Colluson n Capacty Under Irreversble Investment Thomas Fagart November 2017 Stellenbosch Unversty Centre for Competton Law and Economcs Workng Paper Seres WPS01/2017 Workng Paper Seres Edtor: Roan Mnne 18331521@sun.ac.za
Colluson n Capacty Under Irreversble Investment Thomas Fagart September 14, 2017 Abstract Under mperfect competton, frm s ncentve to devate from a collusve agreement s usually short term: the frm ncreases ts mmedate proft but reduces ts future profts due to the upcomng compettors reacton. When frms have to nvest to ncrease ther producton, and that nvestment s rreversble, a frm may devate to preempt ts compettors and obtan a domnant poston on the market. When frms are more patent, preempton s more proftable, and colluson may thus be harder to sustan. Ths result, contrastng wth the lterature on colluson, suggests that folk theorems may be napproprate to study colluson n Markovan frameworks. Keywords: Capacty nvestment and dsnvestment. Dynamc games. Markov perfect equlbrum. Real optons. Colluson. JEL classfcaton: D43 L13 L25 Unversty Cergy Pontose. Emal: fagart.thomas.research@gmal.com Ths research was funded by the labex MME DII. 1
Contents 1 Introducton 3 2 A Framework wth Irreversble nvestment 7 3 Collusve agreement for status quo 11 4 Concluson 20 5 Appendx 21 References 28 2
1 Introducton In many ndustres, the producton of frms s lmted by ther nfrastructure,.e. amount of factores or equpment that they own. Ths producton capacty may evolve n tme, as frms nvest n new nfrastructure. However, the cost of buldng a new capacty s usually sunk and nvestments are thus at least partally) rreversble. These features of capacty lmtaton and rreversble nvestment mpact the way competton works. Ths s the case n a famous example of colluson: the lysne cartel. Lysne s an essental amno acd that stmulates growth and lean muscle development n hogs, poultry, and fsh. It has no substtute. By the late 1960s, Japanese botechnology frms had dscovered a bacteral fermentaton technque that transformed the producton of lysne. The process nvolves the fermentaton of dextrose nto lysne and requres a specfc chemcal nfrastructure. The cost of ths nfrastructure s therefore sunk. 1 the Ths technque s consderably cheaper than conventonal extracton methods. By the end of the 1980s, there were three major players n the lysne market: Ajnomoto and Kyowa Hakko based n Japan, and Sewon based n South Korea. In 1988, ADM acqured a fermentaton technque for lysne and began buldng the world s largest lysne factory n 1989. ADM s entry was effectve n February 1991. Ths entry of ADM nto the lysne market led to the creaton of one of the best-known cartels n hstory. The consortum ran from June 1992 to June 1995, date at whch the FBI raded the headquarters of the partcpatng frms. It was the frst global prce-fxng conspracy to be convcted by US or EU anttrust authortes n 40 years, and the fnancal penaltes amounted to $305 mllon. The cartel started just after Archer-Danels-Mdland Company ADM) entered nto the lysne busness. However, between the end of 1992 and early 1993, ADM nvested massvely to ncrease ts producton capacty, whch rose from 60 000 tons per year to 113 000 tons. At the same tme, ADM started to cheat on the collusve agreement, and a prce war began n March 1993. The prce war ended n November 1993, and the cartel carred on peacefully untl the nterventon of the FBI. 2 1 There s some possblty of transformng a lysne plant to produce another amno acd, however ths s costly. 2 For more nformaton on the lysne cartel, see the books by Echenwald 2000) and Connor 2000), and 3
Ths cartel case emphaszes the followng pont: capactes can evolve durng perods of colluson. Several papers focus on the mpact of constant capacty constrant on colluson, but very few deal wth the strategc evoluton of capacty. Ths work attempts to fll that gap. More precsely, ths work shows the exstence of long-run ncentves for devaton. Indeed, to devate from a collusve equlbrum, a frm has to nvest n new unts of capacty n order to ncrease ts producton. In dong so, the devatng frm commts to ts new capacty level due to the rreversblty of nvestment. Ths reduces ts opponent s ncentve to nvest n order to punsh the devatng frm. In some cases, t even prevents the opponent from mplementng any punshments. A frm may thus ncrease ts long-run proft by devatng from the collusve equlbrum. Ths contrasts wth the standard theory of colluson, whereby devaton s only effect s to ncrease frm s short-run profts, whle reducng long run profts due to punshment. Ths s consstent wth the lysne case, where ADM s devaton n 1993 and the buldng of new capacty allowed the frm to ncrease ts market share n the Amercan market from 44 percent n 1992 to 57 percent n followng years. In the lterature on colluson and capacty constrant, Compte et al 2002) focus on the case of prce competton wth nelastc demand under fxed capacty constrants, and show that larger frms have the most ncentves to devate. Under Cournot competton, lnear demand and soft capacty constrant, Vasconcelos 2005) fnds that the smallest frms have the most ncentves to devate. Under a smlar framework, Bos and Harrngton 2010) show that, when a cartel s not nclusve, devaton may be drven by medum-szed frms. In all of these works, asymmetry between frms capacty hnders colluson. Fabra 2006) and Knttel and Lepore 2010) focus on the mpact of capacty constrant on colluson n presence of demand evoluton, and show that a collusve prce can be counter-cyclcal. Garrod and Olczak 2014) show that, under mperfect montorng, capacty constrants may help to detect a devaton. All of these works assume that capactes are fxed and study the mpact of capactes on colluson. 3 To my knowledge, only two papers focus on the evoluton of capacty durng the artcles by Connor 2001), Roos 2006) and Connor 2014). 3 Compte et al 2002) and Vasconcelos 2005) study the mpact of a change n the capacty dstrbuton due to a merger, and Knttel and Lepore 2010) endogenze the choce of capacty at the begnnng of the 4
