A test of collusive behavior based on incentives

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1 A tst of collusv bhavor basd on ncntvs Rcardo Cabral Workng Papr January 28 Abstract hs papr proposs a novl colluson tst basd on th analyss of ncntvs facd by ach frm n a colludng coalton. In fact, onc colluson s n ffct, ach colludng frm facs th ncntv to scrtly dvat from th agrmnt, snc t thrby ncrass ts profts, although th colludng frms jont proft dcrass. hus, n a colludng coalton ach frm has margnal rvnus, calculatd wth Nash conjcturs, whch ar largr than ts margnal costs. h colluson tst s basd on th rjcton of th null hypothss that th frm margnal rvnus wth Nash conjcturs ar qual to or lss than ts margnal costs. JEL classfcaton: L4, L13 Kywords: Colluson tst, Imprfct Comptton, Cartls; Compttvnss Unvrsdad da Madra, 9-39 Funchal, Portugal, -mal: rcabral@uma.pt. h papr bnftd from a dscusson wth Johan Stnnk. I apprcat commnts by Hnry Chappll. h rsponsblty for any rrors rmans solly mn.

2 1. Introducton h varty and audacty of scrt ndustry cartls dos not cas to surprs. Cartls ar found n many ndustrs and rgons such as th ntrnatonal vtamn makrs, th US lctrc turbn manufacturrs, th Grman cmnt makrs, th Frnch mobl tlcommuncatons oprators, just to ct a fw. 1 Dspt th thrat of larg fns f th cartls ar dtctd, rcnt cass attst to colluson s ndurng appal: US Dpartmnt of Justc anttrust nvstgatons nto allgd prc fxng practcs at th world s ladng flat panl dsplay makrs and of th $2 trllon muncpal bond markt, and nvstgatons by th Grman cartl offc and th Europan Commsson nto th Grman and Europan lctrcty markts. Importantly, a larg numbr of cartls ar dscovrd only through dnuncaton by on of th cartl mmbrs, whch suggsts rgulators, whn unabl to obsrv conduct, do not sm abl to dtrmn msconduct by analyss of structur or prformanc to a standard of crtanty that wll stand th scrutny of th courts. A larg body of rsarch has proposd dffrnt tsts of conduct consstnt wth collusv bhavor (Furstn, 25; Harrngton, 25; Portr, 25). It s gnrally accptd that collusv conduct cannot alon b nfrrd from markt prformanc ndcators such as hgh proftablty, hgh prc-cost margns, or hgh prcs (Dmstz, 1973; Schrr and Ross, 199; Harrngton, 25). 2 h structuralst hypothss that markt structur s, pr s, th fundamntal xplanaton for frm conduct s unsatsfactory snc t mpls th lnkag s mchancal, and snc t fals to consdr th potntal for rvrs causalty (Cabral, 2), th contstabl markts hypothss (Baumol t al, 1982), and to rflct th dynamc natur of olgopolstc comptton. Approachs that sk to modl frm conduct xplctly (Brsnahan, 1 Handlsblatt, Apr. 14, 23, Zmnt-Bußgldr übr 66 Mllonn Euro vrhängt ; Fnancal ms, Adam Jons, Dc. 1, 25, Frnch top mobl groups fnd ; Fnancal ms, Song Jung-a, Dc. 12, 26, Samsung, LG Phlps and Sharp ht by prob ; and Bloombrg, Wllam Slway and Martn Braun, Dc. 7, 26, U.S. Crmnal Prob Rattls $2 rllon Muncpal Bond Markt. 2 hr ar apparntly plnty of othr conducts avalabl to frms to rstrct comptton whch do not ncssarly translat n ncrass n prcs or rductons n output rlatv to th ntal 2

3 1989; Parkr and Röllr, 1997) rqur a dgr of ratonalty, nformaton, and forsght that may not b consstnt wth ralty, mak strong assumptons about dmand and cost functons, and fac sgnfcant stmaton dffcults. S Corts (1999) for an xcllnt ovrvw of th problms wth such modls. Fnally, dynamc modls of stratgc ntracton (Harrngton, 25), by mphaszng th possblty of rtalatory bhavor du to rpatd ntracton, offr th possblty of nfrrng varatons n frm conduct ovr tm. Howvr, frm conduct may not b tm-consstnt nor always consstnt wth ratonal bhavor n th contxt of a stratgc gam. In addton, such modls assum som dgr of markt stablty ovr tm, whch mans thy offr unsatsfactory prdctons n markts whr dmand fluctuats strongly or markts that fac rapd tchnologcal chang (Harrngton, 25). Pror work dvlopd modls of frm conduct that focusd on th dffrnc btwn prcs and margnal costs as th masur of markt prformanc (Stglr, 1964; Cowlng and Watrson, 1976; Brsnahan, 1981; Kamn and Schwartz, 1983; Hall, 1988; Parkr and Röllr, 1997). hs papr proposs a novl colluson tst that sdstps masurs of markt prformanc and nstad focuss on th ncntvs frms fac n an mprfctly compttv sttng. Economc ncntvs affct agnts conduct and prformanc, ncludng n aspcts as mportant as lf and dath. Dckrt-Conln and Chandra (1999) show how US Fdral prsonal ncom tax ncntvs contrbut to th antcpaton of chld brths from January to Dcmbr, and Gans and Lgh (26) show how 1979 nhrtanc tax changs n Australa contrbutd to th dlay of th dath of 5 ndvduals by a wk, quvalnt to mor than half of th ndvduals who would b subjct to th nhrtanc tax. hrfor, t s only logcal to sk to dtct colluson by lookng at th ncntvs ndvdual frms fac rathr than by attmptng to masur, aftr-th-fact, th ffct of colluson on markt prformanc varabls. hs s th ratonal for th tsts proposd n ths papr. qulbrum. For xampl, prstgous frms such as Intl or Mcrosoft hav n th past succssfully 3

