Internatonal Conferene on Apple Eonoms ICOAE 29 551 Cartelsaton of Olgopoly Jaek Prokop 224 Abstrat Ths paper onsers possbltes of artelsaton n the olgopolst nustres. We stngush between two mportant ssues. The frst one, tratonally susse n the lterature, s the problem of artel stablty. The other one, muh less analyse, s the proess of artel formaton. Both aspets have mportant reperussons for the market struture an the role of regulatory agenes. Even though the proft-maxmzng frms have nentves to restrt ompetton by formng artels, the olluson has often been broken ue to the prsoner s lemma. Therefore, t oul have been lame that anttrust agenes o not nee to ntervene. Further analyss shows, however, that there are markets where stable artels may be reate an persst. We present a moel of ollusve preleaershp an show the exstene of stable artels. Next, we nvestgate the artel-formaton proess n a game-theoretal framework. Our analyss allows to formulate general guelnes for the regulator s nterventon to prevent artelsaton. JEL oes: D43, L13, L49 Key Wors: anttrust poly, artels, ollusve behavour 1. Introuton The proft-maxmzng frms n olgopolst markets have strong nentves to restrt ompetton by reatng artels. They reognze that oornaton of output or prng strateges oul help them nrease profts. In the most extreme ase when all ompanes n the nustry form a artel, they an uplate the monopoly outome, an share the largest pool of profts that an be generate. The nrease proft s aheve at the expense of onsumers, an usually leas to a reuton n total welfare. Therefore, artels are llegal an anttrust authortes atvely engage n fghtng them. Even though there have been many suess stores of governmental atvtes leang to eteton an punshment of frms formng artels 225, there s no oubt that t s not an easy task, an atonal tools of analyss shoul be evelope to help elmnate llegal ooperaton. Followng the unversal rule that t s better to prevent than ure, anttrust agenes oul mprove ther effetveness n elmnatng artels from the markets by entfyng the rumstanes n whh artelsaton s most lkely to our. It s the obetve of ths paper to etermne ontons uner whh the antrust authortes shoul get atvely nvolve n orretng the market, an to show the bas retons of the nterventon. The frst naton of possble artelsaton n a gven nustry s the theoretal exstene of a stable artel. 226 Suh exstene s not guarantee n every market struture. In some nustres, the artel stablty s estroye by the ourrene of the prsoner s lemma. In these rumstanes, artelsaton oes not onsttute a maor threat. However, the markets, for whh the theoretal exstene of a stable artel an be shown, shoul be the target of a loser analyss by the anttrust ageny. The theoretal exstene of a stable olluson among frms s not an mmeate argument for an atve market nterventon. We shoul analyse the nustry stuaton to etermne whether there s a hane for the potentally vable artel to form. [Selten: 1973] an [Prokop: 1999] have shown that the susson about the proess of artel reaton s 224 Professor of Eonoms, Warsaw Shool of Eonoms, Al. Nepoleglos 162, 2-554 Warsaw, POLAND, e-mal: proko@sgh.waw.pl. 225 Several nterestng examples of suessful artel eteton by the anttrust authortes are presente, e.g., n [Pepall - Rhars - Norman: 22, pp. 386-4]. 226 For referenes on artel stablty see, e.g., [Martn: 1993].