colluson. Paha 2016) adapts the model devsed by Besanko and Doraszelsk 2004) to smulate the nvestment behavor of frms when they collude n prce but not n capacty. Besdes hs result on the effect of uncertanty on cartel formaton, Paha presents counterntutve evdence that a low dscount rate can mpar colluson. In a duopoly model wth lnear demand and partally rreversble nvestment, Feuersten and Gersbach 2005) studes a grm trgger strategy n whch the cooperatve behavor for each frm nvolves nstallng half of the monopoly capacty, and the puntve behavor nvolves nvestng untl the margnal value of capacty equals the prce of nvestment. They show that these strateges can form an equlbrum, but at a lower dscount rate compared to the case of fully reversble nvestment. In the present paper, I study a Cournot duopoly wth rreversble nvestment n capacty n a dscrete tme settng. More precsely, at each pont n tme, the producton of each frm s determned by ts level of capacty. Frms may decde to ncrease ther capacty through buyng assets. The prce of nvestment s lnear. The results depend on the perod of tme, whch can be nterpreted as the degree of flexblty of the frms nvestment decsons. When the perod s long, t takes tme for the frm to punsh a devaton, whereas when the perod s close to zero, frms react nstantly to any devaton. Ths model can thus be vewed as a generalzaton of Feuersten and Gersbach 2005). In ths framework, frms nvest only at the begnnng of the game n the non-cooperatve equlbrum. If both frms start wth suffcently low ntal capacty, then they nvest to the same Cournot level of capacty. When one of the frms starts out wth an ntal capacty that s hgher than the Cournot level, then the frm s commtted to ths level of capacty due to the rreversblty of nvestment, and thus ts opponent reacts by nvestng at a capacty level lower than the Cournot level. The frm wth the hgh ntal capacty does not nvest. To study colluson, I focus on a partcular knd of strategy,.e. the grm-trgger strategy for status quo. Ths conssts n the frm keepng ts ntal capacty as long as ts opponent also keeps ts own ntal level of capacty. When one of the frms devates, the other responds by nvestng as n the non-cooperatve equlbrum. However, f the devatng frm has nstalled a capacty larger than the Cournot outcome, then t has commtted tself to ths capacty for the rest of the game, and the non-cooperatve equlbrum s for the punshng frm to nvest less than the Cournot outcome. Ths s the long run effect of the devaton: the devatng game. However, at the tme when colluson s mplemented, the capactes are fxed. 5
frm can gan a sze advantage durng the devaton, whch perssts for the rest of the game. The usual short-run effect s also present: there s a perod, between the mplementaton of the devaton and the mplementaton of the punshment, when the devatng frm s the only frm to have a capacty larger than the collusve capactes. Both short-run and long-run effects are mportant to determne a frm s ncentves to devate. Focusng on the lmt case n whch the perod of tme shrnks to zero allow to solate the long run effect. In such case, there s a parallel between the decson of the Stackelberg leader and the optmal devaton. Indeed, the punshng frm decdes on ts capacty takng nto account the capacty already nstalled by the devatng frm. Colluson s then only possble f the Stackelberg proft s under half of the monopoly proft. A low dscount rate s thus harmful for colluson, as more patent frms are more lkely to nvest, and then prefer becomng the Stackelberg leader rather than obtanng half of the monopoly proft. When the perod of tme s postve, the short-run effect may compensate the long-run one and the mpact of the dscount rate on colluson wll depend on whch effect domnates. These fndngs provde ntuton on the results of Paha 2016) and Feuersten and Gersbach 2005). In Paha 2016), the counter-ntutve effect of a low dscount rate on the possblty of colluson comes from the domnaton of the long-run effect over the short-run effect. Feuersten and Gersbach 2005) fnds the opposte result on dscount rate) because the Stackelberg leader s proft wth lnear demand equals half of the monopoly proft. Therefore, the long-run effect does not permt colluson to be mplemented wth lnear demand), and colluson s more dffcult wth rreversble nvestment than wth reversble nvestment as the short-run effect must compensate the negatve long-run effect. Ths depends on the form of proft assumed; the ntroducton of a smple quadratc producton cost leads to a dfferent result. Secton 2 presents the model and the non-cooperatve equlbrum. Secton 3 focuses on the collusve equlbrum for status quo, wth the short-run and long-run effects of devaton. Secton 4 concludes. 6
