Upstream Competition with Complex and Unobservable Contracts

A iyas, Izak; Doganoglu, Toke ; Inceoglu, Fi a A icle — Published Ve sion Ups eam Compe i ion wi h Complex and Unobse able Con ac s Re iew o Indus ial O ganiza ion P o ided in Coope a ion wi h: Sp inge Na u e Sugges ed Ci a ion: A iyas, Izak; Doganoglu, Toke ; Inceoglu, Fi a (2020) : Ups eam Compe i ion wi h Complex and Unobse able Con ac s, Re iew o Indus ial O ganiza ion, ISSN 1573-7160, Sp inge US, New Yo k, NY, Vol. 58, Iss. 3, pp. 399-429, h ps://doi.o g/10.1007/s11151-020-09766-y This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/288645 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/4.0/ Vol.:(0123456789) Re iew o Indus ial O ganiza ion (2021) 58:399–429 h ps://doi.o g/10.1007/s11151-020-09766-y 1 3 Ups eam Compe i ion wi hComplex andUnobse able Con ac s IzakA iyas1· Toke Doganoglu2 · Fi a Inceoglu2 Published online: 1 July 2020 © The Au ho (s) 2020 Abs ac This pape examines si ua ions whe e wo e ically in eg a ed i ms conside sup‑ plying an inpu o an independen downs eam compe i o ia p i a ely obse ed con ac s. We iden i y equilib ia whe e compe i ion in he ups eam ma ke eme ges— he downs eam compe i o ge s supplied—as well as when he down‑ s eam i m does no ecei e he inpu and is excluded om he ma ke . The likeli‑ hood o he ou come in which he downs eam i m does no ge supplied depends no only on demand pa ame e s, bu also on con ac ual lexibili y and obse abili y. We show ha when con ac s a e unobse able, downs eam en y will occu less o en. Fu he mo e, ou esul s sugges ha pe mi ing con ac s ha enable he con‑ ac ing pa ies o coo dina e hei beha io in he downs eam ma ke may imp o e wel a e by inc easing he likelihood ha he downs eam i m is supplied. Keywo ds Collec i e e usal o supply· Fo eclosu e· Unobse able con ac s· Ups eam compe i ion JEL Classi ica ion L13· L40· L42 * Toke Doganoglu oke .doganoglu@uni‑wue zbu g.de Izak A iyas izak@sabanciuni .edu Fi a Inceoglu i a .inceoglu@uni‑wue zbu g.de 1 Facul y o A s andSocial Sciences, Sabanci Uni e si y, O hanlı, Tuzla, 34956Is anbul, Tu key 2 Depa men o Economics, Uni e si y o Wü zbu g, Sande ing 2, 97070Wü zbu g, Ge many 400 I.A iyas e al. 1 3 1 In oduc ion A lack o ups eam compe i ion may a ise in a a ie y o con ex s. Fo example, in se e al o he Eu opean mobile elecommunica ions ma ke s he mobile ne wo k ope a o s (MNOs) e used o supply wholesale ai ime o i al mobile i ual ne ‑ wo k ope a o s (MVNOs). In he alloca ion o spec um, he possibili y ha i ms acqui e and hoa d excess spec um o p e en access by compe i o s aised con‑ ce ns o a collec i e e usal o supply.1 While egula o y and compe i ion policy app oaches ha e implici ly assumed ha his si ua ion is likely o eme ge as a esul o coo dina ion, ecen esea ch has es ablished ha a lack o ups eam compe i ion may a ise as he equilib ium o a s a ic oligopoly game. This li e a u e, which is discussed a leng h in Sec .1, in es iga es he incen i es o e ically in eg a ed i ms (VIFs) o p o ide access o independen downs eam compe i o s and mainly assumes ha con ac s a e obse able o all pa ies.2 Addi‑ ionally, hey ocus on a gi en es ic ed con ac ual o m: e.g. a linea wholesale p ice. Howe e , he deg ee o sophis ica ion o con ac s and hei obse abili y ha e signi ican empi ical and p ac ical ele ance.3 In line wi h hese ac s, we con‑ side con ac s which a e: (i) p i a e; and (ii) a e “complex” o “sophis ica ed” in ha hey allow he con ac ing pa ies o maximize hei join p o i s. Ou model consis s o wo VIFs ha compe e o supply an inpu o an independ‑ en compe i o in he downs eam ma ke , whe e p oduc s a e di e en ia ed. We show ha he unobse abili y o con ac s makes i mo e likely ha he independen compe i o is o eclosed. Fu he mo e, ups eam compe i ion eme ges—o , equi ‑ alen ly in ou con ex , he downs eam i m ge s supplied— o a la ge egion o pa ame e s, when mo e sophis ica ed con ac s a e allowed. We also p o ide p ac i‑ cally ele an and pe missible examples o such con ac s. Ou esul s imply in e es ing complica ions o compe i ion policy. Some exam‑ ples o su icien ly sophis ica ed con ac s en isage a high deg ee o coo dina ion among con ac ing pa ies. Hence hey may aise compe i ion conce ns and may be owned upon by an i‑ us au ho i ies. Howe e , we show ha he e a e si ua ions whe e an i‑ us o ien ed es ic ions on con ac complexi y may ac ually make ups eam compe i ion less likely o eme ge, and he by esul in lowe wel a e. In he nex sec ion, we discuss ela ed li e a u e. The basic model is in oduced in Sec .3. In Sec .4, we desc ibe he se o equilib ia ha ob ain wi h linea unobse ‑ able wholesale con ac s. In Sec .5, we elax his es ic ion and cha ac e ize equi‑ lib ia when join p o i maximizing con ac s a e a ailable and p esen he wel a e implica ions o pe mi ing mo e complex con ac ual o ms. Sec .6 concludes. 1 Addi ional ins ances o collec i e e usal o supply and egula o y and compe i ion policy owa ds hem a e discussed ho oughly in Siciliani (2009). 2 Hombe e al. (2019) in one ex ension o hei model conside linea p i a e con ac s. 3 Hausman and Sidak (2007) men ion ha con ac s in he mobile elecommunica ions indus y “a e o ally lacking in anspa ency, o he simple eason ha he MVNO and he MNO nego ia e hem con i- den ially ”. On he deg ee o sophis ica ion o con ac s, hey s a e “[T]he ag eemen s wi h which we a e amilia con ain complex e ms wi h highly nonlinea p ices”. 