Collusive Decisions and Punishment under Demand and Cost Uncertainty

Transcription

1 Collusve Decsons and Punshent under Deand and Cost Uncertanty Chrysovalants Aountzas Hertfordshre usness School Workng Paper (2016) The Workng Paper Seres s ntended for rapd dssenaton of research results, workn-progress, and nnovatve teachng ethods, at the pre-publcaton stage. Coents are welcoed and should be addressed to the ndvdual author(s). It should be noted that papers n ths seres are often provsonal and coents and/or ctatons should take account of ths. Hertfordshre usness School Workng Papers are freely downloadable fro Hertfordshre usness School eploys approxately 200 acadec staff n a state-of-the-art envronent located n Hatfeld usness Park. It offers 17 undergraduate degree prograes and 21 postgraduate prograes; there are about 75 research students workng at doctoral level. The Unversty of Hertfordshre s the UK s leadng busness-facng unversty and an exeplar n the sector. It s one of the regon s largest eployers wth over 2,600 staff and a turnover of alost 235 llon. It ranks n the top 4% of all unverstes n the world accordng to the Tes Hgher Educaton World Rankngs and s also one of the top 100 unverstes n the world under 50 years old. Copyrght and all rghts theren are retaned by the authors. All persons copyng ths nforaton are expected to adhere to the ters and condtons nvoked by each author's copyrght. These works ay not be re-posted wthout the explct persson of the copyrght holders.

2 Collusve Decsons and Punshent under Deand and Cost Uncertanty Chrysovalants Aountzas Hertfordshre usness School, Departent of Accountng, Fnance and Econocs De Havlland Capus, Hatfeld, Hertfordshre, AL10 9A Abstract Ths paper provdes a wder approach to copettve behavour n sectors affected by a slup n deand, based on the contrbuton of the orgnal paper by Green and Porter (1984). In addton to the cooperatve soluton where frs ay enter a reversonary epsode wthout breakng the colluson, the odel of ths study takes nto consderaton two fors of punshent as a result of future uncertanty; a total breakdown n colluson f there s no trust between ts partcpants; or the foraton of a new one by chargng a lower coon prce as a dscplnary act. It s shown that under deand and cost uncertanty frs ay have the ncentves to choose a dfferent short-run soluton copared to the soluton under certanty. Keywords: ertrand Copetton; Colluson; Punshent; Deand uncertanty JEL Classfcaton: D21; D43; L13-1-

3 1. Introducton A sgnfcant atter for both ndustral organzaton and gae theory econosts s the collusve agreeents fored between frs n order to acheve a desrable outcoe. Such collusons are fored whenever arket partcpants consder ths acton necessary n order to axze ther profts. The ajor nterest s focused on the nature and the degree of plct colluson that can be sustaned through strategc nteractons and producton decsons accordng to past and present nforaton about every fr s actons. The reasonng of forng collusons les on the forces of copetton that provoke a sprted debate about the ntentons of frs sgnng such contracts. A nuber of collusons are fored n order to reduce the degree of copetton n sectors that face a great slup n deand. Other contracts are fored as well even under deand peaks n order for frs to extract the axu aount of consuer surplus. Ths ay be acheved by ncreasng ther profts and explotng the axu aount of consupton especally fro goods characterzed by a hgh degree of nelastcty. George Stgler (1964) provded a dynac nterpretaton of olgopoly theory based on Edward Chaberln s theory (1933) of equlbru soluton were frs have to cooperate n order to axze ther jont profts. Nevertheless, the an attrbute of ths paper s the fact that Stgler took nto consderaton the case that colluson cannot be sustaned for too long due to ts nstablty as Nash equlbru. Ths soluton gves brth to ncentves for defecton under certan crcustances. Therefore, n order for such actons to be avoded there ust be an endogenously deterned self-polcng way that wll sustan the sgned contracts between the partcpatng frs. Under product hoogenety and an ndustry structure une to entry, Stgler focused hs nterest on the secret prce cuttng. Ths acton s otvated by specfc fluctuatons n arkets, such as an unexpected ncrease n deand for a gven prce. It provdes suffcent ncentves to partcpants n breakng the contract wth colluson and charge a lower prce to attract a greater porton of consuers and thus, attan ore profts. In order for such behavour to be avoded, Stgler consdered the colluson as a Levathan (followng the noton gven by Thoas Hobbes) whose work s to protect the nteractng -2-

