However, using the relation between external surplus and the level of prodution we an extend the analysis to the ase when q j > q j. Corollary 4 Under the onditions of Proposition 5, if a merger between some firms whose market shares q j > q j redues the prie, the external effet of the merger is positive. The last omment is devoted to the shape of the external surplus. Given that, when q j inreases, prie dereases and onsumers are better off, the threshold q j defines the relationship between onsumer surplus and outsiders profits effets. When q j > q j the effet on onsumers surplus dominates, while outsiders profit effet dominates if q j < q j. Then, Farrell and Shapiro poliy presription holds only when onsumers are worse off; so, if antitrust authorities are more onerned about onsumer protetion, their analysis is not of great help. In the following Proposition we define a very general ondition on the effiieny gains that are suffiient to avoid a redution in onsumer surplus. Assume that m < n firms merge and denote by J the set of these firms, qj the sum of their quantities, and their average marginal ost at the pre-merger equilibrium. Proposition 6 Assuming nondereasing marginal osts, a merger does not rise the market prie if and only if it ahieves a redution in ost, omputed at qj, equal to (m 1) times the pre-merger average MC ratio of the merging firms. Proof. Sine by equation (12) in the proof of Proposition 4 the effet on total output has the same sign of the variation of the quantity of the insiders, all we need is to sign the hange in prodution of the merging firm. By the first order ondition of the new firm M, P (Q M ) P (Q M )q M M (q M ) = 0 where Q M and q M are the total output and the prodution of M in the post-merger equilibrium. Denoting with Q total output in the pre-merger situation, firm M will produe more than qj if and only if P (Q ) P (Q )qj M (qj) > 0 (16) Summing up the first order ondition of the insiders in the pre-merger equilibrium P (Q )qj = [ P (Q ) i (qi ) ] i J = m [ P (Q ) ] where m is the number of merging firms. Then, substituting in (16) P (Q ) m [ P (Q ) ] M (q J) 0 or that ompletes the proof. M (q J ) > (m 1) P (Q ) 18

21 5 Conlusion Williamson s results are not robust to an equilibrium analysis in a Cournot ontext. Effiieny gains needed to offset large prie inrease due to a merger are very large and, generally, larger than the prie inrease. His onlusion on the relevane of the effiieny gains in merger srutiny may be misleading if applied to ases of mergers in oligopoly. In fat, give that ost savings are omputed on the whole prodution of the industry, it is only adequate to ases of mergers for monopoly. 20 Our equilibrium analysis highlights the different and ontrasting interests of the parties involved by a hange in industry struture: outsiders are worse off when the effiieny gains of the merger are so large that the prie dereases and the onsumers are better off, and, vie versa, onsumers are adversely affeted by a merger when outsiders profits inrease. This work is about effiieny defense in the merger srutiny and takes into aount the effet of mergers on the different parties. We have shown how the effet on onsumer surplus is essentially dominated by the effet on outsiders profits of a merger when the merging firms market share is not so large. The external effet proposed by Farrell and Shapiro as a poliy instrument in merger analysis, has, in this ases, is positive only when onsumers are worse off. This seems an argument to disregard their poliy presriptions when antitrust authorities are onerned about onsumer protetion. In terms of onsumer surplus improving merger we provide a general ondition under Cournot ompetition for whih a merger is not harmful for onsumers. But it requires very high levels of effiieny gains. As a result, effiieny defense beomes a weak argument when antitrust authorities are onsumers-oriented. A last remark on fixed osts. We have hosen not to onsider them expliitly in the analysis sine the results are not affeted by them. In fat, if they an be onsidered sunk, then a merger does not ahieve any saving on them. If they are assets that the new merged firm an ombine in order to redue marginal ost, then the analysis arried out in the general Cournot framework of Setion 4 learly applies. If the merger ahieves savings on fixed osts, the overall welfare effet of the merger inreases, but the analysis on external effet, outsiders profits and onsumer surplus is not affeted at all. 6 Appendix Proof of Proposition 1. To define when a two merger inreases total welfare, onsumer surplus and profit in the pre-merger and post-merger situation are ompared. The differene in onsumer surplus is: CS = CS m CS = 1 [(n 1) (a ) + ] 2 2 bn 2 1 n 2 (a ) 2 2 b (n + 1) 2 20 Reall, however, that he assumes also ompetitive pries in the pre-merger situation. 19

