Supply han wth or wthout uptream ompetton? Chryovalantou Mllou * Unverdad Carlo III de Madrd, Department of onom, Getafe Madrd 890, Span Feruary 00 Atrat We nvetgate a fnal good produer' nentve to engage n an exluve relaton wth one of two ompetng nput uppler n an envronment where oth market de undertake qualty-enhanng nvetment and argan over ther term of trade. Although the nvetment ompatlty full only under exluvty, we tll fnd that the nvetment under exluvty an e lower than that under non-exluvty. We alo fnd that there ext ae n whh although the nvetment are hgher under exluvty, the fnal good produer hooe non-exluvty. Fnally, we fnd that the fnal good produer hoe of exluvty n equlrum never welfare detrmental. JL lafaton: L; L; L; L5 Keyword: xluve Dealng; Supply Chan; Qualty-enhanng Invetment; Compatlty; Barganng * -mal: mllou@eo.um.e. I am grateful to Mamo Motta, mmanuel Petrak and Karl Shlag for ther valuale omment and duon. I would alo lke to thank Vnenzo Denoló and Margaret Slade for ther helpful uggeton. Full reponlty for all hortomng mne.
. Introduton Why do ome fnal good produer develop exluve relaton wth ther nput uppler whle other tend to hop around among a large numer of uppler? What are the prvate and the oal ot and eneft of an exluve upply han truture relatve to a non-exluve one? At frt glane the two dtnt upply han truture dffer n ther level of uptream ompetton. Aordngly, ome would argue that a upply han truture wth an exluve nput uppler nreae the uptream monopoly power, and thu, t not only antompettve, ut t alo underale from the fnal good produer pont of vew. Contrary to the aove reaonng and to what t would have een expeted n a world n whh tehnology ha onderaly dereaed tranaton and earh ot, there growng evdene that frm do not tend to hop around among a large numer of uppler aed purely on pre. What ntead oerved that large manufaturng frm n the U.S. and elewhere tend to retrt the uptream ompetton y developng exluve partnerhp wth ther nput uppler. One of the mot promnent example of th trend oerved wthn the une-to-une BB e-ommere. Many frm ntead of otanng ther nput from 'pul' BB e-marketplae, n whh they have the alty of tradng wth a large numer of partpatng uppler, they hooe ntead to reate ther own 'prvate' e-marketplae, n whh they trade wth ther exluve uppler. One of the reaon ommonly ued to explan th trend that frm are plang an nreaed empha on produt qualty and that they develop a etter oordnaton of ther qualty-enhanng nvetment y dealng wth a ngle uppler. The etter oordnaton omned wth the fat that a uppler enjoy a hgher hare of the upply han urplu under an exluve relaton rather than under a non-exluve one may n turn nreae the level of the qualty-enhanng nvetment. The ojetve of th paper to nvetgate a fnal good produer' nentve to adopt a upply han truture haraterzed y an exluve uyer-uppler relaton. We onder the followng model. A downtream monopolt - an nput uyer - dede at the egnnng of the game whether or not t wll engage n an exluve relaton wth one of two potental nput uppler. After the form of the uyer-uppler relaton ha een deded, oth the uyer and the uppler undertake nvetment that enhane the qualty of ther produt. Fnally, after the frm have undertaken ther nvetment, ut
efore the uyer ell t fnal produt n the downtream market, arganng over the term of a two-part tarff ontrat take plae etween the uyer and the uppler. We aume that that the ompatlty of the uyer and the uppler nvetment full only under exluvty. Th aumpton apture the fat that under exluvty, the relaton etween the uyer and t exluve uppler are tghter, and thu, the oordnaton of ther nvetment hgher than that under non-exluvty. Although the ompatlty of the nvetment full only under exluvty, we tll fnd that the nvetment under exluvty an e lower than that under non-exluvty. In partular, th hold oth for the uyer and the uppler nvetment when the uyer arganng power uffently low. The ntuton for th reult a follow. Under non-exluvty, the uyer doe not enjoy the full ompatlty of t nvetment ut t doe enjoy a ompenaton for t outde opton. Whle the lak of full ompatlty ha a negatve mpat on the uyer nentve to nvet, t ompenaton for the outde opton ha a potve mpat ne t nvetment nreae the value of t outde opton. Under exluvty, the outde opton aent ut the uyer enjoy the full ompatlty of t nvetment whh n turn nreae t nentve to nvet. When the uyer arganng power low, the effet of the outde opton domnate and the uyer nvetment are hgher under non-exluvty than under exluvty. Th o eaue when the uyer arganng power low, the uyer reeve a hgher hare of t outde opton under non-exluvty, and thu, t nentve to nvet under non-exluvty eome even tronger. Strateg omplementarty etween the uyer and the uppler nvetment lead to a mlar ehavor of the uppler nvetment. Regardng the equlrum upply han truture, we fnd that the uyer opt for exluvty only when t arganng power uffently hgh. It not urprng that th reult due to a g extent to the ehavor of the qualty-enhanng nvetment. In partular, when the uyer opt for exluvty, oth the uyer' and the uppler' nvetment, a well a the total effetve nvetment.e. the produt' total qualty level are hgher under exluvty than under non-exluvty. What though urprng that there ext ae n whh the uyer hooe non-exluvty, although the nvetment are hgher under exluvty. In other word, the qualty-enhanng nvetment are not the only fore at work. The uyer' deon alo affeted y the In an extenon of the a model, nluded n Seton 6, we endogenze th aumpton and provde ondton under whh hold.