4 ebers fro exogenous shocks and pose penaltes to anyone who would try to devate fro ther contract. Ths whole reasonng supports that strategc prcng decsons are dependent on detaled arket condtons specfcaton, lke the fact that the nuber of sellers s very sall whle the nuber of heterogeneous buyers s qute large. Also, an unsustanable equlbru soluton can be attaned under soe restrctons posed by the colluson. The frs can avod nfnte reversonary epsode, such as Cournot copetton, whch wll result n less dscounted profts n the long run copared to the ones that can be attaned under the act of cooperaton. Accordng to ths reasonng, two forulatons of the cartel proble treat noncooperatve colluson n a rgorous way. Osborne (1976) provded a reacton functon equlbru n whch frs respond to changes n output by other frs n order to antan ther ntal share of ndustry output. Knowng that copettors wll choose the sae optal strategy, each fr wll realze that t does not pay to devate fro the collusve output level, due to short-ter or even long-ter losses. Fredan (1971), on the other hand, outlned a strategy n whch frs respond to suspected cheatng. Such actons nfer fro a fall n the arket prce below the pre-agreed level. If future proft flows are dscounted suffcently slowly, then a fr would reduce the dscounted value of ts returns by falng to collude. Instead of focusng on the characterstcs that defne onopoly power as entoned above, Green and Porter (1984) ndcated a drect expanson of Stgler s orgnal paper by rentroducng the assupton of perfect nforaton. They presented a odel where prce cuttng s a ratonal choce for frs under specfc crcustances wthout defectng fro ther contractng agreeent. They argued that under deand uncertanty, optal ncentve equlbru ay nvolve an epsodc recourse to a short-ter unproftable soluton (.e. prce war), but t cannot be clearly defned f the explanng outcoe s the sae wth the one under certanty. They showed that collusve conduct ght be characterzed by repeated epsodes that ay result n prce and proft falls, a fact whch s trggered by a decrease n the observed prce for ther goods. Ths outcoe leaves no place for the vew of an ndustry n whch frs are actng on abortve attepts to for colluson. Therefore, f any colluson s fored under deand uncertanty, then no fr wll ever defect due to the lack of nforaton that cannot allow a cost-beneft analyss of expected future returns. Nevertheless, when low prces are observed, ths sgnals an ncrease n deand for a certan product, by renderng the -3-

5 partcpaton n reversonary epsodes a ratonal choce for every fr. The fnal observatons fro these results show that prce nstablty wll be ntense between noral and reversonary perods due to the stable pattern of prces when frs have decded to collude. A generalzed verson of ths paper has been presented by Abreu, Pearce and Stacchett (1986) that ntroduced an optal pure strategy syetrc equlbru of a class of gaes that expand the Green-Porter odel. They found that a constraned effcent soluton s descrbed by two "acceptance regons" n the sgnal space and two actons. Ths eans that n the effcent equlbru, players wll choose to produce the output of colluson as long as the value of the sgnal falls n ths set. Otherwse, they wll swtch to ther reversonary strategy and keep playng that strategy as long as the sgnal falls n the other set. A larger set of strateges and less severe punshent wll generally result n a loss of effcency due to frs constraned ablty to dscrnate between cooperaton and defecton. Thus, after one perod of the best equlbru, players wll be nstructed ether to choose the worst or to restart the best equlbru strategy. The objectve of the present paper s to provde a general nterpretaton of how collusons work based on the odel presented by Green and Porter (1984). The an ntenton s to present a pont of vew, under the fact that threat condtons are regarded as credble based on the arket power that every fr possess; the hgher that power, the ore endurable a fr wll be by trggerng copetton as a for of punshent for the ones who have defected fro the contracts of colluson. These restrctons can transfor ths soluton nto a sustanable one by changng the payoffs of every player and provdng the an effcent outcoe copared to the one acqured by the non-cooperatve soluton. The an arguent has two parts; the frst one provdes a descrpton of the colluson structure that s about to be studed n ters of ndustry conduct. The second shows that even f collusve conduct results n reversonary epsodes as a ratonal choce n whch prce and proft levels sharply decrease, frs ay prefer to bestow a for of punshent upon colluson partcpants. Ths acton occurs based on a hgh degree of future uncertanty that ay render the unavalable to undertake such reversonary actons due to cost eleents, even durng the perod of copetton. Ths eans that ther costs ght unexpectedly rse by leadng to a forced prce ncrease as a result of a argnal cost ncrease. As a result, t wll render the products of such frs unattractve, gven that ther copettors contnue to charge a copettve prce level. -4-