2 A Framework wth Irreversble nvestment Model I consder a market wth two frms A and B) competng à la Cournot wth a homogenous product, n a dynamc tme settng smlar to that of Sannkov and Skrzypacz 2007). Tme s contnuous and the horzon s nfnte, but frms only take decson at dscrete tmes t = {0, τ, 2τ,...}. The game s therefore dscrete. 4 The contnuous tme framework makes t possble to vary τ, the duraton of tme perods between whch nvestment decsons are made. More precsely, at each tme t = nτ n N) frms smultaneously decde whether extend ther capacty through buyng assets. Frms start wth an ntal capacty k0 for {A, B}) and the capacty of frm at tme t s thus kt = kt τ + It, where It denotes the nvestment of frm at tme t It 0). Durng the tme nterval [t, t + τ[ the quantty produced by frm s determned by ts capacty leadng to a producton constrant: for all s [t, t + τ[, q s = k t. 1) The prce of the good s a functon of the total quantty produced, P q A t + q B t ) and the producton cost of frm s cq t). The prce of nvestment, p +, s lnear, and frms face the same dscount factor δ ]0, 1[. The proft of frm made durng the tme nterval [t, t + τ[ s then fully determned by the frms capacty at tme t: t+τ t δ [ s P ) qs + qs q s cq ) ] ds = δ t 1 δτ ) [ ) P k ln 1/δ) t + k j t k t ckt) ] 2) where k j t s the capacty of the opponent of frm. For a vector of capacty k t = k t, k j t ), the nter-temporal proft of frm s then gven by: Π = t=0,τ,.. δ t [ 1 δ τ ) π k t ) p + I t], 3) 4 Even f frms decsons are taken at a dscrete date, ths game s not a repeated game strcto sensu,.e. the stage games are not ndependent as the capacty nstalled at tme t s present at tme t + τ. 7
where π k t ) s defned by: π k t ) = P ) kt + kt k t ckt). 4) ln 1/δ) π s the nter-temporal proft of frm when there s no nvestment durng the game. I assume that frms strateges are Markovan, meanng that the nvestment at tme t s a functon of the capactes of the ndustry at tme t τ, I t = Î k t τ ). 5 The proft of frm can then be rewrtten: Π k) = 1 δ τ ) π k + Îk) ) p + I k) + δ τ Π k + Îk) ). 5) Î k) s then a Markovan equlbrum f and only f, for each {A, B}, { ) )} Î k) = arg max 1 δ τ ) π k + I, k j + Î j k) p + I + δ τ Π k + I, k j + Î j k), I where Î j k) s the strategy of frm j, and the equlbrum profts, Π, are defned by: Π k) = 1 δ τ ) π k + Î k) ) p + Î k) + δ τ Π The followng assumpton ensures the exstence of Markovan equlbra. 6) ) k + Î k). 7) Assumpton A: The cost functon, c.), s a twce-dfferentable postve functon such that c 0, c 0. P.) s also a twce-dfferentable postve functon, wth P < 0, P < 0 when P s strctly postve. Non-cooperatve Equlbrum As usual n dynamc competton wth an nfnte horzon, there exst a multplcty of equlbra. The ssue s to determne whch equlbrum s the non-cooperatve one. In a classc model of repeated Cournot competton, the non-cooperatve equlbrum s gven by the repetton of stage game equlbrum. However, n the present model, perods are lnked 5 For smplcty of notaton, a varable wth a hat wll be a functon of the state varable, a varable wth a captal ndex wll ndcate a functon of the compettor s capacty. 8
by frms capacty, and the stage game depends on the past. There s then no clear noncooperatve equlbrum. 6 The objectve of ths sub-secton s to defned one of the Markovan equlbrum as the non-cooperatve one. There exsts a Markov perfect equlbrum n whch the frms strateges are to nvest n the frst perod, and not n the rest of the game. Ths Markov perfect equlbrum s defned as the non-cooperatve one. There are three reasons for ths. Frst, the frm s strateges are smple, wthout any punshment scheme as the frm does not nvest after the frst perod). Second, when frms start wth the same level of capacty, ths equlbrum corresponds to the classc equlbrum of repeated Cournot competton. Fnally, let us defne a fnte game wth the same payoff functons but n whch frms can only take decsons for the frst T perod of the game T > 1). Ths game features a unque Markov perfect equlbrum that exhbts the same nvestment behavor as the equlbrum defned to be the non cooperatve one. 7 In order to state proposton 1, let k BR be the best response capacty of frm,.e. the level of capacty that frm wll nstall f frm j nstalls a level k j, and frm has no ntal capacty: π k k BR, k j ) = p +. 8) Indeed, when there s no nvestment, Π = π. As the profts are symmetrc for both frm, k A BR.) = kb BR.) and the best response capacty wll be wrtten k BR.) n the followng. The Cournot level, k c, s then gven by: k BR k c ) = k c. 9) The Cournot level s the equlbrum of the game when both frms have no ntal capactes. When frms do have ntal capactes, the equlbrum s gven by proposton 1. Proposton 1 The followng strateges defne a Markov perfect equlbrum: {ˆk } Î k) = max k) k, 0, 10) 6 Unlke n standard statonary games, here there exsts an nfnty of statonary states, and the equlbrum depends on the path chosen by the frms, and on ther ntal capactes. 7 Ths s also the unque sub-game perfect equlbrum. 9
where ˆk k) s defned by: k ˆk BR k j ) f k j > k c k) =. 11) k c f k j k c {Î } All nvestments are made at the begnnng of the game, and the equlbrum path s k 0 ), 0, 0,... Ths equlbrum wll be defned as the non-cooperatve one. k t B H k C k Irr k t A 0,0 k C k Irr k t B k t A Fgure 1: non-cooperatve equlbrum Proposton 1 presents the three possble scenaros dependng on the ntal capactes. If ntal capactes are low enough, both frms nvest, and the equlbrum s the Cournot equlbrum. 8 If one of the frms has a hgh ntal capacty and the other frm a low one, the bggest frm wll want to reduce ts capacty but wl be constraned by the rreversblty of 8 Wth a margnal cost of nvestment p +. 10
