COORDINATED EFFECTS MERGER SIMULATION WITH LINEAR DEMANDS Peter Davis April 2006

Transcription

1 COORDINATED EFFECTS ERGER SIUATION WITH IAR DEANDS Peter Davis Aril 2006

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3 COORDINATED EFFECTS ERGER SIUATION WITH IAR DEANDS Peter Davis SE, STICERD, CEPR and Alied Economics td This drat: Aril 2006 Astract This aer rovides an examle o a methodology or evaluating the otential coordinated eects o mergers in dierentiated roduct marets. Seciically, I examine coordinated eects merger simulation when demand systems are linear and marginal costs are constant. I show that these assumtions are suicient to ensure that analytic solutions or the required collusive, Nash and deection ricing strategies are availale. Consequently the methodology outlined here is easy or antitrust authorities (and others to imlement. A ull imlementation o the methodology using data rom the Networ server maret and est ractice demand structures is rovided in Davis, Huse and Van Reenen (2006. Here I rovide numerical examles, illustrating the techniques and demonstrating the coordinated eects o simulated mergers. While the numerical results are secial to the articular arameters o the demand system considered, the analytic results ensure that the calculations can easily e erormed or essentially any linear demand structure and any ownershi structure in the maret. The numerical examles demonstrate clearly that mergers may enhance the lielihood o collusion ut they also show that mergers will sometimes mae collusion more diicult to sustain, in articular when mergers create highly asymmetric maret structures. Along the way I show (i that a ol theorem does not hold in general dierentiated roduct games, and (ii that it can e the small irm who is hardest to induce to collude. Dr Peter Davis: R58 STICERD, Deartment o Economics, ondon School o Economics, Houghton Street, ondon, WC2A 2AE, United Kingdom. Tel: +44 ( davis@lse.ac.u. We: Thans are due to ohn Davies, ar Ivaldi, Paul Klemerer, Adam and, Cristian Huse, ohn van Reenen, Helen Weeds and Alerto Salvo, as well as seminar articiants at Oxera, the Association o Cometition Economists winter meeting and the Networ o Industrial Economists annual meeting or helul comments and suggestions. The usual disclaimers aly. In articular, these do not urort to e the views o any o the aove organizations and the author ears sole resonsiility or any remaining errors.

4 Contents. Introduction Unilateral eects and the single eriod game Unilateral eects merger simulations A numerical examle: The reeated game The ayo to deection Coordinated eects merger simulations A irst coordinated eects merger simulation Asymmetry and coordinated eects Conclusions...7 Reerences

5 . Introduction This aer reorts on a methodology or emirically determining whether irms have incentives to tacitly collude in dierentiated roduct marets. The techniques I illustrate here will e articularly useul in merger control where olicymaers must decide whether or not to allow roosed mergers. Currently, merger control authorities can and do loc mergers on the asis that the smaller numer o irms ost-merger may mae tacit collusion more liely. In merger evaluation, this is nown as the theory o coordinated eects. Indeed coordinated eects were the rimary reason given or locing mergers until at least the early 990 s when unilateral eects, the idea that when irms roducing close sustitutes merge static equilirium rices may go u, also emerged as a maor source o concern. Recent examles in which antitrust authorities have invoed the theory o coordinated eects include the Nestle-Perrier, Kali and Salz, Gencor-enrho and Airtours cases in the EU urisdiction and Saeway in the UK. (See Dic (2003 or a discussion o the large numer o recent coordinated eects cases in the US. When locing mergers on the grounds o coordinated eects, the authorities ollow a rich theoretical economics literature emhasizing that oligoolists who meet regularly in the maretlace may tacitly collude on higher rices. Chamerlin (929 argued this oint inormally while Stigler (964, Friedman (97 and a large numer o susequent authors ormalized this intuition in the theory o reeated games (see Aumann (986,989 and ertens (987 or surveys. I merge this theoretical literature on reeated games with the emirical literature on the analysis o ricing games in dierentiated roduct marets. In articular, I use the recent literature on the evaluation o the unilateral eects o mergers (Werden and Froe (994, Hausman et al (994 and Nevo (200. In doing so I hoe to rovide techniques which will e useul sulements to the quantitative techniques currently used in ractice to evaluate coordinated eects; see Scheman and Coleman (2003. In order to decide whether a merger would result in an increased lielihood o tacit collusion, antitrust authorities roceed y considering the resence or asence o conditions that acilitate collusion. Seciically, in order to sustain collusion irms must e ale to (i come to an agreement (which can e diicult when roducts are comlex and dierentiated, (ii monitor each others ehavior (in order to detect cheaters and (iii enorce collusive ehavior collectively y unishing those incument irms who cheat (internal staility and deterring entry y new otential rivals or exansion y a cometitive ringe (external staility. In this aer, we examine when irms would e ale to sustain tacitly collusive outcomes using the most asic enorcement mechanism suggested y the theory o reeated games, grim strategies. When using a grim strategy, a irm lays a collusive action to egin with and continues to do so as long as (she never detects that a rival has cheated. I cheating is detected, she lays the static Nash equilirium strategy in all susequent eriods. Grim strategies rovide an enorcement mechanism or tacitly colluding irms ecause they unish cheating against a collusive arrangement y ensuring that a cheater will sacriice their share o uture collusive roits in return or short run gains today. Provided irms are suiciently atient, Friedman (97 showed that, in homogeneous roduct settings, grim strategies will suice to sustain tacitly collusive equiliria. This result is nown as Friedman s ol theorem. One contriution o this aer is to rovide an examle demonstrating that, or grim strategies, such a ol theorem does not aly in dierentiated roduct marets. 3