6 The paper s organzed as follows: secton 2 presents the an assuptons of the odel; secton 3 provdes the forulaton of the odel; secton 4 presents the soluton of the odel; and secton 5 offers the concludng rearks. 2. Colluson Decsons under Deand and Cost Uncertanty The odel that wll be studed s based on the fact that deand and cost fluctuatons are not drectly observed by other frs whch ay lead to unstable ndustry perforance. The an structure reflects a arket sector n whch deand s deteroratng due to a slup n aggregate consupton. When frs notce that ther profts are rapdly decreasng, they wll choose to undertake an act of colluson n order to both secure ther short-ter profts and axze ther long-ter expected returns as well, over the te horzon. The odel conssts of a super gae defned by frs actons accordng to ther ncentves and the sgnals they receve fro the arket envronent. They choose to copete under ertrand behavour by dentfyng a trgger quantty whch ay otvate frs to enter a reversonary epsode. The te horzon ncludes k=0,1,2, K te perods and t=0,1,2, T te sub-perods. Sub te perods denote whether colluson s n a noral or a reversonary state, whle te perods denote the decsons of colluson. Such decsons ay refer to an outcoe slar to Green and Porter s or an outcoe of punshent based on future uncertanty. Specfcally, Green and Porter argued that t s optal for all frs to enter a reversonary epsode whch s trggered by an observed prce reducton (Cournot behavour) as long as the argnal return to a fr fro ncreasng ts producton n noral perods s offset exactly by the argnal ncrease n rsk of sufferng a loss n returns by trggerng a reversonary epsode (1984: p.93). In addton, snce product hoogenety along wth an accurate realzaton of copettors cost functons hold, then there wll be no need for punshent. The an ntenton of ths chapter s to overcoe the assupton of fully observng copettors cost functons and by addng the eleent of uncertanty and speculaton, to render the opton of punshent credble. The ndustry that ths odel ght approprately descrbe s characterzed by four features. Frst, the ndustry s assued to be stable over te by renderng any expectaton ade to be ratonal based on the avalable nforaton that frs are called to use. Ths assupton s necessary n order for ths odel to result n teporary stablty. -5-

7 Second, the decson varable s the relatve prce set by frs whch leads to ertrand copetton 1. Thrd, there s prvate nforaton about cost decsons whch sets uncertanty as a crucal factor of forng or devatng fro colluson 2. Therefore, an accurate dea can be fored regardng only the producton costs of ther copettors. Fourth, the set of nforaton that frs use to ontor whether the colluson s n a collusve (noral) or reversonary state has to be perfectly correlated wth ther conduct. Ths eans that no drect coplance s allowed because reverson would never occur. 3. The odel As entoned above, a gae of K perods and T sub-perods s assued that ncorporates the decsons and strategc nteractons of the partcpatng frs. The gae starts at k=0, t=0 when t s assued that the partcpants for colluson and charge a prce level whch axzes ther jont profts. Consder an olgopoly of N frs whch produce a dfferentated product n a statonary and te separable envronent, lke the one descrbed by Fredan (1971). It s assued that f =1,2,,N ndcates the nuber of frs, then π : R 2 + R s the return functon of fr and π =π (p,q ), where p s the set of prce decsons and q s the output produced correspondng to quantty deanded for a certan prce level (expressed n logarths). If β s the dscount factor and frs are assued to be rsk neutral, then they are called to axze ther long-run value functon E[ The observed deand functon s gven by β k k=0 t=0 β t π (p, q )]. 1 d Pt b ( ) Mt ( ) Zt Q ( ) t A P I (1) P P t t t t where: A, s a constant that captures any shock n deand P t : R + R +, s the relatve prce charged by frs 1 It s assued that qualty proveent durng ths gae reans the sae due to restrctons n nvestent, but dfferentaton n products exsts, because of the set of actons undertaken over the perods pror to the slup n deand. 2 Despte the fact that the Nash equlbru assupton presupposes that frs have an accurate dea of ther copettors cost functons, prvate knowledge renders very dffcult for varables such as qualty nvestent or labltes to be observed. -6-

8 P t : R + R + s the ndustry s aggregate prce level M t ( ) s the wealth effect or the realzaton of lqudty fro the publc I P t t Zt ( ) s the expected/undertaken nvestent n product qualty P t In ths pont, as denoted by Green and Porter (1984), t s assued that frs choose ther strateges fro an nfnte sequence s =(s 0, s 1, s 2,.) where s 0 s a deternate ntal prce level p 0, and s t+1 : R t+1 + R + deternes 's prce level at te t+1 as a functon of past output produced by s t+1 (q 0,,q t )=p t+1. Also, t s assued that a prce decson taken at te t s dependent on past prcng decsons fored by both j copettors and fr, thus confrng the assupton of ratonal choces, where p t =p t (p 0,p 1,.,p t-1,p j0, p j1,., p jt-1 ) and p jt ndcates the prcng decsons of copettors. A strategy profle (s 1,, s n ) deternes recursvely a stochastc process of output, whch n turn nduces a probablty dstrbuton on the space of nfnte sequences of such varable. Expectatons wth respect to ths dstrbuton wll be denoted by E that a Nash Equlbru s a strategy profle (s 1 *,, s n *) that satsfes s1,..., sn. Ths eans E s1,,s n [ E s1,,s n [ t=0 n β k k=0 β t π (s t (q 0,, q t 1 ), q t )] t=0 β t π k (s t (q 0,, q t 1 ), q t ) E s1,,s n [ β t π (s t (q 0,, q t 1 ), q t )] t=0 E s1,,s n [ β k k=0 β t π k (s t (q 0,, q t 1 ), q t )] t=0 (2) k for all frs and feasble strateges s t, where π ndcates the proft level at te k. On ths bass, frs start ther producton at k=0, t=0 under a coonly accepted prce when the slup n deand perssts. The reasonng behnd ths process s based on the degree of nfluence each fr possesses. The hgher that degree s, the hgher the aount of output produced by that fr wll be. As long as quantty deanded reans under a threshold q k (the value of q at te k) whch s coonly accepted by all partcpants as the trgger quantty that wll result n ertrand copetton, colluson s sustaned and frs keep on chargng a coon prce level. If for soe reason, ths threshold s overcoe due to proveent n deand condtons, lke an expansonary polcy that bolsters aggregate -7-