22 that redues to the following polynomial expression in terms of and (a ): CS = (n + 1) (n + 1) 2 (n 1) (a ) ( 2n 2 1 ) (a ) 2 2bn 2 (n + 1) 2 (17) The differene in profits is given by the differene for the merging firms and for the outsiders. The expression for the former is given by π m = π m m 2π i = [(a ) + (n 1) ]2 2 (a )2 bn 2 b (n + 1) 2 Reduing it in the same polynomial form of (17) we obtain: π m = (n 1)2 (n + 1) (n + 1) 2 (n 1) (a ) bn 2 (n + 1) 2 + The differene in profits for all the outsiders is: [ ] [(a ) ] 2 (a )2 (n 2) π o = (n 2) bn 2 b (n + 1) 2 that redues to (n 2) π o = (n 2) (n + 1)2 2 2 (n 2) (n + 1) 2 (a ) bn 2 (n + 1) 2 To have a welfare inreasing merger W = CS + π m + (n 2) π o 0 + ( n 2 2n 1 ) (a ) 2 bn 2 (n + 1) 2 (18) (n 2) (2n + 1) (a )2 bn 2 (n + 1) 2 (19) Summing up the three polynomial equation and ordering with respet to the relevant variable, the welfare hange redues to W = (n + 1)2 ( 2n 2 2n 1 ) (n + 1) 3 (a ) 2bn 2 (n + 1) 2 + (2n + 1) (a )2 2bn 2 (n + 1) 2 0 (20) The denominator is always positive so what matters is the numerator that is a seond degree polynomial in. The roots are: 1,2 = (n + 1) 3 (a ) (n + 1) 2 (2n 2 2n 1) (n + 1) 6 (a ) 2 + (n + 1) 2 (2n 2 2n 1) (2n + 1) (a ) 2 (n + 1) 2 (2n 2 2n 1) 20

23 Beause the solution of inequality (20) alls for values external to the interval of the roots, the negative root doesn t matter. The solution redues to [ a n ] n 2 + 8n + 4 (n + 1) 2 n + 1 2n 2 (21) 2n 1 Computing the MC ratio at the pre-merger equilibrium and defining P = a (n + 1) γ (n) n n 2 + 8n + 4 (n + 1) 2 2n 2 2n 1 the effiieny gains needed to have a welfare improving merger are then γ (n) P that ompletes the proof. Note that γ (0) = 1 and that γ (n) is dereasing but always positive as shown in Figure 3. (22) Figure 3: γ(n) as a funtion of the number of firm in the pre-merger situation. It is the ost saving needed to keep the welfare onstant as a proportion of pre-merger MC ratio. Effiieny gains in terms of elastiity, Table 3. the elastiity of demand at the pre-merger equilibrium η = 1 p P Q = 1 b a + n n b (n + 1) n (a ) = Computing (the absolute value of) a + n n (a )