fat that there ompetton among the uppler only n the non-exluvty ae. Due to the uppler' ompetton, the uyer alway ompenated for t outde opton. In other word, for the ame level of total effetve nvetment n the two ae, the uyer ha effetvely hgher arganng power durng the ontrat term negotaton n the ae of non-exluvty where t ha the outde opton to deal wth an alternatve uppler, than n the ae of exluvty where there no outde opton. Regardng welfare, we fnd that there ext ae n whh although the uyer hooe non-exluvty, welfare not hgher under non-exluvty. However, we alo fnd that there ext no ae n whh the uyer hoe of exluvty n equlrum welfare detrmental. Hene, from an anttrut poly perpetve, although our reult ndate that the oal and the prvate nentve do not alway onde, they tll provde an argument agant the vew that exluve dealng an antompettve prate, n the ae at leat that the exluvty ntated y downtream fnal good produer. There an extenve lterature that examne the nentve to undertake nonontratle nvetment n lateral monopoly ettng, that, n ettng wth one uyer and one uppler e.g. Wllamon, 985, Trole, 986, Groman and Hart, 986, Hart and Moore, 988. Although lateral monopoly not the only tuaton where trade our, the analy of nentve n ettng n whh uppler do not have the monopoly power n the uptream market ha not attrated adequate attenton. The ame hold for the analy of the hoe among upply han truture haraterzed y ether exluve or non-exluve relaton. Segal and Whnton' 000 paper, to the et of our knowledge, the only formal theoretal attempt that examne the ondton under whh uyer ntated exluve ontrat may e prvately and oally valuale for protetng nonontratle nvetment. Ther man fndng that when only one of the uppler undertake nvetment, the exluvty ha no mpat on t nvetment level when the latter do not affet the urplu generated y the uyer and the other uppler. Our paper dffer from ther on everal ground. Frt, n the paper of Segal and Whnton the ven n ae where there lateral monopoly, that monopoly wll e often reated y a hoe etween alternatve uppler n a pror perod. For an nformal duon of the potental mpat of exluvty on nvetment ee Klen 988 and Klen et al. 978. The ame dfferene apply alo n the omparon of our analy wth that of De Meza and Selvagg 00 whh foue on the revere market truture: an uptream nput monopolt and two potental downtream nput uyer.
exluvty provon telf an e renegotated ex pot, that, after the nvetment have een undertaken. In our paper, we fou ntead n the ae that the exluvty provon an not e renegotated. 5 Seond, whle we onder a arganng game over the term of trade n whh the uyer and the uppler make take-t-or-leave-t offer wth proalte equal to ther repetve arganng power, Segal and Whnton ue a ooperatve oluton onept for the mult-party arganng game. Thrd, we onder a novel dtnton etween the ae of exluvty and non-exluvty, the ompatlty of the uyer' and uppler' nvetment n the qualty enhanement of ther produt. Our work alo related to the vertal retrant lterature on exluve dealng. Mot of th lterature ha foued on uppler ntated exluve dealng ontrat. That, t ha motly analyzed the uppler' deon whether or not to offer exluve dealng ontrat to potental uyer of ther produt, takng nto aount the effet of uh a deon on the uyer and/or the uppler nvetment ee e.g. Marvel, 98, Beanko and Perry, 99, Bernhem and Whnton, 998. In other word, th lterature ha not analyzed the ae of fnal good produer' ntated exluve dealng ontrat. Our fou n the ae n whh the downtream frm a produer of one fnal produt and not a mult-produt frm n harp ontrat wth th trend of the lterature n whh under non-exluvty, the downtream frm are mult-produt retaler, ellng the ompetng fnal produt of all the uptream frm. Undoutedly th lterature ha hed ome lght on the anttrut ue arng n ae wth uppler ntated exluvty ontrat, ut ha not examned the anttrut mplaton of uyer ntated exluvty ontrat. The remander of the paper organzed a follow. In Seton, we dere our a model. In Seton, we analyze the non-exluvty ae. In Seton, we analyze the exluvty ae and du the mpat of exluvty on the nvetment nentve. In Seton 5, we analyze the uyer' deon regardng exluvty and examne the welfare mplaton of our model. In Seton 6, we extend the model y onderng the ae n whh ompatlty an e the outome of the uppler' trateg hoe. Fnally, n Seton 7, we onlude and propoe avenue for further reearh. All the proof are nluded n the Appendx. 5 A mple jutfaton for our approah the ame approah alo adopted n the vertal retrant lterature on exluve dealng that an exluve dealng ontrat an alo nlude a tehnologal ommtment that not only affet the ompatlty of the uyer' and t exluve uppler' nvetment ut t alo make trade wth the alternatve uppler not pole, e.g. the uyer dede to loate t plant far away from alternatve uppler and next to t exluve uppler.