552 Internatonal Conferene on Apple Eomons ICOAE 29 at least as mportant as the researh on artel stablty tself. 227 Followng that lterature, we present a moel of artelformaton proess, an formulate the key ontons that make the artel reaton hghly probable. We emonstrate that n an nustry ompose of more than fve frms the hane of havng a artel s qute small ue to the free-rng problem among the market partpants. In the ase of nustres wth up to fve frms, we shoul expet a artel to be reate for sure; there are strong nentves for the frms to ollue. These results are use to prove the guane for an nterventon of the anttrust authorty to prevent nustry artelsaton. When the ynams of the market struture leas to olluson among frms, the anttrust ageny shoul step n; suh stuaton takes plae for nustres wth the number of ompettors fallng below sx. Inustres wth a larger number of frms (more than fve) shoul not be of large onern by the anttrust authortes; the market tself makes the artelsaton ffult. The rest of ths paper s organze as follows. In seton 2, the artel stablty s efne, an the lassal example of prsoner s lemma s gven to llustrate the lak of stable olluson n some olgopoles. Next, we present a moel of an nustry haraterze by ollusve pre-leaershp where the exstene of a stable artel s shown. In seton 3, we offer the artel-formaton game an suss the equlbrum behavour of nustry partpants as well as the role of the anttrust ageny. The onlusons are n seton 4. 2. Cartel Stablty 2.1. The Defntons We onser an olgopolst nustry ompose of n > 1 ental frms proung a homogeneous output. Any number of frms n the nustry may form a artel,.e. agree to maxmze ont proft. Suh ollusve agreement s usually ame at restrtng the ont output n orer to rase the pre, an thereby nrease profts. For any artel to stay n the market the olluson among frms must be stable. In orer to efne artel stablty, we nee to stngush between nternal an external stablty. A artel s sa to be nternally stable f t s not proftable for a member frm to efet from the ollusve agreement. A artel s externally stable f t s not proftable for any non-member frm to on the ollusve agreement. A stable artel s a artel whh s both nternally an externally stable. It has been long known that artels have a teneny to break ue to the prsoner s lemma among the ooperatng frms (lak of nternal stablty). The most typal behavour of artel members an be llustrate by the followng example of the two-frm game gven n Table 1. Table 1: Prsoner s lemma game of artel members. Cooperate Frm 2 Defet Frm 1 Cooperate (25, 25) (15, 35) Defet (35, 15) (2, 2) There s only one Nash equlbrum of ths game; n the equlbrum both frms ee to evate from the artel agreement.e. (Defet, Defet). Even though, t s Pareto omnant for the frms to ooperate, there s lak of nternal stablty of the artel ue to the strong nentves for the nvual members to efet (to heat). It has been argue that by takng nto aount an nfnte horzon of the ompetton the frms are enourage to ooperate, an to avo heatng. 228 Unfortunately, the mperfetons of nformaton oul estroy the olluson even n an nfnte horzon nteratons between ompanes. When heatng by artel members s unobservable, the partpatng frms may not be sure about the soure of ther poor performane. The low proft oul result ether from not obeyng the artel agreement, or from a reuton n the overall market eman. 229 Hene, the exstene of the prsoner s lemma makes artels not to funton too smoothly. Ths result proves ustfaton for many observers to onlue that government nterventon s not requre beause ompettve fores keep estroyng the artels. 227 Selten [1973], Prokop [1999], an Morash [2] moel the artel formaton proess an show that not all proftable an stable artels wll form n a subgame perfet equlbrum. 228 See, e.g., [Trole: 1997, pp. 258-259]. 229 For a susson of the lterature on seret pre uts see, e.g., [Trole, 1997, pp. 251-253, 262-265].
Internatonal Conferene on Apple Eonoms ICOAE 29 553 The prsoner s lemma game however oes not arse n every olgopolst market. Table 1 aptures, for example, an nustry when ompanes wth lnear ost funtons are engage n a Cournot-type ompetton when the artel agreement fals.23 Sne the ooperatve (artel) output levels o not onsttute a par of best responses, eah frm has a stronger proft nentve to efet or to heat upon the ooperatve agreement than to stk wth t. Uner fferent assumptons about the ost funtons an the type of ompetton among frms, t an be shown that a stable artel exsts. An example of the exstene s susse below. 2.2. The Exstene of Stable Cartels Let us assume that the market eman for our nustry s prout s gven as (1) D(p) = 1 p. Eah frm has the followng ost funton: (2) 1 2 C( q ) q, 2 where q s the output of frm. Further suppose that ths market s haraterze by pre leaershp of a omnant frm or artel. The non-omnant frms are pre takers; they onsttute a ompettve frnge. 231 Denote the number of artel members by k. Then, there are n k remanng frms n the ompettve frnge. The latter frms take the artel pre p(k) as gven an hoose ther prouton level by settng margnal ost equal to ths pre. The artel behaves as a monopolst wth respet to the resual eman urve,.e. the market eman at pre p(k) reue by the supply of the ompettve frnge. Sne all ompanes are ental, the prouton wthn the artel wll be sprea unformly among ts members. We o not allow any restrbuton of profts among the frms. In ths market, the proft of eah frm n the artel s (3) 1 ( k) 5, 2 2 ( n 1) k an the proft of eah frm n the ompettve frnge s 2 (4) ( ( n 1) k) 5. 