Kou andi, F ago; Be as, Nikolaos
A icle
Dynamic e ical o eclosu e wi h lea ning-by-doing
p oduc ion echnologies
Games
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Kou andi, F ago; Be as, Nikolaos (2024) : Dynamic e ical o eclosu e wi h
lea ning-by-doing p oduc ion echnologies, Games, ISSN 2073-4336, MDPI, Basel, Vol. 15, Iss. 2, pp.
1-23,
h ps://doi.o g/10.3390/g15020009
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Ci a ion: Kou andi, F.; Ve as, N.
Dynamic Ve ical Fo eclosu e wi h
Lea ning-by-Doing P oduc ion
Technologies. Games 2024,15, 9.
h ps://doi.o g/10.3390/g15020009
Academic Edi o s: Kons an inos
Se es, Ul ich Be ge and
Kaniska Dam
Recei ed: 31 Decembe 2023
Re ised: 20 Feb ua y 2024
Accep ed: 27 Feb ua y 2024
Published: 29 Feb ua y 2024
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games
A icle
Dynamic Ve ical Fo eclosu e wi h Lea ning-by-Doing
P oduc ion Technologies
F ago Kou andi 1and Nikolaos Ve as 2,3,*
1Depa men o Economics, Na ional and Kapodis ian Uni e si y o A hens, 1 So okleous S .,
10559 A hens, G eece; [email p o ec ed]
2Depa men o Economics, A hens Uni e si y o Economics and Business, 76 Pa ision S .,
10434 A hens, G eece
3Cen e o Economic Policy Resea ch, London EC1V 0DX, UK
*Co espondence: [email p o ec ed]
Abs ac : He e, we s udy e ical o eclosu e in a dynamic se up wi h lea ning-by-doing p oduc ion
echnologies. The e is a downs eam monopoly and an ups eam duopoly, whe e manu ac u e s p o-
duce di e en ia ed p oduc s and can gain p o iciency h ough he accumula ion o hei p oduc ion.
We s udy he dynamic in e ac ions in he e ical chain when he monopolis se s he p ices; we ind
ha cus ome o eclosu e may a ise in equilib ium when he p oduc s a e close subs i u es and be
wel a e-enhancing. The a e o lea ning is lowe han he social op imal and a social planne would
end o impose exclusi i y mo e o en compa ed o he downs eam monopolis .
Keywo ds: dynamic in e ac ions; lea ning-by-doing; exclusi i y
JEL Classi ica ion: L42; L13; L14; L11; L81
1. In oduc ion
How p oduc ion is o ganized wi hin and ac oss i ms c ucially a ec s he wel a e
o he inal consume s and is, na u ally, he co e conce n o indus ial o ganiza ion and
ela ed ields. One o he ac o s ha shape he key ma ke ou comes is he in ensi y o
compe i ion and he a ie y o a ailable p oduc s. In his ega d, di e en ins ances o
exclusiona y p ac ices in e ically ela ed indus ies, be ween ups eam and downs eam
i ms, ha e d awn he a en ion o egula o s and an i- us au ho i ies and ha e been he
ocus o in luen ial academic esea ch, wi h he o eclosu e o some p oduce s iewed
wi h suspicion in mos o he cases. Ye , he e a e also o he ac o s ha play a c ucial ole
in de e mining ma ke ou comes and wel a e, including e iciency in p oduc ion. While
exclusiona y p ac ices ha e been ex ensi ely s udied in he li e a u e, less a en ion has
been gi en o he ole and na u e o exclusi i y in dynamic en i onmen s.
In his pape , we in oduce dynamic in e ac ions in a e ical chain h ough lea ning-
by-doing in p oduc ion. This is pe o med in a simple way: he e is an ups eam duopoly,
whe e he p oduc ion o each i m may become mo e e icien acco ding o he accumula ion
o p oduc ion o e ime. The ups eam i ms p oduce ei he complemen s o subs i u es,
wi h ho izon al p oduc di e en ia ion aking he o m o “lo e o a ie y”. A downs eam
monopolis can ecei e he p oduc s o he ups eam i ms and has he powe o se linea
wholesale con ac e ms. C ucially, he game las s o wo pe iods. In each pe iod,
he e aile se s he con ac e ms o each o he ups eam i ms, which can ei he accep o
ejec hei con ac o e s. Following ha s age, he e aile sells he p oduc s o he inal
consume s and pays he ups eam i ms acco ding o he con ac s ha ha e been ag eed.
When he game p oceeds o he second pe iod, he cos s o he ups eam i ms may ha e
been educed based on hei i s pe iod p oduc ion le el.
Games 2024,15, 9. h ps://doi.o g/10.3390/g15020009 h ps://www.mdpi.com/jou nal/games
Games 2024,15, 9 2 o 23
In his se ing, we explo e he ensions be ween assu ing access o compe ing a ie ies
and lowe ing p oduc ion cos s ia lea ning-by-doing. Exclusion a ises implici ly when
he dominan e aile o e s e y disad an aged con ac e ms o an ups eam i m, and,
hus, o ces i o s ay ou o he ma ke . Thus, exclusi i y does no a ise by a dominan
ups eam i m’s denial o supply some downs eam i ms wi h he essen ial inpu ha
i p oduces.
1
Ins ead, he e aile is a la ge playe , and he ups eam i ms can each he
inal consume s only ia his e aile . Ou analysis can be applicable o a mul ip oduc
i m whe e p oduc ion is cha ac e ized by he lea ning hypo hesis o o a s a egic buye
ha can manipula e he lea ning p ocess h ough hei p oduc choice. The ole o he
in e media y is s udied by examining whe he he exis ence o his i m in ensi ies he
lea ning p ocess compa ed o he case whe e he e is no in e media y in he ma ke , bu he
ups eam i ms compe e di ec ly in he inal ma ke .
Wi hin such a amewo k, in e es ing ques ions a ise: How does he inal ma ke
ou come depend on he in ensi y o lea ning-by-doing and p oduc di e en ia ion? Does
he e aile choose o ca y only one p oduc o in ensi y he lea ning p ocess? Unde wha
condi ions do exclusi i y and, hus, cus ome o eclosu e eme ge? Unde wha condi ions
may hey be bene icial o he inal consume s and he i ms? Does he ma ke compe i ion
a o an ou come wi h lowe p ices o wi h mo e a ie y in he ma ke ? Is he p esence o
he e aile in he ma ke necessa y o in ensi y he lea ning p ocess?
The ood indus y is a leading mo i a ing example, co esponding o such e ical
s uc u e models and one ha has been a ac ing he a en ion o policy make s and e en
he popula p ess. In he las couple o decades, la ge supe ma ke s ha e eme ged and
become powe ul in hei ansac ions wi h ups eam supplie s in many indus ies. O he
examples include la ge ou ope a o s ading wi h ai lines and ho els o gene al e aile s
such as Wal-Ma . The issue o exclusiona y p ac ices is also cen al in manu ac u ing, like
in he au omobile sec o (see, e.g., B enke s and Ve bo en, 2005) [1].
The ac ha lea ning-by-doing can be an impo an ac o in p oduc ion p ocesses
has been documen ed in a numbe o business uni s and sec o s, such as he ag icul u e
indus y whe e sus ainable a ming elies on lea ning om he expe ience gained o e
mul iple g owing seasons. Fu he mo e, his is he case in manu ac u ing con ex s, such
as in he au omobile indus y whe e Toyo a P oduc ion Sys em is based on he concep o
“Kaizen”, i.e., con inuous imp o emen , h ough lea ning om expe ience. O he leading
examples o lea ning-by-doing in p oduc ion include he cons uc ion indus y and he
so wa e de elopmen indus y.