. The Model We onder an ndutry ontng of a downtream frm - nput uyer, denoted y B, and two uptream frm - potental nput uppler, eah denoted y S, wth,. 6 There an one-to-one relaton etween the nput and the fnal produt produed y the uyer. ah of the two nput uppler fae a ontant margnal ot of produton, denoted y. We analyze a full nformaton four-tage game ee Fg.. In the frt tage, the uyer dede whether or not t wll engage n an exluve relaton wth one of the uppler. The exluve relaton an e etalhed through the ue of an exluve dealng ontrat that pefe a prohtve ompenaton that the uyer mut pay to t exluve uppler n ae t otan the nput from the non-exluve uppler. 7 In the eond tage, the uyer B and t potental uppler S and S multaneouly and ndependently hooe ther nvetment level,, and repetvely. ah frm nvetment lead to an nreae n the qualty of t own produt. We aume that the hgher the qualty of the nput ued n the fnal produt, the hgher the latter qualty. Moreover, we aume that onumer have a hgher wllngne to pay for produt of hgher qualty. In partular, the nvere demand funton for the fnal produt : p a ˆ q, a > 0 where q and p are repetvely the quantty and the pre of the fnal produt. The urpt, ndate the uppler from whh the uyer otan the nput. The parameter ˆ apture the degree of ompatlty of the uyer and t nput uppler nvetment. Low value of ˆ reflet low ompatlty of the outome of ther reearh projet e.g. ad mathng due to lak of oordnaton. We aume that ompatlty full only under exluvty. In partular, ˆ under exluvty and ˆ, wth 0 <, under non-exluvty. The nvetment of oth the uyer and the 6 A t wll eome lear n the model oluton, we would otan the ame reult f we had aumed ntead that the numer of upple n, wth n. 7 ote that the exluve dealng ontrat doe not nlude any other term ede the exluvty provon. In partular, t pefe nether the future nvetment level nor the term of future trade. In th ene t an 'nomplete ontrat'. A jutfaton for th aumpton that the ontratual arrangement for an exluve uyer-uppler relaton ha longer run haratert than ther pef term of trade, gven that the latter ould e eaer hanged. For addtonal jutfaton of th type of ontrat ee Groman and Hart 986, and Hart and Moore 988. 5
uppler are ujet to dmnhng return to ale, aptured y the quadrat form of ther ot funton: and,,. In the thrd tage, arganng over a two-part tarff ontrat, ontng of a wholeale pre w and a franhe fee F, take plae among the uyer and t potental nput uppler. Under exluvty the uyer argan only wth t exluve uppler, whle under non-exluvty t argan multaneouly wth oth S and S. In modelng the arganng game, we adopt the approah ued y Chemla 00 and Rey and Trole 00. In partular, n the exluvty ae, a take-t-or-leave-t offer over w and F made wth proalty y the exluve uppler and wth proalty - y the uyer. Smlarly, n the non-exluvty ae, take-t-or-leave-t offer over w and F, are made multaneouly and ndependently y S and S wth proalty and wth proalty - y the uyer. The parameter, 0<<, denote the uppler arganng power. In the lat tage of the game, the uyer hooe the quantty of t fnal good and produe t ung the nput otaned aordng to the term of trade pefed n the prevou tage. We derve the ugame perfet ah equlra n pure tratege of the aove fourtage game. Sne the uptream frm are dental, there are two eond-tage ugame to onder, the ugame wth non-exluvty and the ugame wth exluvty. In what follow, we tart y analyzng the two ugame eparately and then we move to the analy of the frt tage.. on-xluvty In th eton we derve the equlrum for the non-exluvty ae, that, the ae n whh the uyer free to otan t nput from any of the two uppler. We proeed y akward nduton. In the fourth tage, the uyer hooe the output that maxmze t gro proft: π w,,, q a q w q B The urpt, mply pefe the uppler from whh the uyer otan t nput. 8 From the frt order ondton of wth repet to q, we otan the equlrum 8 We aume w.l.o.g. that the uyer alway uy all t nput quantty from one uppler. When the uyer ndfferent etween purhang from any one of the two uppler, we an dtnguh among two ae. Frt, f the uppler offer dfferent nput qualte, the uyer wll alway uy from the hgh qualty uppler th a reaonale te-reakng rule. Seond, f the two uppler offer the ame nput qualty and the ame term of trade, t make no dfferene for our analy f the uyer uy all the nput quantty from one of them or f t plt th quantty etween the two n any artrary way. 6
quantty of the fnal good: a w q w,, In the thrd tage of the game, where the arganng take plae multaneouly among the uyer and t potental uppler, we dtnguh among the followng two ae, for, j, and j: a j 0: When the uppler offer the ame nput qualty, ompetton among them reult not only n oth of them makng the ame ontrat offer, ut alo n makng an offer that leave them wth zero proft. Formally, eah S make an offer that maxmze the uyer proft ujet to the ontrant that t own proft are nonnegatve: Max w, F a w F a w.t. w F 0 The ontrant n ndng, and thu, the uppler' maxmzaton prolem equvalent to the maxmzaton of the uyer and uppler jont proft. A a reult, oth uppler end up offerng wholeale pre whh are equal to the margnal ot of produton, w w j, and franhe fee whh are equal to zero, F F j 0. When the uyer make the ontrat offer, t hooe w and F n order to maxmze t proft ujet to the ontrant that S ' proft are non-negatve. In other word, the uyer prolem equvalent to. A a reult, the uyer offer the ame ontrat term wth the uppler. It follow that the expeted net proft of the two uppler are zero n the ae that they have not undertaken any nvetment, and negatve otherwe. > j 0: When one of the uppler offer a hgher nput qualty than t ompettor, then the two uppler fae two dfferent maxmzaton prolem. The hgh nput qualty uppler maxmze t proft ujet to the ontrant that the uyer wll have no nentve to uy from the low nput qualty uppler,.e. Max. t. a w w F w, a w a j w j F F F j 5 7