401 1 3 Ups eam Compe i ion wi hComplex andUnobse able Con ac s 2 Rela ed Li e a u e The ea lies pape o no e is Dewen e and Haucap (2006). They employ ixed‑ ee con ac s: Nei he VIF can al e he ma ginal cos o a downs eam compe i o and all downs eam compe i o s a e symme ic. Wi h a model o di e en ia ed‑p oduc p ice compe i ion in he downs eam ma ke , hey in es iga e he incen i es o one VIF—among M ac i e VIFs— o in i e a downs eam compe i o and show ha no downs eam compe i o will be in i ed i p oduc s a e close subs i u es o one ano he . Hö le and Schmid (2008) analyze a se up ha is simila o ou s bu do no con‑ side he possibili y ha he ups eam i ms choose no o supply a downs eam i al, and hey allow o only obse able linea wholesale con ac s. They show ha en y by a new downs eam compe i o need no educe downs eam p ices. Howe e , hei esul only holds ue when one o he VIFs is exogenously assigned o sup‑ ply he downs eam i m and/o when downs eam compe i ion is spa ial. When he VIFs compe e in he ups eam ma ke , he equilib ium ha hey de i e in a model wi h linea demands en ails ma ginal cos wholesale p ices.4 Bou eau e al. (2011) also examine he na u e o compe i ion be ween VIFs and non‑in eg a ed downs eam compe i o s. They show ha an equilib ium whe e bo h VIFs e use o supply he inpu —“comple e o eclosu e”—is a possibili y when all o he i ms in he downs eam ma ke a e symme ic. The exis ence o his ype o equilib ium depends on he deg ee o subs i u abili y be ween downs eam p oduc s. In de i ing hei esul s, Bou eau e al. (2011) employ linea wholesale p ice con‑ ac s ha a e obse able o all pa ies. Hombe e al. (2019) s udy e ical me ge wa es ha may esul in insu icien ups eam compe i ion o supplying inpu s o he emaining unin eg a ed down‑ s eam i ms. The mos ele an pa o hei pape is an ex ension o hei model whe e hey conside p i a e linea wholesale p ice con ac s. They ocus on monop‑ oly‑like equilib ia in which one ups eam i m supplies he independen downs eam compe i o while all o he in eg a ed i ms do no make compe ing o e s; his is a si ua ion ha esul s in pa ial o eclosu e. Fo his esul o ob ain howe e , hey equi e ha he in eg a ed i ms a e mo e e icien in he downs eam ma ke . O do e and Sha e (2007) s udy how wo VIFs espond o demand o access o hei ups eam p oduc s by a non‑in eg a ed downs eam i al. They ocus on obse able linea wholesale p ice con ac s. They demons a e by nume ical simula‑ ions ha he e may be equilib ia whe e nei he VIF o e s he inpu o sale. How‑ e e , he unde lying mechanism ha yields his equilib ium in hei model in ol es asymme ic cannibaliza ion o he downs eam p oduc s o he wo VIFs.5 4 They do no conside he equilib ium whe e bo h VIFs e use o supply he inpu , which we do below. 5 On he o he hand, when he downs eam compe i o ’s p oduc p opo ionally cannibalizes bo h VIFs p oduc s, hey ind ha ups eam compe i ion a ises. I is impo an o no e ha hei nume ical analysis, in his case, conside s only p oduc s ha a e su icien ly di e en ia ed. As such, hey do no ecognize he comple e o eclosu e equilib ium ha we ind wi h unobse able con ac s. 402 I.A iyas e al. 1 3 Calcagno and Gia dino‑Ka linge (2019) s udy a model whe e wo VIFs collude on he ups eam and downs eam ma ke s and decide whe he o supply a mo e e i‑ cien independen downs eam compe i o . Fo eclosu e o he downs eam compe i‑ o a ises because he VIFs canno commi no o unde cu he downs eam com‑ pe i o a e he ups eam wholesale con ac is signed. They poin ou ha ce ain con ac ual p o isions may alle ia e his hold‑up p oblem and lead o downs eam en y. This esul is simila o one o he main poin s o ou pape —mo e complex con ac ual a angemen s can lead o downs eam en y—bu he unde lying mecha‑ nism is di e en . Ho s mann e  al. (2017) conduc an expe imen whe e downs eam o eclosu e a ises e en hough he pa ame e ange ha is chosen o he expe imen p edic s compe i ion a he ups eam segmen as a unique equilib ium when con ac s a e obse able. The se up o hei expe imen uses eal‑ ime mo es, which mo e accu‑ a ely desc ibes a scena io in which ups eam ag eemen s a e no (immedia ely) obse able. In his espec , hei indings con i m ou hypo hesis ha unobse abil‑ i y o ups eam con ac s makes o eclosu e mo e likely. Ou pape is also ela ed o he li e a u e on he implica ions o obse abili y o con ac s in e ical ela ionships. The seminal wo k by Ha and Ti ole (1990) examines he incen i es o ups eam inpu supplie s o in eg a e wi h downs eam i ms unde he assump ion o p i a e con ac s. McA ee and Schwa z (1994; see also Rey and Ti ole (2007)) examine an ups eam monopolis ha supplies an inpu o downs eam i ms and show he impo ance o di e en o ‑equilib ium belie s on equilib ium p edic ions. In ou analysis, we adop o ‑equilib ium belie s ha a e simila o he wa y belie s ha a e in oduced by McA ee and Schwa z (1994) and a e s udied la e in mo e de ail in Rey and Ve gé (2004). Pagnozzi and Piccolo (2012) examine he e ec o con ac obse abili y on manu ac u e s’ incen i es o sell h ough inde‑ penden agen s, a he han selling o he public di ec ly. To ou knowledge, ou s is he i s pape o examine he e ec s o unobse able con ac s and hei complexi y on he incen i es o e ically in eg a ed i ms o sup‑ ply o an independen downs eam compe i o .6 3 The Model Conside an indus y wi h wo e ically in eg a ed i ms (VIFs): i=1, 2 . The VIFs p oduce a homogenous inpu ha is needed o he p oduc ion o a downs eam p oduc . The e is a hi d downs eam compe i o — i=3 — ha can compe e wi h he VIFs i i can ob ain he inpu . Fu he mo e, bo h VIFs p oduce he homogeneous inpu a a ma ginal cos o cI . One uni o he inpu is hen used in p oducing one 6 In a di e en se ing, Pagnozzi e al. (2016) conside he e ec o con ac complexi y on he equilib‑ ium ma ke s uc u e. 403 1 3 Ups eam Compe i ion wi hComplex andUnobse able Con ac s uni o he inal good a a ma ginal cos o cO , which is assumed o be he same o all h ee i ms.7 We assume ha he p oduc s in he downs eam ma ke a e di e en ia ed. This implies ha an addi ional a ian in he downs eam ma ke no only inc eases com‑ pe i ion bu also he su plus as consume s alue a ie y. In u n, he VIFs will ha e incen i es o o e con ac s o a po en ial downs eam compe i o wi h he hopes o app op ia ing some o his addi ional su plus. I u ns ou ha he cha ac e iza ion o he esul ing equilib ia is no easily achie ed in a gene al se ing. We will, he e o e, in oduce a commonly used linea demand o mula ion ha is due o Shubik