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 :
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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 ‑
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2015 o hei aluable commen s.
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Appendix
Duopoly Equilib ium
When nei he VIF supplies he inpu , hey compe e in he downs eam ma ke as
duopolis s. 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
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Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published
maps and ins i u ional a ilia ions.