In he s a ic e sion o ou model, he e aile always chooses o ca y bo h p od-
uc s. In a one-pe iod se ing, he e is no lea ning and exclusi i y educes he a ie y in
he ma ke wi hou educing he p oduc ion cos s o he p ices. In he dynamic model,
exclusi i y a ises in equilib ium when p oduc s a e close subs i u es. In con as , when
p oduc s a e complemen s o no close subs i u es, bo h a e pu chased in bo h pe iods.
This esul ollows om wo opposi e e ec s. The e is a ade-o be ween lowe cos s
due o lea ning and mo e p oduc a ie ies in he ma ke . When p oduc di e en ia ion
is low, he “lea ning” e ec domina es he “p oduc a ie y” e ec , and he o al p o i s
o he chain and he consume ’s su plus a e highe when one p oduc is excluded om
he ma ke . I ollows ha exclusi i y is wel a e-enhancing. Fu he mo e, ela i e o he
non-exclusi i y case, he p oduc p ices in bo h pe iods a e lowe when exclusi i y is
imposed by he e aile . Howe e , he a e o lea ning is no equal o he social op imum
and he social planne would impose exclusi i y mo e o en compa ed o he e aile . We
also main ain ha he p esence o he downs eam in e media y is necessa y in in ensi ying
he lea ning-by-doing p ocess when p oduc s a e complemen s o when p oduc s a e close
subs i u es, and he e aile imposes exclusi i y.
Ou pape con ibu es o wo di e en ields in he li e a u e wi hin indus ial o ga-
niza ion: e ical con ac ing and lea ning-by-doing and business o ganiza ion. Each o
hese con ains e y impo an pape s and i is oo as o su ey he e. We only e e o
wo k ha is mo e closely ela ed o ou analysis. Fi s , nume ous con ibu ions highligh
Games 2024,15, 9 3 o 23
he impac o e ical con ac ing and, in pa icula , he impac o exclusiona y beha io on
ma ke compe i ion. Fo a e iew and some key esul s on e ical o eclosu e, see Rey and
Ti ole (2007) [
2
]. K a enmake and Salop (1986) [
3
] a gue ha con ac s wi h inpu sup-
plie s can be e ile g ound o aising compe i o ’s cos and Aghion and Bol on (1987) [
4
]
demons a e ha con ac s be ween buye s and selle s will be signed o en y-p e en ion
pu poses. Ma hewson and Win e (1987) [
5
], Besanko and Pe y (1993) [
6
], and O’B ien and
Sha e (1993) [
7
] de i e ha exclusi e a angemen s can ha e desi able wel a e p ope ies.
2
The oppo unis ic beha io is aced h ough exclusi i y acco ding o McA ee and Schwa z
(1994) [
8
]. We ob ain ha exclusi i y is wel a e-imp o ing, compa ed o non-exclusi i y,
due o lea ning in he p oduc ion.
Rey and S igli z (1995) [
9
] ob ain main ain ha , when goods a e close subs i u es,
p oduce s’ p o i s a e highe unde exclusi e e i o ies. We main ain ha when goods
a e close subs i u es, e aile ’s p o i s a e highe unde exclusi i y since he lea ning e ec
domina es he p oduc a ie y e ec . Fo eclosu e has also ecen ly been s udied in e i-
cal con ac ing amewo ks wi h e ically in eg a ed i ms by Reisinge and Ta an ino
(2015) [
10
] and Kou andi and Pinopoulos (2023) [
11
]. We depa om e ical in eg a ion
and s udy how lea ning a ec s u u e p oduc ion e iciency and, hus, he ma ke exclusion
o ups eam p oduce s. A dynamic se ing wi h in e empo al linkages and inpu o eclo-
su e is s udied by Fumagalli and Mo a (2020) [
12
] and Sandiumenge i Boy (2023) [
13
]. We
ocus on cus ome o eclosu e.
Fu he mo e, he e is a well-known ield in he li e a u e on common agencies ha
deal wi h exclusi i y. See, o example, Ma imo (1996) [
14
], Be nheim and Whins on
(1998) [
15
], and Segal and Whins on (2000) [
16
]. Fu he , Ma x and Sha e (2007) [
17
] adop
up on paymen s wi h ba gaining powe in he downs eam le el and ind ha exclusi i y
a ises in equilib ium. In ou se up, when p oduc s a e close subs i u es, he bo leneck
e aile coo dina es he pu chases o he wo p oduc s and akes ad an age o he lea ning
p ocess in a mo e e icien way compa ed o he case whe e ups eam p oduce s di ec ly sell
hei p oduc s o he inal consume s. Ne e heless, ela i ely ew pape s ha e examined
e ical chain in e ac ions in a dynamic amewo k. An exemp ion is Chen (2005) [
18
], who
s udies e ical disin eg a ion in a dynamic model wi h lea ning-by-doing and downs eam
compe i ion.
3
In ou dynamic model wi h lea ning-by-doing, we ocus, ins ead, on p oduc
di e en ia ion and cus ome o eclosu e.
The second ela ed ield in he li e a u e examines s a egic pu chases in one- ie
indus ies in a dynamic se ing unde he lea ning-by-doing hypo hesis. Spence (1981) [
21
]
ob ains he dynamic ou pu pa h o a single- and hen a mul i- i m model and s udies he
open and closed loop equilib ia. Fudenbe g and Ti ole (1983) [
22
] de i e he p ecommi -
men and pe ec equilib ia wi h linea demand and lea ning unc ions. A p ice-se ing
di e en ia ed duopoly wi h an in ini e sequence o he e ogeneous buye s and unce ain
demands is analyzed by Cab al and Rio dan (1994) [
23
], whe e hey ind he Ma ko pe ec
equilib ium and s udy he concep o ma ke dominance. In an en iched e sion o he
Cab al and Rio dan model, Besanko, Do aszelski, and K yuko (2019) [
24
] use compu a-
ional me hods o analyze he Ma ko Pe ec equilib ium beha io . Lewis and Yildi im
(2002a) [
25
] s udy he ade-o be ween expe ience and compe i ion in an indus y wi h
p i a ely cos unc ions and a single buye . Meanwhile, Lewis and Yildi im (2002b) [
26
]
in es iga e how incen i e egula ion should be designed o encou age supplie s o de-
elop and adop cos -sa ing echnologies. Recen ly, Swee ing e al. (2022) [
27
] s udied
dynamic p ice compe i ion when selle s bene i om lea ning-by-doing and buye s a e
long-li ed and s a egic, while cap u ing, wi h a single pa ame e , he ex en o which each
buye in e nalizes u u e buye su plus. They ind ha e en mode a e deg ees o o wa d-
looking buye beha io may elimina e he mul iplici y o equilib ia and ha equilib ia wi h
compe i ion a e mo e likely o su i e.