and Le i an (1980). Le a ep esen a i e consume o he downs eam p oduc s ha e he ollowing quasi‑linea u ili y unc ion:8 whe e qj , j=1, 2, 3 , a e he quan i ies o he h ee downs eam p oduc s consumed and y is a composi e good whose p ice is no malized o uni y. I is s aigh o wa d o de i e he co esponding demand unc ions: The smalle is he alue o 𝛽 , he mo e di e en ia ed a e he p oduc s. When 𝛽=0 , he p oduc s a e independen o one ano he and each i m becomes a monopoly in i s downs eam p oduc ma ke ; as 𝛽 → ∞ he p oduc s become pe ec subs i u es. In he e en ha only he wo VIFs a e ac i e in he downs eam ma ke , he demand unc ions ha hey ace a e de i ed by maximizing he u ili y gi en in equa‑ ion (1) wi h he cons ain ha q3=0 : Gi en hese demand unc ions, i is necessa y ha we ha e 1−cI −cO>0 so ha p oduc ion akes place; his is an assump ion ha we make o he es o he pape . We assume ha he VIFs make ake‑i ‑o ‑lea e‑i con ac o e s which con‑ ain an exclusi i y clause.9 (1) U (q1,q2,q3,y;𝛽)= 3 � j=1 qj−1 2 � 3 � j=1 qj � 2 −3 2(1+𝛽) ⎡⎢⎢⎢⎣ 3 � j=1 q2 j− �∑ 3 j=1qj � 2 3 ⎤⎥⎥⎥⎦ + y (2) q j(p1,p2,p3)= 1 3 [ 1−pj−𝛽 ( pj−1 3(p1+p2+p3) )] j= 1, 2, 3. (3) q i(p1,p2)= 1+𝛽 3+2 𝛽[ 1−pi−𝛽 3(pi−pj) ] i≠j∈{1, 2} . 7 The assump ion o cons an ‑ e u ns‑ o‑scale p oduc ion echnology in bo h he ups eam and down‑ s eam segmen s allows us o ocus on he s a egic na u e o he con ac s and abs ac om con ac ual ela ionships ha may a ise due o cos sa ing mo i es. 8 The same u ili y unc ion is employed in Hö le and Schmid (2008), Bou eau e al. (2011), as well as Hombe e al. (2019). 9 Since he ups eam inpu is homogeneous, Fi m 3 would accep he o e s o bo h VIFs only when hese in ol ed nega i e ixed ees o we e con ex in quan i y. These o e s hen would no be made in 404 I.A iyas e al. 1 3 The sequence o ac ions in ou model is as ollows: S age 1: The VIFs o e p i a ely obse ed exclusi e ups eam sales con ac s o i m 3. S age 2: Fi m 3 decides which, i any, o e o accep . The accep ance decision is publicly obse ed—al hough he e ms o he con ac emain p i a e in o ma ion o he con ac ing pa ies. S age 3: All i ms ha ha e/ob ain he inpu compe e in he downs eam p oduc ma ke in p ices. The ac ha ups eam con ac s a e only p i a ely obse ed plays an impo an ole: Wi h an obse able ups eam con ac he con ac ing pa ies (one VIF and i m 3) can in luence he beha io o he ou side VIF—which is akin o he i s ‑ mo e ad an age o a S ackelbe g leade : Obse abili y allows he con ac ing pa ‑ ies o use he con en s o he con ac as a commi men de ice. By con as , wi h a p i a e con ac his de ice is no longe a ailable and he ou ‑ side VIF needs o o m a belie wi h ega d o he ea u es o he con ac ; hus i s beha io is independen o he ac ual con en s o he con ac . As we will also see in he speci ic cases la e on, unobse abili y weakens he commi men powe o an ups eam con ac and makes supplying he inpu less a ac i e o a VIF. Gi en ha we ocus on unobse able con ac o e s, we will use Pe ec Bayes‑ ian Equilib ium (PBE) as ou solu ion concep . The se o PBE is po en ially la ge due o he la i ude one has in speci ying o ‑equilib ium belie s. We adop a e sion o wa y belie s in ou se ing. Wa y belie s we e i s in oduced by McA ee and Schwa z (1994) and de eloped u he by Rey and Ve gé (2004). In ou se up, o ‑equilib ium belie s need o be speci ied only ollowing a de ia‑ ion whe e he iden i y o he con ac ing VIF changes. Following such a de ia ion, he VIF ha inds i sel being he ou side —con a y o he expec ed equilib ium ou come—needs o o m a belie abou he ag eed‑upon con ac by he o he wo i ms in o de o o mula e i s downs eam esponse. We assume ha he ou side VIF belie es ha he o he wo i ms ha e ag eed upon a con ac ha maximizes he payo o he con ac ing VIF condi ional on being accep able o i m 3. Mo eo‑ e , he wo con ac ing i ms belie e ha he ou side VIF o ms i s belie s in his ashion. No e ha his se up and he co esponding in o ma ion s uc u e di e s om hose in McA ee and Schwa z (1994). Howe e , he unde lying idea ha agen s base hei o ‑equilib ium belie s on he expec a ion ha he de ia ing pa ies design a new con ac o he la e ’s maximum bene i emains he same. Because o his, we will s ill call hese “wa y belie s”. We explici ly de i e and demons a e he implica‑ ions o his speci ica ion o o ‑equilib ium belie s in each case ha we s udy below. Foo no e 9 (con inued) equilib ium. Exclusi e o e s enable he VIFs o include nega i e ixed ees in hei con ac s, which a e necessa y o be e align he in e es s o he con ac ing pa ies. See oo no e16 o mo e de ails. 405 1 3 Ups eam Compe i ion wi hComplex andUnobse able Con ac s 4 Ups eam Compe i ion wi hLinea Wholesale P ices In his sec ion, we ollow mos o he li e a u e ha s udies compe i ion be ween VIFs by using linea obse able con ac s o supply a downs eam compe i o . How‑ e e , we ocus on linea p i a e con ac s.10 We s a ou analysis by in es iga ing possible equilib ia in which i m 3 is com‑ ple ely o eclosed. In such an equilib ium, bo h VIFs a e expec ed o make unac‑ cep ably high wholesale p ice o e s o i m 3 and will ea n duopoly p o i s. In all o he cases ha we in es iga e la e on, his ou come ep esen s he ou side op ion o a VIF ha con empla es making a con ac o e o i m 3 when i s i al is expec ed o make an unaccep able o e . The de i a ion o he equilib ium p ices and p o i s o his case—based on he model ha we p esen ed in Sec .3 and deno ed by {p1,p2} and  𝛱1(p1,p2)=  𝛱2(p1,p2)=𝛱D , espec i ely—is s aigh o wa d and is p o ided in he Appendix. Fo o eclosu e o a ise in equilib ium, nei he VIF should ha e an incen i e o de ia e unila e ally and make an accep able o e o i m 3. Suppose, in con as o he equilib ium expec a ions, VIF i has made an accep able o e . In he con inua ion game ha ollows his de ia ion, he h ee i ms compe e downs eam. The ac ha VIF i and i m 3 signed a con ac is obse ed by VIF j, j≠i . Le us s a by compu ing he op imal wholesale p ice om he pe spec i e o VIF i ha esul s in he highes de ia ion p o i s when VIF j is expec ed o espond o his de ia ion by a downs eam p ice gi en by pj . Fo a gi en alue o he linea wholesale p ice w and a belie wi h ega d o pj and he e ail p ice o i m 3— p3 — he p o i unc ion o VIF i is gi en by The bes esponse o VIF i in downs eam compe i ion is gi en by11 A he same ime, i m 3’s bes esponse p ice—gi en w, pj , and i s belie as o he e ail p ice o VIF i— pi is de e mined by One can sol e hese wo bes esponses— pi=pi(pj,p3(pi,pj,w),w) and p3=p3(pi(pj,p3,w),pj,w) — o ob ain he p ices ha VIF i and i m 3 would cha ge as unc ions o w and pj only, which esul in pi(pj,w) and p3(pj,w) . Suppose ha a 𝜋i(pi,pj,p3,w)=(pi−cI−cO)qi(pi,pj,p3)+(w−cI)q3(pi,pj,p3). p i (p j ,p 3 ,w)=a gmax pi 𝜋 i (p i ,p j ,p 3 ,w). p 3 (p i ,p j ,w)=a gmax p3 𝜋 3 (p i ,p j ,p 3 ,w) ≡ a gmax p3 (p 3 −w−c O )q 3 (p i ,p j ,p 3 ). 