6 In sum, our aim in this aer is to tae seriously the idea that dynamic oligooly models can e taen to data in dierentiated roduct marets. As Kuhn and otta (2004 note, the udgment y the Court o First Instance in annulling the decision o the Euroean Commission to loc the Airtours merger has clariied that oint dominance analysis in Euroean mergers has to e treated as coordinated eects analysis and thus has to e consistent with collusion theory. Thus, our aim in this aer is to rovide a natural next ste in develoing merger simulation into a ractical toolox, one that can eectively allow cometition authorities to consider whether cometition concerns arise rom either otential unilateral or otential coordinated eects o mergers. The methods resented here are most directly useul to evaluate (i internal staility and (ii one element o external staility; the incentive or the cometitive ringe to exand in the ace o a cartel. Simulation o the other ey element o external staility, whether cartel roits would attract new entrants to the industry, requires additional inormation, or examle an evaluation o the costs o entry into the industry, and would involve a non-trivial extension o the model. Our wor is related to a numer o rior literatures. The most directly related emirical literature has attemted to evaluate the existing conduct o irms using game-theoretic ricing models. Seciically, authors ollowing Gollo and Roerts (979, Bresnahan (982, au (982, Roerts (983, Porter (983, Suslow (986, Bresnahan (982,987, Gasmi, aont and Vuong (990, 992, Nevo (998, 200, Slade (2004 and Salvo (2004 have attemted to evaluate whether oserved equilirium rices are more consistent with collusive or Nash equilirium ricing. 2 In contrast to this ranch o the literature, I ocus rimarily on evaluating whether collusion is sustainale as an equilirium. In the theoretical literature, some recent related theoretical contriutions have egun to study collusion in settings with asymmetric maret structures. The act that the analysis must consider cases where irms are asymmetric ost-merger maes theoretical analysis in this area challenging. Nonetheless, there is some recent rogress. In articular, Comte, enny and Rey (2002 examine coordinated eects in the context o a Bertrand-Edgeworth homogeneous goods model with caacity constraints and calirate their model with the data rom the Nestle-Perrier case. In their model, caacity constraints mean it is the large irms who can oth e temted to cheat and also have the aility to unish their rivals. Thus, it is the irm with large caacities who must e induced to collude, erhas y giving smaller irms sare caacity and thus an aility to unish their larger rivals. We will ind that the oosite can also e true that the small irms can e the ones who will e diicult to induce to collude. The reason is undamentally the standard oservation rom merger models in dierentiated roduct marets that small irms lie concentrated rivals ecause the larger or more collusively their rivals act, the higher the rices charged in Nash equilirium. That act however maes it diicult to induce small irms to collude, ecause their share o the collusive ie (at least in a world without side ayments will e small while their reedom to undercut their rivals rices is more valuale the more concentrated their rivals are. Kuhn and otta (2004 rovide the irst aer to study asset transers in dierentiated roduct marets. The intuition or their conclusions regarding asymmetry is identical to 2 Salvo (2004 rovides the most recent contriution to the literature, examining the determination o conduct when the otential or imort sustitution constrains the ehaviour o incument irms to charge high rices. 4