In hese models, lea ning-by-doing can be unde s ood ei he as cos - educing inno-
a ions o as he esul o economies o scale ac oss mul iple ma ke segmen s, whe e he
di e en ime pe iods play he ole o di e en ma ke segmen s. Ou se up adds an ex a
Games 2024,15, 9 4 o 23
ie in he e ical supply chain. P oduc s a e eached by he consume s ia a e aile ha
may decide o educe p oduc a ailabili y. We explo e he condi ions unde which he
lea ning e ec domina es he p oduc a ie y e ec in a e ical chain and ind, consis en ly
wi h he lea ning-by-doing li e a u e, ha equilib ia wi h ei he mo e (non-exclusi i y,
i.e., compe ing supplie s) o less (exclusi i y, i.e., monopoly supplie ) a ie y can eme ge
depending on he le el o p oduc di e en ia ion. Ye , ou model is kep simple, especially
in he sense ha he powe o se wholesale p ices is gi en o he e aile . Thus, while he
analysis cap u es he main ension be ween p oduc a ie y and dynamic cos educ ion,
he iche model is necessa y o add ess mo e complex issues ha would a ise i ba gaining
powe was mo e e enly dis ibu ed ac oss ups eam and downs eam i ms.
The emainde o he pape is p esen ed as ollows. Sec ion 2cha ac e izes he equilib-
ium when he game las s only one pe iod. The dynamic model is analyzed in Sec ion 3.
Consume ’s su plus and o al wel a e a e calcula ed in Sec ion 4. In Sec ion 5, we de i e
he social planning solu ion and compa e i o he equilib ium. Sec ion 6s udies he ole o
he in e media y in he lea ning p ocess and Sec ion 7concludes.
2. The S a ic Model
We begin ou analysis by s udying he po en ial o exclusi i y o a ise in a simple
se ing whe e i ms compe e o a single pe iod. We conside wo manu ac u e s, A and
B, who each p oduce a single p oduc wi h hese p oduc s o be ei he subs i u es o
complemen s. P oduc di e en ia ion is ho izon al and akes he o m o “lo e o a ie y”.
Manu ac u e s supply he downs eam ma ke and ha e he same cons an ma ginal
p oduc ion cos ,
c
, since we ocus on he possibili y o exclusi i y o eme ge as a esul o
s a egic in e ac ion and no due o cos asymme ies in he ups eam indus y.
The downs eam ma ke is monopolized by a e aile , R, ha has he powe o se he
con ac e ms which ake he o m o linea wholesale p ices.
4
One uni o he manu ac-
u e s’ p oduc becomes one uni o inal good a ze o ma ginal cos . The in e se inal
demand unc ions ake he linea o m
pi=a−qi−bqj
, whe e
pi
is he e ail p ice and
qi
is he quan i y o p oduc
i
,
i=A
,
B
,
a≥c
and
b∈(−
1, 1
)
is he p oduc di e en ia ion
pa ame e .
5
When
b
equals ze o, he wo p oduc s a e independen . As
b
app oaches he
uni y (
b→
1) he wo goods become close subs i u es and as
b
app oaches minus one
(
b→ −
1) he wo goods become close complemen s. When
b
is equal o one, he goods a e
homogeneous and when
b
is equal o minus one, he wo goods a e pe ec complemen s.
The e is no unce ain y and all con ac s a e obse able. The s a ic amewo k is p esen ed
in Figu e 1.
1
c c
A B
R
wAwB
pApB
inal consume s
pA=α-qA-bqB
pB=α-qB-bqA
b
del a
-10
1.0
0.5
1.0
0.5 0
0.0
10
0.0
Figu e 1. The s a ic model.
Games 2024,15, 9 5 o 23
We conside a h ee-s age game. Fi s , he e aile se s he con ac e ms. Then,
he manu ac u e s ei he accep o ejec he con ac o e s and inally he e aile sells he
p oduc s o he inal consume s and pays he manu ac u e s acco ding o hei con ac s.
The game is sol ed backwa ds, using subgame pe ec ion as he equilib ium concep .
In he i s s age o he game, he e aile can se e y disad an aged e ms o one o
he wo supplie s (say B) o o ce his supplie o ejec his o e and s ay ou o he ma ke .
The e aile implici ly denies access o he inal consume s o ha i m’s p oduc by o e ing
a wholesale p ice lowe han he p oduc ion cos o ha manu ac u e .
6
Fi s , we conside
he case whe e bo h p oduc s a e ca ied by he e aile (non-exclusi i y) and hen we s udy
he co ne case whe e he e aile chooses o ca y only one p oduc (exclusi i y).
Non-exclusi i y: Sol ing backwa ds, in he inal s age o he game, he e aile max-
imizes i s p o i s,
ΠR= (pA−wA)qA+ (pB−wB)qBs
.
.
pi=a−qi−bqj
, in he inal
goods ma ke . The op imal quan i ies a e de i ed by he i s -o de condi ions.
Then, he e aile se s he wholesale p ices
wi
,
i=A
,
B
, gi en he indi idual a io-
nal cons ain s o he manu ac u e s. The manu ac u e s accep he o e i hey ob ain
non-nega i e p o i s, since a he downs eam le el he e is no al e na i e e aile , which
leads o ze o ou side op ion o he manu ac u e s. Thus, he downs eam monopolis
se s he wholesale p ices equal o he minimum possible le el, ha is,
wi=c
o bo h
manu ac u e s.7
Lemma 1. In he s a ic model wi h bo h p oduc s ca ied by he e aile , we ob ain:
qi=a−c
2(1+b),pi=a+c
2,wi=c,Πi=0, ΠR=(a−c)2
2(1+b).
The inal p ices a e equal o he monopoly p ices. The chain p o i s a e maximized
and cap u ed by he e aile .
Exclusi i y: Assume now ha , in he i s s age o he game, he e aile chooses o
dis ibu e only one p oduc , say A. Now, i maximizes i s p o i s,
ΠR= (pA−wA)qAs
.
.
pA=a−qA, and ob ains he op imal quan i y by he i s -o de condi ion.
The e aile hen de e mines he wholesale p ices
wi
o he manu ac u e A o accep
he o e and o B o ejec he o e . The e aile cha ges a wholesale p ice equal o he
uni p oduc ion cos o manu ac u e A,
wA=c
, and a wholesale p ice lowe han he
p oduc ion cos o manu ac u e B,
wB<c
, o exclude his p oduc om he ma ke . The
monopoly esul s a e eached wi h he e aile o ex ac he e ical chain’s p o i s.
Lemma 2. In he s a ic model, wi h one p oduc in he ma ke , we ob ain:
qA=a−c
2,pA=a+c
2,wA=c,wB<c,ΠA=0, ΠR=(a−c)2
4.
When p oduc s a e pe ec subs i u es (
b=
1), he esul s om he wo subgames
coincide, apa om he ac ha he o al quan i y is equally spli o he wo manu ac u e s
when hey a e bo h selling. Unde impe ec subs i u abili y (
b∈(
0, 1
)
), he inal p ices
a e equal in he wo subgames. Howe e , he o al quan i y and he p o i ob ained by
he e aile inc ease as
b
dec eases unde non-exclusi i y, since consume s p e e o ha e
bo h a ie ies a ailable. When p oduc s a e complemen s (
b∈(−
1, 0
)
), inal quan i ies and
p o i s unde non-exclusi i y a e u he inc eased.
The consume ’s su plus when bo h p oduc s a e a ailable is equal o
(a−c)2
4(1+b)
, and when
only one p oduc is dis ibu ed i is equal o
(a−c)2
88
Since p o i s and consume ’s su plus
unde non-exclusi i y a e highe , o al wel a e is also highe . In he s a ic model, exclusi i y
ha ms bo h consume s and i ms.
Games 2024,15, 9 6 o 23
P oposi ion 1. In he s a ic model, exclusi i y does no a ise in equilib ium. The o al p o i s o he
chain, he consume ’s su plus, and he o al wel a e when bo h p oduc s a e pu chased a e highe
han when one p oduc is excluded om he ma ke .
3. The Dynamic Model
He e, we depa om he s a ic model by assuming ha he game las s o wo pe iods.