2 2 [( n 1) k ] 2 From (3) an (4), t s easy to see the followng: Proposton: The proft of a artel member s smaller than the proft of a ompettve-frnge partpant,.e., ( k) ( k) for all k. An mmeate mplaton of ths proposton s that t s strtly preferre for any frm to be n the ompettve frnge rather than n the artel. Ths preferene, however, shoul not be onfuse wth the frm s nentves to leave the artel for the ompettve frnge. As t wll be emonstrate, suh a eparture s not neessarly proftable. Now, let us turn to the artel-stablty ontons. A artel wll be nternally stable f k 1 an (5) ( k 1) ( k), External stablty of a artel takes plae when k n 1 an (6) ( k 1) ( k). Thus for a artel to be (nternally an externally) stable both ontons (5) an (6) must hol. Usng the proft funtons (3) an (4), a straghtforwar algebra shows that: 232 (a) for an nustry ompose of n 5 frms, the stablty ontons (5) an (6) are satsfe only for k=n, (b) for an nustry ompose of n > 5 frms, the stablty ontons (5) an (6) hol only for k=3. Thus, n the olgopolst nustres susse n ths subseton there exsts a unque stable artel. 233 When the number of nustry partpants oes not exee fve, the artel onssts of all of them. For nustres wth more than fve ompanes, the sze of the stable artel oes not epen on the number of frms, an s always equal to three. 234 23 For a etale numeral example see, e.g., [Pepall - Rhars - Norman: 22, pp. 353-357]. 231 Other types of ompetton n the nustry haraterze by olluson may be foun n [Martn: 1993]. For example, a artel may at as a Stakelberg leaer, an the frnge frms follow the Cournot behavour. 232 For omparson see [Daskn: 1989]. 233 The exstene of a stable artel n the nustres wth ollusve pre leaershp has been frst shown n [ Aspremont - Jaquemn - Gabszewz - Weymark: 1983]. Further susson of the exstene of stable omnant artels an be foun n [Donsmon: 1985], [Donsmon - Eonomes - Polemarhaks: 1986], an [Noke: 1999]. A fferent approah s offere by [Yong: 24] who argues
554 Internatonal Conferene on Apple Eomons ICOAE 29 The exstene of a stable artel oes not automatally mean that suh artel an be forme. The formaton proess may fae some obstales that prevent the reaton of artels. Some of the ffultes arse from the fat that artels are prohbte by the law of most of the emorat ountres. Even n the ase when these hurles oul be overome, however, there are also other reasons why nepenent frms n some nustres may not be able to ollue espte the theoretal exstene of a stable artel n these markets. In the next seton, we propose a moel of artelformaton proess an analyse ts potental outomes. 3. The Proess of Cartel Creaton 3.1. The Cartel-Formaton Game Conser agan an nustry wth n > 1 ental frms. The market eman for the nustry s prout an the ost funtons of the frms are as esrbe n the prevous subseton. The artel-formaton game s playe n two stages. 235 In the frst stage, the frms smultaneously an nepenently ee whether to partpate n a artel or reman n the ompettve frnge. In the seon stage, the members of the artel hoose the pre that maxmzes ther ont proft, an the frms n the ompettve frnge take ths pre as gven an hoose ther proft-maxmzng prouton level. Eah frm N={1,, n} has two pure strateges, to stay n the ompettve frnge, an 1 to on the artel. Denote the strategy spae {, 1} of frm by S. Then S 1 S n s the outome spae. When eah frm hooses ts strategy s an the omnant artel s forme, the payoff to frm s etermne. Frm that hooses to stay n the ompettve frnge obtans a proft of π (k), where k=σ N s. We assume ths proft to be the payoff to frm f t ees to stay n the ompettve frnge. The payoff to frm from onng the omnant artel s π (k). Thus the payoff funton of frm s gven as (7) s, N f s h ( s) h ( s1,..., sn ) s 1. N f s In aton to pure strateges, we allow the frms to play mxe strateges. Denote the set of mxe strateges of frm by T,.e. T = {p : p 1} for N, where p s the probablty of onng the omnant artel by frm ; we all t the partpaton probablty. Defne the expete payoff to frm by H ( p) p p (1 p ) ( S 1) S N { } S S N { } (8) (1 p ) p (1 p ) ( S ) S N { } S S N { } for p ( p1,..., pn ) T1... Tn. The frst braket n (8) s the expete payoff to frm from onng the omnant artel, an the seon braket s the expete payoff from stayng n the ompettve frnge. We apply the Nash equlbrum onept to our game. A strategy n-tuple pˆ ( pˆ,..., ˆ 1 pn ) s alle a Nash equlbrum f for all N, H pˆ ) H ( pˆ, p ) for all p T, ( where pˆ ( ˆ,..., ˆ, ˆ,..., ˆ p1 p 1 p 1 pn ). Sne t s expete that ental frms wll behave n a smlar way, we restrt our analyss to the symmetr Nash equlbra. Denote the symmetr equlbrum strategy of a frm by ˆp. In the symmetr ase, t follows from (8) that the equlbrum payoff of frm s n1 n 1 k n1k (9) H ( ˆ ) ˆ ˆ (1 ˆ p p p p ) ( k 1) k k (1 pˆ ) n1 k n 1 k pˆ (1 ˆ p ) k n1k ( k), that the Nash noton of stablty use n the lterature s too weak, beause t only permts evatons by nvual frms,.e. group evatons are not allowe. 234 It s nterestng to observe that for many other spefaton of eman an ost funtons the stable artel s usually ompose of 3 ompanes when the number of nustry partpants s greater than fve. See, for example, omments by [Daskn: 1989, pp. 7-12]. 235 Compare [Prokop: 1999].