7 ours while their setting is ar more comlex it is one o imerect inormation. Unortunately, to study asset transers theoretically they must mae rather strong assumtions, ones that mae their model diicult to consider directly as a asis or emirical wor. For examle they consider only the case when the rice or every good sold y each irm is the same. 3 For that reason, the numerical and simulation aroach ollowed in this aer seems liely to e useul. The closest aer to our own is rovided y Saatini (2004 o the Italian Antitrust Authority who has recently, and indeendently, suggested a similar aroach to coordinated eects merger simulation. 4 Throughout the aer I tae the stance that merger simulation is a useul and interesting exercise. Some authors warn that simulation results are oten sensitive to the details o the model (see or examle the discussion in Waler (2005. Such oservations however are not necessarily critiques we actively want merger investigations to have dierent outcomes deending on the nature o, say, sustitution atterns etween goods. In any modeling exercise, one must always e careul to seciy the emirical and theoretical comonents o the model aroriately and merger simulation exercises are no dierent in this regard. The good news is that given enough data, we will e ale to tell models aart rom one another given the right data variation (see in articular Bresnahan (982 and Nevo (998. Even without rich datasets, simulation allows investigators to descrie the way in which their conclusions are, or are not, sensitive to aseline modeling assumtions. A range o lausile scenarios can thereore e evaluated. In the case o coordinated eects merger simulation we have exanded the set o models o irm ehaviour eing considered, rom static to dynamic rice setting models. While extending the set o models eing considered is advantageous, it does not come without a rice. For examle, there are numerous alternatives to using grim strategies to sustain collusive arrangements. The good news is that the techniques outlined here can, at least in rincile, e amended to also allow their evaluation. In articular, the results rovided in Areu (988 rovide an algorithm or checing whether more sohisticated simle enal codes are su-game erect Nash equilirium strategies. On the other hand, we choose to ocus on grim strategies ecause they are simlest, are well understood, do not assume that comanies can unish otimally and will generally rovide a coherent enchmar against which to udge whether there are liely to e increased incentives or tacit collusion. However, we do not retend that this is the last word on the toic and we intend to return to more sohisticated treatments in susequent aers. An area o articular concern in model seciication is the underlying demand system (see Waler (2005. Again, given enough data, such modeling choices can usually e tested against one another. inear and log-linear demand seciications or examle can e distinguished using a Box-Cox test (see any good econometrics textoo. This aer studies only the case in which demand curves are linear and marginal costs are 3 Their aer is, however, richer than this one in the sense that they study collusion under asymmetric inormation ollowing the aers y Areu, Pearce and Stachetti (986, I than ar Ivaldi or ringing Saatini s interesting woring aer to my attention. Pierluigi Saatini and I anticiate comining our resective research eorts into a single next version o our aer(s i we can agree a common est ractice aroach. At resent, Saatini avours a notion due to Friedman (97 called a alanced temtation equilirium, a related ut somewhat dierent aroach to that outlined here. 5

8 constant in outut. For this secial class o models, analytic results are derived or each comonent o the model, acilitating comutation enormously. The techniques are illustrated here using a articular and simle numerical examle. A ull imlementation o our coordinated eects methodology using data rom a real maret, the Networ Server (comuter maret, as well as est ractice demand structures is rovided in Davis, Huse and Van Reenen (2006. In that aer, we use random coeicient discrete choice demand structures ollowing Berry (994, Berry, evinsohn and Paes (995 and Nevo (200. The aer roceeds as ollows. Section 2 develos the single eriod game that we assume is layed y our irms and introduces a numerical examle. The material in this section will e amiliar rom the unilateral eects merger simulation literature. Section 3 introduces the dynamic game and shows that, to analyze it, we need only calculate one additional item eyond those required or the unilateral eects merger simulation literature, namely the ayo to deection. Section 4 continues the numerical examle, resenting some coordinated eects merger simulations. In articular, an examle is rovided that demonstrates that when maret structures are made more asymmetric y a merger, collusion can ecome more diicult to sustain ater a merger, rather than less. These numerical results are thereore consistent with the recent theoretical results rovided y Comte, enny and Rey (2002 and Kuhn and otta (2004 and add weight to those arguing against a strict structural (maret share or Herindahl ased test or an evaluation o whether a merger increases the lielihood o collusion. In section 5 I conclude and suggest some directions or uture research. 2. Unilateral eects and the single eriod game This section resents the stage-game o the dynamic model. The stage game is simly a standard dierentiated roduct Bertrand ricing game, identical to that used in the unilateral eects merger simulation literature y Werden and Froe (99, Berry (994, Hausman et al (994, Berry, evinsohn and Paes (995 and Nevo (200. In articular, I consider the case o linear demand systems; this acilitates the rovision o analytic solutions or use in undertaing oth unilateral and coordinated eects merger simulations. Doing so rovides a set o results or antitrust authorities that are articularly simle to imlement. Seciically, suose that demand or roduct {,.., } I may e written as a linear unction o the rices o all the goods in the maret: D a + + i a 0, 2,..., 0 otherwise ( with demand intercet arameter a and sloe arameter descriing the change in demand or roduct when good s rice increases y. Consider the ricing game wherein each irm roduces a suset o the availale roducts, I I, and chooses the rices o those roducts to maximize its roits: 6