At the ame tme, the low nput qualty uppler maxmze the uyer proft ujet to t own proft eng non-negatve. Jut lke n ae a, th tranlate nto optmally ettng w j and F j 0. Due to th and to the fat that the ontrant n 5 ndng, the maxmzaton prolem of the hgh nput qualty uppler redue to: Max w w a w a w a j Th equvalent to the maxmzaton of the uyer and the hgh nput qualty uppler nremental jont proft.e. thoe aove the uyer 'outde opton' and t eay to ee that t lead agan to w. However, t doe not lead to a zero franhe fee, t lead ntead to: F a a j ote that when the uppler wth the hgh nput qualty make the ontrat offer, t annot extrat through the franhe fee all the uyer proft. Intead, t ha to ompenate the uyer for t 'outde opton', that, for the proft that the uyer would make n ae t aepted the ontrat offered y the other uppler. 9 When the uyer make the ontrat offer, t maxmze t proft ujet to the ontrant that S ' proft are non-negatve. In other word, the uyer maxmzaton prolem agan equvalent to. A a reult, B offer to S a ontrat n whh w and F 0. It follow that the expeted net proft of the low nput qualty uppler are zero n the ae that t ha not undertaken any nvetment, and negatve otherwe. Intead, the expeted net proft of the hgh nput qualty uppler are: S a a j 8 From the aove analy of the two ae we an onlude that n the eond tage of the game only one of the two uppler wll nvet n the qualty mprovement of t produt. 6 7 9 Bolton and Whnton 99 how that an alternatng offer arganng game wth three player alo dental to the equlrum of an outde opton arganng game etween the parte wth the larget jont urplu where the party wth the alternatve tradng partner ha an outde opton of tradng wth t le preferred partner and otanng the entre urplu from that trade. It well known that n the oluton to the outde opton arganng game the uyer not only otan the orrepondng to t arganng power hare of the larget urplu ut t alo ompenated for the urplu t ould get from t outde opton ee Runten, 98. 8
Lemma : Under non-exluvty, only one of the uppler undertake qualty enhanng nvetment. Baed on Lemma and on the dervaton nluded n equaton to 8, we haraterze n the followng Lemma the equlrum outome under non-exluvty. Lemma : Under non-exluvty, the level of nvetment hoen y the uyer and the uppler, a well a ther repetve expeted net proft, for, j, and j, are: a ; a ; j 0 9 6 a 8 8 B 0 S a ; S j 0 From the npeton of the equlrum value n 9 t follow mmedately that an nreae n the ompatlty of nvetment ha a potve effet oth on the uyer and the uppler nvetment. The effet though of an nreae n the arganng power on the nvetment not o traghtforward and t nluded n the followng Propoton. Propoton : Under non-exluvty, there ext, nreang n, uh that an nreae n ha a potve mpat oth on and f <. Whle f t ha a potve mpat only on. In aordane wth our expetaton, an nreae n the uppler arganng power lead to an nreae n the uppler nvetment. Contrary to th and to our expetaton, the ame doe not hold for the uyer nvetment. In partular, an nreae n the uyer arganng power lead to a dereae n the uyer nvetment, provded that the uyer arganng power uffently hgh. The ntuton ehnd th urprng reult a follow. Under non-exluvty, the uyer get ompenated for t outde opton. Whle the value of t outde opton nreang n the uyer nvetment, the uyer hare of the outde opton dereang n the uyer arganng power. Gven thee, a dereae n the uyer arganng power ha two oppote effet on the uyer' nvetment nentve. On the one hand, t dereae the uyer' nentve eaue the uyer wll approprate a maller hare of t own proft n the arganng game. On the other hand, t nreae the uyer' nentve eaue y 9
undertakng hgher level of nvetment, t wll nreae t ompenaton for t outde opton. Provded that the arganng power of the uyer uffently hgh, the 'outde opton effet' domnate the frt effet, and thu, a dereae n the uyer arganng power ha a potve mpat on the uyer' nvetment.. xluvty We turn now to the analy of the exluvty ae, aumng wthout any lo of generalty that the uyer award exluvty to uppler S. The lat tage of the game the ame a under non-exluvty wth only one dfferene, the ompatlty of nvetment now aumed to e full. Formally, n the fourth tage the uyer hooe t output n order to maxmze t gro proft: π w,,, q a q w q B From the frt order ondton of wth repet to q, we otan the equlrum quantty of the fnal good: a w q w,, In the thrd tage, the arganng game take plae only among uyer B and t exluve uppler S. Gven that S ' offer the only offer reeved y B, S olve the followng maxmzaton prolem: Max a w w, F w F a w. t. F 0 The ontrant ndng, and thu the maxmzaton prolem of uppler S turn out to e equvalent to the maxmzaton of the uyer and uppler jont proft: Max w w a w a w 5 From the frt order ondton of 5 wth repet to w, t follow that w, and thu that the franhe fee : a F 6 ote that the franhe fee equal to the uyer gro proft. In other word, when the exluve uppler make the ontrat offer, t extrat through the franhe fee all the uyer proft ne the latter ha no outde opton. 0
In the ae that B make the ontrat offer to S, B hooe w and F n order to maxmze t proft ujet to the ontrant that S ' proft are non-negatve: Max a w w, F F a w.t. w F 0 Sne the ontrant ndng, the uyer prolem redue to 5. Hene, B alo offer a wholeale pre whh equal to the margnal ot, w. Settng w n the ontrant n 7, t follow that the franhe fee offered y B equal to zero, F 0. In the eond tage, S and B hooe and repetvely n order to maxmze ther expeted net proft: 0 S a, ; B 7 a, 8 The equlrum value under exluvty derved from equaton to 8 are nluded n the followng Lemma. Lemma : Under exluvty, the level of nvetment hoen y the uyer and t exluve uppler, a well a ther repetve expeted net proft are: a ; a 9 B a ; S a 0 An npeton of the equlrum value under exluvty reveal that ontrary to the non-exluvty ae, oth the uyer and the exluve uppler nvetment and proft nreae n ther own arganng power. Havng n hand the equlrum nvetment level for oth ae, we an now ompare them, and thu, we an du the effet of exluvty on oth the uyer' and the uppler' nvetment. Our man fndng are ummarzed n the followng Propoton. Propoton : There ext, and e, all dereang n and wth lm 0, lm 0 and lm 0 uh that, e > f and only f < 0 ote that t follow mmedately from our analy that the non-exluve uppler wll not undertake any nvetment.