4 h man charactrstc of a collusv outcom s that t s nhrntly unstabl snc t s not a prfct Nash qulbrum and ach frm has an ncntv to dvat from th agrd output or prc (Stglr, 1964; Cabral, 2). 3 hat s, for vry possbl tact or xplct collusv agrmnt, ach colludng frm s margnal rvnu (wth Nash conjcturs) s largr than ts margnal cost, snc ach frm s not maxmzng ts ndvdual proft for th bnft of th colludng coalton jont profts. Undr comptton a la Cournot ach frm n th colludng coalton has an ncntv to ncras output n ordr to ncras ts profts. Lkws, undr comptton a la Brtrand, ach frm n th colludng coalton has an ncntv to lowr prcs blow agrd lvl so as to ncras ts profts. Furthr, th ncntv to scrtly dvat from th collusv agrmnt s largr th largr th prc-cost margns ar (Stglr, 1964; Harrngton, 25; Portr, 25). In fact, th papr shows that a tst of collusv conduct can b accomplshd by comparng frm margnal rvnus (wth Nash conjcturs) wth ts margnal costs, rjctng th hypothss of no colluson f frm margnal rvnus ar statstcally sgnfcantly largr than ts margnal costs. o drv th thorms and proofs, I us th standard mprfct compttv markt assumptons, wak-concav and downward slopng dmand, lnar costs, and Nash conjcturs. I dvat from standard olgopoly thory only n that I offr a mor prcs dfnton of ndustry markt powr, and ths dfnton s not ncssary for drvng th rsults. It can b droppd wthout affctng th proofs. hs papr s organzd as follows. Scton 2 focuss on th dualty of th Cournot and Brtrand modls, to motvat th utlzaton of Nash conjcturs n dtctng collusv bhavor, and as a bass for th futur xtnson of th thorms and proofs to th Brtrand modl. Usng a Cournot comptton modl, Scton 3 drvs th thorms and proofs that thratnd supplrs or clnts of a compttor, to th dtrmnt of th compttor s busnss. 3 In th lat 199s th Economst pontd out that th chatng that th OPEC cartl countrs dd on thr allocatd producton quotas wr th caus for th low ol prcs, although thr was a modcum of colluson (th Intrnatonal Enrgy Agncy stmatd a complanc of 75%). Accordng to th Economst, quotng Vahan Zanoyan of th Washngton-basd Ptrolum Fnanc Company, Whn prcs ar strong, th tmptaton to b th frst chat s mpossbl to rsst (s Lyng low, h Economst, July 2nd 1998; s also Stll kckng?, h Economst, March 25th 1999 ). 4

5 show that a tst of colluson can b quatd wth a tst of th proft maxmzaton condton wth Nash conjcturs. Scton 4 spcfs th on-tald tst hypothss consstnt wth th thorms stablshd n Scton 3. Scton 5 dntfs ssus for furthr rsarch. Scton 6 concluds. 2. Cournot and Brtrand Dualty and mportanc of Nash conjcturs As s wll known, Cournot and Brtrand olgopoly modls dffr n what s thought to b th dcson varabl of th frm, quantty and prc, rspctvly. As a consqunc, Cournot and Brtrand qulbra offr radcally dffrnt prdctons for th outcoms of mprfctly compttv markts. If th numbr of frms s largr than on, Cournot comptton s always mor monopolstc than Brtrand comptton, and th lattr always achvs qulbra whr prcs qual margnal costs (Sngh and Vvs, 1984). A larg body of ltratur has sought rconcl th prdctons of ths modls. hs s accomplshd through, for xampl, th ntroducton of product dffrntaton, capacty constrants, or a tmporal dmnson n th Brtrand modl, or consstnt conjctural varatons and barrrs to ntry n th Cournot modl. Lss wll mphaszd s th fact that, apart from th dcson varabl, both modl framworks ar dntcal. Both ar basd on an dntcal st of blfs about othr frms ractons, assumd not to chang whn th frm changs ts own dcson varabl, an assumpton whch I dsgnat hr by Nash conjcturs but Kamn and Schwartz (1983) dsgnat by zro conjctural varatons. As a rsult of th assumpton of Nash conjcturs, Cournot and Brtrand qulbra ar prfct commtmnt Nash qulbra, as no frm has th ncntv to unlatrally chang ts dcson varabl (Sngh and Vvs, 1984; Fudnbrg and rol, 1986). Arsng from ths st of blfs, Grossman (1981) argus that undr Brtrand frms cannot mak bndng prc contracts, whl undr Cournot, output contracts ar always bndng. Furthrmor, undr lnar dmand and costs, Cournot and Brtrand substtut and complmntary goods qulbra hav bn shown to b dual (Sngh and Vvs, 1984). 5

6 Pror ltratur (Chambrln, 1933; Grossman, 1981; Brsnahan, 1981; Kamn and Schwartz, 1983) has crtczd Cournot Nash conjcturs as a not approprat modl of frm conduct. Among th argumnts offrd, contrary to th assumptons undrlyng Cournot Nash conjcturs, othr frms do ract to changs n th frm s output dcsons; blfs undrlyng Cournot Nash conjcturs ar logcally nconsstnt, as frms ract to changs n othr frms outputs whl assumng that ts rvals do not do so; Cournot Nash conjcturs ar nconsstnt wth tact colluson (Chambrln, 1933); t s argud that prc, not output, s th rlvant stratgc varabl n mprfctly compttv markts wth homognous goods; and, fnally, mprcal vdnc for a varty of ndustry s nconsstnt wth Cournot Nash conjcturs (Kamn and Schwartz, 1983). Many othr authors hav also pontd out that th non-coopratv on-shot gam rflcts ralty poorly. hs crtcsms hav bn oftn msntrprtd as crtcsms of th Cournot Nash qulbrum, rathr than of th st of blfs,.., th undrlyng Cournot Nash conjcturs. Morovr, as was sn abov, most of th abov argumnts apply also to Brtrand Nash conjcturs. Nonthlss, thr s a strong cas for usng Cournot and Brtrand Nash conjcturs for solvng th frm proft maxmzaton problm (Daughty, 1985; Corts, 1999). Formost, th powr of ths st of blfs n dscrbng frm ntractons n on-shot prsonr dlmma typ gams or whn output and prc dcsons by frms ar not prfctly obsrvabl by compttors. In addton, Nash conjcturs nsur that th Cournot and Brtrand Nash qulbra ar prfct commtmnt qulbra (Fudnbrg and rol, 1986). If othr frms, for som rason such as rratonalty or mprfct nformaton, dvat from thr optmal rspons, a proft maxmzng frm can us Nash conjcturs to drv th optmal off-path qulbrum rspons, whch rsults n th so-calld racton functon. Fnally, ths papr argus that Nash conjcturs ar partcularly approprat for nfrrng tact and xplct collusv bhavor. Gvn that colluson s takng plac, f a colludng frm volats th collusv agrmnt, t dos so snc t assums that ts rvals cannot ract or snc t assums ts rvals wll not ract, for xampl, bcaus th rvals do not obsrv that th frm s volatng th collusv agrmnt. Altrnatvly, vn f output dcsons ar 6