10 No e ha he ocus on linea wholesale p ices when con ac s a e unobse able is no wi hou p ob‑ lems. I i expec s i s i al o make a linea con ac o e , a VIF can ind a mo e p o i able nonlinea con‑ ac . Thus, linea con ac s canno be pa o an equilib ium—absen some ou side es ic ion/ egula ion. We s udy his case o he sake o compa abili y wi h he li e a u e and o es ablish a benchma k agains which he ou comes wi h mo e complex con ac s can be discussed. 11 The echnical de i a ions in his and he ollowing sec ions can be ound in he Appendix. 406 I.A iyas e al. 1 3 hese p ices, i m 3 ob ains a non‑nega i e p o i : 𝜋3(pi(pj,w),pj,p3(pj,w),w)≥0 , so ha i m 3 inds he o e ed wholesale p ice w accep able. VIF i will choose w o maximize subjec o he cons ain ha i m 3 accep s i , which yields he op imal wholesale p ice: w(pj) . Le us u n o he p icing decision o VIF j. In o de o o mula e i s e ail p ice, VIF j needs o o m a belie wi h ega d o he linea wholesale p ice ha has been nego ia ed by he con ac ing pa ies. Gi en his belie as o he wholesale p ice, w , VIF j is in a posi ion o ind he bes esponse p ices o VIF i and i m 3 as a unc‑ ion o hei belie abou VIF j’s e ail p ice, pj . Acco dingly, VIF j chooses i s p ice pj o maximize No e ha he e VIF j inco po a es he in o ma ion ha a linea con ac is signed be ween VIF i and i m 3 in o mula ing i s belie s wi h ega d o he downs eam e ail p ices o i s i als, pi(pj,w) and p3(pj,w) . Maximizing 𝜋j(pi(pj,w),pj,p3(pj,w)) wi h espec o pj esul s in he bes esponse p ice o VIF j, pj(pj,w) —when i expec s he i al i ms o ag ee upon a wholesale p ice o w and when hey expec VIF j o cha ge pj . Deno e he equilib ium p ices o he con inua ion game ha s a s wi h a de ia ‑ ing o e o wd wi h ( p d i ,p d j ,p d 3) . In his equilib ium, i mus be ha wd =w(p d j) , pd i =pi(p d j ,wd ) , p d j =pj(p d j ,wd ) and pd 3 =p3(p d j ,wd ) . The de ia ion p o i o VIF i is gi en by 𝜋 i(p d i ,p d j ,p d 3 ,w d) . A de ia ion is no p o i able whene e 𝜋 i(p d i ,p d j ,p d 3 ,w d )<𝛱 D , which we show o hold whene e he subs i u abili y be ween he downs eam p oduc s is su icien ly la ge; speci ically, whene e 𝛽>15.09 . As a esul , whene e downs eam p oduc s a e close subs i u es, he e exis s a pe ec Bayesian Nash equilib ium wi h wa y belie s, in which nei he VIF makes an accep able o e o i m 3, which is he eby o eclosed. Nex , we conside equilib ia in which i m 3 ecei es accep able o e s. F om he discussion abo e, i is clea ha when he downs eam p oduc s a e su icien ly di ‑ e en ia ed— o 𝛽<15.09 —a i m will ha e incen i es o ex end an o e o i m 3 when i s i al does no make an accep able o e . Rega dless o he le el o subs i‑ u ion in he downs eam ma ke , he i m whose o e is accep ed ends up wi h a highe p o i han i s i al VIF. Thus, expec ing he o he VIF o make an o e , each VIF has an incen i e o make a compe ing o e o i m 3 o all possible alues o downs eam subs i u abili y. In such a si ua ion, i m 3 aces wo wholesale p ice o e s— w1 and w2 —in he beginning o s age 2. Wi h wa y belie s, i is always op imal o i m 3 o accep he lowe wholesale p ice o e . When VIFs make he same wholesale p ice o e , we b eak he ie in a o o VIF 1 and assume ha i m 3 pa onizes VIF 1. Gi en his decision ule and he ac ha he con ac ing VIF ea ns a leas as much as he ou ‑ side VIF, in equilib ium, bo h VIFs unde cu one ano he un il hey no longe can 𝜋 i(pj,w)=(pi(pj,w)−cI−cO)qi(pi(pj,w),pj,p3(pj,w))+(w−cI)q3(pi(pj,w),pj,p3(pj,w)), 𝜋j(pi(pj,w),pj,p3(pj,w))=(pj−cI−cO)qj(pi(pj,w),pj,p3(pj,w)). 413 1 3 Ups eam Compe i ion wi hComplex andUnobse able Con ac s Howe e , he e is ano he equilib ium whe e bo h VIFs make compe ing o e s. When mul iple equilib ia exis , om he poin o iew o he VIFs, he o eclosu e equilib ium domina es he equilib ium whe e hey compe e wi h each o he o sup‑ ply i m 3. As we discuss below, in con as , a u ili a ian social planne would p e e he equilib ium whe e ups eam wholesale compe i ion o supply he downs eam i m eme ges. On he o he hand, when downs eam p oduc s a e su icien ly poo subs i u es, ups eam wholesale compe i ion should na u ally eme ge.16 The equilib ium ou comes in P oposi ion2 a e simila o hose o P oposi ion1. In o de o compa e he wo cases, we p esen in Table1 he c i ical alues o he subs i u abili y pa ame e 𝛽 , abo e which he VIFs do no supply he downs eam compe i o . As can be seen, when compa ed wi h obse able con ac s, he c i ical subs i u abili y le el is lowe o bo h ypes o unobse able con ac s. On he o he hand, a compa ison o he subs i u abili y pa ame e s along he con ac complexi y dimension shows ha in bo h cases complex con ac s inc ease he possibili y ha ups eam wholesale compe i ion eme ges. Thus, he e may be a p ocompe i i e ole o complex con ac s when con ac s a e unobse able. Whene e 15.09 <𝛽≤32.89 , ups eam wholesale compe i ion a ises as a unique equilib ium i i ms ha e access o su icien ly sophis ica ed con ac s, while he e is an equilib ium in which nei he VIF supplies he inpu o he downs eam i m when con ac s a e es ic ed o linea a i