capacty. The smaller frm then adjusts ts capacty to the ntal capacty of ts opponent, nvestng untl ts best response level. Fnally, when the ntal capacty of both frms s too hgh, no one nvests and the frms keep ther ntal capactes forever. 3 Collusve agreement for status quo Agreement for status quo In ths secton I consder a specfc collusve strategy,.e. the collusve agreement for status quo. Ths conssts n each frm keepng ts ntal capacty as long as the other frm also keeps ts ntal capacty unchanged, and n nvestng as n the non-cooperatve equlbrum f the other frm has nvested. There are several reasons for usng ths partcular grmtrgger strategy. In real cartel agreements, frms usually agree on quanttes proportonal to the capacty owned at the tme of the agreement. Whle the grm-trgger strategy s the most consdered strategy n the lterature, some recent studes focus on other equlbra wth renegotaton. However, n ths case, the rreversblty of nvestment prohbts any renegotaton after the punshment, as the frms cannot reduce ther nvestment or ther producton). More precsely, the grm-trgger strategy for status quo s defned by: 0 f k = k 0 Î coll k) = Î k) elsewhere. 12) The objectve of ths secton s to characterze the set of ntal capactes such that the grm-trgger strateges 12) form a Markovan equlbrum: Ψ = { k 0 R 2 + I coll verfes 6) and 7) }. 13) I assume n the followng that the ntal capactes are suffcently low for both frms to nvest n the non-cooperatve equlbrum as defned n the prevous secton). If both capactes are hgh and no frms nvest at the non-cooperatve equlbrum, then the agreement for status quo s obvously an equlbrum, as t concdes wth the non-cooperatve equlbrum. 11
In that case, there s no real colluson, as frms behave exactly n the same way n both equlbra. If one of the frms has a capacty hgher than the Cournot capacty, gven by 9), and the other frm has a low enough capacty to nvest n the non-cooperatve equlbrum, then there s no possblty for a status quo agreement. Indeed, when the largest frm keeps ts capacty constant, the smallest frm s best response s to nvest up to the capacty level 8). Ths corresponds to the non-cooperatve equlbrum behavor, and for the smallest frm, the best response to the status quo agreement s to devate from the collusve agreement. Assume that frm devates from the agreement for status quo by nvestng n a level of capacty I d. At the tme of the nvestment decson, ts compettor does not know that frm s devatng and therefore keeps ts ntal capacty. When the nvestment s made, the opponent observes the devaton and the non-cooperatve equlbrum s played out. However, at ths pont n tme, frm s capacty s no longer ts ntal capacty, but ts capacty of devaton, kd = k 0 + I d. The non-cooperatve equlbrum played after the devaton may be mpacted by ths evoluton of capacty. The form of the non-cooperatve equlbrum mples two propertes of the optmal devaton. If the capacty of devaton s below the Cournot level kd < k c), then for both frms the non-cooperatve equlbrum s to nvest untl the Cournot level. In that case, by nvestng drectly up to the Cournot level, frm does not change the non-cooperatve equlbrum establshed after carryng out the devaton, and t ncreases ts profts durng the tme when the devaton capacty s nstalled, but not the punshment capacty. The optmal devaton should then be hgher than the Cournot level. Ths also mples that the devatng frm does not nvest after the devaton. Furthermore, there exsts a level of capacty, k BR ) 1 k0), j such that, f the devatng capacty s greater than ths level of capacty, the punshng frm does not nvest after the devaton. Indeed, due to the rreversblty of nvestment, the devatng frm s commtted to a capacty suffcently hgh to prohbt the other frm from nvestng at the non-cooperatve equlbrum. If the devatng capacty s greater than ths level, then reducng the devatng capacty does not change the reacton of the punshng frm, but enables greater profts for the devatng frm. 9 The optmal devaton should then be lower than ths level of devaton. 9 Indeed, when the capacty of frm s k BR ) 1 k j 0 ) and the capacty of frm j s kj 0 the non-cooperatve 12
These features are summarzed n the followng Lemma. Lemma 1 The optmal devaton verfes k d [ k c, k BR ) 1 k j 0) ]. In the followng, I focus on devatons whch verfy the condton gven n Lemma 1, n order to fnd the optmal devaton. In that case, the proft of frm s: Π = 1 δ τ ) π kd, k0) j p )) + kd k0) + δ τ π k d, k BR k d. 14) The frst term of the proft s obtaned when the devatng frm has nstalled ts capacty, but before the punshment s mplemented. The second term s smply the nvestment cost of the devaton. The last term occurs durng the rest of game, when frms reach the noncooperatve equlbrum. It depends on the capacty nstalled durng the devaton. The devaton therefore has two mpacts: t ncreases the short-run proft and t also reduces the capacty nstalled by the compettors n the long-run, durng the punshment phase. 10 Infntely short tme perod Ths sub-secton focuses on a case where the tme perod goes to zero τ 0). As tme s contnuous, the devaton s nstantly detected and both the devaton capacty and the punshment capacty are nstantly nstalled. The devatng frm does not make any proft before the mplementaton of ts capacty or durng the tme between the devaton and the punshment. Indeed, when τ 0, the devatng frm s proft becomes: Π = π k d, k BR k d ) ) p + k d k 0). 15) Therefore, devatng from the collusve equlbrum has no short-run mpact. It only nfluences the long-run dstrbuton of capacty. In ths lmt case, there s a clear parallel between the choce of the optmal devaton and the Stackelberg game. Due to the rreversblty of nvestment, when a frm devates, t equlbrum that no frm nvests. Frm has then no nterest to nvest more than k BR ) 1 k j 0 ). 10 Assumpton A mples that the capacty nstalled n punshment, determned n 8) s decreasng. 13