7 obsrvabl by rvals and th colludng coalton truly nforcs dscpln n th cas of volatons to th collusv agrmnt by a frm, th unconstrand ncntv to dvat that ach frm facs s stll calculatd as f th rvals could not obsrv and ract to th frm s output changs,.., as f ach colludng frm wr opratng undr Nash conjcturs. hus, Cournot and Brtrand Nash conjcturs, vn f not rprsntatv of actual frm practc (Kamn and Schwartz, 1983), provd an mportant rfrnc pont as to th optmal slfntrstd and non-coopratv conduct of frms, f lft to thr own dvcs. DEFINIION 1. In an mprfctly compttv ndustry, dfn ndustry markt powr as th xtnt to whch markt prc (output) xcds (falls short of) th non-coopratv Nash qulbrum prc (output) 4, whr maxmum markt powr s consstnt wth monopoly proft maxmzaton, and mnmum markt powr s consstnt wth non-coopratv Nash qulbrum. Markt powr s tradtonally dfnd as th xtnt to whch prcs dffr from margnal costs, for xampl through masurs such as th Lrnr ndx (Motta, 24; Schrr and Ross, 199; rol, 1988). Undr comptton a la Brtrand, Dfnton 1 concds wth th tradtonal ltratur dfnton, as Nash qulbrum s such that prc quals margnal cost. Howvr, undr comptton a la Cournot, Nash qulbrum prc s largr than margnal cost, and output s smallr than prfct comptton output (Sngh and Vvs, 1984). In offrng ths altrnatv dfnton, I buld on pror mprcal work that ndrctly stmat th dgr of colluson by th xtnt obsrvd prc-cost margns dffr from thos undr th Cournot Nash qulbrum (Parkr and Röllr, 1997). S also Corts (1999) for an ovrvw of smlar approachs n th mprcal ltratur. h motvaton for ntroducng a novl dfnton of markt powr s that th rfrnc prfct Nash qulbrum consstnt wth optmal olgopoly prformanc wll dffr dpndng 4 h non-coopratv Nash qulbrum dffrs dpndng on whthr comptton s a la Cournot or a la Brtrand. 7

8 on what s consdrd as th corrct stratgc varabl and undrlyng modl (Cournot or Brtrand). Furthrmor, wth Dfnton 1, th masur of ndustry of markt powr compars th sam dcson varabl, ndustry output wth non-coopratv Nash qulbrum output, and markt prc wth non-coopratv Nash qulbrum prc, consstnt wth th stratgc varabl of th undrlyng Cournot or Brtrand modl, rathr than always compar prc wth margnal cost. Fnally, gvn th dual way n whch markt powr s dfnd, t s always possbl to masur ndustry markt powr, vn whn frm costs ar asymmtrc. DEFINIION 2. Colluson occurs whn frms cooprat (xplctly or tactly) to ncras ndustry markt powr rlatv to th non-coopratv Nash qulbrum. Undr ths dfnton, colluson also comprhnds th cass whr tact coopraton occurs, although tact colluson s not unlawful n most countrs. Hraftr, I focus on th Cournot modl wth n frms n qulbrum. 3. Modl and horms Usng a Cournot comptton modl, assum homognous output and an olgopolstc ndustry structur wth n frms n qulbrum, ach wth constant and dntcal margnal costs, thn: n = 1 n n q π ( Q ) = π ( q ) = p( Q ) C ( q ) = p( Q ) Q C ( q ) ( 1 ) = 1 = 1 whr π ar th ndustry s total profts, π ar frm s profts, q s frm s output, Q s total ndustry output, p(q ) s markt dmand functon, and C s frm s total cost of producton of output q. h ndustry s total costs, C s gvn by: n = 1 = n 1, q2,..., qn ) mc q = = 1 C ( q mc Q ( 2 ) Each frm s frm proft maxmzng condton s gvn by: dq q = p Q + p Q + π ( ) ( ) ( ) 1 q = mc dq ( 3 ) 8

9 dq whr Q - s th sum of th outputs of all othr frms n th ndustry, γ = s th dq conjctural varaton whch rflcts th xpctaton of frm about how othr frms ract n trms of thr (jont) output lvl, Q -, to a chang n frm s output, q. In a non-coopratv qulbrum ach frm chooss output to maxmz ts own profts dq gvn othr frms optmal output dcsons and has Cournot Nash conjcturs ( γ = =), dq.., assums absnc of racton by othr frms. hus, xprsson (3) rducs to th Cournot qulbrum: whr o mr, q and n a Cournot qulbrum, rspctvly. mr o mc = ( 4 ) Q ar th frm margnal rvnus, frm output, and th ndustry output Hraftr, followng th argumnts prsntd n Scton 2, I dstngush btwn th frm margnal rvnu calculatd wth Nash conjcturs, th Cournot Nash qulbrum, but dffrs from (4) for all othr possbl olgopoly outcoms. mr, and frm margnal rvnu at o mr. mr concds wth (4) at th Cournot Nash qulbrum, ASSUMPION 1. For any Q > such that p(q )>, assum that: () Dmand s downward slopng, p (Q )< () Dmand s wakly concav, p ( ) Q It follows from Assumpton 1 that th ndustry proft functon s strctly concav,.., π ( Q ) p ( Q ) Q + 2 p ( Q ) < ( 5 ) = and that frm s proft functon wth Cournot Nash conjcturs, γ = dq dq =, s also strctly concav: π ( q ) = p ( Q ) q + 2 p ( Q ) <, 1,..., n ( 6 ) 9