9 max { I } I s. t. ( 0 c D ( or I where,,..., and c is the marginal cost o roduct, assumed constant. Notice ( 2 that or the linear demand and constant marginal cost case, this oective unction is a quadratic unction o rices while the constraints are linear unctions o rices so that the rolem may easily e solved numerically using standard otimization tools or quadratic rograms. 5 Alternatively, rovided equilirium rices o all goods in the maret are ositive and all goods are sold in ositive quantities as is universally assumed in the existing emirical literature (and so the constraints or this rogram do not ind in equilirium, we may solve this rolem analytically y examining the irst order conditions to the unconstrained rolem: D ( D ( + ( c 0, or all I I where, with linear demands, D (. At this oint, the literature on unilateral eects games has ound it useul to introduce an ownershi matrix to standardize these irst order conditions. Seciically, deine the (x matrix with, th element:, roduced y same irm 0 otherwise where, y construction or all, I. Notice that changing the ownershi structure in unilateral eects merger simulations reduces solely to changing this ownershi indicator matrix. Using the ownershi indicators, irm s irst order condition may e simly rewritten as: D ( D ( + ( c 0 ie, a + + ( ( c 0 Notice that there is one o these irst order conditions rom irm s oective unction or every I. Since every roduct is owned y some irm, under the ehavioural assumtions that each irm rices its roducts to maximize its roits rom the stage game, we otain a total o irst order conditions one or every roduct. We may then stac u the irst order conditions. To do so it is useul to introduce some matrix notation. Deine the (x vector a ( a,..., a ' and also the matrices: 5 atla, Gauss, athematica or ale can all solve this ind o rolem easily and quicly. Constraints requiring that equilirium rices and quantities e non-negative may also e added easily using those quadratic rogramming tools. 7

10 B' and B (where B is nown as the Hadamard or element y element or dot roduct. In the Aendix I show that given these deinitions, we can write the demand system or all goods as D ( a + B', while the vector o irst order conditions can also e written very comactly as a + B' + ( B( c 0. The solution to this set o equations is the Nash equilirium vector o rices, (,..., since, y construction, each irm is choosing the rices o its roducts to maximize its roits given the rices charged y other irms. Rearranging the irst order conditions gives an analytic matrix exression or the Nash equilirium rices o all roducts: ( a + ( B c ( B' + ( B The eauty o the linear demand curve seciication is that every oect o interest can e comuted easily or any ownershi structure,. For instance, given the exression or the Nash equilirium rices aove, equilirium demands or each roduct are given y the vector D ( a + B' and roits derived rom each roduct are ( c D( where reresents the element y element multilication o the two x vectors ( c and demands D ( a + B' resectively. The roits o each irm can thereore e calculated y adding across the owned roducts. I irm owns roduct ' and is the th row o the ownershi matrix, then irm s Nash equilrium roits. ' Π ( may e calculated using Π ( ( c ( a + B' Unilateral eects merger simulations In a unilateral eects merger simulation, where irm s set o roducts and costs eore PRE POST PRE POST and ater the merger are resectively I and I, and and or all PRE POST I and I resectively. These ownershi structure changes are catured in the matrix, so a unilateral eects merger simulation with linear demands amounts to calculating the Nash equilirium rices in each case: PRE PRE ( a + ( B c, PRE PRE ( B' + ( B c c and POST POST ( a + ( B c, POST POST ( B' + ( B. Naturally, oten data will not e availale on marginal costs. I not, then the method suggested y the unilateral eects literature is to estimate the marginal costs using remerger rices and an assumtion, which may e tested, aout the nature o rice 8

11 cometition in the re-merger eriod. Seciically, given the oserved re-merger rices PRE PRE and the ownershi matrix, we may solve the irst order conditions or remerger marginal costs: 2.2 A numerical examle: PRE PRE ( a + ( B' + ( B PRE PRE c ( B In this susection we will consider a unilateral eects merger simulation, ut we will use the same examle demand structure and cost assumtions throughout the aer. Seciically, consider the case where there are a total o six roducts,2,...,6 in the maret and where there are no eiciencies that result rom the merger, so that PRE c POST c. Suose urther that the linear demand structure in a maret and where the st roduct s demand equation is given y: q and the others, or 2,..,6 are symmetrically deined with the coeicient 2 on own rice and the coeicients 0.3 on rival roducts rices. In our ormulae derived aove thereore, we set each a 0 and construct the matrix B to have 2 s along the diagonal and all o-diagonal elements tae the value 0.3. Tale reorts the redicted static equilirium rices under a variety o ownershi structures. For examle, the irst column reorts the ownershi structure where every roduct is owned y a dierent irm while the sixth column reorts the case where there is a single irm owning all six roducts. The intermediate columns reort intermediate maret structures so or examle, (4,2 indicates that the irst irm owns our roducts and the second irm owns two. Tale 2 reorts the resulting equilirium roits. TABE Predicted static equilirium rices or each roduct under a variety o maret structures. The shaded cells reresent the roducts roduced y the largest irm under each ownershi structure. aret structure Product (,,,,, (2,2,2 (3,3 (4,2 (5, 6(Cartel