s. Fo mode a e subs i u abili y be ween downs eam p oduc s, allowing sophis i‑ ca ed con ac s—e en hose ha coo dina e he downs eam p ices o he con ac ‑ ing pa ies—may lead o compe i ion in he ups eam ma ke , which esul s in highe wel a e when compa ed wi h he o eclosu e ou come, as we show below.17 Indeed, a simple compa ison o he equilib ium p ices unde he duopoly and he h ee‑ i m oligopoly s uc u es e eals ha — o mode a e subs i u abili y be ween downs eam p oduc s—consume s a e unambiguously be e ‑o when downs eam en y occu s. The nex p oposi ion sums up ou indings. P oposi ion 3 (Wel a e compa ison) Assume ha when he e a e mul iple equilib- ia, he VIFs coo dina e on he o eclosu e equilib ium ha esul s in highe p o i s. When we compa e he ou comes be ween linea a i s and sophis ica ed con ac s, o “mode a e le els o subs i u abili y”— o 15.09 ≤𝛽<32.89 —downs eam 17 In he Appendix we p esen a simila analysis wi h a es ic ion ha he VIFs can only make wo‑ pa a i con ac o e s. In ou se ing hese con ac s a e no su icien ly sophis ica ed since a single wholesale p ice is no su icien o con ol he wo downs eam p ices. Wi h wo‑pa a i con ac s, he o eclosu e equilib ium exis s o 22.13 <𝛽≤32.89 . On he o he hand, ups eam compe i ion is he unique equilib ium ou come wi h su icien ly sophis ica ed con ac s o he same 𝛽 ‑ alues. We conclude ha he mo e complex a e he easible con ac s, he highe is he likelihood ha he VIFs compe e in he ups eam ma ke . 16 The assump ion o exclusi e o e s is essen ial o his esul . When he VIFs compe e ups eam, hey make posi i e paymen s o i m 3 in equilib ium. Absen he exclusi i y equi emen , his equilib ium will collapse, as i m 3 will ha e incen i es o accep bo h o e s. An icipa ing his, he VIFs will no make such o e s in he i s place. The analysis o he s a egic in e ac ion when exclusi e deals a e no allowed is an in e es ing p oblem in i sel , howe e , i is beyond he scope o ou cu en analysis. 414 I.A iyas e al. 1 3 en y occu s only unde he la e . Fu he mo e, downs eam en y leads o lowe p ices o all p oduc s and leads o highe consume and o al wel a e. P oo S aigh o wa d algeb aic calcula ions ha p o e hese s a emen s a e p o‑ ided in he Appendix. ◻ P oposi ion 3 indica es ha o mode a e le els o subs i u abili y o eclosu e is an equilib ium when con ac s a e linea , bu no when con ac s a e complex. Fo his ange o subs i u abili y, s a ing a a si ua ion whe e nei he VIF makes an accep able con ac o e , a p o i able de ia ion exis s when con ac s a e su icien ly sophis ica ed, bu no wi h linea con ac s. We now y o p o ide he in ui ion o his esul . Compa ed o he case wi h linea con ac s, h ee ac o s make a de ia ion om a o eclosu e ou come p o i able unde complex con ac s ( emembe ha hese con ac s allow he con ac ing pa ies o beha e as i hey ha e me ged): Fi s , his elimina es he double ma ginaliza ion p oblem ha a ises wi h linea con ac s. Sec‑ ond, ac ing as a me ged en i y educes he compe i ion be ween he con ac ing VIF and i m 3 in he downs eam ma ke . These wo ac o s inc ease he join p o i s o he de ia ing VIF and i m 3. Thi d, he de ia ing VIF is able o cap u e he down‑ s eam p o i s o i m 3 by means o a ixed ans e . Such oppo uni ies do no exis when con ac s a e linea . Since a VIF ha de ia es can inc ease i s p o i s by making an accep able o e o i m 3, o eclosu e canno be an equilib ium o hese mode a e le els o subs i‑ u abili y when su icien ly sophis ica ed con ac s a e a ailable. The only possible equilib ium is one whe e bo h VIFs make accep able o e s o i m 3, and i m 3 en e s. The esul ing highe le el o compe i ion in he downs eam ma ke inc eases consume wel a e, as compa ed o he case whe e ups eam compe i ion o supply he downs eam i m does no eme ge due o linea con ac s. Do sophis ica ed con ac s always lead o highe consume wel a e compa ed o he case whe e only linea con ac s a e allowed? The answe is no. Speci ically when 0<𝛽<15.09 , mo ing om linea o sophis ica ed con ac s ac ually educes consume wel a e.18 To see why, we ha e o compa e he equilib ia unde linea and Table 1 The alue o he subs i u abili y pa ame e , 𝛽 , abo e which he VIFs e use o supply he downs eam compe i o Con ac Complexi y Su icien ly Linea Sophis ica ed In o ma ion Obse able 26.77 ∞ S uc u e Unobse able 15.09 32.89 18 Fo 𝛽 = 0 , all downs eam i ms ac as independen monopolis s and, in equilib ium, ups eam com‑ pe i ion always a ises. Fu he mo e, wi h linea and sophis ica ed con ac s he equilib ium downs eam p ices a e iden ical. 415 1 3 Ups eam Compe i ion wi hComplex andUnobse able Con ac s sophis ica ed con ac s. In bo h cases, he only possible equilib ium is one whe e ups eam wholesale compe i ion ob ains. Unde linea con ac s, he wholesale p ices a e equal o ma ginal cos , as shown in P oposi ion1.19 Equilib ium down‑ s eam p ices eplica e hose ha would a ise unde compe i ion be ween h ee sym‑ me ic i ms. By con as , unde sophis ica ed con ac s, because he con ac ing VIF and i m 3 ac as me ged en i ies, hey coo dina e hei p ices in he downs eam ma ke , which leads o highe p ices. Due o s a egic complemen a i y he ou side VIF esponds wi h a p ice inc ease as well, which esul s in highe downs eam equilib ium p ices when compa ed wi h equilib ium p ices unde linea con ac s. As a esul , o low le els o subs i u abili y, compa ed o he case o linea con ac s, equilib ium unde sophis ica ed con ac s leads o highe p ices and lowe consume wel a e. To summa ize: Whene e ups eam compe i ion o supply he downs eam i m will occu in any e en , a swi ch om linea o sophis ica ed con ac s educes con‑ sume su plus and wel a e. Howe e , o mode a e le els o subs i u abili y, he e a e wo equilib ia and i is easonable o assume ha he VIFs a o he o eclosu e equilib ium. In his case, he downs eam compe i o canno en e absen sophis‑ ica ed con ac s. We can, he e o e, conclude ha allowing complex con ac ual ag eemen s wo ks o he a o o consume s and