10 Industry profts undr colluson ar gvn by: n π ( Q ) = π ( q ) ( 7 ) = 1 whr maxmum ndustry profts undr colluson ar always qual to or lss than monopoly profts, and frm and ndustry outcoms undr colluson ar hraftr dntfd wth an astrsk suprscrpt. hus, th followng condton can b addtonally mposd: ASSUMPION 2. For any collusv outcom wth ndustry output Q >, assum that: Q π ( ) ( 8 ) Snc, from Assumpton 1, ndustry profts ar strctly concav, Assumpton 2 mpls that, undr colluson, ndustry output s qual to or largr than monopoly output, snc t would b sub-optmal for th colludng coalton to rduc output blow monopoly output lvls, whr jont ndustry proftablty s ncrasng n output. LEMMA 1. A proft maxmzng frm colluds f colluson s a proft nhancng actvty. Proof. I want to show that () collusv bhavor by a frm mpls an ncras n th frm profts rlatv to th x-ant cas whr th frm dos not collud. h transpos s that () f colluson rsults n profts that ar qual to or smallr than th non-collusv cas, thn th frm dos not collud. () Proof by transposton and rducton ad absurdum. Suppos that th frm colluds and that profts undr colluson ar qual to or smallr than th x-ant non-coopratv profts. Consdr frst th cas of xplct colluson. It s costly to xplctly coordnat actons wth othr frms rathr than act ndpndntly. Morovr, thr s a, howvr small, rsk of punshmnt by authorts n th cas of dscovry of xplct coordnaton, whch has a ngatv mpact on xpctd profts. In addton, thr s also th rsk of scrt volatons of th collusv agrmnt by compttors, also wth a ngatv mpact on xpctd profts. 1

11 hrfor, th frm could rduc costs and ncras xpctd profts by not ngagng n xplct colluson, ctrs parbus. A smlar rasonng can b appld to th cas of tact coordnaton. Although tact coordnaton s not subjct to punshmnt wth ngatv xpctd mpact on profts, thr ar no a pror rasons why tact coordnaton should b prfrrd to no coordnaton, unlss thr ar som ncntvs to do so. act coordnaton rqurs at last as much frm ffort and rsourcs as non-coopratv proft maxmzaton. hus, tact coordnaton s not lss costly than non-coopratv proft maxmzaton. Morovr, thr s also th rsk of chatng by compttors wth a ngatv mpact on xpctd profts. hrfor, th frm could ncras profts by not ngagng n tact colluson. Snc th frm s proft maxmzng, f colluson s not a proftabl actvty thn t follows that th frm wll not collud, contradctng our ntal hypothss. Q.E.D. Not that f colluson s n ffct pror to th partcpaton of th frm, thr would not b an ncntv for th frm to jon n th colluson ffort x-post, as th x-ant non-colludng frm profts would b largr than undr colluson. Altrnatvly, th colludng coalton mght ngag n punshng bhavor to nduc th non-colludng frm to jon n. AXIOM 1. Colluson s proftabl f and only f all low-cost frms collud and f low-cost frms hav a suffcntly larg combnd shar of th markt. A formal proof of ths Axom falls outsd th scop of ths papr, and thrfor t s offrd hr as an Axom rathr than as a horm. h xplanaton of th Axom s lad out blow, whch bulds on mrgr ltratur and on an xampl. 11

12 Obvously, mor than on frm must collud n ordr for colluson to b proftabl. 5 h ssu s whthr t s possbl to charactrz th mnmum colludng coalton sz rqurd n ordr for colluson to b proftabl. Snc thr s a contnuum of possbl colludng outcoms, t s not possbl to dtrmn th mnmum colludng coalton sz. Howvr, snc prfct colluson achvs, at bst, a prformanc (.., profts) dntcal to that achvd through th mrgr of th frms, th mnmum proftabl mrgr sz s th lowr bound for th mnmal coalton sz. hus, t s usful to buld on th horzontal mrgr ltratur that sks to dtrmn th condtons undr whch mrgrs among frms ar proftabl (Salant t al, 1983; Prry and Portr, 1985; Farrll and Shapro, 199; Chung, 1992; Faulí-Ollr, 1997, 22). Salant t al (1983) smnal papr showd that followng a mrgr among symmtrc frms wth lnar costs and homognous goods, th parts to th mrgr (nsdrs) would hav th ncntv to rduc output, whras th parts xtrnal to th mrgr would ncras output. Morovr, ndustry output would dcras and prc would ncras. Buldng on ths rsult, Chung (1992) shows that th mrgrs ar always unproftabl f thy nvolv lss than 5% of th frms n th ndustry, but may b proftabl f thy nvolv mor than that prcntag of frms, usng strctly concav ndustry proft functon such as that dfnd n quaton (5) of ths papr. Faulí-Ollr (1997) gnralzs th modl to show that th mnmal markt shar rqurd to nsur that th mrgr s proftabl s ncrasng n th dgr of concavty of dmand and n th x-ant numbr of frms n th ndustry. hus, th horzontal mrgr ltratur ndcats that th condton for succssful colluson s that th numbr of frms n th ndustry s not too larg, and that a larg prcntag of frms partcpat n th collusv agrmnt,.., colluson would b qut dffcult to achv for a symmtrc olgopoly. hs rsult s somwhat paradoxcal as t 5 If only frm colluds, t follows that frm profts undr colluson ar lowr than ts profts undr Cournot qulbrum snc frm s proft functon s strctly concav and, pr dfnton, th Cournot qulbrum lvl s proft maxmzng for frm, gvn all othr frms optmal rsponss. hus, frm could ncras profts by not ngagng n colluson, thus volatng Lmma 1. 12