We s udy he dynamic in e ac ions ha eme ge in he e ical amewo k by in oducing
he lea ning-by-doing hypo hesis. O e ime, ups eam p oduce s gain p o iciency h ough
he epe i ion o hei p oduc ion. The uni p oduc ion cos dec eases as he p oduce gains
mo e expe ience; ha is, he uni cos unc ion is a dec easing unc ion o pas accumula ed
p oduc ion. In e es ing issues a ise in his dynamic en i onmen : Does exclusi i y eme ge
in equilib ium and unde wha condi ions does i eme ge? Does ma ke compe i ion a o
an ou come wi h lowe p ices o wi h mo e a ie ies in he ma ke ?
In ou model, p oduc ion is cha ac e ized by he linea lea ning-by-doing hypo hesis.
The uni p oduc ion cos o he manu ac u e s in he second pe iod educes p opo ionally
wi h he p oduc ion o he i s pe iod. Fi ms lea n om hei own p oduc ion. Speci ically,
he uni cos unc ions in he second pe iod a e gi en by:
ci2=c−λqi1i qi1<c
λ
0 i qi1≥c
λ o i=A,B. (1)
The i s subsc ip e e s o he manu ac u e and he second o he ime pe iod (
=
1, 2),
while λis he lea ning pa ame e .
The iming o he game is he same as in he s a ic game, wi h he di e ence being ha
i ms now in e ac o wo pe iods. In each pe iod, he e aile i s se s he con ac e ms;
hen, he ups eam i ms ei he accep o ejec he o e s and, subsequen ly, he e aile sells
he p oduc s o he inal consume s and pays he manu ac u e s. Two cases a e examined in
each pe iod: bo h p oduc s a e pu chased by he e aile o only one p oduc is pu chased
ei he in pe iod one o in pe iod wo. Thus, in e ms o p oduc a ailabili y, ou al e na i e
cases may a ise in he wo-pe iod model.
C ucially, he e is in e dependence be ween he wo pe iods due o he lea ning-by-
doing p ocess. The uni cos s in he second pe iod a e a ec ed by he quan i ies p oduced
in he i s one. The e o e, he e aile maximizes he p esen alue o i s p o i s in he i s
pe iod:
ΠR1+δΠR2
, whe e
δ∈(
0, 1
)
is he discoun ac o . We p oceed backwa ds o sol e
o he subgame’s pe ec equilib ium.
3.1. Pe iod Two
P oduc a ailabili y and quan i ies pu chased in he i s pe iod shape he p oduc ion
cos s in he second pe iod. As a ma e o no a ion, i he e aile chooses o ca y one
p oduc in he i s pe iod, his will be p oduc A. So, in he second pe iod, he mo e
cos -e icien i m (i any) is i m A wi h
cA2≤cB2
. Taking as gi en he p oduc ion cos s
cA2
,
cB2
, we conside i s he case whe e he e aile does no s a egically exclude an
ups eam supplie by o e ing disad an aged con ac e ms and hen he case whe e he
e aile pu chases only one p oduc in pe iod wo.
Non-exclusi i y: We begin om s age h ee. The e aile maximizes i s p o i s:
max
qA2,qB2
ΠR2= (pA2−wA2)qA2+ (pB2−wB2)qB2
s. .pi2=a−qi2−bqj2wi h i=A,Band i=j.
Simila ly o he s a ic model, he ups eam i ms accep he o e s and he e aile se s he
wholesale p ices equal o he uni p oduc ion cos s. wi2=ci2. We ob ain
Games 2024,15, 9 7 o 23
qi2=a
2(1+b)+bcj2−ci2
2(1−b2),pi2=a+ci2
2,wi2=ci2,Πi2=0,
ΠR2=a(2a−cA2−cB2)
4(1+b)+(a−cA2)(bcB2−cA2) + (a−cB2)(bcA2−cB2)
4(1−b2).
No e ha
qA2≥
0 always holds since
b≤
1
≤a−cA2
a−cB2
and
cA2≤cB2
, bu
qB2≥
0 holds only
when
b≤a−cB2
a−cA2≤
1, whe e he cos asymme y in he second pe iod is no high enough.
E en i he e aile does no s a egically exclude one supplie by o e ing a wholesale p ice
lowe han he p oduc ion cos , p oduc B may be excluded om he ma ke in pe iod
wo when i m B has a p oduc ion cos oo high compa ed o he i al’s cos , i.e., when
b>a−cB2
a−cA2. Taking in o accoun his possibili y o a co ne solu ion, we ob ain:9
qA2=
a
2(1+b)+bcB2−cA2
2(1−b2)i b ≤a−cB2
a−cA2
a−cA2
2i b >a−cB2
a−cA2
,qB2=
a
2(1+b)+bcA2−cB2
2(1−b2)i b ≤a−cB2
a−cA2
0i b >a−cB2
a−cA2
,
pA2=a+cA2
2,pB2=(a+cB2
2i b ≤a−cB2
a−cA2
−i b >a−cB2
a−cA2
,(2)
wA2=cA2,wB2=(cB2i b ≤a−cB2
a−cA2
−i b >a−cB2
a−cA2
,
ΠA2=0, ΠB2=(0i b ≤a−cB2
a−cA2
−i b >a−cB2
a−cA2
,
ΠR2=
a(2a−cA2−cB2)
4(1+b)+(a−cA2)(bcB2−cA2)+(a−cB2)(bcA2−cB2)
4(1−b2)i b ≤a−cB2
a−cA2
(a−cA2)2
4i b >a−cB2
a−cA2.
Exclusi i y: Now, assume ha he e aile pu chases only one p oduc in pe iod wo (p oduc
A) by o e ing o supplie B e y disad an aged con ac e ms. The e aile maximizes:
max
qA2
ΠR2= (pA2−wA2)qA2s. .pA2=a−qA2.
The ups eam i m A accep s he o e and i m B ejec s he o e whe e
wA2=cA2
and
wB2<cB2. The e o e:10
qA2=a−cA2
2,qB2=0 (3)
pA2=a+cA2
2,wA2=cA2,wB2<cB2
Πi2=0, ΠR2=(a−cA2)2
4.
Ou come in pe iod wo: Gi en he possible alues o he wo p oduc ion cos s in pe iod
wo, he e aile decides i i will impose exclusi i y in pe iod wo o no , by compa ing i s
p o i s in his pe iod o he wo al e na i e cases abo e (by Exp essions (2) and (3)). When
−
1
<b≤a−cB2
a−cA2
, we ind ha i is p o i able o he e aile o pu chase bo h p oduc s in he
inal pe iod o he game.
11
The second pe iod is he las pe iod o he game and excluding
one supplie om he ma ke by o e ing disad an aged e ms only educes he a ie y in
his pe iod wi hou op ing o u u e cos educ ion.
When 1
>b>a−cB2
a−cA2
, he e aile ’s p o i s in (2) o (3) a e iden ical wi h p oduc B
no a ailable in pe iod wo. In (2), p oduc B canno su i e in he ma ke due o high
cos asymme y; meanwhile, in (3), p oduc B is s a egically excluded by he e aile ia
disad an aged wholesale p icing.