social wel a e only when, in hei absence, an independen downs eam i m is unable o ob ain he ups eam inpu and en e he downs eam ma ke . The ange o 𝛽 alues whe e su icien ly sophis ica ed con ac s esul in highe wel a e is a guably ele an . Gi en he s ylized na u e o ou model, i is impos‑ sible o epo di ec e idence in suppo o ou claim. We can, howe e , p o ide some indi ec e idence ha shows ha he ange o 𝛽 ‑ alues ha a e conside ed in P oposi ion3 co espond o a mode a e deg ee o ma ke powe . In a ecen pape De Loecke e  al. (2020) measu e he ma kups in a wide ange o indus ies and epo ha in 2016 he ma kups—de ined as he a io o p ice o ma ginal cos — anged be ween oughly 0.93 and 2.64. Clea ly, his measu e is no independen o ma ginal cos s. In Fig. 1, we p esen he ma kups ha a e gene a ed by ou model o 𝛽∈[15.09, 32.89] — he ele an ange o P oposi ion3—and o se e al possible alues o ma ginal cos s. The le panel p esen s ma kups ha would a ise in a duop‑ oly ( o eclosu e equilib ium); he igh panel p esen s he ma kups o he con ac ‑ ing VIF ha co espond o he equilib ium wi h su icien ly sophis ica ed con ac s and downs eam en y. 19 This also means ha , in con as o he case whe e we conside a de ia ion om he o eclosu e ou ‑ come abo e, since he wo VIFs compe e away he ups eam ma gins, double ma ginaliza ion does no exis in his equilib ium. 416 I.A iyas e al. 1 3 As can be seen in he igu e, ma kups decline wi h 𝛽 and he le el o ma ginal cos s as expec ed. Mo e impo an : Fo bo h ypes o equlib ia and o all alues o he ma ginal cos s ha we conside , ou model gene a es ma kup le els ha a e compa able o hose ound in DeLoecke e al. (2020). The e o e, we p esume ha he ange o he 𝛽 ‑ alues ha a e conside ed in P oposi ion3 o be qui e common in p ac ice. Fu he mo e, gi en ha he ma kups ha we ha e compu ed a e all la ge han 1.2, o he 𝛽 alues ha a e in conside a ion, i ms ha e mode a e ma ke powe . In u n, his indica es ha he 𝛽 alues whe e P oposi ion3 applies can be conside ed o be mode a e. 5.2 Su icien ly Sophis ica ed Con ac s inP ac ice The use o su icien ly sophis ica ed con ac s allows close coo dina ion o beha ‑ io among he con ac ual pa ne s. As we ha e shown abo e, while such con ac s a e o en owned upon by compe i ion au ho i ies, allowing hem may be wel a e‑ enhancing. Ne e heless, simila join p o i ‑maximizing ou comes can also be implemen ed by con ac s ha a e deemed accep able by compe i ion au ho i ies. As an example, we p esen an in o mal discussion showing ha maximum RPM con‑ ac s a e su icien ly sophis ica ed. A o mal p oo is p o ided in he Appendix. ▪ Maximum RPM Con ac s: Maximum RPM con ac s a e no conside ed o be ha m ul pe se and a e pa o a block exemp ion in Eu ope,20 while in he US hey a e e alua ed on a ule‑o ‑ eason basis.21 A maximum RPM con ac in ou con ex speci ies a maximum p ice— pmax 3 — ha i m 3 can cha ge in he downs eam ma ‑ ke , oge he wi h a pe ‑uni wholesale p ice w3 and a ixed ee 3 . 1.2 1.5 2.0 2.3 15.0920.00 25.0030.00 32.89 β ma kup Duopoly Compe i ion 1.2 1.5 2.0 2.3 15.0920.00 25.00 30.00 32.89 β ma kup Downs eam Compe i ion wi h Su icien ly Sophis ica ed Con ac s c I +c O =0. 1c I +c O =0. 15 c I +c O =0. 2c I +c O =0. 25 Fig. 1 Ma kups de ined as P ice Ma ginal Cos . The le panel p esen s duopoly ma kups, he igh panel p e‑ sen s ma kups when h ee i ms compe e downs eam wi h sophis ica ed con ac s 20 Eu opean Commission Regula ion No 330/2010. 21 S a e Oil Co. . Kahn, 522 U.S. 3, 15 (1997) 417 1 3 Ups eam Compe i ion wi hComplex andUnobse able Con ac s Wi h a maximum RPM con ac , he con ac ing VIF—say VIF 2—has an addi‑ ional ins umen : By se ing he p ice ceiling o he downs eam p ice o i m 3 equal o pe 3 and choosing he wholesale p ice o cap u e all o i m 3’s downs eam ma gin— pe 3 −cO —VIF 2 can ully in e nalize he e ec s o i s own downs eam p ice choice on i s ups eam p o i s. As a esul , i op imally selec s he join p o i ‑ maximizing p ice o i s downs eam p oduc — pe 2 —as well. Gi en his wholesale p ice and he maximum p ice es ic ion, i m 3 would ac u‑ ally p e e o cha ge a highe downs eam p ice han pe 3 , bu is unable o do so. The maximum p ice limi induces i m 3 o cha ge he downs eam p ice ha he in e‑ g a ed i m would p e e . In addi ion, he choice o he wholesale p ice allows VIF 2 o commi o a downs eam p ice as i VIF 2 and i m 3 a e in eg a ed. The wo ins umen s—maximum p ice and he wholesale p ice— he e o e play a c ucial ole in implemen ing he p ices ha would be chosen by an in eg a ed i m. Absen one o hese ins umen s—as is he case wi h a wo‑pa a i con ac — his ou come canno be implemen ed. The implica ion o he p eceding a gumen is ha —by using a maximum RPM con ac —al hough VIF 2 and i m 3 make hei downs eam decisions indepen‑ den ly, hey will se downs eam p ices ha a e equal o hose ha would be se by an in eg a ed i m. VIF 1—co ec ly an icipa ing ha VIF 2 and i m 3 will en e a join p o i ‑maximizing con ac ual ela ionship— esponds o hese p ices. The e‑ o e, he equilib ium ou come when i m 3 en e s in o a con ac ual a angemen ha is based on RPM wi h one o he VIFs is equi alen o he ou come whe e he con ac ing pa ies me ge. Hence, maximum RPM con ac s a e su icien ly sophis i‑ ca ed and ou esul s om P oposi ions2 and3 apply. One policy implica ion o his esul is ha when p oduc s a e su icien ly close subs i u es—in ou model, when 𝛽<15.09 —maximum RPM con ac s will educe consume wel a e. Gi en ha compe i ion au ho i ies usually ega d maximum RPM con ac s a o ably, i seems impo an o make a dis inc ion o when he ups eam i m is also ac i e in he downs eam ma ke .22 6 Conclusion We s udy compe i ion be ween wo VIFs in an ups eam ma ke and ex end he exis ‑ ing li e a u e in wo di ec ions: Fi s , we ocus on unobse able con ac s. Second, we allow o mo e sophis ica ed con ac s, which, we a gue, would a ise na u ally when wo pa ies nego ia e in sec e . Rega dless o con ac complexi y, we ind ha he e always