13 suggsts that both mrgrs but also tact and xplct colluson would tnd to occur vry scarcly, a rsult contrary to th ancdotal and mprcal vdnc avalabl. hr man approachs suggst ways n whch thr can b proftabl mrgrs wth smallr substs of frms, namly cost convxty (Prry and Portr, 1985), Stacklbrg ladrshp (Daughty, 199), and cost asymmtry (Faulí-Ollr, 22). Stll, Hywood and McGnty (27) pont out that thr s a scond mrgr paradox snc, vn whn th mrgr s proftabl, frms qut oftn fac an ncntv not to mrg, as frms that do not mrg oftn gan mor from th mrgr than th frms that mrg. A smlar prncpl appls to colluson, why collud f th frm can fr-rd and bnft from th collusv fforts of othrs, wthout ncurrng th costs and rsks assocatd wth collusv bhavor. Cost asymmtrs provd th bst ln of xplanaton for ths paradoxs. Faulí-Ollr (22) shows that a mrgr btwn a low- and a hgh-cost frm can b proftabl, as th mrgd frm swtchs producton to ts low cost faclts. 6 hs rsult s not, n tslf, hlpful for colluson analyss snc, undr colluson, t s not fasbl to swtch producton among frms f thr ar no sd-paymnts. Instad, ancdotal vdnc of xplct colluson cass suggsts that colludng coaltons narly always nvolv all th larg frms n a markt,.., th frms wth larg markt shar. In a Cournot sttng, frms wth larg markt shars hav low margnal costs. hus, colluson wll only b proftabl f a suffcntly larg coalton of frms wth low margnal costs, and as a rsult, a hgh combnd shar of th markt, partcpat n th collusv ffort. As an xampl consdr th cas wth lnar dmand gvn by P ( Q) A B Q = ( 9 ) 6 Faulí-Ollr (22, p.83) ncorrctly argus that mrgr among larg frms s always not proftabl. hs s probably a poorly wordd statmnt, gvn that hs own pror work (1997) charactrzd condtons undr whch mrgrs could b proftabl vn wthout cost ffcncy ffcts. Nonthlss, both of hs contrbutons to mrgr thory hav not yt, n my vw, gand thr wll dsrvd rcognton. 13

14 and asymmtrc lnar costs, mc Low and mc Hgh. Assum thr ar n Low frms wth low margnal costs, and n Hgh frms wth hgh margnal costs. hn, t can b shown that th Cournot Nash qulbrum s gvn by: q Hgh = A ( n Low + 1) mc ( n Low + n Hgh Hgh + n Low + 1) B mc Low ( 1 ) q Hgh = A ( n Hgh + 1) mc ( n Low + n Low Hgh + n Hgh + 1) B mc Hgh ( 11 ) whr q Hgh and q Low ar th output of th hgh- and low-cost frms, rspctvly. If th cost asymmtrs ar suffcntly larg, thn th low cost frms can hav a combnd vry larg markt shar of th markt, and t can b proftabl for thm to collud vn f th hgh cost frng frms do not collud. As a rsult of colluson among larg frms, thr combnd markt shar falls sgnfcantly, whl th markt shar and profts of th frng (hgh cost) frms rs substantally. Howvr, f a low cost frm dos not collud, thn t s not proftabl for th rmanng low cost frms to collud, a rsult smlar to that prdctd by Salant t al (1983), rgardlss of whthr or not th hgh cost frm colluds. hus, colluson must nvolv all larg (low cost) frms. For xampl, f th abov modl s solvd wth th followng paramtr valus: mc Low = 2, mc Hgh =2, n Low = 5, and n Hgh =1, A=2, B=1, thn th combnd markt shar of th larg frms undr th Cournot Nash qulbrum s 92%. Each larg (low cost) frm markt shar s 16.4% and th frng (hgh cost) frm has a 7.6% markt shar. If all 5 low cost frms collud prfctly, thn t s proftabl to collud, but combnd markt shar of colludng frms falls to 57%. Howvr, f on of th larg frms dos not collud, thn colluson s no longr proftabl as th combnd markt shar of colludng frms falls to 37.5%. hs Axom addrsss both mrgr paradoxs and s consstnt wth th pror ltratur. Colluson s proftabl f all larg frms partcpat, thrfor thr s an ncntv for larg frms to partcpat. On th othr hand, no larg (low-cost) frm can fr-rd on th 14

15 collusv agrmnt, snc t suffcs that thr s a larg fr-rdr frm to dstroy any collusv ffort. If a larg frm fr-rds on th collusv ffort, by maxmzng profts undr Cournot Nash conjcturs, t maks th collusv ffort unproftabl for th colludng frms, and thrfor th othr larg frms would hav no ncntv to collud. Colluson s an all or non proposton for larg frms. Fnally, Axom 1 suggsts a ky dffrnc btwn th colluson and mrgr ltratur. Collusv ffort ams at nvolvng a larg combnd markt shar of th frms n th markt. On th othr hand, as partly argud by Faulí-Ollr (22), mrgr ffort ams at achvng cost ffcncs, and thus only occurs btwn larg (low-cost) and small (hghcost) frms. hs happns bcaus only mrgrs that am for cost ffcncs ar both proftabl and fasbl n th currnt rgulatory nvronmnt. Mrgrs btwn two larg frms ar not proftabl snc, typcally, thy do not achv th mnmum markt shar thrshold ncssary to nsur that th mrgr s proftabl a la Salant t al (1983). Mrgrs btwn all larg frms, whl proftabl, ar not socally accptd nor fasbl n th currnt rgulatory contxt. HEOREM 1. Colluson occurs f and only f ndustry profts ar abov th lvl of th noncoopratv Nash qulbrum. Proof. h proof s agan for comptton a la Cournot. I want to show that () colluson mpls that ndustry profts ar abov th non-coopratv Cournot qulbrum lvl; and () th fndng of qulbra wth ndustry profts abov Cournot qulbrum lvls mpls that thr s colluson. () Proof by rducto ad absurdum. From Axom 1, assum that at last two frms collud by dvatng from th Cournot qulbrum output lvl (rprsntd wth th suprscrpt ) and that ndustry profts n th nw qulbrum (suprscrpt ) ar lss than or qual to th non-coopratv qulbrum lvl. From Lmma 1, t follows that a proft maxmzng frm colluds bcaus colluson s a proft nhancng actvty. hn, 15