12 TABE 2 Predicted static equilirium roits or each irm under a variety o maret structures aret structure Firms (,,,,, (2,2,2 (3,3 (4,2 (5, 6(Cartel Industry roits The reeated game The next ste is to consider the aove analysis as a stage game within the roader context o an ininitely reeated game. Following the reeated game literature, each irm is assumed to maximize the net resent value (NPV o its roits, and we require that at each oint in the game tree the irm maes choices which are otimal given that it reached that node o the game tree so that we study su-game erect equiliria o the reeated game (Selten 965. Following Friedman (97, we will consider the easiility o sustaining a candidate collusive equilirium using grim strategies. Friedman (97 demonstrated that i each layer adoted grim strategies, and was suiciently atient, then there can e a large numer o su-game erect equiliria o the dynamic game, sometimes including the outcome that irms choose to rice in a way that maximizes industry roits in each eriod o the game, ie, each stage-game. In contrast to Friedman, who considered the homogeneous roducts case, we will consider the dierentiated roduct game. To do so, we must irst introduce some notation. Denote the one eriod Nash equilirium collusion and collusive ayos to irm as π and π resectively. These are exactly the ayos reorted in Tale 2 aove. Similarly denote the one eriod gain to irm rom deection deection when all other irms are laying collusively as π. We will discus how to comute extensively in section 3. elow. deection π When rivals are laying grim strategies a deector earns his one eriod deection ayo and then susequently receives only his Nash equilirium roits. Thus, the net deection deection δ anticiated return to deection today is π V ( δ π + or irm, while her δ ayo to collusion today and in all susequent eriods given that rivals continue to collusion collude is given y Collusion π V ( δ. Hence, layer has no incentive to deviate δ rom collusive ricing rovided that: collusion collusion deection π deection δ π ( δ > ( δ > π +. δ δ V V 0

13 In order to examine the incentives to collude using grim strategies, we must thereore consider the returns achieved y each irm in the three ricing scenarios collusion, Nash equilirium ricing and deection. We have already demonstrated using unilateral eects analysis how to calculate Nash equilirium roits and also the returns to (erect collusion. It remains thereore only to calculate the ayo to deection. Beore roviding exressions or each comonent in this incentive comatiility constraint, we note that the only comonents in this equation that are not already deection evaluated in a unilateral eects merger simulation are (i the ayo to deection π and (ii the discount actor δ. The ormer, lie the Nash and Collusive equilirium ayos, deends directly on the nature o the static roit unction or each irm and thereore may e easily calculated using the methodologies develoed or the analysis o data generated y static ricing games (see elow. In an antitrust case, the discount actor could usually e taen rom internal documents seciying the comany s required rate o return. Alternatively, i comanies are listed, CAP or another rate o return model could e used to iner an aroriate discount rate or ayos. Thirdly, more closely aralleling the theoretical literature, we can reort the range o discount actors or which collusion could e sustained under any given industry structure. Since I examine a numerical examle, here I tae the latter aroach. 3. The ayo to deection Following the theoretical literature on reeated games, deine the ayo to irm rom deection as the maximum amount o roit that could e achieved given its rivals rices (i.e. treating them as ixed. In the case most directly o interest, where irm is deciding whether or not to deect rom the tacitly collusive agreement, other irms will e choosing their rices to e the collusive rices, and so the static ayo to irm when deecting is: π deection s. t. D (, max { I } I collusion 0 ( c D (, collusion and or I. collusion Where denotes the vector o rices o goods roduced y irm, denotes the collusive rices o other irms and where the last set o constraints enorce caacity constraints on the deecting irm. For the case o linear demand equations, this nonlinear maximization rolem is again a quadratic oective unction suect to linear constraints and so is easy to solve numerically, even or large rolems, using standard methods such as the quadratic rogramming toolox rovided as a standard element within atla or Gauss. In other cases, it must e solved using more general non-linear otimization techniques ut even in those cases since it involves only an otimization, it is a simler mathematical oect to evaluate than the Nash Equilirium that must e comuted in unilateral eects merger simulations. Figure descries the calculation o a deection rice in the context o an examle with two single roduct irms. Solving or the deection rices simly involves inding the location on the deviating irm s est resonse unction at the oint where its rival is charging collusive roits.