9 ncoe or deand, then at least one fr wll reduce ts prce to the copettve level, by leavng no other opton to the rest but to follow such acton. In ths odel, the eleent of uncertanty does not provde an outcoe based on utual trust. In fact, three cases eerge after the ncrease of the observed quantty deanded above the trgger threshold; the frst s the one where the trust of colluson s not broken and thus, frs return to chargng the ntal prce level; the second declares a cruble n the relatonshp of the colluson ebers that leads to a new colluson under whch the prce level charged s lower than the ntal one; the last reflects a coplete lost n trust whch leads to an nfnte ertrand copetton for k=1,2,, K where the strongest fr(s) wll survve. Intally, assue that p k ={p k 1,.,p k N } s a profle of onopolstc prcng choces for each fr and p k ={p k 1,.,p k N } s a ertrand prcng profle. For splcty, the case for k=0 wll be consdered. An output level q s chosen along wth a length of te t to be noral f () t=0 or () t-1 was noral and q k q dk dk t-1 or () t-t was noral and q t-t >q k,where q dk t = q dk t (p k t ) ndcates the observed deand functon for te k. For any other case, defne t to be reversonary. Each fr faces a prcng strategy set p t = p, f t s noral under no punshent n effect p 1, f t s noral under punshent n effect p, f t s reversonary It s optal for frs to charge a fxed coon prce p k3 n noral perods and p k n reversonary perods. The analyss starts fro the frst colluson. The jont expected profts that frs have to axze for k=0 are gven by π t = N =1 γ = p q d t t ( p, t z t ) c t (q p t, l t ) N =1 (3) where p t = p t = N ψ N p =1 ψ t =1 (4) 4 q t p = ψ N q =1 ψ t (5) 3 The coon prce charged by colluson at te k=0 s denoted by p. 4 See Roteberg (1982a). -8-

10 q t = q t d (p t, z t ) = N p =1 q t (6) The varable q p t corresponds to the quantty produced by fr ; q d t = q d t (p t, z t ) refers to the observed deand functon of the colluson at k=0, t=0; q t s the observed quantty deanded for prce p ; z t denotes a vector of deternants of colluson s deand curve; l t refers to a vector of cost deternants for each ndvdual fr, and ψ reflects the nfluence that fr possesses n the operatng sector. Therefore, for the last factor holds that N ψ N =1 =1 ψ = 1 (7) where θ = ψ N ψ =1 s the weghted average of ndvdual producton n colluson 5. The expected profts of ndvdual fr partcpatng n colluson are gven by γ t (p ) = p ψ N =1 ψ q t d (p, z t ) c t ( ψ N =1 ψ q t, l t ) (8) However, f producton s ncreased beyond the threshold pont by an ndvdual fr (due to an ncrease n observed deand), then that fr wll start chargng a copettve ertrand prce p by forcng the reanng N-1 frs to follow such strategy as well. Under ertrand copetton, the expected profts for each fr are gven by δ t (p t ) = p t q d t (p t, z t ) c t (q p t, l t ) (9) where q p t corresponds to the quantty produced by fr when chargng p. 4. Defnton of Value Functons Let V ( p ) be the expected dscounted present value of fr f p = p n noral perods. Let also Pr(.) denote probablty wth respect to the dstrbuton of θ whch follows the sae propertes as ψ, dependent on deand shocks. Also, Prb(.) denotes the probablty wth dscrete densty that defnes the volue of output produced n every sub-perod t. If t s also assued that γ (p )>δ (p ), the value functon for each fr satsfes the followng 5 Ths factor can also be vewed as the degree of arket power of fr. -9-