Summing up, he equilib ium ou come in pe iod wo is gi en by Exp ession (2).12
Games 2024,15, 9 8 o 23
3.2. Pe iod One
The i s ques ion now is whe he he e aile has an incen i e o exclude one ups eam
supplie in he i s pe iod o in ensi y he lea ning p ocess by pu chasing highe quan i y
by only p oduce A. The second ques ion is, gi en ha he e aile chooses o ca y only
p oduc A in he i s pe iod, whe he i will pu chase high enough quan i y by p oduce A
o make his p oduce e icien enough in pe iod wo as well (which will lead o high cos
asymme y in he nex pe iod). Wi hou exclusion in he i s pe iod, p oduce s a e equally
cos -e icien in he u u e, in con as o he case whe e exclusi i y is imposed in he i s
pe iod. The e a e wo al e na i e cases: one whe e he e aile pu chases bo h p oduc s and
one whe e he e aile pu chases only one p oduc in pe iod one by o e ing disad an aged
con ac e ms o p oduce B.
Non-exclusi i y: Gi en ha he supplie s ini ially ha e equal cos s, he equilib ium
in pe iod one will be symme ic and he supplie s equal e icien in he second pe iod.
The e aile maximizes he p esen alue o i s p o i s
ΠR1+δΠR2
, whe e
ΠR2
is gi en
by Exp ession (2). The quan i ies pu chased in pe iod one a ec he p oduc ion cos s
in pe iod wo and, subsequen ly, he p ices and he p o i s ob ained a e also a ec ed.
The e aile sol es:
max
qA1,qB1
ΠR1+δΠR2= (pA1−wA1)qA1+ (pB1−wB1)qB1
+δa(2a−cA2−cB2)
4(1+b)+(a−cA2)(bcB2−cA2) + (a−cB2)(bcA2−cB2)
4(1−b2)
s. .ci2=c−λqi1 o i=A,B.
By he ac ha he ups eam i ms accep he o e s and he wholesale p ices a e se o he
ma ginal cos wi1=c, we ha e:13
Lemma 3. In he dynamic model wi hou exclusi i y in pe iod one, we ha e:14
ΠNE
R1+δΠNE
R2=2(a−c)2((δ+1)(b+1)+λδ)
8b−λ2δ+4b2+4. (4)
Exclusi i y: Assume now ha , in he i s pe iod, he e aile chooses o dis ibu e
only p oduc A. Then, manu ac u e A will be mo e cos -e icien in pe iod wo (
cA2<cB2
)
since i has bene i ed om he lea ning-by-doing (
qA1>
0 and
qB1=
0). This cos educ ion
de e mines whe he p oduc B will be p oduced in pe iod wo, depending also on he
p oduc di e en ia ion pa ame e . The e aile maximizes:
max
qA1
ΠR1+δΠR2=
(pA1−wA1)qA1+δa(2a−cA2−cB2)
4(1+b)+(a−cA2)(bcB2−cA2)+(a−cB2)(bcA2−cB2)
4(1−b2)i b ≤a−cB2
a−cA2
(pA1−wA1)qA1+δ(a−cA2)2
4i b >a−cB2
a−cA2
s. .cA2=c−λqA1and cB2=c.
The wholesale p ice o manu ac u e A is se o be equal o he p oduc ion cos
in pe iod one and o manu ac u e B lowe han he p oduc ion cos , making his i m
no ope a e in he i s pe iod. So, we ha e
wA1=c
and
wB1<c
. Choosing he op i-
mum le el o
qA1
now a ec s he subsequen cos asymme y and, hus, he subsequen
p oduc a ailabili y.
When p oduc s a e ei he complemen s o no so close subs i u es,
−
1
<b≤a−cB2
a−cA2
,
o equi alen ly when he quan i y o i m A in pe iod one is no high enough,
qA1≤(a−c)(1−b)
λb
, bo h p oduc s a e pu chased in pe iod wo. The e aile ’s maximand is
Games 2024,15, 9 15 o 23
WE−S
1+δWE−S
2=
(a−c)2(λ2δ2(2b−2λ+2bλ+2λ2δ+3b2−5)+(b+1)(b−1)2(b+4δ+4λδ+1))
2((1−b2)−δλ2)2i b ≤1−δλ2
1+λ
(a−c)2(−2λ3δ2−λ2δ2+4λδ+2δ+1)
2(1−δλ2)2i b >1−δλ2
1+λ.
(9)
Again, he e he social planne in ensi ies he lea ning p ocess by se ing p ices lowe han
he p oduc ion cos s in he i s pe iod and, hus, incu ing some losses. In he second
pe iod, p ices a e equal o he p oduc ion cos s. We ind ha exclusi i y is imposed in he
i s pe iod o he game which also leads o exclusi i y in he second pe iod o he game
when
b>1−λ2δ
1+λ
; howe e , when
b≤1−λ2δ
1+λ
, he quan i y o p oduc A pu chased in he
i s pe iod o he game is no high enough o exclude p oduc B in he second pe iod.
Cha ac e iza ion o he social op imum: To ully cha ac e ize he social planne ’s
solu ion we ha e o check when he social planne excludes an ups eam supplie in he
i s pe iod o he game o manipula e he lea ning p ocess. We compa e he p esen
alue o he o al wel a e using Exp essions (8) and (9) o e e y p oduc di e en ia ion
pa ame e .
∆W=WE−S
1+δWE−S
2−WNE−S
1+δWNE−S
2=
(a−c)2(λ2δ2(2b−2λ+2bλ+2λ2δ+3b2−5)+(b+1)(b−1)2(b+4δ+4λδ+1))
2((1−b2)−δλ2)2−(a−c)2(b+δ+bδ+2λδ+1)
(1+b)2−λ2δi b ≤1−δλ2
1+λ
(a−c)2(−2λ3δ2−λ2δ2+4λδ+2δ+1)
2(1−δλ2)2−(a−c)2(b+δ+bδ+2λδ+1)
(1+b)2−λ2δi b >1−δλ2
1+λ.
When
∆W
is posi i e and
b≤1−δλ2
1+λ
, exclusi i y is imposed only in pe iod one and in
pe iod wo bo h p oduc s a e pu chased. Howe e , when
∆W
is posi i e and
b>1−δλ2
1+λ
,
exclusi i y is imposed in bo h pe iods. Nume ically, we ind ha :
Lemma 6. Fo low alues o he p oduc di e en ia ion pa ame e , he social planne chooses o ha e
bo h p oduc s in he ma ke in bo h pe iods. Fo in e media e alues o he p oduc di e en ia ion
pa ame e p oduc B is excluded in he i s pe iod bu no in he second pe iod. Finally, o high
alues o he p oduc di e en ia ion pa ame e , p oduc B is excluded in bo h pe iods.
Fu he mo e, we ha e:
Lemma 7. The social planne ends o impose exclusi i y mo e o en han he e aile .
To be e unde s and hese esul s, we p esen a nume ical example. Fo gi en
lea ning and discoun ac o pa ame e s, we plo in Figu e 4( o each
b
): he di e -
ence in he p esen alue o he wel a e (
∆W=WE
1+δWE
2−WNE
1+δWNE
2
, he ob-
jec i e unc ion o he social planne ) and he di e ence in he p esen alue o he
p o i s
(∆Π =ΠE
R1+δΠE
R2−ΠNE
R1+δΠNE
R2
, he objec i e unc ion o he e aile om
Sec ion 3)
. We obse e ha , when he p oduc s a e close complemen s, he social planne
ne e imposes exclusi i y and his also holds o he e aile (
∆W<
0 and
∆Π <
0 o low
and nega i e
b
). Howe e , when he p oduc s a e no close complemen s (
b
is nega i e
bu no e y low), he social planne excludes p oduc B only in he i s pe iod, which
is ne e an equilib ium choice o he e aile . The e aile ne e excludes p oduc B in
he i s pe iod when he p oduc s a e complemen s (close o no ,
∆Π <
0 o
b<
0).