commts to ts new level of capacty. When the opponent observes the devaton, t reacts to ths new choce of capacty. After ths pont, frms mantan ther capacty forever. As the tme perod goes to zero, frms only make a proft n the long run. The devatng frm s then n the poston of a Stackelberg leader, whereas ts opponent, whch reacts to the devaton, behaves as a Stackelberg follower. Let ks be the Stackelberg capacty, as defned by the frst order condton of 15): π ) k k s, k BR k k BR ) s ) + k π ) k s k j k s, k BR k s ) = p +. 16) Lemma 2 descrbes the optmal devaton. Lemma 2 When τ 0, the optmal devaton s to nstall a capacty: ˆk d = k s f k j 0 k BR k s) k BR ) 1 k j 0) f k j 0 > k BR k s). 17) If the punshng frm starts wth a low ntal capacty, the punshng frm wll nvest after the devaton. Knowng that ts opponent wll nvest after the devaton, the optmal strategy of the devatng frm s to nstall the Stackelberg capacty. In such case, the condton for frm to accept the collusve agreement s to make more profts wth ts ntal capacty than f t nstalls the Stackelberg capacty and ts opponent nvests to the best response level of the Stackelberg capacty: If k j 0 k BR ks), ) π k 0 ) π ks, k BR k s ) p + ks k0). 18) When the ntal capacty of the punshng frm s suffcently hgh, the devatng frm nvests less than the Stackelberg capacty n order to take nto account the fact that the punshng frm s commtted by ts ntal level of capacty. In such case, the punshng frm does not nvest after the devaton. The condton for frm to accept the collusve agreement s thus to make more proft wth ts ntal capacty than f t ncreases ts capacty to the best response level: If k j 0 > k BR ks), π k 0 ) π kbr ) 1 ) k0), j k j 0 p + k BR ) 1 ) k0) j k0. 19) 14
Fgure B presents the optmal devaton and the punshment whch follows n functon of the ntal capactes. k t B H k C Optmal devatons of frm A Punshment of frm B k Irr k t A 0,0 k C k S k Irr k t B k t A Fgure 2: optmal devaton The condtons 18) and 19) allow us to determne whch frm has the more ncentves to devate from the collusve agreement. Lemma 3 When τ 0, the frm wth the most ncentves to devate from the collusve agreement s the frm wth the lowest ntal capacty: If 18) and 19) holds for and k 0 < k j 0, then 18) and 19) holds for frm j. Assume that both frms are small, so that the devaton nvolves nstallng the Stackelberg capacty, even when the larger frm devates k 0 k BR k s) for both = A, B). In such case, 15
the total capacty after the devaton s the sum of the leader and the follower capacty of Stackelberg ks + k BR ks)). The prce after the devaton s then the same whchever the devatng frm. In that case, the dfference between the ntal capacty and the Stackelberg capacty s greater for the smaller frm than for the larger one. The smaller frm then has more ncentve to devate, as t gans more capacty. If both frms are large k0 k BR ks) for one ) so that the devaton nvolves prohbtng the other frm from nvestng, as descrbed by 19), the prce after the devaton s hgher when the small frm devates than when the larger frm devates. Indeed, as the margnal proft of nvestment decreases wth the frm s capacty, the smaller frm has more ncentve to nvest for the same total capacty n the non-cooperatve equlbrum. Therefore, to prohbt the other frm from nvestng after the devaton, the larger frm has to nstall a greater capacty than the smaller frm. Ths prce reducton makes the devaton less proftable for the larger frm than for the smaller one. The same reasonng apples when one frm s small and behaves as n 18) and the other one s large and behaves as n 19). Fnally, Lemma 2 and 3 allow us to determne the ntal capactes for whch there s a possblty of colluson. Proposton 2 When τ 0, the set of ntal capactes such that the agreement for status quo s a Markov perfect equlbrum s: { } Ψ = k R 2 + 18) and 19) holds for = arg nf {A,B} {k 0}. 20) Furthermore, let k m be the frms capacty that maxmzes the jont proft, k m = arg max { π A k m, k m ) + π B k m, k m ) 2p + k m } 21) then, f k M < k Irr k S ), there s a possblty of colluson f and only f the Stackelberg proft s under half of the jont proft: π ks, k BR k s )) p + k s π Ak m, k m ) + π B k m, k m ) 2p + k m 2 = π k m, k m ) p + k m. 22) The parallel between the optmal devaton and the choce of the Stackelberg leader allows us to obtan a clear condton for the possblty of colluson. When half of the monopoly 16