Internatonal Conferene on Apple Eonoms ICOAE 29 555 where π ( ) an π ( ) are gven by (3) an (4). In the next seton, we use the above efntons to esrbe the equlbrum strateges of the market players. 3.2. The Equlbrum Behavour of Frms The followng theorem haraterses the symmetr equlbrum of our game: 236 Theorem: The artel formaton game has exatly one symmetr equlbrum pont. At ths pont: p ˆ 1 for n 5, pˆ 1 for n 5. Aorng to theorem, n an nustry ompose of not more than fve frms, the equlbrum partpaton probablty s equal one,.e. the artel-formaton game leas to the reaton of a stable artel. The probablty of onng the artel by an nvual frm falls below one when the number of frms s greater than fve. The atual hane of formng a omnant artel n the ase of n>5 an be seen only by alulatng the exat levels of the equlbrum partpaton probabltes. Table 1 shows the smulaton results for varous numbers of frms n the nustry. Observe that the nvual probablty of onng the artel s elnng wth an nreasng number of frms n the nustry. The explanaton of ths outome s not ffult. The stable artel onssts of three frms n the nustres wth more than fve ompettors. Hene t s beomng a oornaton problem to reah the ooperatve agreement. The frms have a teneny to free re by watng for others to form the artel. Table 1: Symmetr equlbrum of the artel formaton game. Number of frms n the nustry (n) 2 3 4 5 6 7 8 9 1 11 12 13 14 15 2 5 Partpaton probablty p ˆ ) ( 1, 1, 1, 1,,4282,3159,2582,226,1932,1723,1555,1418,133,126,879,335 The above outomes have very mportant onsequenes for the behavour of frms n the market where a artel plays the role of the pre leaer. In the ase of nustry ompose of not more than 5 frms we shoul expet all of the frms to reate a artel. The artel wll beome a monopolst n ths market. Thus, from the pont of vew of the total welfare, suh market struture s neffent, an there shoul be some nvolvement of the anttrust ageny to orret t. When the number of frms n the market exees fve than the partpaton probablty suenly an sgnfantly elnes to the values muh below,5. It means that the probablty of reatng a stable artel beomes small, an t s rather har to expet that t wll be forme. Base on the presente results, n general, the anttrust ageny shoul take measures to blok mergers an aqustons that lea to a reuton of the number of frms below sx n any gven nustry. Moreover, t woul be benefal to stmulate an nrease of the number of frms n any nustry. A relatvely large number of ompanes (more than fve) serves as the best guarantee of a low probablty of artel reaton. 236 The proof of theorem follows retly from the reasonng presente n [Prokop: 1999, pp. 25-251].
556 Internatonal Conferene on Apple Eomons ICOAE 29 Sne the nustres wth a small number of frms (less than sx) are expete to be artelse, an atve role of the anttrust ageny n ths ase s of great mportane. One of the suggeste atons that oul be unertaken by the anttrust authortes s to break up large ompanes (e.g. Mrosoft) nto few smaller ones n orer to nrease the number of nepenent nustry partpants. Another possblty s to enable other frms to enter the market. Otherwse, the nustres omprse of up to fve ompettors shoul be the target of a lose wath to etet sgns of artel behavour, beause all the nentves work here for the market partpants to ollue. In summary, only the nustres ompose of less than sx frms shoul be of maor onern to the anttrust authortes. The markets wth at least sx frms may be vewe as qute ompettve an requre muh less attenton from the regulator s se. 4. Conlusons Any regulatory measures mpose by anttrust authortes make sense only when the market mehansm leas to some unesre results. Therefore, on the one han, researh shoul be onute towars entfaton of ontons uner whh unertakng any regulatory measures s reunant. On the other han, t s mportant to esrbe the rumstanes when some nterventon of anttrust s nspensable. The urrent paper ame at ontrbutng to the lterature on artelsaton of nustres n orer to prove guane on the nees for governmental regulaton of some markets. The analyss of markets haraterze by pre leaershp of one frm or a group of frms shows that there are nustres n whh the theoretal exstene of a stable artel an be prove. However, suh potental exstene of olluson among frms oes not neessarly mean that a stable artel an be