14 In the case where caacity constraints do not ind and rices and quantities are ositive in equilirium, we can rovide an analytical solution to this rolem when demand curves are linear or aritrary ownershi structures. To do so we derive the irst order conditions or this rolem, which are ust those comuted or a single irm in the unilateral eects simulations evaluated when rival irms charge collusive rices. FIGURE 2 2 Collusion 2 Deection 2 * R ( 2; c Collusive rices * 2 R2( ; c2 * R ; 2 2( c2 Prices ater deection y irm 2 Collusion Considers the case with two single roduct irms. Each line shows a irm s reaction unction, * and * resectively, descriing the rice that R( 2; c 2 R 2 ( ; c 2 maximizes the irm s roits, given the rice charged y the rival irm. Where these reaction unctions intersect descries the Nash equilirium rices. Collusive rices, those which maximize industry roits are also descried. Each o these rices is calculated in a unilateral eects merger simulation. The deection rice is that which maximizes irms roits given that the rival is charging the collusive rice. Here that can e ound or each layer y evaluating their reaction unction rice when their rival is charging the collusive rices. In act, the irst order condition or this rolem with a linear demand system may e written in terms o matrices and this acilitates the rovision o an analytic solution to this rolem. To do so, it hels to introduce some notation. Deine or any (x matrix A, the su-matrix A[, ] which corresonds to ust the rows and columns o A rom roducts owned y irm. Similarly, deine A to e the rows o A corresonding to roducts [, ] owned y irm and the columns o A corresonding to roducts owned y other irms and the su-matrix A [,.] which means the rows o A or s roducts and all columns o A. Similarly, or any (x vector a, deine a simly to e the rows o the vector a corresonding to roducts that irm roduces. The irst order conditions rom the roit maximization rolem that is deined aove or the caacity unconstrained version o the deection rolem can e written as: 2

15 D ( D ( + ( c 0 or each I Deection collusive evaluated at (, where collusive is arametric and Deection is the rice which solves this system o irst order equations. We can stac u these moment ' 6 conditions and otain the vector equation: a + B + B ( c 0. [., ] ( [,.] Breaing u this exression so that we can write it in terms o rices o goods owned y irm and rices o goods owned y rivals gives: a Deection ' Collusive ( [, ] + ( B + [ B] [, ] 0 ' [ B] c + B + [ B] [,.] [, ] [, ] which can in turn e rearranged to give: ' Deection ' ( B + [ B] [, ] ( a [ B] [,.] c B + [ B] [ [, ] (, ] [, ] and hence we can solve or irm s deection rices analytically as: Deection ' ' ( B + [ B] ( a [ B] c B + [ B] [, ] [, ] ( ( [,.] [, ] Collusive [, ] Collusive. Having rovided the deinitions o the three core elements o the incentive comatiility constraint, we are now in a osition to summarize the methodology and rogress to our numerical examles o coordinated eects merger simulations. 4. Coordinated eects merger simulations In this section we will resent a numerical examle o our methodology or evaluating the coordinated eects o mergers. Beore doing so I summarize the methodology which involves taing the six stes reorted in Tale 3. 6 Note that the notation ', denotes the matrix transosed. I the roducts are suitaly ordered, we can write B B and [., ] B [., ] [., ] B so ' [, ] B. B[, ] ' ' B [., ] [ B[, ] B[, ] ] B [ B ] [., ] [., ] 3

16 TABE 3 The rocedure to undertae a coordinated eects merger simulation Ste no. Descrition Estimate the dierentiated roduct demand system 2 Use the re-merger eriod data to iner marginal costs or each roduct y using an aroriate assumtion aout the nature o re-merger rices (usually that it is a static Bertrand Nash equilirium. 3 Calculate the static Nash and collusive equilirium rices and ayos, (, π collusion collusion and (, π resectively 4 deection deection Calculate the deection rices and roits, (, π 5 Evaluate each comonent o the incentive comatiility constraint: 7 collusion collusion deection π deection δ V ( δ > V ( δ > π + δ 6 Evaluate ranges o δ needed to sustain collusion under grim strategies. π δ In order to erorm a coordinated eects merger simulation, the only new element that deection must e actively comuted is the deection ayo, π. Tale 4 reorts the deection ayos or our numerical examle. They are comarale directly to the numers reorted in Tale 2 where static Nash equilirium roits and collusive collusion π roits were reorted. For comleteness, note that in the case o a cartel, the total ayo is so that each single roduct irm in this symmetric situation earns the ayo, collusion π π TABE 4 This tale reorts the calculated deection ayos or each irm under each maret structure in our numerical examle. Payos are shown when irm is the deecting irm. aret structure: Firm (,,,,, (2,2,2 (3,3 (4,2 (5, 6(Cartel Having calculated each o the required oects rom the stage game, 7 Some authors reer to ut ayments at the end o eriods whereas here they are imlicitly laced at the eginning o the eriod. 4