> for all when 0 0. 89 and f and only f < when 0.89 < < > for all when 0 0. 766 and f and only f < when 0.766< <. Aordng to the frt part of Propoton, exluvty ha a negatve mpat on the uyer nvetment nentve only when t arganng power uffently low Fg. demontrate the reult. The ntuton for th reult the followng. Under nonexluvty, the uyer doe not enjoy the full ompatlty of t nvetment ut t doe enjoy a ompenaton for t outde opton. Whle the lak of full ompatlty ha a negatve mpat on the uyer nentve to nvet, t ompenaton for the outde opton ha a potve mpat ne t nvetment nreae the value of t outde opton. Under exluvty, the outde opton aent ut the uyer enjoy the full ompatlty of t nvetment whh n turn nreae t nentve to nvet. When the uyer arganng power uffently hgh, the effet of the ompatlty domnate and the uyer nvetment are hgher under exluvty than under nonexluvty. When the uyer arganng power low, the effet of the outde opton domnate and the uyer nvetment are hgher under non-exluvty than under exluvty. Th o eaue when the uyer arganng power low, the uyer reeve a hgher hare of t outde opton under non-exluvty and thu t nentve to nvet under non-exluvty eome even tronger. Aordng to the eond part of Propoton, exluvty ha a negatve mpat on the uppler' nvetment only when oth the degree of ompatlty and the uppler' arganng power are hgh Fg. demontrate the reult. The ntuton for th lat reult a follow. When take value loe to.e. hgh degree of ompatlty under non-exluvty, the nvetment ompatlty doe not dffer gnfantly aro the two ae. Moreover, a we aw aove, when the uppler arganng power uffently hgh the outde opton effet domnant and thu the uyer' nvetment are hgher under non-exluvty than under exluvty. Strateg omplementarty etween the uyer and uppler nvetment mple that the hgher uyer nvetment under non-exluvty lead to hgher uppler nvetment nentve. It nteretng to ompare alo the total 'effetve' nvetment, that, the fnal produt total qualty level, whh are equal to under exluvty and to e
under non-exluvty. Th omparon wll e ueful n the analy of the uyer deon regardng exluvty. A tated n the thrd part of Propoton, t turn out that the omparon of the total effetve nvetment mlar to that of the uppler' nvetment. 5. xluvty v. on-xluvty In th eton we analyze the uyer deon regardng exluvty and t welfare mplaton. ote that n the ae that the uyer hooe exluvty n the frt-tage, and thu, t dede to offer an exluve dealng ontrat, the ontrat wll alway e aepted y at leat one of the nput uppler. Th o eaue whle an exluve uppler alway enjoy potve n expeted term proft, one of the uppler under non-exluvty alway make zero proft. Propoton : There ext, dereang n and wth lm 0, uh that the uyer prefer exluvty to non-exluvty f and only f < and < 0.707. Aordng to Propoton, a neeary ondton for the uyer to engage n an exluve uyer-uppler relaton that t arganng power uffently hgh, and n partular, t larger than 0.9 Fg. dept the reult nluded n Propoton. When t arganng power low, t alway hooe non-exluvty. The ntuton ehnd th reult lear. When the uyer arganng power low, the uyer approprate a mall hare of t own proft under oth exluvty and non-exluvty. However, reall from Propoton that when the uyer arganng power low, the total effetve nvetment, and thu, the fnal good qualty level lower under exluvty than under non-exluvty. Gven that a hgher produt qualty lead to hgher ale, t follow that when the uyer arganng power low, t own net proft not even takng nto aount t ompenaton for t outde opton are greater under non-exluvty than under exluvty. Interetngly enough the area under whh the uyer hooe non-exluvty larger than the area under whh the total effetve nvetment are hgher under nonexluvty than under exluvty ee Fg. 5. The ntuton that under nonexluvty, the uyer argan wth two uppler, and thu, t alway ompenated for t outde opton. Hene, for the ame level of total effetve nvetment n the two
ae, exluvty and non-exluvty, the 'effetve' arganng power of the uyer n the ae of non-exluvty hgher than that n the ae of exluvty. ext, we turn to a welfare omparon of the two upply han truture. Defnng welfare a the um of produer and onumer urplu, we fnd the followng. Propoton : There ext W, dereang n and wth lm 0 W, uh that welfare alway hgher under exluvty than under non-exluvty when 0 0.78 and f and only f < when 0.78 < <. W Propoton tate that when the ompatlty of nvetment n the ae of nonexluvty dealng uffently low, exluvty alway preferale from a oal pont of vew. The ame hold for hgh degree of ompatlty a long a the arganng power of the uppler uffently low. Th welfare reult, to a great extent, due to the ehavor of the total effetve nvetment. Th eome lear from an npeton of Fg. 6. In Fg. 6, the old lne repreent the rtal for welfare value of the uppler' arganng power, W. In the area to the left of th lne welfare under exluvty exeed that under non-exluvty, whle the oppote hold n the area to the rght of the lne. The dahed lne n Fg. 6 repreent the rtal for the total effetve nvetment value of the uppler' arganng power, e. To the left of the dahed lne the total effetve nvetment are hgher under exluvty than under nonexluvty, whle the oppote hold to the rght of the dahed lne. A t an e ealy een the two lne are qute loe to eah other. Thu, the total effetve nvetment and the oal welfare are hgher under exluvty than under non-exluvty for qute mlar parameter onfguraton. Havng n hand oth the uyer hoe and the welfare omparon we an now anwer the followng queton: doe the uyer hooe the upply han truture that preferale from the oal pont of vew? The anwer to th queton not alway and t nluded n the followng tatement whh a Corollary of Propoton and. Corollary : When 0 0. 78 and > 0. 707 the uyer hooe non-exluvty whle welfare hgher under exluvty than under non-exluvty, the uyer hooe non-exluvty. Corollary mply tate that there ext ae n whh although the uyer hooe non-exluvty, welfare not hgher under non-exluvty. In partular, for all the