Fu he mo e, he social planne excludes p oduc B om he ma ke only in he i s pe iod
when p oduc s a e no close subs i u es and excludes p oduc B om he ma ke in bo h
pe iods when p oduc s a e close subs i u es. In con as , he e aile ne e excludes p oduc
B only in he i s pe iod, i excludes p oduc B in bo h pe iods o high and posi i e alues
o b.
Games 2024,15, 9 16 o 23
To summa ize, as
b
inc eases ( om
−
1 o 1), he social planne i s ca ies bo h p od-
uc s in bo h pe iods, hen, ca ies only p oduc A in he i s pe iod and bo h p oduc s in he
second pe iod and, inally, i ca ies only p oduc A in bo h pe iods. Howe e , he e aile ,
i s , ca ies bo h p oduc s in bo h pe iods and hen ca ies only p oduc A in bo h pe iods.
No e also in he igu e ha exclusi i y occu s mo e o en in he social op imum se ing
compa ed o he equilib ium ou come whe e he e aile imposes exclusi i y. The a ea o
he pa ame e bwhe e ∆Wis posi i e is g ea e han he one whe e ∆Π is posi i e.
-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 0.6
-3
-2
-1
1
2
3
b
To be e unde s and hese esul s, we p esen a nume ical example. Fo gi en lea ning
and discoun ac o pa ame e s, we plo in Figu e 4 ( o each b) he di¤e ence in he p esen
alue o he wel a e (W=WE
1+WE
2WNE
1+WNE
2, he objec i e unc ion o he
social planne ) and he di¤e ence in he p esen alue o he p o… s ( = E
R1+E
R2
NE
R1+NE
R2, he objec i e unc ion o he e aile om Sec ion 3). We obse e ha ,
when he p oduc s a e close complemen s, he social planne ne e imposes exclusi i y and
his also holds o he e aile (W < 0and <0 o low and nega i e b). Howe e ,
when he p oduc s a e no close complemen s (bis nega i e bu no e y low) he social
planne excludes p oduc B only in he … s pe iod which is ne e an equilib ium choice o
he e aile . The e aile ne e excludes p oduc B in he … s pe iod when he p oduc s a e
complemen s (close o no , <0 o b < 0). Also, he social planne excludes p oduc B
om he ma ke only in he … s pe iod when p oduc s a e no close subs i u es and excludes
p oduc B om he ma ke in bo h pe iods when p oduc s a e close subs i u es. In con as ,
he e aile ne e excludes p oduc B only in he … s pe iod, i excludes p oduc B in bo h
pe iods o high and posi i e alues o b.
To summa ize, as binc eases ( om -1 o 1) he social planne … s ca ies bo h p oduc s
in bo h pe iods, hen, ca ies only p oduc A in he … s pe iod and bo h p oduc s in he
second pe iod and, …nally, i ca ies only p oduc A in bo h pe iods. Howe e , he e aile ,
… s , ca ies bo h p oduc s in bo h pe iods and hen ca ies only p oduc A in bo h pe iods.
No e also in he …gu e ha exclusi i y occu s mo e o en in he social op imum se ing
compa ed o he equilib ium ou come whe e he e aile imposes exclusi i y. The a ea o he
pa ame e bwhe e Wis posi i e is g ea e han he one whe e is posi i e.
By di ec compa ison o he quan i ies in he … s pe iod unde he planning solu ion and
23
ΔΠ
ΔW
Figu e 4. Wel a e and p o i compa isons o λ=1, δ=0.5.
By di ec compa ison o he quan i ies in he i s pe iod unde he planning solu ion
and he equilib ium ou come o Sec ion 3, we conclude o he ollowing p oposi ion.
P oposi ion 5. The a e o lea ning in equilib ium is lowe han he social op imum. The p oduc ion
cos s in he second pe iod a e lowe unde he social op imal due o he highe quan i y p oduced in
he i s pe iod, compa ed o he equilib ium ou come wi hou a social planne .
6. The Role o he In e media y
Finally, we s udy he ole o he e aile in he lea ning p ocess. Is he p esence o such
a e aile indeed necessa y o in ensi y his p ocess? O can i ms ake ad an age o he
lea ning p ocess equally e icien ly wi hou he downs eam monopolis ? Fi s , we analyze
he ole o he e aile in he s a ic model whe e he e is no lea ning and hen we s udy he
dynamic model.
We compa e he equilib ium ou come when he ups eam i ms sell hei p oduc s o
he inal ma ke indi ec ly, ia he e aile , o he equilib ium ou come when he ups eam
i ms dis ibu e hei p oduc s di ec ly o he inal consume s. When he e is no in e medi-
a y, he ou come is he Cou no duopoly ou come wi h di e en ia ed p oduc s and when a
e aile exis s, he ou come is he monopoly ou come (gi en in Sec ion 2). The key is he
p oduc di e en ia ion pa ame e . When he p oduc s a e subs i u es (
b∈(
0, 1
)
) and he e
is no in e media y, he inal p ices and o al p o i s a e lowe and he consume ’s su plus
and o al wel a e a e highe compa ed o he case whe e a e aile exis s.
Rema k 4. In he s a ic model, when he p oduc s a e subs i u es, he p esence o he in e media y
hu s he ma ke , since he e aile gains he monopoly p o i s by inc easing he inal p ices and, hus,
lowe ing he consume ’s su plus and o al wel a e. In con as , when he p oduc s a e complemen s,
he ole o he e aile in he ma ke is posi i e, since i in e nalizes he ex e nali ies om he wo
complemen goods. The p ices a e lowe and he consume ’s su plus, p o i s, and o al wel a e a e
highe when he e exis s a e aile in he downs eam le el.
Games 2024,15, 9 17 o 23
Then, we sol e o he dynamic Cou no di e en ia ed duopoly model wi h lea ning-
by-doing echnology and no in e media y. Bo h p oduc s a e pu chased in bo h pe iods
and he equilib ium quan i ies o he i s pe iod a e gi en by26
qC
i1=(a−c)(2−b)(2+b)2+4δλ
(2−b)(2+b)3−4δλ2 o i=A,B.
We compa e hese quan i ies o he quan i ies in ou equilib ium model wi h he
p esence o he e aile :27
q∗
A1=
(a−c)(2(1+b)+δλ)
4(1+b)2−δλ2NE in bo h pe iods i b ∈−1, b
(a−c)(2+δλ)
4−δλ2E in bo h pe iods i b ∈b, 1
and ob ain ha :
Rema k 5. Fo
b∈(−1, 0)
we ha e
qC
A1<q∗
A1
, o
b∈0, b
we ha e
qC
A1>q∗
A1
and o
b∈b, 1we ha e qC
A1<q∗
A1.
The p esence o he in e media y in ensi ies he lea ning p ocess when he wo p od-
uc s a e complemen s o when he p oduc s a e close subs i u es and exclusi i y is imposed
by he in e media y (i.e.,
b∈(−1, 0)∪b, 1
). The e aile coo dina es he pu chases o he
wo p oduc s and akes ad an age o he lea ning p ocess in a mo e e icien way compa ed
o he case whe e ups eam p oduce s sell hei p oduc s di ec ly o he inal consume s.
The a e o lea ning is close o he social a e o lea ning ( o hese pa ame e alues o
b
)
when he e is an in e media y in he ma ke .