proft s greater than the Stackelberg leader s proft, there s a possblty of colluson. Half of the monopoly proft s the best collusve proft the frms can make and the Stackelberg s leader proft s the devatng proft derved from the monopoly capacty. In the usual theory of colluson, when frms are more patent.e. when ther dscount rate, δ, ncreases), the sze of the proft at the tme of devaton relatve to the future proft of punshment decreases, and the colluson s easer to sustan. When nvestment s rreversble, t s possble to dfferentate the condton 18) and 19) wth respect to the dscount rate to determne ts mpact on colluson. Lemma 4 When τ 0, f the dscount rate ncreases, t becomes harder to sustan colluson: If δ > δ, Ψ δ Ψ δ 23) When frms are more patent, ther wllngness to nvest ncreases, as they are ready to pay more n the short run to gan proft n the long run. In the case wth nfntesmal tme perods, devatng from the agreement only mpacts frms long-run profts. Therefore, when frms are more wllng to nvest, ther ncentves to devate ncrease. Ths result dffers from the usual theory. The reason s that the present case, wth no tme-to-buld, focuses on the long-run proftablty of devaton, whereas the usual theory focuses on the short-run proftablty of devaton. General case Ths sub-secton generalzes the results of the prevous secton when the tme perods are postve τ > 0). In such case the ncentve for devaton s a combnaton of short-run and long-run effects. Indeed, the devatng frm makes a postve proft durng the tme between the nstallaton of the devaton capacty and the nstallaton of the punshng capacty. After the nstallaton of the punshng capacty, the devatng frm makes ts long-run proft as n the prevous sub-secton. 17
In order to state the optmal devaton, let k D kj 0) be the capacty that the devatng frm wshes to nstall f ts compettor nvests after the devaton, as defned by: 1 δ τ ) π [ ] k k D, k0) j + δ τ π ) k k D, k BR k k BR ) D ) + k π ) k D k j k D, k BR k D ) Ths optmal capacty of devaton s below the Stackelberg capacty k D < k S ). 11 = p +. 24) Indeed, the fact that the devatng frm makes a short-run proft reduces ts ncentves to nvest, as the capacty whch maxmzes the short-run proft of devaton s below that whch maxmzes the long-run proft. Lemma 5 descrbes the optmal devaton. Lemma 5 The optmal devaton s to nstall a capacty: kd = ) ) kd kj 0) f k j 0 k BR k D k j 0 k BR ) 1 k j 0) f k j 0 > k BR k D k j 0 ) ). 25) As n the case wth nfntesmal tme perods, there are two possble reactons for the devatng frm. If the ntal capacty of the punshng frm s low, the punshng frm wll nvest after the devaton. In such case, the optmal devaton s the maxmzng one 24), and frm accepts the collusve agreement f t makes more proft wth ts ntal capacty than t does by devatng: ) If k j 0 k j BR k D k j 0), π k 0 ) 1 δ τ ) π k D k j 0 ), k j 0 ) ) )) + δ τ π k D k j 0, kbr k D k j 0) ) p + kd k j 0) k 0. When the ntal capacty of the punshng frm s suffcently hgh, the devatng frm nvests less than the capacty defned by 24) n order to take nto account the fact that the punshng frm s commtted by ts ntal level of capacty and wll not nvest after the devaton. 26) Frm accepts the collusve agreement f t wll make more profts s by mantanng ts ntal capacty than t wll by ncreasng ts capacty to the best response level: If k j 0 > k j BR k D k j 0) ), π k 0 ) π kbr ) 1 k j 0), k j 0) p + k BR ) 1 k j 0) k 0). 27) 11 The proof of ths fact s n appendx. 18
As prevously, the frm wth the lowest ntal capacty s the one that has the most ncentves to devate from the status quo agreement. Lemma 6 The frm wth the most ncentves to devate from the collusve agreement s the one wth the lowest ntal capacty: If 26) and 27) holds for and k 0 < k j 0, then 26) and 27) holds for frm j. Lemma 6 allow us to determne the ntal capactes for whch there s a possblty of colluson. Proposton 3 The set of ntal capactes such that the agreement for status quo s a Markov perfect equlbrum s: { } Ψ = k R 2 + 26) and 27) holds for = arg nf {A,B} {k 0}. 28) The condton ensurng that the exstence of a possble colluson meanng that Ψ s not restrcted to the Cournot equlbrum) has no general clear mathematcal formula. However, when condton 22) s vald, colluson s always possble when τ s small enough. The mpact of the dscount factor on the possblty of colluson s ambguous. Indeed, the two ncentves for devaton coexst n the general case. The short-run one, present n standard colluson models, comes from the addtonal proft made by the devatng frm before ts compettor mplements punshment. The long-run ncentve, prevously descrbed n sub-secton 3.2, comes from the frst mover advantage of the devatng frm due to the rreversblty of nvestment, and s a drect result of the rreversblty of capacty nvestment. When the dscount factor s very low, frms are mpatent, and the short run ncentve domnates: an ncrease of δ augments the possblty of colluson. However, when the dscount rate s hgh, frms are patent, and the long-run ncentve prevals. In such case, an ncrease of δ makes the frms more patent, and wllng more to nvest and devate from the collusve agreement n order to obtan a domnant poston n the market. 19
To llustrate ths, let H be the possblty of colluson, as defned by: Hk 0 ) = π k0, k0) j [ 1 δ τ ) π kd, k0) j p ) ] + kd k0) + δ τ π kd, k BR k D ). When Hk 0 ) 0 a possblty for colluson exsts,.e. the grm-trgger strateges for status quo defned by 12) form a Markov perfect equlbrum when Hk 0 ) 0. Fgure 3 presents the possblty of colluson n functon of the dscount factor, n the case of lnear demand and quadratc producton cost. H δ) 0.2 0.4 0.6 0.8 1.0 δ -0.05-0.10-0.15 Fgure 3: Possblty of colluson Wth parameter values: DQ) = 1 Q, cq ) = 0.1 q ) 2, p + = 0.3, τ = 0.1, k A 0 = k B 0 = 0.15. 4 Concluson Ths work shows that the rreversblty of nvestment creates a long-run ncentve for the devaton. Indeed, when a frm wshes to devate from a collusve agreement, t has to 20