11 equaton V (p = p ) = γ (p ) + βpr (q d q )V (p ) +Pr (q d > q )(Prb(q p = q p )) T 1 Prb(q p = θ q T 1 )[ t=1 β t δ (p ) + β T V ] +Pr (q d > q )(Prb(q p = q p )) T 1 Prb(q p = θ 1 q 1 T 1 )[ t=1 β t δ (p ) + β T V 1 ] +Pr (q d > q )(Prb(q p = q p )) T 1 Prb(q p = q p T )[ t=1 β t δ (p )] (10) The frst ter of the rght hand sde reflects the returns that every fr expect to receve f the agreeent for chargng a fxed prce level p perssts, as long as the quantty deanded threshold s not overcoe. The reanng three ters capture the plcatons of devatng fro the pre-agreed prce level due to an ncrease n observed deand. Specfcally, the second ter reflects the assupton presented by Green and Porter where Cournot (ertrand n ths case) copetton perssts for T-1 sub-perods and n te T colluson reverts back n chargng the ntal onopolstc prce level. The thrd ter provdes the frst for of punshent; after copetng n ertrand ters for T-1 sub-perods, ost of the frs beleve that such behavour wll be repeated. In order to punsh such actons by nzng nterteporal expected occurrng losses, they agree n forng another colluson under whch they charge a prce level p 1 < p. Ths acton ateralzes because even f at least one fr starts chargng p, the partcpants wll not be able to dentfy that fr because all of the wll adopt the sae strategy alost nstantaneously. Ths assupton ay not accurately correspond to realty, but t s of great help to ths analyss for eergng ts dynac eleents. If frs could observe the one who would be devatng every sngle te, then they could adopt varous strateges. They could ether bestow penaltes on ths fr, or f the devatng fr had hgher arket power than the rest, all of the would be forced to charge copettve prces, where n the long-run only the strongest fr would survve. Ths effect s captured by the last ter of ths equaton. It ndcates a coplete breakdown n colluson agreeents and gves the sgnal for an all-out copetton aong partcpants, thus renderng any agreeent about future colluson possble. Another dfference fro the orgnal paper concerns the probablty that deternes the volue of output produced. It wll be set as r k+1 = (Prb(q p = q k )) T 1 the probablty whch shows how long ertrand copetton wll last. In the orgnal paper, t s assued that -10-

12 r 1 (Prb(q p = q pk )) = 1 and thus, the duraton of chargng a copettve prce s deterned only by Pr (q d > q ). In the present case, the duraton of such copetton s deterned by r 1 (Pr(q d > q )) and accordng to fr decsons of how they wll respond n te T, ther strategy s gven by Prb(q p = θ q ) f they choose to return to the ntal colluson; Prb(q p = θ 1 q 1 ) f they choose to for another colluson; and Prb(q p = q p ) f they choose not to cooperate. For ths reason holds that r 1 [Prb(q p = θ q ) + Prb(q p = θ 1 q 1 ) + Prb(q p = q p )] = 1 (11) In ths pont, by takng logarths of (1) at k=0, t=0, t follows that d 1 t ( t t ) ( t ) q a p z p (12) In ths equaton, t s seen that the real prce effect of ths sector s not taken nto consderaton snce every fr charges the sae nonal prce level and acts lke a onopolst whose products do not have any substtutes 6. Ths eans that elastcty b wll be fxed respondng to the agreed prce level and won t pose any changes n deand for the output of colluson. Intutvely, ths outcoe s consstent wth the assuptons of ths odel because (12) ndcates that a change n deand wll take place only f there s a change n μ or ζ 1 that can occur due to fluctuatons n the lqudty capacty of the publc or the qualty of nvestent. Ether way, a change n observed deand does not result fro a change n the elastcty of deand wth respect to nonal prce. Snce t has been assued that producton always corresponds to the level of observed deand, t holds that q t d = q t p = θ q t q t = q t d (p, z t ) = a +μ ( t I t λ p )+ζ 1 (zt p ) θ (13) ased on the assuptons ade for Pr(.), t holds that Pr(q d q ) = Pr ( a +μ ( t I t λ p )+ζ 1 (zt p ) q θ ) = 1 F( q d t ) (14) q The last eleent of ths analyss corresponds to the ncentves of punshent. In order for such acton to be credble, all frs ust abandon a Pareto optal condton and choose a dfferent one, less preferable than the ntal. Ths eans that the expected profts and the 6 N It holds because =1 θ p t = p t (see equaton 4). -11-