17 collusion deection ( π, π, π or each irm we can rogress to an evaluation o the incentives or collusion catured in our ey ormula: collusion deection V ( δ > V ( δ collusion π δ > π deection δ π + δ 4. A irst coordinated eects merger simulation We egin with the returns to tacit collusion or the single-roduct irms in our numerical examle where maret structure is (,,,,, and the situation ollowing three mergers so that the maret structure ecomes (2,2,2. The advantage o choosing to loo at three mergers in this examle is that, oth eore and ater the merger, all irms are symmetric. That, in turn, acilitates resentation o the results. Tale 5 reorts the returns to collusion and also the returns to cometition in the numerical examle. The shading in the tale shows the larger side o the incentive comatiility constraint or each maret structure. First notice that collusion is easier to sustain at higher discount actors or any given maret structure; within each maret structure the shading is on the collusion side at higher discount rates. Second, notice that in this examle, the set o discount actors which can sustain collusion gets roader. I we deine a critical discount actor aove which collusion is sustainale or any given maret structure, in this examle we can * * write δ > 0. > δ. (,,,,, 6 ( 2,2,2 TABE 5 The returns to collusion and cometition under the two symmetric maret structures (,,,,, (2,2,2 δ V Collusion V Cometitio n V Collusion V Cometitio n Asymmetry and coordinated eects In this susection I resent an examle o a situation where concentration actually reduces the lielihood o collusion. This is consistent with the recent theoretical literature which has suggested that coordination may e harder to sustain under asymmetry; see or examle the reerences and discussion in the very nice survey aer rovided y Ivaldi et al (

18 Tale 6 reorts the results or two asymmetric mergers. In articular, consider mergers rom the aseline maret structure (4,, to the alternate maret structures (5, and (4,2 resectively. This could, or examle, e the choice acing an antitrust authority when deciding whether to allow a small irm to merge with its larger or similar sized cometitor. In this examle, the let hand anel o the tale reorts that, with a maret structure (4,,, collusive equiliria can e sustained with discount actors aove aroximately 0.8; each small irm and the large irm are oth willing to suort collusion aove this oint. However, notice that the columns comaring the two small irms incentives to collude relative to their incentive to cheat, clearly suggest that they are willing to cheat or a ar wider range o discount actors than the irm roducing our roducts. Thus, it is the small irms incentive comatiility constraints which ind in sustaining the collusive equiliria in this case. This oservation is central to understanding the incentives to collude ollowing the merger. In articular, when deciding whether to deect, it is the single roduct irms who have the incentive to reeride most on multi-roduct rivals or essentially the same reasons as induces them to cut rices most in the unilateral eects situation. The irst merger examined is the merger o the two small irms, so that the maret structure goes rom (4,, to (4,2. In that case, the range o discount actors that can sustain collusion changes little. In act, the only sustantive change is in the range o discount actors that will rovide the large irm with an incentive to collude which increases rom around 0.4 to around 0.5. This however is not the inding incentive comatiility constraint since it is the small irm who has the most to gain y deecting rom a collusive arrangement and selling as much as it can. TABE 6 Reorts a coordinated eects merger simulation rom the initial maret structure (4,, to two alternate maret structures (4,2 and (5,. The irst merger has little imact on the range o discount actors liely to sustain collusion while the latter maes collusion ar less liely ater the merger ecause the merger increases maret asymmetries. δ (4,, (4,2 (5, Firm with 4 roducts IC Firms with Firm with 4 Firm with 2 Firm with 5 Firm with constraint roduct roducts roducts roducts roduct Coll. Cheat Coll. Cheat Coll. Cheat Coll. Cheat Coll. Cheat Coll. Cheat , ,805, ,805, ,256, The second merger creates a more concentrated ut also more asymmetric maret structure since it creates one very large irm owning ive out o six o all o the roducts sold in the maret. Following this merger, the aility to sustain collusion actually reas down or all ossile discount actors! Simly, the small irm s ayo rom articiating in the collusive arrangement, given its very narrow roduct line, is too small relative to its incentive to deviate. Indeed, its ayo in the Nash equilirium is actually higher than its 6