exis s an equilib ium in which ups eam compe i ion eme ges. When downs eam p oduc s a e close subs i u es, he e is ano he equilib ium whe e he VIFs e use o sell he inpu . This la e equilib ium yields highe p o i s o bo h VIFs, so i is na u al o expec ha he VIFs coo dina e on his ou come. 22 We hank he edi o o his obse a ion. 418 I.A iyas e al. 1 3 This cha ac e iza ion o equilib ium ou comes esembles quali a i ely hose ound in he li e a u e ha ocus on obse able and linea wholesale con ac s. How‑ e e , wi h unobse able con ac s, he o eclosu e ou come ob ains o a wide se o pa ame e s. In a wo ld whe e con ac s a e nego ia ed in p i a e, an ou come whe e VIFs collec i ely e use o sell o a downs eam compe i o can a ise mo e o en. Fu he mo e, we show ha when i ms can use con ac s ha maximize he join p o i s o he con ac ing pa ies, i is mo e likely ha ups eam compe i ion o sup‑ ply he downs eam i m eme ges. Ou indings ha e some implica ions o compe i ion and egula o y policy: Basing egula ion on a logic ha ela es obse ed collec i e e usal o supply o epea ed in e ac ions is incomple e in ha a collec i e e usal o supply may occu as a esul o non‑collusi e coo dina ion in a s a ic game ha has mul i‑ ple equilib ia. Absen a equi emen o an obliga ion o deal, compe i ion pol‑ icy should be ca e ul in e alua ing con ac ual ela ionships be ween VIFs and independen downs eam compe i o s. Ou esul s indica e ha pe mi ing ag ee‑ men s be ween con ac ing pa ies—which a e ou come‑equi alen o hose whe e con ac ing pa ies coo dina e hei beha io in he downs eam ma ke —may be wel a e‑imp o ing. Acknowledgemen s Open Access unding p o ided by P ojek DEAL. We would like o hank he Edi‑ o , Law ence J. Whi e, and anonymous e e ees as well as he semina pa icipan s a Sabanci Uni e ‑ si y, Izmi Uni e si y o Economics, Copenhagen Business School, Uni e si y o Mannheim, and EARIE 2015 o hei aluable commen s. 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They bo h ea n The equilib ium o he p ice compe i ion game is symme ic and yields he ollow‑ ing equilib ium p ices  𝛱i (p 1 ,p 2 )=(p i −c I −c O )q i (p 1 ,p 2 )i∈{1, 2} . 419 1 3 Ups eam Compe i ion wi hComplex andUnobse able Con ac s which in u n yield equilib ium p o i s o P oo o P oposi ion1 Le us s a by de i ing he equilib ium choices ha he con ac ing VIF i and he downs eam compe i o , i m 3, would make when hey ha e ag eed upon a whole‑ sale p ice w, and hey belie e VIF j will se a downs eam p ice o pj . In his case, i m 3 maximizes 𝜋3(pi,pj,p3,w) , while VIF i maximizes 𝜋i(pi,pj,p3,w) . Sol ing he i s ‑o de condi ions o he wo i ms simul aneously, we ind and Subs i u ing pi=pi(pj,w) and p3=p3(pj,w) in 𝜋i(pi,pj,p3,w) and maximizing wi h espec o w, yields he op imal choice o w condi ional on pj and he accep ance decision by i m 3. Fo mally, w(pj) sol es 𝜕 𝜕w 𝜋i(pi(pj,w),pj,p3(pj,w),w)= 0 and yields On he o he hand, VIF j maximizes 𝜋j(pi,pj,p3,w) ; howe e , gi en he ac ha i is awa e o he p ocess pi and p3 will be se , i inco po a es (4) and ( 5) in o i s maximiza‑ ion p oblem. Consequen ly, he op imal choice o VIF j sol es, and is gi en by We a e now in a posi ion o de i e a ious equilib ium condi ions when i ms a e es ic ed o con ac s ha a e based on linea wholesale p ices. Fi s conside he equi‑ lib ium whe e bo h VIFs a e expec ed o make unaccep able o e s o i m 3. Conside he de ia ion by VIF i o wd , which is hen accep ed by i m 3. Gi en ha we ha e de i ed abo e he op imal choice o w o VIF i condi ional on pj , and since we assume  p 1=p2=cI+cO+ 3(1−c I −c O ) 𝛽+6,  𝛱 1(p1,p2)=  𝛱2(p1,p2)=  𝛱D=3(1+𝛽)(3+𝛽)(1−cI−cO) 2 (6+𝛽) 2 (2+3𝛽). (4) pi(pj,w)= 1 2+𝛽 +1 3 (3+2𝛽) 2+𝛽 (cI+cO)+ 𝛽(3+2𝛽) (2+𝛽)(6+5𝛽) (w−cI)+1 3 𝛽p j (2+𝛽), (5) p3(pj,w)= 1 2+𝛽 +1 3 (3+2𝛽) 2+𝛽 (cI+cO)+ 6+8𝛽+3𝛽 2 (2+𝛽)(6+5𝛽) (w−cI)+1 3 𝛽p j (2+𝛽). w (pj)=cI−1 6 (6+5𝛽)(36 +42𝛽+13𝛽2) (24 +32𝛽+11𝛽 2 )(3+2𝛽)( cI+cO− 3+𝛽p j 3 + 𝛽). 𝜕 𝜕 pj 𝜋j(pi,pj,p3,w) | | | | p i =p i (p j ,w),p 3 =p 3 (p j ,w) = 0 p j(pj,w)= 6+5𝛽 2(2+𝛽)(3+2𝛽) +6+5𝛽 6(2+𝛽) (cI+cO)+ 𝛽(1+𝛽) 2(2+𝛽)(3+2𝛽) (w−cI)+ 𝛽 2 pj 3(2+𝛽)(3+2𝛽). 420 I.A iyas e al. 1 3 VIF j holds wa y belie s, in he equilib ium o his con inua ion game we mus ha e wd =w(p d j) and pd j =pj(p d j ,w d) . Sol ing hese condi ions we ind he op imal whole‑ sale p ice in case VIF i de ia es as and he downs eam equilib ium p ice ha is cha ged by he non‑con ac ing VIF ha holds wa y belie s in he con inua ion game is gi en by Subs i u ing hese alues in equilib ium p ice policies o VIF i and o i m 3 esul s in p i(p d j ,wd ) and p 3(p d j ,wd ) . Subs i u ing all o he equilib ium alues in he p o i unc ion o VIF i esul s in equilib ium p o i s in he con inua ion game ha ollows a de ia ion: I is hen s aigh o wa d o e i y ha 𝜋 i(p d i ,p d j ,p d 3 ,w d )<𝛱 D whene e 𝛽>15.09 , which implies ha o su icien ly high alues o subs i u abili y be ween he down‑ s eam p oduc s, he decision by bo h VIFs no no supply i m 3 is a PBE wi h wa y belie s. Now conside he si ua ion whe e VIFs 1 and 2 make compe ing o e s. Suppose w1≤w2 is a PBE and he o e o VIF 1 is accep ed in his equilib ium. This implies ha all i ms when hey choose hei downs eam p ices know ha i m 3 will ace a wholesale p ice gi en by w1 . In his case, he Nash equilib ium downs eam p ices o all h ee i ms a e A hese p ices 𝜋3(p1(w1),p2(w1),p3(w1),w1) is a posi i e‑ alued quad a ic con ex unc ion o w1 and achie es i s minimum alue o ze o when Fo w1≤wacc , 𝜋3(p1(w1),p2(w1),p3(w1)) is dec easing; mo eo e , only in his case is he demand ha is aced by i m 3 posi i e. Thus, i m 3 inds he o e accep able as long as he linea wholesale p ice is less han wacc . Fu he mo e, when i m 3 is w d=cI+(6+5𝛽)(36 +42𝛽+13𝛽 2 ) 432 +936𝛽+732𝛽 2 +237𝛽 3 +25𝛽 4(1−cI−cO ) pd j=cI+cO+3(2+𝛽)(36 +51𝛽+19𝛽 2 ) 432 +936𝛽+732𝛽 2 +237𝛽 3 +25𝛽 4(1−cI−cO) . 