parameter value etween the lne and W n Fg. 6, although welfare hgher under exluvty, the uyer hooe ntead non-exluvty. From an anttrut poly perpetve, although our reult ndate that the oal and the prvate nentve do not alway onde, they tll provde an argument agant the vew that exluve dealng an antompettve prate, n the ae at leat that the exluvty ntated y the downtream produer. In fat our welfare analy how whenever the uyer hooe exluvty, welfare alo hgher under exluvty. Th an e een ealy n Fg. 6 where the alway le to the left of the W lne. In other word, there ext no ae n whh the uyer hoe of exluvty n equlrum welfare detrmental. 6. Compatlty of Invetment So far we have aumed that ˆ n the ae of exluvty, whle ˆ, wth 0 <, n the ae of non-exluvty. In th eton, we relax th aumpton y onderng a model n whh full ompatlty an tem a the outome of an nput uppler' trateg hoe. The ompatlty etween the produt of the uppler and the uyer now depend on the uppler' deon to open a pef lne of reearh for the uyer. If a uppler, e.g. uppler S, open a pef lne of reearh for B then the ompatlty etween t nvetment and thoe of the uyer full, ˆ, otherwe ˆ. Gven that ometme the nreae n the ompatlty, that, the openng of a pef lne, mght e otly, we aume that n order for a uppler to aheve full ompatlty wth the uyer, t ha to nur a fxed ot, denoted y A > 0. In partular, we analyze the ame game a n the a model, modfyng t only y deompong the frt tage of the game nto two utage, tage a and tage. Stage a exatly the ame a tage of the a model. In tage, after the hoe among exluvty and non-exluvty ha een made, eah uppler, S and S, multaneouly and ndependently dede whether or not t wll open a pef lne of reearh for B. xamnng the uppler' nentve to open a pef lne of reearh oth under exluvty and non-exluvty, we otan the followng reult. We would have otaned qualtatvely mlar reult under an alternatve model n whh n tage the uyer dede how many pef lne t wll open gven than n the ae that t doe not open any ˆ for oth uppler, whle when t open a pef lne only for S, S j produt ha no value for B. 5
Propoton 5: There ext A > 0 and A > 0, wth A > A when uffently mall, uh that under exluvty the exluve uppler open a pef lne of reearh f and only f open a pef lne of reearh f A < A, and under non-exluvty none of the uppler A > A. Fg. 7 dept the reult nluded n Propoton 5. In partular, n the area elow the urve the rtal fxed ot value elow whh the uppler open a pef lne under exluvty exeed the repetve rtal value aove whh none of the uppler open a pef lne under non-exluvty. The oppote hold n the area aove the urve. It follow from Propoton 5, that there ext a range of value of the fxed ot uh that only under exluvty a uppler open a pef lne for the uyer. Formally: Corollary : If A < A < A, then ˆ under exluvty and ˆ under nonexluvty. Aordng to Corollary there ext a range of value of the ot of openng a pef reearh lne, uh that our a model wth t ompatlty aumpton an e jutfed a a redued form of the more general model analyzed here. It follow that n th range our prevou analy apple. Fnally, t mportant to examne whether the ae that the uyer hooe exluvty n the a model, orrepond to the ae that ompatlty an e full only under exluvty n the extended model. In partular, we know from the a model that the uyer opt for exluvty when t arganng power uffently hgh, that, n the area elow the urve n Fg. 8. In addton, we know from the extended model analyzed n th eton that ompatlty ould, under ome rumtane, turn out to e full only under exluvty n the area elow the A urve n Fg. 8. It follow that exluvty wth full ompatlty ould emerge n equlrum n the ntereton of the area, provded however that the ot of openng a pef lne of reearh take ome ntermedate value, that, provded that A < A <. A 7. Conluon In th paper, we have ondered two dtnt upply han truture, an exluve upply han truture and a non-exluve one. Moreover, we have examned a fnal 6
good produer hoe among thee two upply han truture, n an envronment where oth de of the market, uptream and downtream, undertake qualty-enhanng nvetment and argan over ther term of trade. We have found that although the ompatlty of the uyer and uppler nvetment full only under exluvty, the nvetment under exluvty may not exeed thoe under non-exluvty. We have alo found that the uyer wll opt for exluvty only when t arganng power uffently hgh. Th ugget that the oerved extene of oth exluve and non-exluve upply han truture ould e alo due to dfferene n the fnal good produer arganng poton relatve to ther nput uppler. When the uyer hooe exluvty, oth the uyer' and the uppler' nvetment a well a the total effetve nvetment are alway hgher under exluvty than under non-exluvty. However, the oppote not alway true n the ae that the uyer hooe non-exluvty. Th mean that although the nvetment play a rual role n the uyer' deon whether or not t wll opt for exluvty, they are not the only fore at work. The uyer' deon alo affeted y the fat that the ompetton among the uppler hgher n the ae of non-exluvty relatvely to that n the ae of exluvty. From a welfare perpetve, we have found that there ext no ae n whh the uyer hoe of exluvty n equlrum welfare detrmental. Hene, our reult provde an argument agant the vew that exluve dealng an antompettve prate, n the ae at leat that the exluvty ntated y the downtream fnal good produer. In um, we have provded a mple theoretal foundaton for the frequently oerved uyer ntated exluve relaton n upply han. Our paper jut a frt tep toward th dreton. In future work we plan to extend our analy y onderng unoervale and/or dfferent degree of ompatlty for the two nput uppler. Moreover, we plan to analyze the trateg nentve for exluvty n a ettng wth downtream ompetton. 7
Appendx Proof of Lemma Cae a, the ae wth j 0, annot e an equlrum eaue one of the uppler wll alway have nentve to devate. In partular, when j 0 oth of the uppler have zero proft and one of them ha alway nentve to devate and undertake potve nvetment level eaue y dong o t wll earn potve proft. Smlarly, when j > 0 oth of the uppler make negatve proft and one of them alway ha nentve to devate and undertake zero nvetment level o that t proft are equal to zero. Gven that one of the uppler wll undertake hgher nvetment than the other and thu that t wll offer a hgher qualty nput, we an onlude that the uppler wth the lower qualty nput wll undertake zero nvetment, otherwe t wll make negatve proft. Proof of Lemma We know from Lemma that the equlrum wll take the followng form:,,,,0, wth, j,, j and > 0. W.lo.g. we aume that S j the uppler that undertake the potve nvetment level. In order to fnd the equlrum level of and we proeed n the followng way. We tart y aumng that S devate and hooe >. If >, then n aordane wth ae, n the thrd tage, w and the franhe fee wth proalty wll e equal to: F a a The repetve expeted proft of the devatng uppler wll e: S,, a a A A From the frt order ondton of A w.r.t. t follow that the proft of S n ae of devaton wll e maxmzed y hoong the followng level of nvetment : * a A In order for S not to have nentve to devate, t uffent that greater or equal to the value of gven y equaton A aove. Th o eaue when greater or equal to the aove value then the devaton proft of S are negatve. The lat thng for determnng the equlrum n the eond tage to fnd the level of nvetment that 8