13 expected value fro an acton of punshent ust be less than the ntal expected returns fro colluson. For ths reason, snce t has already been assued that γ (p )>δ (p ), t ust also hold that holds that 1 1 ( ) ( ) 0 V p V p. If (10), (11) and (14) are substtuted n ths nequalty t V 1 1 p p p ( p ) d d q q 1 1 ( ) F( ) r1 F( ) Pr( q ) ( ) ( ) ( ) q q d q 1 r1 F( ) Pr( q ) q [ ] ( p ) d d q q (1 ){1 ( ) F( ) r1 F( ) Pr( q )} q q (15) The frst and the second ter of the rght hand sde s the sae as n the odel of Green and Porter. They ndcate the sngle-perod gan n returns to colludng plus the expected dscounted value of fr n ertrand envronent. Ths was the su of the value of fr when there was no punshent. In ths odel, expresson (15) reflects the fact that the value functon ncludes an extra eleent; the expected gan n enterng an nfnte ertrand copetton for ore than one perods or for the rest of the gae. If the rght hand sde s greater than the for of punshent under whch frs create a new colluson wth lower prce (.e. V 1 (p 1 )), then frs wll have the ncentves not to choose ths new for of punshent. Specfcally, (15) provdes the an outcoe of ths paper. The act of punshent ndcates the rsk that frs ay be wllng to take n order to dscplne ther colluson. If ost of the are deterned to sustan such colluson n the long-run to secure and for a strong arsenal aganst future uncertanty, then they ay also be wllng to force such punshent upon the colludng frs. Ths acton can nze any unnecessary losses and keep the frs on operatng by both ensurng ther survval and effort to recover ther losses after the eerged slup n deand. In addton, the act of punshent s a way of explotng the weakest frs by revealng ther cost eleents through ther nablty to keep on operatng under a lower prce level. Ths way, the reanng ebers of the colluson and especally the ones on the argn of operatng under the new prce level, wll be forced to abde by the contracted rules. -12-

14 Therefore, f frs ntend to pose a for of punshent, they wll have to axze the gap between the two fors of colluson at k=0 and k=1 and thus, by substtutng (2) n (15) t s obtaned V ( p ) V ( p ) V ( p ) V ( p ) (16) The frst-order partal dervatve for (16) s θ[v (p ) V 1 (p 1 )] θp =0 for every fr. So, t holds that d d q ' 0= q [(1 ) ( ) F( ) r1 F( ) Pr( q )] ( p ) q q [( ) r Pr( q ) ]( ( p ) ( p )) 1 d d 1 q q ( p ) [( r1 Pr( q ) )(1 ( ) F( ) r1f ( ) Pr( q )] q q 1 q q ( p ) d 1 [( r1f ( ) Pr( q ))(( ) r1 Pr( q ) )] 1 θv 1 (p 1 ) θp (17) d 1 ' q 1 where t zt F ( )[ ( 1) ( 1)] and γ (p ) = θγ q q p p θp Equaton (17) states that the argnal return to a fr fro reducng ts prce n noral perods (γ (p )) ust be equal to the su of () the argnal ncrease n rsk of sufferng a loss n returns (γ (p ) δ (p )), () the dscounted expected profts fro antanng an nfnte ertrand copetton ( δ (p ) 1 β )and () the argnal value fro enterng a new colluson by trggerng a reversonary epsode 7. Wthout () and (), ths equaton reflects the ncentves of chargng a lower prce level when observed deand s 7 If ths equaton holds, then the partcpants wll be ndfferent n choosng between the alternatve fors of punshent. -13-

15 ncreased beyond the trgger quantty and subsequently, return to the ntal colluson. In the present odel, t reflects the ncentves of partcpants to punsh any defecton fro the ntal colluson. The only ter that has further to be defned s the expected argnal value of a new colluson under a change n p. If a slar functon lke V s assued, then the decson varable that would affect V 1, would be the prce set p 1 under whch frs are called to set a new fxed prce p 1 < p. Equaton (17) reflects the set of strateges that frs have n ther dsposal n order to explot the benefts of colluson and axze ther nterteporal gans. The for of punshent n forng new collusons wll not be adopted, only when ndvdual profts fro colluson k are equal or slghtly less than the ones under ertrand copetton (for k=0, the ntal onopoly holds). Ths eans that frs wll stop adoptng the frst for of punshent as long as p k k = p and γ k (p k ) δ k (p k ) p k θ k q k c (q p ) p k q k c (q p ) p k c (q p ) c (q p ) q k θ k q k (18) The rght hand sde of (18) ndcates the dfference between the rsk n average cost that fr wll undertake under ertrand copetton and the beneft n average cost that fr faces f the choce of producng θ k q k s antaned wthout defectng fro colluson. As long as the prce choce falls below that dfference, then t pays no ore to use as a ethod of punshent (or sustan) the foraton of a new colluson by chargng a lower coon prce because p k p = c, thus sgnalng negatve profts. 5. Concludng Rearks Accordng to such results, there are two fnal observatons about the foral odel of colluson under deand and cost uncertanty. Frst, the hgher the operatng cost of ndvdual fr, the lower the ncentves of devatng fro colluson wll be. However, gven the fact that colluson cannot observe the sequence of frs that cause a reversonary epsode, then f a for of punshent s chosen, the weakest frs wll be the frst to face the consequences. On the other hand, f soe of the frs wth a hgh ψ value are expected to devate, then ertrand copetton wll be chosen. Ths happens because the degree of dstrust aong -14-