7. Conclusions and Fu he Resea ch
Lea ning-by-doing echnologies can play a signi ican ole in e ical chains. In his
pape , we ha e examined how he lea ning-by-doing p ocess a ec s he inal ma ke
ou come in a e ical amewo k. Ups eam i ms p oduce di e en ia ed p oduc s, ei he
subs i u es o complemen s, and he downs eam monopolis se s he linea con ac e ms.
The uni p oduc ion cos o he ups eam i ms educes wi h he accumula ed p oduc ion.
We s udy how he dynamic in e ac ions be ween ups eam and downs eam i ms a ec
he exclusi e dealing decisions in a wo-pe iod game. The exis ing li e a u e has ei he
examined e ical con ac ing wi hou dynamic in e ac ions due o lea ning e ec s o
he lea ning-by-doing p ocess in an oligopolis ic indus y wi hou e ical conside a ions.
Ou pape is he i s ha s udies a dynamic e ical o eclosu e wi h lea ning-by-doing
p oduc ion echnologies.
In he s a ic model, bo h p oduc s a e ca ied by he e aile . Exclusi i y ne e a ises
in equilib ium, since inal consume s like p oduc a ie y. When we in oduce dynamic
conside a ions, he decision on imposing exclusi i y depends on he p oduc di e en ia ion
pa ame e . Close subs i u abili y leads o exclusi i y in he dynamic model, since he
“lea ning” e ec (leading o lowe p ices) domina es he “p oduc a ie y” e ec . In con-
as , complemen a i y o a lack o close subs i u abili y leads o an equilib ium whe e
he e aile pu chases bo h p oduc s in bo h pe iods. Mo e a ie ies in he ma ke a e
p e e ed o lowe inal p ices. Consume ’s su plus and o al wel a e a e also highe
unde exclusi i y when i is imposed by he e aile compa ed o he case whe e exclusi -
i y is no imposed ( o example, due o egula o y es ic ions). The e o e, exclusi i y is
wel a e-imp o ing. Ne e heless, he equilib ium a e o lea ning is no equal o he social
op imum. A social planne would mo e o en impose exclusi i y and would u he educe
he p oduc ion cos s in he second pe iod compa ed o he e aile . Finally, when p oduc s
a e complemen s o close subs i u es he p esence o he e aile is necessa y o coo dina e
he lea ning p ocess.
Games 2024,15, 9 18 o 23
The e is o en in ense c i icism o a ious exclusiona y p ac ices, which unde lies
economic policy decisions. O en, hese p ac ices a e an i-compe i i e, bu some imes can
be p o-compe i i e. In his pape , we cha ac e ize condi ions unde which exclusi i y is
bene icial o i ms and consume s in a simple dynamic e ical amewo k wi h lea ning-
by-doing p oduc ion echnologies. Ou analysis en iches, hus, he se o esul s conce ning
exclusiona y p ac ices by highligh ing he ole o possible e iciencies h ough cos educ ion.
Fu he esea ch can, o cou se, shed ligh on addi ional aspec s o he issue. In pa -
icula , ou model is kep e y simple ega ding he alloca ion o ba gaining powe and
endows he downs eam i m wi h he abili y o se linea wholesale p ices. In a mo e
complex se ing, whe e he ba gaining powe may be mo e e enly dis ibu ed ac oss he
ups eam and downs eam i ms, addi ional ensions will likely a ise, pa ly e lec ing
esul s om ea lie esea ch on downs eam ‘bo leneck’ models. I he ups eam i ms
ac i ely pa icipa e in he p icing decisions, he implici coo dina ing ole o he e aile
would end o be diminished, pe haps also leading o a educed abili y o mo i a e and
exploi lea ning-by-doing. In pa allel, compe i ion be ween he ups eam i ms may end
o be mo e in ense in he i s s age o he game, so ha hey s a egically imp o e hei
e iciency in he second s age, and pa ly bene i om his cos educ ion. How he cos
educ ion e sus p oduc a ie y ension will play ou in such a ich se ing, and how
p o i s will be alloca ed among he i ms, would o cou se la gely depend on he ype o
con ac s ha can be employed. Finally, i would be in e es ing o ex end he p esen model
by conside ing al e na i e demand speci ica ions and lea ning-by-doing p ocesses.
Au ho Con ibu ions: Bo h au ho s con ibu ed o he s udy concep ion, design and w i ing equally.
All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding: This esea ch ecei ed no ex e nal unding.
Da a A ailabili y S a emen : No new da a we e c ea ed o analyzed in his s udy.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Appendix A
All exp essions o Lemma 3:
qNE
i1=(a−c)(2(1+b) + δλ)
4(1+b)2−δλ2,qNE
i2=a−c
2(1+b)+λ(a−c)(2(1+b) + δλ)
2(1+b)(4(1+b)2−δλ2),
pNE
i1=2(b+1)2(a+c)−aλ2δ−λδ(a−c)(b+1)
4(1+b)2−δλ2,
pNE
i2=2(b+1)2(a+c)−aλ2δ−λ(a−c)(b+1)
4(1+b)2−δλ2,
wNE
i1=c,wNE
i2=4c(1+b)2−λ2δ−2λ(a−c)(1+b)
4(1+b)2−δλ2,
ΠNE
i1=ΠNE
i2=0,
ΠNE
R1=2(a−c)22(1+b)2−λ2δ−λδ(1+b)(2b+λδ +2)
(4(1+b)2−δλ2)2,
ΠNE
R2=2(a−c)2(2b+λ+2)2(1+b)
(4(1+b)2−δλ2)2.
All exp essions o Lemma 4:
Games 2024,15, 9 19 o 23
qE
A1=
(1−b)(a−c)(2b+λδ+2)
4(1−b2)−δλ2i b ≤4−λ2δ
2(λ+2)bo h in = 2
(a−c)(2+δλ)
4−δλ2i b >4−λ2δ
2(λ+2)only A in = 2 ,
qE
A2=
(a−c)(2(1−b)+λ)
4(1−b2)−δλ2i b ≤4−λ2δ
2(λ+2)
(a−c)(2+λ)
4−δλ2i b >4−λ2δ
2(λ+2)
,qE
B2=
(a−c)(4−δλ2−2b(λ+2))
2(4(1−b2)−δλ2)i b ≤4−λ2δ
2(λ+2)
0i b >4−λ2δ
2(λ+2)
,
pE
A1=
2(1−b)(b+1)(a+c)−δλ((1−b)(a−c)+aλ)
4(1−b2)−δλ2i b ≤4−λ2δ
2(λ+2)
c(λδ+2)+a(2−δλ(λ+1))
4−δλ2i b >4−λ2δ
2(λ+2)
,
pE
A2=
2(1−b)(b+1)(2(a+c)−λ(a−c))−δλ2(2a−b(a−c))
2(4(1−b2)−δλ2)i b ≤4−λ2δ
2(λ+2)
c(λ+2)+a(2−λ(λδ+1))
4−δλ2i b >4−λ2δ
2(λ+2)
,pE
B2=
a+c
2i b ≤4−λ2δ
2(λ+2)
−i b >4−λ2δ
2(λ+2)
,
wE
A1=c,wE
B1<c,wE
A2=
2(1−b)(b+1)(2c−aλ+cλ)−δλ2(a−b(a−c))
4(1−b2)−δλ2i b ≤4−λ2δ
2(λ+2)
c(2λ+4)−aλ(λδ+2)
4−δλ2i b >4−λ2δ
2(λ+2)
,wE
A2=c,
ΠE
A1=ΠE
A2=ΠE
B2=0,
ΠE
R1=
(a−c)2(1−b)(2(1−b)(b+1)−δλ(1−b+λ))(2(1+b)+λδ)
(4(1−b2)−δλ2)2i b ≤4−λ2δ
2(λ+2)
(a−c)2(2+δλ)(2−δλ−δλ2)
(4−δλ2)2i b >4−λ2δ
2(λ+2)
,
ΠE
R2=
(a−c)2(λ4δ2−8λ2δ(1−b)(b+1)+4(1−b)(b+1)(λ(λ+4(1−b))+8(1−b)))
4(4(1−b2)−δλ2)2i b ≤4−λ2δ
2(λ+2)
(a−c)2(2+λ)2
(4−δλ2)2i b >4−λ2δ
2(λ+2)
.