16 , j 1,..., n π ( q ) > π ( q ) π ( q ) > π ( q ) ( 12 ) that s, at last two frms hav profts hghr than th non-coopratv lvl. From ths rsult and th ntal hypothss that ndustry profts ar lss than or qual to non-coopratv Cournot profts, t follows that th sum of all othr n-2 frm profts must b lowr than thr Cournot qulbrum lvl, and f so thn j j j j k k, j 1,..., n π ( q ) < π ( q ) ( 13 ) Dfnton 1 and 2 ndcat that colluson mpls a dcras n ndustry output and an ncras n markt prc,.., k k k thn from (13) and (14) t follows that Q p ( Q ) > p( ) ( 14 ) k q k q < ( 15 ) Snc from Assumpton 1, frm profts ar strctly concav, and non-coopratv Nash qulbrum profts ar maxmzd wth: t follows that k k k π ( q ) = p ( Q ) q + p( Q ) mc = ( 16 ) k q k hus, frm k could ncras profts by ncrasng output, so π ( ) > ( 17 ) qulbrum for frm k. Furthrmor, a countr xampl shows that snc: q k s not a non-coopratv Nash q k q Q Q ( 18 ) k thn frm k could achv profts hghr than th non-coopratv lvl by ncrasng output to q k, contradctng (13) and mplyng that th ndustry profts undr colluson cannot b qual to or lss than th non-coopratv Cournot qulbrum lvl. Q.E.D. 16

17 () Proof by rducto ad absurdum. Suppos thr s a sustanabl qulbrum (dntfd wth th suprscrpt ) whr ndustry profts ar abov th non-coopratv Cournot qulbrum and thr s no colluson,.., frms n th ndustry do not cooprat xplctly or tactly. Snc from Assumpton 1 t follows that th ndustry proft functon s strctly concav thn π ( Q ), Q QM, whr Q M s optmal monopoly output, and: π ( Q ) > π ( Q ) Q < Q p( Q ) > p( Q ) ( 19 ) snc p ( ) <. hn, thr (a) all frms hav profts abov th non-coopratv lvl or (b) Q at last on frm has profts blow th non-coopratv lvl. If (a) appls, thn th noncoopratv Cournot qulbrum cannot b a Nash qulbrum bcaus thr s anothr noncoopratv qulbrum, whch mprovs profts for all frms n th ndustry; f (b) appls thn for th frms that hav profts blow th non-coopratv lvl t follows that: from (19) t follows that ( p( Q ) mc) q < ( p( Q ) mc) q ( q ) < π ( q ) π ( 2 ) ( p( Q ) mc) > ( p( Q ) mc) ( 21 ) thus, from (2) and (21) for frm to hav profts blow th non-coopratv lvl t must b that q < q ( 22 ) but thn a smlar rasonng to part () of th proof can b usd to show that frm could achv profts at last as hgh as th non-coopratv lvl by ncrasng output to th Cournot qulbrum lvl, contradctng our ntal hypothss (b). hus, th outcom wth hghr ndustry profts would not b a sustanabl qulbrum, contrary to our ntal hypothss. Q.E.D. 17

18 COROLLARY 1. If ndustry profts ar largr than th non-coopratv Nash qulbrum ndustry profts, thn th profts of any subst of th frms n th ndustry ar largr than th profts of th sam subst of frms undr a non-coopratv Nash qulbrum. Proof. Both colludng and non-colludng frms bnft from colluson. horm 1 shows that f ndustry profts ar largr than th non-coopratv Nash qulbrum lvl thn thr s colluson. Lmma 1 shows that frms collud f and only f colluson s proftabl. hus, th profts of any subst of colludng frms s largr than th profts of th sam subst of colludng frms undr th non-coopratv Nash qulbrum. Morovr, Salant t al (1983), Hywood and McGnty (27), and othrs show that frms xtrnal to mrgrs bnft from a mrgr, oftn mor so than frms that mrg. From Assumpton 1, t follows that th ndustry proft functon s strctly concav. Assumpton 2 addtonally mans that hghr ndustry profts can only b achvd through hghr prc and lowr output lvl. As a rsult, ach and vry non-colludng proft maxmzng frm, xprncs a shft to th rght of ts rsdual dmand functon, and achvs hghr profts than undr th non-coopratv Nash qulbrum. hus, th jont proft of any subst of noncolludng frms s also largr than th profts of th sam subst of frms undr th noncoopratv Nash qulbrum. Q.E.D. hs rsult s rlvant f t s only possbl to obtan frm data for a subst of th ndustry. For xampl, t mayb th cas that t s only possbl to obtan data for th largst frms n th ndustry. Axom 1, horm 1, and Corollary 1 ndcat that data on th largst frms would suffc to nvstgat whthr th ndustry has or not collusv bhavor. HEOREM 2. Industry profts ar abov non-coopratv Nash qulbrum lvls f and only f margnal rvnu wth Nash conjcturs s largr than margnal cost for at last on frm. Proof. I want to show that: () th xstnc of ndustry profts largr than non-coopratv lvl mpls that margnal rvnus ar largr than margnal costs for at last on frm; and 18

19 () f at last on frm has margnal rvnus largr than margnal costs thn ndustry profts ar mprovd rlatv to th non-coopratv Nash qulbrum. Condtons () and () can b xprssd as: > π ( Q ) > π ( Q ) 1,..., n : mr ( q ) mc ( 23 ) 1,..., n : mr ( q ) > mc π ( Q ) > π ( Q ) ( 24 ) rspctvly, whr th suprscrpt rprsnts markt outcoms undr colluson, and th suprscrpt Cournot qulbra. () Proof by rducto ad absurdum. Suppos thr s a sustanabl qulbrum (dntfd wth th suprscrpt ) whr ndustry profts ar abov th non-coopratv Cournot qulbrum and margnal rvnus ar smallr than or qual to margnal costs for all frms n th ndustry,..: from (25) and (8) t follows that π ( Q ) > π ( Q ) mr ( q ) mc, 1,..., n ( 25 ) thus, from Assumpton 1() t can b shown that Q < Q p( Q ) > p( Q ) ( 26 ) mr ( q ) = p ( Q ) q + p( Q ) > p ( Q ) q + p( Q ) ( 27 ) It suffcs to show that thr s on cas whr (25) s volatd. hus, consdr th cas of a symmtrc qulbrum whr q Q Q = q = ( 28 ) n n t follows that = mr ( q ) > mr mc ( 29 ) whch contradcts our ntal hypothss. Q.E.D. 19