𝜋 i(pd i,pd j,pd 3,wd)= 1 3 (3+2𝛽)(6+5𝛽)(3+𝛽)(11𝛽 2 +32𝛽+24)(23𝛽 2 +69𝛽+54) (432 +936𝛽+732𝛽 2 +237𝛽 3 +25𝛽 4 ) 2(1−cI−cO)2 . p 1(w1)= 3 2(3+𝛽)+1 2 3+2𝛽 3+𝛽cO+1 2 18 +18𝛽+5𝛽 2 (6+5𝛽)(3+𝛽)cI+1 2 𝛽(9+5𝛽) (6+5𝛽)(3+𝛽)w1, p 2(w1)= 3 2(3+𝛽)+1 2 3+2𝛽 3+𝛽cO+1 2 18 +24𝛽+7𝛽2 (6+5𝛽)(3+𝛽)cI+1 2 𝛽(1+𝛽) (6+5𝛽)(3+𝛽+3)w1 , p 3(w1)= 3 2(3+𝛽) +1 2 3+2𝛽 3+𝛽 cO+3 2 𝛽(2+𝛽) (6+5𝛽)(3+𝛽) cI+1 2 18 +21𝛽+7𝛽2 (6+5𝛽)(3+𝛽) w1. w 1=wacc ≡cI+ 5+5𝛽 (6+𝛽)(1+𝛽) (1−cI−cO) . 421 1 3 Ups eam Compe i ion wi hComplex andUnobse able Con ac s aced wi h wo accep able o e s, i ea ns mo e when i con ac s wi h he VIF ha o e s he lowe wholesale p ice. I he iden i y o he i m ha makes he accep ed o e changes, he equilib ium p ices ha he VIFs cha ge in he downs eam ma ke will be as we ha e w i en abo e wi h changing he iden i ies o he VIFs. The e o e, i m 3 will accep he lowe o e , as long as w<wacc . Nex , we can compu e he di e ence in equilib ium p o i s o he con ac ing VIF and i s i al VIF. In he case we a e conside ing, his is whe e 𝛥(w1)=9(2+𝛽)(6+5𝛽)(1−cO−w1)−𝛽(36 +48𝛽+13𝛽2)(w1−cI) . Since 𝛥(w1) is a dec easing unc ion o w1 , his di e ence is quad a ic conca e, and posi i e whene e cI≤w1≤wmax whe e wmax is de ined by 𝛥(wmax )=0 . I is easy o e i y ha No e ha wmax >wd since Mo eo e , i is also easy o show ha Le us s a by es ablishing ha he e is no equilib ium in which bo h VIFs compe e and he accep ed wholesale p ice o e w sa is ies wmax <w<wacc . Wi hou loss o gene ali y, le us con inue wi h he p oposed equilib ium abo e in which w1≤w2 and he o e o VIF 1— w1 —is accep ed by i m 3. When wmax <w1<wacc , VIF 2 will no ha e an incen i e o unde cu , VIF 1. Howe e , gi en ha VIF 2 is expec ed o eac wi h p2(w1) in his p oposed equilib ium, one needs o check whe he VIF 1 and i m 3 ha e incen i es o de ia e. Gi en ha such a de ia ion does no change he iden i y o he VIF ha se es i m 3, he e is no need o VIF 2 o change i s downs eam p ice. We ha e de i ed he op imal wholesale p ice abo e in he e en ha VIF 2’s o e is unaccep able, condi ional on he expec ed p ice o VIF 2: w(p2) . Whene e w1>wd , we ha e w1>w(p2(w1)) . Hence VIF 1 has an in e es o educe he wholesale p ice ha i o e s below w1 , hus upse ing he p oposed equilib ium. The e o e, in ou se ‑ ing, a PBE wi h pa ial o eclosu e as is ound in Hombe e al. (2019) canno exis . These a gumen s imply ha wholesale p ices ha sa is y wmax <w<wacc canno be pa o an equilib ium. 𝜋 1(p1(w1),p2(w1),p3(w1),w1)−𝜋2(p1(w1),p2(w1),p3(w1)) = (w 1 −c I )(1+𝛽)𝛥(w 1 ) (6+5𝛽) 2 (3+𝛽), w max =cI+ 9(2+𝛽)(6+5𝛽) 108 +180𝛽+93𝛽 2 +13𝛽 3(1−cO−cI) . w max −wd=(3+𝛽)(3+2𝛽)(6+5𝛽)(108 +162𝛽+72𝛽 2 +7𝛽 3 )(1−cO−cI) (108 +180𝛽+93𝛽 2 +13𝛽 3 )(432 +936𝛽+732𝛽 2 +237𝛽 3 +25𝛽 4 ) . w acc −wmax = 4𝛽2(𝛽+3)(5𝛽+6) ( 1−cI−cO ) (𝛽+6)(𝛽+1) ( 13 𝛽3+93 𝛽2+180 𝛽+108 ) > 0. 422 I.A iyas e al. 1 3 Nex conside wholesale p ices in he ange wd ≤ w1 ≤ wmax . When w2>w1 in equilib ium, VIF 1 ea ns a highe p o i han VIF 2. Thus, VIF 2 has an incen i e o make a compe ing o e and he iden i y o he i m supplying i m 3 will change. This will a ec he equilib ium p ices ha he i ms will cha ge in he downs eam ma ke . Fi s , we need o pin down he belie s o VIF 1 a e i obse es ha he o e o VIF 2 is accep ed. I belie es ha VIF 2 de ia es in a ashion o maximize i s p o i s and ha VIF 2 and i m 3 belie e ha i will o m i s belie s in his ashion. Gi en ha i m 3 accep s o e s wi h lowe wholesale p ices, and wd<w1 , he op imal de ia ion by VIF 2 will be o se i s wholesale p ice such ha w2=wd , and make he sales o i m 3. Mo eo e , VIF 1, ha ing wa y belie s, will consequen ly se i s downs eam p ice equal o p 1=p d j , which we de i ed abo e. These a gumen s sugges ha wd<w<wmax canno be pa o a PBE in which wo VIFs make compe ing o e s. Finally, conside he wholesale p ices in he ange cI<w ≤ wd . Le us s a again om a p oposed equilib ium in which VIF 1 cha ges c ≤ w1<wd and VIF 2 cha ges w2>w1 . Whene e w1>cI , we ha e es ablished abo e ha VIF 1 ea ns a highe p o i han does VIF 2 in he p oposed equilib ium. This implies ha VIF 2 will ha e an incen i e o de ia e; bu gi en he alue o w1 , he op imal de ia‑ ion will be o unde cu VIF 1’s o e sligh ly. In he equilib ium o he con inu‑ a ion game ollowing such a de ia ion, i m 3 will ea n almos he same as in he p oposed equilib ium, while VIFs 1 and 2 will swi ch oles, meaning ha in he con inua ion game VIF 2 will ea n mo e han VIF 1. Hence, whene e w1>cI , such a de ia ion is p o i able. The e o e, he p oposed equilib ium whe e whole‑ sale p ices exceed he ma ginal cos o p oducing he inpu good canno exis . The only emaining possibili y is ha bo h i ms make compe ing o e s wi h wholesale p ices equal o he ma ginal cos o p oducing he inpu , namely w=cI . As a esul , all i ms compe e in he downs eam ma ke wi h a pe uni cos o cI+cO and he downs eam equilib ium is symme ic wi h equilib ium p ices gi en by No ice ha , o his equilib ium o exis we do no need o make a es ic ion on he subs i u abili y be ween he downs eam p oduc s. Thus, such an equilib ium in which bo h VIFs make compe ing o e s wi h ma ginal cos wholesale p ices always exis s, as is s a ed in P oposi ion1. P oo o P oposi ion2 When i m i supplies he inpu o i m 3 while i m j≠i , i,j∈{1, 2} , does no , and i m i and 3 a e able o selec join p o i maximizing p ices downs eam, i is as i i m i and 3 ha e me ged. Su icien ly sophis ica ed con ac s emula e his ou come and he downs eam equilib ium p ices in a con inua ion game a e inde‑ penden o he na u e o hese con ac s. Expec ing i m j o cha ge pj , he esul ing equilib ium p ices o VIF i and i m 3 a e gi en by pl 1=pl 2=pl 3=cI+cO+ 3 2 1−c I −c O 3+𝛽 . 429 1 3 Ups eam Compe i ion wi hComplex andUnobse able Con ac s Rey, P., & Ve gé, T. (2004). Bila e al con ol wi h e ical con ac s. The RAND Jou nal o Economics, 35(4), 728–746. Shubik, M., & Le i an, R. (1980). Ma ke s uc u e and beha io . Ha a d: Ha a d Uni e si y P ess. Siciliani, P. (2009). Collec i e dominance and e usal o supply: Closing he gap in a icle 82? The An i- us Bulle in, 54(3), 683–748. 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