ncrease ts producton capacty. In dong so, t benefts from a short-run ncrease n profts before ts opponent starts to boost ts producton as a punshment, but t also commts to ths capacty for the rest of the game, due to the rreversblty of nvestment. Ths commtment reduces ts opponent s ncentve to nvest n a punshment regme, and permts the devatng frm to gan a long-run advantage n terms of capacty. Ths s consstent wth the lysne example, n whch nvestment made by the entry frm, ADM, durng the frst perod of colluson 1992-1993), lead to the demse of the cartel. Durng the ensung prce war, ADM s ncreased capacty allowed the frm to ncrease ts market share at the expense of ts compettors. ADM retaned ts new market share durng the second collusve perod that began at the end of 1993. Ths work s based two hypotheses, the fact that frms always produce at full capacty, and that ndustry starts out below capacty whch can be due to an entry, or postve demand shock). These opens up to two dfferent drectons for future studes. Frst, frms can have unused capacty that can be observed by ther compettors, and used to enforce cartel agreements. One promsng topc for future research would be to determne how ths unused capacty facltates colluson and how t affects frms nvestment n capacty. Second, frms often face uncertan market condtons. Market developments may have an mpact on capacty colluson, both n relaton to the preservng a cartel, and to ts creaton. An ndustry dealng wth an ncrease n demand may prefer to collude rather than ncrease ts capacty, n partcular f the demand shock s temporary. New research s requred to understand the lnk between cartels n capacty and market evoluton. 5 Appendx Proof of Proposton 1: The pattern of ths proof s the followng: frst t characterzes a frm s best response, dependng on the nter-temporal proft, then t determnes ths nter-temporal proft at the equlbrum, and fnally t gves the equlbrum strateges. Let I j k) be the strategy of frm j. The proft of frm Π s then: Π k) = max { 1 δ τ ) π k + I, k j + I j k) ) + δ τ Π k + I, k j + I j k) ) p + I }. 29) I 21
Let V k) be defned by: V k) = 1 δ τ ) π k) + δ τ Π k). Equaton 29) can then be rewrtten as: Π k) = max I { V k + I, k j + I j k) ) p + I } Assumng that V s dfferentable and concave n the frst varable), the optmal nvestment of frm s thus determned by the mplct equaton: V k k + I, k j + I j k) ) = p +. 30) As V k., k j + I j k)) s decreasng, the best response of frm s defned by 30) f V k k, k j + I j k) ) < p + and 0 elsewhere. Therefore, f both frms follow the best response strategy defned prevously, no frms nvest f V k k, k j ) p + and V j k k, k j ) p + and, n that case, k, k j ) s a steady pont as nvestment s rreversble and there s no decrease n capactes, a steady pont s defned as a pont where no frms nvest at the equlbrum). By 29), the nter-temporal proft functon at a steady state s gven by: Π k, k j) = π k, k j). 31) By 30), frms reach a steady pont after ther frst nvestment s made. Ths allow us to rewrte 29), the proft of frm for a gven I j k): Π = 1 δ τ ) π k + I, k j + I j k) ) + δ τ π k + I, k j + I j k) ) p + I, 32) and gves the equlbrum strateges of the frms: do not nvest f and, f not, nvest at level I defned by the mplct equaton: π k k, k j + I j k) ) p +, 33) π k k + I, k j + I j k) ) = p +. 34) Note that the above reasonng mples that there s no other Markovan equlbrum such that the nter-temporal proft functon wth equlbrum strateges) s dfferentable and concave. However, other Markovan equlbra may exsts wth less regular nter-temporal 22
profts secton 3 shows an example of strateges that are not contnuous n the state varables, and that create jumps n the proft functon). Lemma 1 s shown n the text. The results of secton 3.2, when there s no tme-to-buld, are the lmt case of the results of secton 3.3, when there s tme-to-buld, and therefore are shown after the results of secton 3.3. Proof of footnote number 8: By assumpton A, π s concave, and π k Therefore: Equaton 24) mples that p + 1 δ τ ) π k k D, k 0 ) = δ τ < 0. In partcular: π k k D, k 0 ) < p +. 35) δ τ p + < p + 1 δ τ ) π k k D, k 0 ). 36) [ π k k D, k BR k D )) + k BR k k D ) π ] k k D, k j BR k D )), 37) and equaton 16) mples that [ δ τ p + = δ τ π k k s, k BR k s )) + k BR k k s ) π ] k k s, k j BR k s )). 38) Then, π k k s, k BR k s ))+ k BR k k s ) π k k s, k j BR k s )) < π k k D, k BR k D ))+ k BR k k D ) π k k D, k j BR k D )), 39) and, as π., k k BR.)) + k BR.) π k., k k j BR.)) s decreasng by assumpton H and by 8), k s > k D. Lemma 5 s the result of the maxmzaton of 14) under the constrant k d k BR) 1 k j 0), comng from Lemma 1. Proof of Lemma 6: 23