16 frs overcoes the degree of proft loss due to uncertanty. In equlbru, the frequency of a reversonary epsode to occur s gven by F( q d q ). Second, frs know that a hgher observed deand level does not reflect sultaneous low prcng strateges by copettors. Consequently, t s ratonal for the to partcpate n reversonary epsodes as long as there s belef that no punshent wll occur 8. A reversonary epsode s just a teporary swtch to Nash equlbru n non-contngent strateges. It would not pay any fr to devate unlaterally fro ts Nash strategy n ths teporary stuaton as was presented by the orgnal paper. Ths behavour s expressed by equaton (17) and as long as t s satsfed, frs wll be able to choose a for of punshent as the optal reacton. Ths strategy ay be adopted even f enterng a reversonary epsode was the optal choce, as t would be suggested f the ters of punshent δ and V 1 β were excluded. 1 (p 1 ) p The structure of ths odel has tested a ore general case, as well as provded a general outcoe copared to the one of Green and Porter by tryng to provde a degree of convergence between theory and realty. Soe of the assuptons ay stll have qute a sgnfcant gap fro realty, but the an pont was to forulate a odel of ratonal strategc choces consstent wth Nash equlbru where punshent s taken nto account. In arked contrast, such actons play an essental role n antanng an ongong schee of collusve ncentves. The tradtonal vews would predct the transence of colluson n a arket arked by these epsodes of prce nstablty, and a breakdown of colluson at the begnnng of copetton by elnatng such effect. However, ths odel suggests that ndustres under certan structural characterstcs wll exhbt deand and ndustry fluctuatons as a feature of a stable, te-statonary pattern of output f the operatng frs are colludng 9. 8 Ths outcoe corresponds to the one proposed by Green and Porter (1984). 9 See Appendx 1 for the stochastc process of output whch arses n the equlbru of the odel. -15-

17 Appendx A slar approach adopted by Green and Porter (1984) s accounted under whch the observed output process {Y t } t N s deterned by three processes; {Q t } t N that reflects the output process when t s noral (f the ndustry sets p onopoly prce for k=0), {Q t } t N the output process whch would ensue k f t s reversonary (f the ndustry s under ertrand copetton by chargng p =c ) and {Q t k } t N the output process that occurs when k>0 and t s noral (f the ndustry sets a new prce set p k <p k-1 <p ) when the foraton of new colluson anfests. As n orgnal odel, t s assued that the te perod ends at k=1 whch shows that up to two new collusons can be fored. Whether the observed output level s obtaned by one of the three sets, t s deterned by a process {W t } t N that specfes the condton the ndustry s under (noral, reversonary or noral after punshng a reversonary epsode). Also, {Y t } t N s the only coponent of the jont process {(W t, Q t, Q t, Q t 1, Y t )} t N whch s observed. In ths pont, defne a swtchng process to be deterned by a probablty space (Ω, β, ), a state space S, a subset N S, and fve sequences of rando varables {W}= {W t : Ω S} t N, {Y}= {Y t : Ω R} t N, {Q }= {Q t : Ω R} t N, {Q }= {Q t : Ω R} t N and {Q 1 }= {Q t 1 : Ω R} t N that satsfy the followng condtons (Q ) ( Q ) (Q 1 ) s a set of ndependent rando varables (I) (Q ) s dentcally dstrbuted wth c.d.f. G, (II) (Q ) s dentcally dstrbuted wth c.d.f. H, (III) (Q 1 ) s dentcally dstrbuted wth c.d.f., J, (IV) (W) s a Markov process wth statonary transton probabltes 10 For k=0 and t, S t N Y t = Q t For k=1 and t, S t N Y t = Q t 1 (V) (VI) (VII) 10 A Markov process s descrbed by eorylesness whch s why the current decsons of prcng strateges have eboded the nteractons of prevous strateges. In ths way, past observatons are not needed and thus, the Markov propertes can be used n order to test the stochastc process of output. -16-