9 S and B hooe n order eah of them to maxmze t proft under the ontrant that *. Formally, S and B olve the followng maxmzaton prolem:,, a a Max S.. a t,, a a Max B From the frt order ondton of the two maxmzaton prolem, we have: ; a a A Solvng the aove ytem of equaton, we otan the nvetment level of B and S gven y equaton 9. It eay to hek that thee are the equlrum nvetment level, ne the value of gven y equaton 9 doe atfy the ontrant *. Fnally, uttutng 9 n the expeted net proft of B and S we otan ther equlrum proft n the non-exluvty ae, gven y equaton 0 and repetvely. Proof of Propoton We dfferentate the equlrum value gven y equaton 9 wth repet to and our reult follow mmedately. Proof of Lemma The frt order ondton of 8 wth repet to and are: ; a a Solvng the aove ytem of equaton, we otan the equlrum level of nvetment gven y 9. Fnally, uttutng thee equlrum value nto proft funton of S and B, we otan ther equlrum expeted net proft nluded n equaton 0. Proof of Propoton Takng the dfferene of equaton 9 and 9, we have: Κ D a A5
0 where D and. The denomnator of the aove expreon, D, alway potve. Regardng the numerator,, ettng t equal to zero and olvng for the rtal value of n term of, we otan: 0 8 5 > W R W R where 5 6 8 5 6 8 R and. 6 66 56 7 96 8 9 0 7 8 6 5 W ext we alulate the dfferene A5 at the extreme value of : 0 0; 0 < > a K lm a K lm It follow from the aove that 0 > K f and only f. < Moreover, dfferentatng K w.r.t. we have: 0 8 6 < D a K Thu, we alo have that 0 / < for all value of. Fnally, n order to how that 0 lm, we alulate the. / lm It an e heked that the latter trtly nreang n and that t equal to zero for 0. Takng the dfferene of equaton 9 and 9, we have: K D a A6 where. The denomnator of the aove expreon, D, alway potve. Regardng the numerator,, dfferentatng t w.r.t. we have: 0 < Thu, take t maxmum value when 0 and t mnmum value when. In partular: 0; 0 > lm lm Settng the latter equal to zero and olvng for, we have:
9 9 0.89 Sne lm > 0 f and only f 0 0. 89, t follow that > 0 when 0 0.89 for all value of. Settng equal to zero and olvng for the rtal value of n term of, we otan the followng: Sne we know from the aove that when 0.89 < <, lm > 0 and lm < 0, t 0 follow that when 0.89 < <, > 0 f and only f <. Moreover, dfferentatng. w.r.t. we have: 6 < 0 It follow from the aove that take t mnmum value when. Sne lm 0, t follow that > 0 when 0.89 < <. Takng the dfferene of the effetve total nvetment: e a K A7 D where. e Dfferentatng K w.r.t. we otan: K 8 Moreover, we have: < 0 lm K > 0; lm K 0 The latter potve f and only f 0 0.766. Thu, when 0 0. 766, K >0. Settng K equal to zero and olvng for the rtal value of n term of, we otan the followng: e
Sne we know from the aove that when 0.766 < <, lm K > 0 and lm K < 0, t 0 follow that when 0.766 < <, then we have K > 0 f and only f <. Moreover, dfferentatng e. w.r.t. we have that for 0.766 < < : e < 0 Fnally, we alulate lm 0. Proof of Propoton e Takng the dfferene of equaton 0 and 0, we have the followng: where a K A8 D B B 6 6 8 8 6 8 6 0 Dfferentatng K w.r.t. we otan: 5 6 6 6 5 6 8 6 5 6 6 K 6 a < 0 D Moreover, we have: 6 8. e lm K < 0; lm K 0 The latter negatve f and only f > 0.707. Thu, when > 0.707, we have K < 0. It eay to how that K / 0 when 0 < < 0.707. In addton, we have: < lm / K 6 6 < 0; lm K > 0 0 It follow that when 0 < < 0. 707, there ext > 0 uh that K > 0 f and only f <. Sne K / 0, we alo have that / < 0. Fnally, to how < that lm 0, we take lm B /. It an e heked that the latter trtly nreang n and that t equal to zero for 0. Proof of Propoton Calulatng welfare n the exluvty ae and n non-exluvty ae, we have: B
W a A9 W a 6 A0 Takng the dfferene of A9 and A0, we have: W D W W A W a a K DW where > 0 W 6. It to hek that K 5 > 0 when 0 0. 78 for all. Moreover, we have: lm K 5 6 9 6 < 0; lm K 5 0 5 > 0 8 In order to defne the rtal value of, W, for 0.78 < <, we et 0. Takng the total dervatve of 0, we otan: d / d W / /. W W Suttutng 0 n the latter, one an hek, after ome manpulaton, that t W alway negatve. It follow that when 0.78 < <, there ext W > 0 uh that K 5 > 0 f and only f > and that W trtly dereang n. Fnally, to W how that lm 0, we take the lm W / W. It an e heked that the latter W trtly nreang n and that t equal to zero for 0. Proof of Propoton 5 In the ae of exluvty when S open n tage a pef lne for B, the ontnuaton of the game exatly the ame a the one nluded n eton. Thu, the proft of S are gven y the dfferene of equaton 0 and the fxed ot A: A S a A A When S doe not open a pef lne for B n tage, we follow exatly the ame proedure a the one nluded n eton wth the only dfferene that we no longer aume that ˆ. Dong o, we otan the proft of S when t doe not open the pef lne: S a A W
Takng the dfferene of equaton A and A, ettng t equal to zero and olvng for A, we fnd: 6 A a A Sne the proft gven y equaton A are alway lower than that gven y equaton 0, t follow that S open a pef lne of reearh for B, when A < A. In the ae of non-exluvty when none of the uppler open a pef lne, the analy exatly the ame a the one nluded n eton. Thu, the proft of S j are zero whle thoe of S are potve and are gven y equaton 0. In order for th to e the equlrum, that, n order none of the uppler to open a pef lne t uffent to how that S j doe not have nentve to devate and open a pef lne. W.lo.g. we aume for the ret of the proof, that n the ae where none of the uppler open a pef lne, S the uppler wth the zero proft and S the uppler wth the potve proft. In ae that S devate and nur A, then the ontnuaton of the game mlar to that n eton. The only dfferene that the degree of ompatlty now aymmetr for the two uppler, that, ˆ for the nvetment of S, and ˆ, wth 0 <, for the nvetment of S. ext we provde the ontnuaton of the game n the ae of devaton. In the fourth tage, the uyer hooe t output n order to maxmze t gro proft: π a ˆ q w B q The equlrum quantty of the fnal good : a ˆ w q w,, where the urpt, ndate the uppler from whh the uyer otan the nput. In ae t otan the nput from S, ˆ, whle n the ae t otan t from S, ˆ. In the thrd tage, we dtnguh among the followng three ae: a : Smlarly to the ae wth ymmetr ˆ we have w, F, 0. > : In th ae w w for oth uppler, however whle F 0, F wth proalty equal to: a a F and wth the ret of the proalty equal to zero.