19 ayo under collusion since (i in the Nash equilirium the small irm roduces ar more than under collusion while (ii the large irm has incentives not to rice too low ecause in doing so it must charge low rices on all its roducts. 8 For the small irm, in this instance, the incentive comatiility constraint is never satisied since: 45 Collusion δ 49 V 7 Cheating SmallFirm < + VSmallFirm, or all δ. ( δ δ These results, or examle, suggest that mergers which involve the removal o a small irm, which a merger authority might consider to e a maveric irm, one that is an imediment to eective coordination, need not mae tacit coordination easier. In act, y creating urther asymmetries in the maret, the removal o a small rival may even mae it harder to collude. Note that this result, while similar in lavor to Comte, Rey and enny (2002, in the sense that asymmetry maes collusion harder, is rather dierent in character. In their model, caacity constraints determine oth the aility to unish and in articular the temtation to cheat. These two eects mean that the small irm has oth less aility to unish the large irm and the large irm has more aility to cheat. Thus, in their model, it is the large irm that will e more temted to cheat. That is in contrast to the examle rovided aove where it is the small irm who ree-rides on the act that the large irm will set relatively high rices even in the Nash equilirium while the small irm s share o collusive roits is relatively small. One reason this numerical examle seems imortant is that it demonstrates that, in a general asymmetric dierentiated roduct setting, the conditions required or a ol theorem result do not hold at least under grim strategies. As such, the dierentiated roduct collusion game aears sustantively dierent to the homogeneous roduct game tyically studied. In articular, it is imortant to note that the intuition derived rom the homogenous roduct results rovided y Friedman (97 does not universally extend to the dierentiated roduct context. To inish let me note that, in our examle, the results in Tale 2 demonstrate that the ost merger ayo to the largest irm in the static (one eriod game is 39 under the (4,2 maret structure ut 88.5 under the (5, ost-merger maret structure. Thus the merger o (4,, to (5, would liely generate ar greater unilateral eects concerns than the merger to (4,2 and so our articular simulation exercise suggests the ollowing conclusion: This merger investigation should ocus on the otential unilateral eects o allowing the merger rather than the otential or coordinated eects. 5. Conclusions This aer has shown how to erorm merger simulations to evaluate the resence o coordinated eects o mergers. In doing so we extend the methods develoed or the evaluation o mergers eyond the unilateral eects case. The method roosed here is entirely consistent with collusion theory and only requires a small additional amount o eort on the art o the agency or researcher relative to erorming a unilateral eects 8 Recall rom TABE 2 that in a collusive arrangement the ayo to each single roduct irm is 270.8/6 45 (column headed cartel while the Nash equilirium ayo to the single roduct irm in the (5, maret structure is 49. (See ayo to irm 2 in the (5, column. The reason is that there s a large ree-riding eect availale to the small irm in the (5, static Nash equilirium. 7

20 merger simulation. We elieve the tools and techniques develoed here will e useul or olicy maers, roviding a set o ractical tools or evaluating the coordinated eects o mergers. We rovide analytic results that are useul or a large class o demand structures and aritrary ownershi structures. However, y considering a numerical examle, we have avoided a numer o ractical issues that would need to e addressed eore alying these tools into the ield. First, in ractical settings, we must determine the length o a eriod. Clearly, a eriod is aroriately deined as the amount o time that it would tae or rivals to detect a ricing deviation and change their rices aroriately, which under grim strategies would mean reverting to the Nash equilirium. We elieve it will usually e ossile to iner an aroriate eriod length on the asis o the oserved historical requency o rice changes and timeliness o rival resonses to news in a seciic alication, though deciding on an aroriate eriod length may require some udgment. Second, we must determine the sequence o uture ayos under oth collusion and cometition. Here, we choose to ollow the theoretical literature which treats the susequent games as identical and this is liely a sensile aroach in many instances. In others it may e aroriate to assume that marets will grow at some rate over time. Either method maes strong assumtions, including the act that we assume an asence o entry, roduct sets do not evolve and so on. Such assumtions, though strong, are o course identical to those made oth y the unilateral eects merger evaluation literature and y investigators rogressing without descriing their understanding o the alicale model o the industry. Provided the results are interreted with aroriate caution, and some sohistication, merger simulation rovides a useul set o techniques to hel in the evaluation o oth the unilateral- and now coordinated-eects o mergers. 8

21 APPENDIX This aendix develos the matrix exressions useul when comuting Nash equilirium rices. The results are rovided under the ehavioural assumtion that each irm rices its roducts to maximize its roits rom the stage game ut are suiciently general to coe with an aritrary ownershi structure. We saw in the text that any irm s irst order condition may e written as: + c D D 0 ( ( ( ie, ( + + c a 0 ( Notice that there are one o these irst order conditions rom irm s oective unction or every. Since every roduct is owned y some irm, we otain a total o irst order conditions one or every roduct eing sold. We may then stac u the irst order conditions. To do so it s useul to introduce some matrix notation. Deine the (x vectors and and also the (x matrices: I I a a a c c c B ' and. B where B is nown as the Hadamard or element y element matrix roduct. We may then stac u the irst order conditions to otain: c c c a a a The solution to this set o equations,, rovides the rices at which each irm is maximizing its roits given the rices o others, and hence is the Nash equilirium rice vector to the stage game. The vector o irst order conditions may e written more comactly in matrix terms as,..., ( 0 ( ( ' + + c B B a. Rearranging, we get c B a B B ( ( ' ( + + and hence Nash equilirium rices can e comuted using the matrix ormula: (. c B a B B ( ( ' ( + + 9

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24 Suslow, V (986. Estimating onooly Behaviour with Cometitive Alication to Alcoa. Rand ournal o Economics Vol. 7, Waler, The Potential or Signiicant Inaccuracies in erger Simulation odels. ournal o Cometition aw and Economics, Vol, No. 3, Werden, G and Froe, (994 The Eects o ergers in Dierentiated Products Industries: Structural erger Policy and the ogit odel ournal o aw, Economics, & Organization, Vol. 0,

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26 Cometition Commission