18 For k and t, S t N Y t = Q t (VIII) The specal case of a swtchng process usually studed occurs when S={0,1} and N={0}, where {W} s a ernoull process whch s ndependent of (I). In the case of a collusve output process, G, J and H denote the c.d.f. s noral (under no punshent and punshent actons) and reversonary output dstrbuton when S={0,1,,T-1} and N={0}. The Markov process {W} s defned recursvely by startng wth an arbtrary ntal W 0 : Ω S, and then posng If W T =0 and Q T q, then W T+1 =0 If W T =0 and Q T1 q 1, then W T+1 =1/2 If W T =0 and Q T q or Q T1 q 1, then W T+1 =1 If W T =ν, 1 v< T-1, then W T+1 =ν+1 If W T =T-1, then W T+1 =0 or W T+1 =1/2 (IX) (X) (XI) (ΧΙΙ) (XIII) The process {W} defned by (IX)-(XIII) s a Markov process wth statonary transton probabltes because {Q } s..d, based on (I) and (II). The transton graph of {W} s shown n Fgure 11 as a sequental gae, n whch each arrow reflects a transton probablty. The a s to show that W 0 can be chosen n such a way that {Y} wll be a statonary ergodc process. Condtons (VI)-(VIII) show that f Y t s a functon of (Q t, Q 1 t, Q t ) t wll be suffcent to prove that the jont process {Q, Q 1, Q } s ergodc. As argued n the Appendx of the orgnal paper (Green and Porter, 1984), ths process s ergodc f t s a statonary Markov process havng a unque nvarant dstrbuton, such that f W 1 s defned by (IX)- (XIII), then {Y 0, Q 0, Q 1 0, Q 0 } and {Y 1, Q 1, Q 1 1, Q 1 } have dentcal dstrbutons accordng to rean (Theore 7.18, 1968). -17-

19 Fgure 11: Strateges that frs can choose when a teporary shock n deand occurs. k=0, t=0 θ q, θ j q θ q, θ j q k=0, t=1,,t-1 q, q j k=0, t=t θ q, θ j q θ 1 q 1, θ j 1 q 1 q, q j k=1, t=0 θ 1 q, θ j 1 q θ 1 q 1, θ j 1 q 1 q 1, q j 1 k=1, t=t θ 1 q 1, θ j 1 q 1 θ 2 q 2, θ j 2 q 2 q 1, q j 1 Accordng to ths fgure t s seen that the donant strategy under certanty would be the one where fr axzes ts long-run value functon by axzng ts θ k q. Ths occurs when θ k tends to unty by reflectng that onopolstc power has been acqured by the reanng fr(s), thus preventng any threat of copetton. For ths very reason, ertrand copetton wll be the optal choce for fr only f K k=0 β k T t=0 β t δ k (p k ) K k=0 β k T t=0 β t δ k j (p k j ) (1.1) Ths expresson shows that f the expected value of enterng a ertrand copetton s greater than the expected value of any other copettor, then fr wll have the ncentves to enter an nfnte reversonary epsode and cause a breakdown n colluson n order to acqure onopolstc power. On the other hand, when uncertanty s ntroduced as presented n ths odel, then ertrand copetton wll not be the optal soluton as long as two condtons are et: there s no overconfdence about ndvdual cost functons beng uch lower than the reanng frs ; and there s no total collapse n trust aong the partcpatng frs. For ths reason, as -18-

20 n the odel of Green and Porter, the optal soluton would be the sustanablty of collusve actons and f punshent s necessary, then frs wll have the ncentves to for a new agreeent. The resultng colluson wll be sustaned only n the short-run and return to the ntal (optal) agreeent (p=p ) afterwards, f trust s restored aong the reanng partcpants. Ths eans that chargng a coon prce p 1 fro a prce set p 1 wll be a shortrun soluton snce n noral perods holds that γ k (p ) γ (p ) γ 1 (p 1 ) (1.2) Ths shows that the lower the nuber of the reanng frs n the operatng sector, the greater the ncentves to return to the ntal chargng prce p wll be n order for profts to be axzed under the constrant of uncertanty. As a result, based on nequalty (15), k t wll also hold that V k (p ) V (p ) V 1 (p 1 ) K k=0 β k T t=0 β t γ k (p ) K k=0 β k T t=0 β t γ (p ) K k=0 β k T t=0 β t γ 1 (p 1 ) (1.3) Consequently, the long-run equlbru choces under uncertanty can ether result n frs sustanng a collusve act by chargng p k n the short-run and p n the long-run or by chargng p j k when there are no ncentves n forng a new colluson by at least one fr (through fr s lack of trust or expectatons for elnatng ts copettors). -19-

21 References Abreu, D., Pearce, D., & Stacchett, E. (1986). Optal cartel equlbra wth perfect ontorng. Journal of Econoc Theory, 39(1), rean, L., (1968) "Probablty Readng", Addson-Wesley. Chaberln, E. (1933), The Theory of Monopolstc Copetton. Cabrdge: Harvard Unversty Press. Fredan, J. W., (1971) "A Non-cooperatve Equlbru for Supergaes", Revew of Econoc Studes, 28, Green, E. J., & Porter, R. H. (1984). Noncooperatve colluson under perfect prce nforaton. Econoetrca: Journal of the Econoetrc Socety, Osborne, D. K., (1976) "Cartel Probles", Aercan Econoc Revew, 66, Roteberg, J.J., (1982a) Monopolstc prce adjustent and aggregate output. Revew of Econoc Studes 49, pp Stgler, G. J., (1964) "A Theory of Olgopoly," Journal of Poltcal Econoy, 72, pp