24 The MC ratio at the pre-merger equilibrium an be expressed in terms of elastiity: P = a (n + 1) = 1 nη 1 In this sense, the elastiity at the pre-merger equilibrium summarizes the values of the strutural parameters of the market. So, the effiieny gains needed to keep the welfare onstant an be rewritten in the following way: = γ (n) nη 1 When a merger implement the ost savings that keeps the welfare onstant, the effet on prie is defined by the following expression: P P = P m P [ a + (n 1) P = a + n ] n + 1 a (n + 1) = n n + 1 a + n n (a + n) Substituting for the threshold value of we obtain the prie inrease ( ) P n (1 + η) γ (n) = (n + 1) P nη 1 nη 1 1 ( ) n n(1+η) nη 1 + n = (n + 1) (1 γ (n)) nη 1 = 1 γ (n) n 2 η n 1 ( nη(n+1) nη 1 Combining equations (24) and (25) we obtain the effiieny gains needed to keep the welfare onstant as a funtion of the prie inrease and of the elastiity: ( = 1 1 n 2 η P ) (26) nη 1 P Using this relation and onsidering n as a ontinuous variable, we obtain the results summarized in Table 3. Proof of Proposition 2. To ompute the effiieny gains needed to have a profitable merger, ompare the profits of merging firms before and after the merger ) (23) (24) (25) π m m 2π i [(a ) + (n 1) ] 2 bn 2 2 (a )2 b (n + 1) 2 Being interested only in the positive root of the problem, it is possible to make the square root of both sides and, simplifying the terms ( ) (n + 1) (n 1) 2n n 1 (a ) (a ) n ( 2 1 ) 1 n + 1 n 1 22

25 Defining and using the equation (22) for the MC ratio φ (n) n ( 2 1 ) 1 n 1 To ompare φ (n) and γ (n) φ (n) P φ (n) γ (n) n ( 2 1 ) 1 n n 2 + 8n + 4 (n + 1) 2 n 1 2n 2 2n 1 ( 2n 2n 2 2n 1 ) n ( n 2 n 1 ) n (n 1) n 2 + 8n ( 2n 2 2n 1 ) ( n 2 n 1 ) (n 1) n 2 + 8n + 4 After some algebrai manipulations, the unique real and positive solution is n 1 2 In Figure 4 φ (n) and γ (n) are ompared. ( ) Figure 4: φ (n) and γ (n) as funtions of the number of firms in the pre-merger situation. They define the ost savings as a proportion of the pre-merger MC ratio needed to keep the welfare onstant and to have merging firms profits unhanged, respetively. 23

26 Proof of Proposition 3. The simplest way to ompare onsumer surplus is to ompare the prie before and after the merger. A merger inreases onsumer surplus if P P m a + n a + (n 1) n + 1 n a (n + 1) 0 a n + 1 = P where the last equality omes from equation (22). Referenes Amir, Rabah, Cournot Oligopoly and the Theory of Supermodular Games, Games and Eonomi Behavior, August 1996, 15 (2), , Supermodularity and Complementarity in Eonomis: An Elementary Survey, Southern Eonomi Journal, January 2005, 71 (3), , Effrosyni Diamantoudi, and Liun Xue, Merger performane under unertain effiieny gains, CORE Disussion Paper 2003/38, Université atholique de Louvain, Center for Operations Researh and Eonometris, May Bloh, Franis, Endogenous Strutures of Assoiation in Oligopolies, RAND Journal of Eonomis, Autumn 1995, 26 (3), Davidson, Carl and Raymond Denekere, Horizontal mergers and ollusive behavior, International Journal of Industrial Organization, June 1984, 2 (2), DePrano, Mihael E. and Jeffrey B. Nugent, Eonomies as an Antitrust Defense: Comment, Amerian Eonomi Review, Deember 1969, 59 (5), Farrell, Joseph and Carl Shapiro, Horizontal Mergers: An Equilibrium Analysis, Amerian Eonomi Review, Marh 1990, 80 (1), Faulí-Oller, Ramon, On Merger Profitability in a Cournot Setting, Eonomis Letters, January 1997, 54 (1), Gaudet, Gérard and Stephen W. Salant, The Profitability of Exogenous Output Contrations in Cournot Equilibrium: Impliation for Stikes, Mergers and Export Subsidies, Tehnial Report, University of Mihigan May Gugler, Klaus, Dennis C. Mueller, B. Burin Yurtoglu, and Christine Zulehner, The effets of mergers: an international omparison, International Journal of Industrial Organization, May 2003, 21 (5),