20 () I show (24) through rducto ad absurdum. Suppos that (24) s not tru and: 1,..., n : mr ( q ) > mc π ( Q ) π ( Q ) ( 3 ) Not that ndustry profts n th Cournot Nash qulbrum ar gvn by n π ( Q ) = π ( q ) ( 31 ) whr total ndustry output can b rwrttn as th sum of th frm own output plus th output of all othr frms n th ndustry = 1 o Q + Q ( 32 ) = q o and o q = ) π ( Q ) p( Q ) + p ( Q ) ( q + Q mc ( 33 ) whr π ( ) s frm s proft at th non-coopratv qulbrum charactrzd by (4) for all o Q frms n th ndustry, and π ( ) ar th ndustry non-coopratv Cournot qulbrum q profts. From (4) I know that π ( ) =. It thn follows that: Q = π ( Q ) π ( q ) + p ( Q ) Q = p ( Q ) Q < ( 34 ) > snc p ( ) < and Q. hn, from (8) and (3) t can b shown that π ( Q ) < π ( Q ) Q > Q p( Q ) < p( Q ) ( 35 ) snc Q Q > thn thr must xst at last on frm, say frm, that producs at last as much as undr Cournot qulbrum,.., 1,..., n : q q ( 36 ) hn, from Assumpton 1(), (35), and (36) t follows that th margnal rvnu for frm mr ( q ) p( Q ) + p ( Q ) q < p( Q ) + p ( Q ) q = ( 37 ) snc dmand s wakly concav (Assumpton 1()), t follows that Q p ( Q ) < p ( ) ( 38 ) 2

21 From (37) and (38) t mmdatly follows that mr ( q ) < mr( q ) = mc ( 39 ) contradctng our ntal hypothss (3). Q.E.D. COROLLARY 2. h maxmzaton of ndustry profts (prfct colluson) mpls that frm margnal rvnus ar largr than margnal costs for all frms n th ndustry,.., that ach and vry frm n th ndustry colluds. Proof. I want to show that: () th maxmzaton of ndustry profts (prfct colluson) mpls that frm margnal rvnus ar largr than margnal costs for all frms n th ndustry. Prfct colluson s akn to a stuaton whr th ndustry olgopoly conduct achvs a prformanc dntcal to that obtand f th ndustry wr a monopoly. Condton () can b xprssd as: π ( Q ) = mr ( q ) > mc 1,.., n ( 4 ) whr th suprscrpt rprsnts markt outcoms undr colluson, and th suprscrpt th Cournot qulbrum. Exprsson (4) can b rwrttn as: = p( Q ) + p ( Q ) Q mc ( 41 ) or MR = MC = mc ( 42 ) whr n Q = q = 1 ( 43 ) s th ndustry optmal output undr (prfct) colluson, q s th cartl output allocaton to frm, and MR, MC ar th ndustry margnal rvnus and costs, rspctvly. hat s, 21

22 undr prfct colluson a cartl bhavs as f t wr a monopoly. From (41) and (42) t follows that p ( Q ) p ( Q ) q > p( Q ) + p ( Q ) Q + ( 44 ) snc dmand s downward slopng (Assumpton 1()) and Q q <. hs xprsson can b rwrttn as ( q ) MR mr > ( 45 ) whr mr q ) dsgnats frm s margnal rvnus wth Cournot conjcturs at ( From (45), (42), and (4) t follows that q. mr ( q ) > mc = mr, 1,..., n ( 46 ) Q.E.D. Exprsson (46) ndcats that th collusv proft maxmzaton condton ngats proft maxmzaton at th frm lvl, a wll known rsult whch rol (1988) ascrbs to th ngatv xtrnalty btwn frms: whn frms maxmz ndvdual profts thy tak nto account th ffct of ts output chang on markt prc and ts own rvnus but do not consdr th ffct on total ndustry rvnus. hus, prfct colluson mpls that ach frm partcpatng n th collusv agrmnt has (non-coopratv) ndvdual margnal rvnus that ar largr than th ndustry margnal rvnus undr colluson and than th frm margnal costs,.., ach frm has an ncntv to dvat from collusv outcom and to ncras output n ordr to ncras profts. In ssnc, colluson s akn to hav ach frm n th colludng coalton act as f t had hghr margnal costs. rol (1988) shows that ncrass n a frm margnal costs lad th frm to produc lss and rsult n a shft outwards of th rsdual dmand facd by all othr frms n th ndustry. If vry frm n th colluson coalton chooss a lowr output lvl n a coordnatd mannr, th rsdual dmand of ach frm shfts outward suffcntly so that as a rsult ach frm has a hghr proftablty than undr th non-coopratv qulbrum. 22

23 HEOREM 3. Colluson occurs f and only f th frm margnal rvnu s largr than margnal cost for a suffcntly larg subst of th frms n th ndustry. Proof. Follows from Lmma 1, Axom 1, horm 1, and horm Colluson tst In Scton 3, I showd that f and only f ndustry profts ar largr than Cournot Nash qulbrum profts, thn thr s colluson (horm 1). Altrnatvly, f and only f th frm colluds, thn ts margnal rvnus wth Cournot Nash conjcturs ar largr than ts margnal costs (horm 3). h tst can thus b spcfd as on-tald hypothss tst thr basd on horm 1: H : Π HA : Π, t, t Π Π, t, t > ( 47 ) or altrnatvly basd on horm 3: H : mr HA : mr, t, t mc mc, t, t > ( 48 ) whr Π Π, t,, t ar actual total ndustry profts and hypothtcal total ndustry Cournot Nash qulbrum profts n prod t, rspctvly, and mr mc,, t, t ar frm s margnal rvnus (wth Nash conjcturs) and margnal costs n prod t, rspctvly. In both spcfcatons, rjcton of th null hypothss ndcats that th hypothss of non-collusv bhavor should b rjctd. h frst spcfcaton allows th rjcton of noncollusv bhavor at th ndustry lvl, for xampl, usng ndustry lvl tm srs data. h scond spcfcaton allows th rjcton of non-collusv bhavor at th frm lvl, for xampl, usng frm lvl tm srs data. Not that th purpos of th tst s to quckly flag nstancs of suspctd colluson for furthr nvstgaton by authorts, whch would thn dtrmn whthr t s tact or xplct colluson. h tst dos not, by tslf, consttut vdnc of xplct colluson. 23