< : In th ae w w for oth uppler, however whle F 0, F wth proalty - equal to zero and wth proalty equal to: F a a It follow from the aove that Lemma hold here too. ext, we analyze the ae n whh S the uppler that undertake the potve nvetment level. Later on we wll how that ndeed n equlrum S and not S wll e the uppler that undertake the potve nvetment level. In order to fnd the equlrum level of and we proeed n the followng way. We tart y aumng that S devate and hooe uh that <, that > and then we follow the ame proedure a the one n the proof of Lemma. Dong o, we fnd the followng equlrum level of nvetment: a A5 a The repetve expeted net proft of uppler S are: A6 A S a A A A7 where A 6. Settng A7 equal to zero and olvng for A, we fnd: a A A8 It follow that S doe not open a pef lne of reearh for B, when A > A. Fnally, takng the dfferene A A and ettng t equal to zero, we an mpltly defne A. Sne t mpole to get an analytal expreon for A, n order to how that A > A we need to evaluate ntead the followng lmt: lm 0 A A > It follow from the aove that for uffently mall, we have that A > A. 5
Referene Benako, D. and Perry, M. 99, "qulrum Inentve for xluve Dealng n a Dfferentated Produt Olgopoly", RAD Journal of onom,, 66-67. Bernhem, B. and Whnton, M.D. 998, "xluve Dealng", Journal of Poltal onomy, 06, 6-0. Bolton, P. and Whnton, M.D. 99, "Inomplete Contrat, Vertal Integraton, and Supply Aurane'', Revew of onom Stude, 60, -8. Dagupta, S. 990, ''Competton for Prourement Contrat and Undernvetment'', Internatonal onom Revew,, 8-65. De Meza, D. and Selvagg, M. 00, "Pleae Hold Me Up: Why Frm Grant xluve Dealng Contrat", CMPO Workng Paper Sere. 0/066. Groman, S.J. and Hart, O. 986, "The Cot and Beneft of Ownerhp: A Theory of Vertal and Lateral Integraton'', Journal of Poltal onomy, 9, 69-79. Hart, O. and Moore, J. 988, "Inomplete Contrat and Renegotaton'', onometra, 56, 755-85. Klen, B. 988, "Vertal Integraton a Organzatonal Ownerhp: The Fher Body- General Motor Relatonhp Revted", Journal of Law, onom and Organzaton,, 99-. Klen, B., Crawford, R.G. and Alhan, A.A. 978, "Vertal Integraton, Approprale Rent, and the Compettve Contratng Proe", Journal of Law and onom,, 97-6. Marvel, H.P. 98, "xluve Dealng", Journal of Law and onom, 5, -5. Runten, A. 98, "Perfet qulrum n a Barganng Model", onometra, 50, 97-09. Segal, I. and Whnton, M.D. 000, "xluve Contrat and Proteton of Invetment'', Rand Journal of onom,, 60-. Trole, J. 986, "Prourement and Renegotaton'', Journal of Poltal onomy, 9, 5-59. Wllamon, O. 985, The onom Inttuton of Captalm, ew York: Free Pre. 6
B dede xluvty or on-xluvty Invetment of B, S and S Barganng over w, F Fnal Good Produton Fg. : Stage of the game 0.8 < 0.8 0.6 0.6 0. 0. > 0. 0. > 0 0 0. 0. 0.6 0.8 Fg. : Comparon of uyer' nvetment 0 0 0. 0. 0.6 0.8 Fg. : Comparon of uppler' nvetment 0.8 on-xluve Dealng 0.8 e 0.6 0.6 0. 0. 0. xluve Dealng 0. 0 0 0. 0. 0.6 0.8 0 0 0. 0. 0.6 0.8 Fg. : Comparon of uyer' proft Fg. 5: The rtal value and e
0.8 0.6 W e 0. 0. 0 0 0. 0. 0.6 0.8 Fg. 6: The rtal value W, e and 0.6 0.5 0. A < A 0. 0. 0. A > A 0 0 0. 0. 0.6 0.8 Fg. 7: Comparon of the rtal value of A 0.7 0.6 0.5 0. 0. 0. A 0. 0 0 0. 0. 0.6 0.8 Fg. 8: The rtal value A and