atually forme. We llustrate the problem by offerng the artel-formaton game. It s emonstrate that n the nustres ompose of less than sx frms, the equlbrum outome has all frms formng a omnant artel. The proess of artel formaton wll not be suessful n the markets omprse of more than fve frms. The reason for these results stems from the nentves of frms to free re on the exstene of a artel n the ase of a larger number of ompanes n the nustry. The analyses of both the artel stablty an the formaton proess are rual n the susson about the nvolvement of anttrust ageny n fghtng artels. It has been shown that n the nustres haraterse by ollusve pre leaershp an quarat ost funtons of frms, a stable artel always exsts. Thus, suh nustres shoul be the target of a loser wath by the anttrust ageny. Even though the exstene of a stable artel an be shown, the atual reaton of a artel n ths ase s not automat. We have emonstrate that only n the nustres ompose of up to fve frms, a artel ompose of all of them wll be reate for sure. Thus an atve nvolvement on the part of an anttrust ageny s neessary to fght the ollusve behavour. When there are more than fve frms n the nustry, our moel prets that the free rng among frms wll prevent the reaton of a stable artel. Suh nustres o not requre mmeate regulatory measures by anttrust authortes. Clearly, further researh on the preventon an eteton of artels s neessary. Frst, the ase of possble artelsaton n the nustres wth asymmetr frms shoul be analyse. The stablty of olluson by heterogeneous frms n some nustres has been susse, for example, by [Donsmon: 1985], but the proess of artel reaton stll remans to be onsere. Seon, many other types of ost funtons an forms of ompetton among frms shoul be onsere from the vewpont of both stablty an formaton proess of a potental olluson. Thr, the role of asymmetr nformaton n the artelsaton proess shoul be nvestgate. Fourth, other than the Nash noton of stablty oul be susse. For example, [Yong: 24] suggeste an nluson of group esons n the efnton of stablty. The exstene of suh stablty shoul be supplemente by the analyss of the olluson proess, as well. Ffth, the analyss of artelsaton shoul be extene to a ynam framework. The susson of the ollusve behavour has been onute n the framework of one shot games. A mult-pero analyss woul onsttute a natural extenson.
Internatonal Conferene on Apple Eonoms ICOAE 29 557 Fnally, the empral stues 237 wll be the ultmate uge of the theoretal results of the artelsaton of olgopolst nustres an the effetveness of the anttrust authortes. Referenes Aspremont, C. - Jaquemn, A. - Gabszewz, J.J. - Weymark, J.A. (1983), On the Stablty of Collusve Pre Leaershp, Canaan Journal of Eonoms, 16:17-25. Daskn, A.J. (1989), Cartel Stablty n the Pre Leaershp Moel: Three-Frm Cartels an the Role of Implt Colluson, Workng Paper 89-17, Boston Unversty Shool of Management. Donsmon, M.P. (1985), Stable heterogeneous artels, Internatonal Journal of Inustral Organzaton, 3:451-467. Donsmon, M.P. - Eonomes, N.S. - Polemarhaks, H.M. (1986), Internatonal Eonom Revew, 27:317-327. Martn, S. (1993), Avane Inustral Eonoms, Blakwell: Basl. Morash, K. (2), Strateg Allanes as Stakelberg Cartels-Conept an Equlbrum Allane Struture, Internatonal Journal of Inustral Organzaton, 18, 257-282. Noke, V. (1999), Cartel Stablty Uner Capaty Constrants: The Tratonal Vew Restore, Workng Paper, Nuffel College, Oxfor. Pepall, L. - Rhars, D.J. - Norman, G. (22), Inustral Organzaton: Contemporary Theory an Prate, Seon Eton, South-Western: Cnnnat, Oho. Prokop, J. (1999), Proess of Domnant-Cartel Formaton, Internatonal Journal of Inustral Organzaton, 17:241-257. Selten, R. (1973), A Smple Moel of Imperfet Competton, where 4 are Few an 6 are Many, Internatonal Journal of Game Theory, 4:25-55. Suslov, V. (25), Cartel Contrat Duraton: Empral Evene from Inter-War Internatonal Cartels, Inustral an Corporate Change, 14:75 744. Trole, J. (1997), The Theory of Inustral Organzaton, The MIT Press: Cambrge, Massahusetts. Yong, J.-S. (24), Horzontal Monopolzaton va Allanes, or Why a Conspray to Monopolze s Harer Than It Appears, Workng Paper, MIAESR, Unversty of Melbourne. 237 For some nterestng empral researh on stablty of nternatonal artels see, e.g., [Suslov: 25].
558 Internatonal Conferene on Apple Eomons ICOAE 29