24 5. Opn ssus h dscusson n ths scton dntfs opn ssus for furthr rsarch, and dscusss possbl approachs to mplmntng th proposd colluson tsts act vrsus xplct colluson Consdr th colluson tst spcfcaton gvn by (48). If th null hypothss of no colluson s rjctd wth a gratr dgr of statstcal sgnfcanc, t s lkly that for th obsrvatons of th sampl, frm margnal rvnu s much largr than ts margnal cost,.., th obsrvatons undrlyng th mprcal rsults lkly fall on rght tal of th dstrbuton. For thos obsrvatons, f th frm margnal rvnu s much largr than ts margnal cost, thn th ncntv to scrtly volat th collusv agrmnt s larg. act colluson s mor dffcult to mantan f th ncntv to volat th agrmnt s larg. hus, whn th null hypothss s rjctd at th on prcnt lvl, w ar lkly to fnd a much smallr proporton of tact colluson cass, than whn th null hypothss s rjctd at th fv or tn prcnt lvl. Lkws, f margnal rvnu s only slghtly largr than margnal cost, thn th gans from colluson ar lkly to b rlatvly small. On th on hand, for frms ngagng n xplct colluson, profts from colluson must b largr than th xpctd costs of xplct colluson (s proof Lmma 1). On th othr hand, bttr coordnaton s lkly to allow xplct colludng coaltons to ncras markt powr by mor than fasbl for tact colludng coaltons, and thus n largr dffrncs btwn ach colludng frm margnal rvnus and margnal costs. hus, th proporton of xplct colluson cass, whn margnal rvnu s only slghtly abov margnal cost, s lkly to b much smallr than that whn margnal rvnu s much largr than margnal costs. If ndd so, th proposd tst would hav attractv proprts, as typ I rrors would lkly b tact colluson cass, whch ar not rlvant for rgulatory purposs. 24

25 5.2. Data ssus h man ssu s what varabls can b mor asly obtand by rgulatory authorts n ordr to dtct nstancs of colluson. h data rqurd for th tsts suggstd by thr horm 1 or 3 ar dffcult to obtan, mor so than data on prcs and margnal costs as usd, for xampl, n th Hall (1988) analyss of prc-cost margns for dffrnt US ndustrs. For both tsts, thr s th ssu of th dfnton of th rlvant markt, and data avalablty n th ncssary dtal, partcularly for mult-product frms. A tst basd on comparson of ndustry profts wth Cournot Nash qulbrum profts (horm 1) would fac a numbr of challngs. Frst, thr s a dffrnc btwn conomc and accountng profts. Scond, thr would b fwr obsrvatons and probably mor mssng obsrvatons, as th tst would b basd on a comparson of ndustry lvl data. Stll, vn f, for xampl, t s not possbl to obtan data for small frms, Corollary 1 ndcats that th sam tst could b appld for any subst of th ndustry. Fnally, th rfrnc ndustry profts undr Cournot Nash qulbrum, whch would b usd as bass of comparson, cannot b obsrvd. A possbl approach would b to us th fact that frm markt shars, n a Cournot Nash qulbrum, provd nformaton about ach frm s margnal costs. A tst basd on th comparson of margnal rvnus and margnal costs also facs sgnfcant data challngs. Both varabls ar not drctly obsrvabl. Nonthlss, pror ltratur has lookd at prc cost margns (Cowlng and Watrson, 1976; Hall, 1988; Parkr and Röllr, 1997), and thrfor dffrnt approachs hav bn proposd to masur margnal costs. Structural conduct prformanc paradgm mprcal studs usd accountng masurs of margnal costs. On possbl approach n ths ln, assumng th avalablty of product ln accountng rvnus, costs, and output lvls, could b to drv avrag ncrmntal margnal rvnus and margnal costs from quartrly or yarly sals and cost data. An altrnatv approach, would b to follow mthodologs of th Nw Emprcal Industral Organzaton ltratur (Hall, 1988; Brsnahan, 1989; Parkr and Röllr, 1997) that attmpt to stmat paramtrs of th frm s supply rspons (Corts, 1999). 25

26 6. Concluson It s wll known that colluson outcoms ar unnatural n that colludng frms act contrary to th conomc ncntvs thy fac ndvdually. Colludng frms hav th ncntv to scrtly volat th collusv agrmnt and on of th man ssus assocatd wth th sustanablty of colluson s to prvnt such volatons of th collusv agrmnt from occurrng (Stglr, 1964). h proposd tst of collusv conduct, basd on th analyss of frm ncntvs, sks to dtct frm coopratv bhavor not consstnt wth th prfct Nash qulbrum proft maxmzng condton, a stuaton that occurs undr both tact and xplct colluson. It has svral advantags. For on, masurs of markt prformanc, such frm and ndustry prc-cost margns ar not drctly rlvant to th tst, thus t dos not pnalz ffcnt frms (Dmstz, 1973). Markt structur and th dgr of contstablty of th markt smlarly do not affct th ffcacy of th proposd tst (Cabral, 2; Baumol t al, 1982), snc th tst analyzs frm ncntvs for a gvn frm markt structur, and frm conduct consstnt wth th contstabl markt hypothss wll rsult n th non-rjcton of th null hypothss of no collusv bhavor. Furthr, n th contxt of dynamc comptton, th tst may dtct nstancs of tact colluson whr frms do not optmz profts for far of rtalatory acton by compttors. Fnally, th tst dos not dpnd on th stmaton of a conduct paramtr that n ffct masurs how th ndustry conduct s corrlatd to that of ndustrs undr compttv or monopoly qulbra (and to prc-cost margns). h tst smply masurs whthr th frm has an ncntv to dvat from ts currnt output dcson. h papr rass ntrstng qustons that wll hopfully b subjct of furthr rsarch. hs nclud, among othrs, th analyss of th Brtrand dual cas and th consdraton of product dffrntaton. 26

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