Let αx) be defned by 1 δ τ ) π kd αx) =, kj 0) p + kd k 0) + δ τ π kd, k BR kd )) f x k BR kd x)) π kbr ) 1 x), x ) p + k BR ) 1 x) f x > k BR kd x)). As the devaton s the optmal one, the dervatve of α s see equaton 24) and 8) ): 1 δ τ ) kd x)p kd x) + x) f x k BR kd ln1/δ) α x)) x) = k ) P j 1 ) k BR x) + x j 1. 41) BR) x) f x > k BR kd ln1/δ) x)) Wth ths notaton, frm agrees to collude f and only f: Let h be a functon defned by: 40) π k 0 ) p + k 0 αk j 0) 0 42) hx) = P T )x cx) ln1/δ) p + x + αx), 43) where T s a constant whch verfes T < k D x)+x f x k BR k D x)) and T < k j BR) 1 x)+ x f x > k BR kd x)). Then, the dervatve of h s: P T ) + 1 δ τ ) kd x)p kd x) + x) c x) p + f x k BR k h D ln1/δ) x)) x) = P T ) + P k BR ) 1 x) + x ) k BR ) 1 x) c. x) p + f x > k BR kd ln1/δ) x)) 44) As P T ) > P k D x)+x) f x k BR k D x)) and P T ) > k BR) 1 x)+x f x > k BR k D x)), the dervatve of h s postve. Then, f k j 0 > k 0, P T )k j 0 ck j 0) ln1/δ) p + k j 0 + αk j 0) P T )k 0 ck 0) ln1/δ) By assumng that T = k 0 + k j 0, 45) can be rewrtten: p + k 0 + αk 0) 45) π j k 0 ) p + k j 0 αk 0) π k 0 ) p + k 0 αk j 0). 46) Therefore, f k j 0 > k 0, and frm agrees for colluson, then frm j also agrees, as: π k 0 ) p + k 0 αk j 0) 0 π j k 0 ) p + k j 0 αk 0). 47) 24
Proposton 3 comes from the combnaton of Lemmas 5 and 6. In secton 3.2, Lemma 2 s a corollary of Lemma 5 and Lemma 3 s a corollary of Lemma 6. Proof of Proposton 2: The frst part of proposton 2 comes from the combnaton of Lemmas 2 and 3. To state the second part of proposton 2, note that the ncentve for colluson of frm IC coll ) can be wrtten: π k 0 ) p + k0 π k s, k BR k s )) p + ks ) f k j 0 k Irr k s ) IC coll = π k 0 ) p + k0 π kbr ) 1 ) k0), j k j 0 p + k 1 BR kj 0) ) f k j 0 > k BR k s ). For a gven ntal state, the collusve agreement for the status quo s a Markovan equlbrum f, and only f, ths ncentve for colluson s postve. To show wether or not there s a possblty for colluson, I determne the ntal state maxmzng the ncentve for colluson and study the sgn of IC coll n ths partcular state. As the smallest frm has a greater ncentve to devate than ts opponent, the ntal state maxmzng IC coll s symmetrc. In the followng, k 0 desgns alternatvely the capacty of one of the frms or the vector of capacty, k 0, k 0 ). Frst, assume that k 0 > k Irr k S ). Then, the ncentve to collude s: IC coll = π k 0, k 0 ) p + k 0 π kbr ) 1 k 0 ), k 0 ) p + k BR ) 1 k 0 ) ). 48) The second term, π kbr ) 1 k 0 ), k 0 ) p + k BR ) 1 k 0 ) s the value of the maxmzaton of π k s, k BR k s)) p + k s n the boundary k BR ) 1 k 0 ). Therefore, ts dfferental s postve as k 1 BR k 0) < k S ). The frst term s maxmzed n k m < k BR ks ), as the proft s symmetrc and: k m = arg max { π A k m, k m ) + π B k m, k m ) 2p + k m } = arg max { π k m, k m ) p + k m }. 49) Therefore, the dfferental of π k 0, k 0 ) p + k 0 s negatve, as for the dfferental of IC coll, and the ntal state maxmzng IC coll should be below k BR k s ). Now, assume that k 0 < k BR k s ). Then, the ncentve to collude s: IC coll = π k 0 ) p + k 0 π k s, k BR k s )) p + k s ), 25
whch s maxmzed n k m. Proof of lemma 4: The pattern of proof s to fx a vector of ntal capacty such that frm has an ncentve to devate, and to show that frm has an ncentve to devate for a hgher δ. The proof s decomposed nto two parts, dependng whether k j 0 k BR k s) or not. Before dong so, note that the best response of capacty, k BR x), s an ncreasng functon functon of the dscount rate, δ for all x R + ). Indeed, the mplct equaton defnng k BR, 8), can be rewrtten: P x + k BR x) ) k BR x) + P x + k BR x) ) c k BR x) ) = ln Ths gves, by dfferentaton: k BR x) δ = 1 δ ) 1 p +. 50) δ p + [P x + k BR x) ) k BR x) + 2P x + k BR x) ) c k BR x) )]. 51) By assumpton A, the second dervatve of the frm s payoff s negatve, and therefore k BR x) δ > 0, meanng that a more patent frm nvest a hgher quantty. Assume that k j 0 k BR k s). Then, frm has an ncentve to devate f and only f: whch s equvalent to P k 0 + k 0 π k 0 ) p + k 0 < π k s, k BR k s )) p + k s, 52) ) k 1 0 ck0) < P k s + k BR k s ) ) k0 cks) ln δ By defnton of k s ths can be rewrtten, P k 0 + k 0 ) k 0 ck 0) < max x ) p + k s k 0). 53) { ) 1 P x + k BR x) ) x cx) ln p + x k δ 0) }. 54) The frst term of the nequalty does not depend on δ. To show that an ncrease of δ augments the ncentve to devate, t s enough to show that the dervatve of the second term s postve. Let V δ) = max x R+ { P x + kbr x) ) x cx) ln 1 δ ) p + x k 0) }. Then, the envelope theorem mples that: V δ) = k BR k S ) P k s + k BR k s ) ) k s + 1 δ δ p+ k s k0), 55) 26
by usng 54), ) V δ) = p+ ks ) k0 P k s + k BR k s ) ) k s, δ P k s + k BR k s ) ) k BR k s ) + 2P k s + k BR k s ) ) c k BR k s ) ) 56) thus V δ) = p+ δ k s As P < 0, ) P k s + k BR k s ) ) k BR k s ) + P k s + k BR k s ) ) c k BR k s ) ) p+ P k s + k BR k s ) ) k BR k s ) + 2P k s + k BR k s ) ) c k BR k s ) ) δ k 0. 57) P k s + k BR k s ) ) k BR k s ) + P k s + k BR k s ) ) c k BR k s ) ) P k s + k BR k s ) ) k BR k s ) + 2P k s + k BR k s ) ) c k BR k s ) ) and V δ) > 0 as k s > k 0 by assumpton k 0 < k c ). > 1, 58) Assume now that k j 0 > k BR k s). To smplfy the notaton, let k 1 BR = k BR) 1 k j 0). Then, frm has an ncentve to devate f and only f: P ) k0 + k0 k 0 ck0) < P k 1 BR + kj 0) ) k 1 BR c ) ) 1 k 1 BR ln p + k 1 BR δ ) k 0. 59) Let V δ) = P k 1 BR + ) kj 0 k 1 BR c kbr) 1 ln 1 ) δ p + k 1 BR ) k 0. The dervatve of the second term s gven by: V δ) = k 1 BR δ P k 1 BR + ) kj 0 k 1 BR + P k 1 BR + ) ) kj 0 c 1 k 1 BR ln δ ) p + ) + 1 δ p+ k 1 BR k 0 ). As k 1 BR < k s, P k 1 BR + ) kj 0 k 1 BR + P k 1 BR + ) kj 0 c kbr) 1 ln 1 ) δ p + s postve. Furthermore, as k BR ncreases wth δ, k 1 BR also ncreases wth δ and thus V δ) > 0. 60) 27
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