All exp essions o P oposi ion 19:
q∗
A1=
(a−c)(2(1+b)+δλ)
4(1+b)2−δλ2NE in bo h pe iods i b ∈−1, b
(a−c)(2+δλ)
4−δλ2E in bo h pe iods i b ∈b, 1,
q∗
B1=
(a−c)(2(1+b)+δλ)
4(1+b)2−δλ2i b ∈−1, b
−i b ∈b, 1,
q∗
A2=
a−c
2(1+b)+λ(a−c)(2(1+b)+δλ)
2(1+b)(4(1+b)2−δλ2)i b ∈−1, b
(a−c)(2+λ)
4−δλ2i b ∈b, 1,
q∗
B2=
a−c
2(1+b)+λ(a−c)(2(1+b)+δλ)
2(1+b)(4(1+b)2−δλ2)i b ∈−1, b
−i b ∈b, 1,
p∗
A1=
2(b+1)2(a+c)−aλ2δ−λδ(a−c)(b+1)
4(1+b)2−δλ2i b ∈−1, b
c(λδ+2)+a(2−δλ(λ+1))
4−δλ2i b ∈b, 1,
p∗
B1=
2(b+1)2(a+c)−aλ2δ−λδ(a−c)(b+1)
4(1+b)2−δλ2i b ∈−1, b
−i b ∈b, 1,
Games 2024,15, 9 20 o 23
p∗
A2=
2(b+1)2(a+c)−aλ2δ−λ(a−c)(b+1)
4(1+b)2−δλ2i b ∈−1, b
c(λ+2)+a(2−λ(λδ+1))
4−δλ2i b ∈b, 1,
p∗
B2=
2(b+1)2(a+c)−aλ2δ−λ(a−c)(b+1)
4(1+b)2−δλ2i b ∈−1, b
−i b ∈b, 1,
w∗
A1=c,w∗
B1=
c i b ∈−1, b
<c i b ∈b, 1,
w∗
A2=
4c(1+b)2−λ2δ−2λ(a−c)(1+b)
4(1+b)2−δλ2i b ∈−1, b
c(2λ+4)−aλ(λδ+2)
4−δλ2i b ∈b, 1,
w∗
B2=
4c(1+b)2−λ2δ−2λ(a−c)(1+b)
4(1+b)2−δλ2i b ∈−1, b
−i b ∈b, 1,
Π∗
A1=Π∗
B1=Π∗
A2=Π∗
B2=0,
Π∗
R1=
2(a−c)2(2(1+b)2−λ2δ−λδ(1+b))(2b+λδ+2)
(4(1+b)2−δλ2)2i b ∈−1, b
(a−c)2(2+δλ)(2−δλ−δλ2)
(4−δλ2)2i b ∈b, 1,
Π∗
R2=
2(a−c)2(2b+λ+2)2(1+b)
(4(1+b)2−δλ2)2i b ∈−1, b
(a−c)2(2+λ)2
(4−δλ2)2i b ∈b, 1.
The ele an exp essions o non-exclusi i y unde he social op imum:
qNE−S
i1=(a−c)(b+λδ +1)
(1+b)2−λ2δ,qNE−S
i2=(a−c)(b+λδ +1)
(1+b)2−λ2δ,
pNE−S
i1=c(b+1)(b+λδ +1)−aλδ(b+λ+1)
(1+b)2−λ2δ
<c,
pNE−S
i2=c−λ(a−c)(b+λδ +1)
(1+b)2−λ2δ=ci2,
ΠNE− o al−S
1=−2(a−c)2(b+λδ +1)λδ(b+λ+1)
(1+b)2−λ2δ2<0, ΠNE− o al−S
2=0,
CSNE−S
1=(a−c)2(b+1)(b+λδ +1)2
(1+b)2−λ2δ2≥0,
CSNE−S
2=(a−c)2(b+1)(b+λ+1)2
(1+b)2−λ2δ2≥0,
WNE−S
1=(a−c)2(b+λδ +1)2b−λδ −2λ2δ+b2−bλδ +1
(1+b)2−λ2δ2,
WNE−S
2=(a−c)2(b+1)(b+λ+1)2
(1+b)2−λ2δ2≥0.
The ele an exp essions o exclusi i y unde he social op imum:
Games 2024,15, 9 21 o 23
qE−S
A1=
(a−c)(1−b)(b+λδ+1)
(1−b2)−δλ2i b ≤1−δλ2
1+λbo h in = 2
(a−c)(1+λδ)
1−δλ2i b >1−δλ2
1+λonly A in = 2 ,
qE−S
A2=
(a−c)(1−b+λ)
(1−b2)−δλ2i b ≤1−δλ2
1+λ
(a−c)(1+λ)
1−δλ2i b >1−δλ2
1+λ
,
qE−S
B2=
(a−c)(1−b−bλ−λ2δ)
(1−b2)−δλ2i b ≤1−δλ2
1+λ
0i b >1−δλ2
1+λ
,
pE−S
A1=(c−b2c−aλδ+cλδ−aλ2δ+abλδ−bcλδ
(1−b2)−δλ2<c i b ≤1−δλ2
1+λ
c−aλδ+cλδ−aλ2δ
1−δλ2<c i b >1−δλ2
1+λ
,
pE−S
A2=(c−aλ+cλ−b2c+ab2λ−b2cλ−aλ2δ+abλ2δ−bcλ2δ
(1−b2)−δλ2=cA2i b ≤1−δλ2
1+λ
c−aλ+cλ−aλ2δ
1−δλ2=cA2i b >1−δλ2
1+λ
,
pE−S
B2=(c i b ≤1−δλ2
1+λ
−i b >1−δλ2
1+λ
,
ΠE− o al−S
1=
(a−c)2(b−1)λδ(1−b+λ)(b+λδ+1)
((1−b2)−δλ2)2<0i b ≤1−δλ2
1+λ
−(a−c)2(λδ+1)λδ(λ+1)
(1−δλ2)2<0i b >1−δλ2
1+λ
,
ΠE− o al−S
2=(0i b ≤1−δλ2
1+λ
0i b >1−δλ2
1+λ
,
CSE−S
1=
(a−c)2(b−1)2(b+λδ+1)2
2((1−b2)−δλ2)2i b ≤1−δλ2
1+λ
(a−c)2(λδ+1)2
2(1−δλ2)2i b >1−δλ2
1+λ
,
CSE−S
2=
(a−c)2(λ4δ2+(1−b2)(2λ−2b−2bλ+λ2−2λ2δ+2))
2((1−b2)−δλ2)2i b ≤1−δλ2
1+λ
(a−c)2(λ+1)2
2(1−δλ2)2i b >1−δλ2
1+λ
,
WE−S
1=
(a−c)2((1−b)(b+2δ+2λδ+1)−λ2δ2)
2((1−b2)−δλ2)i b ≤1−λ2δ
1+λ
(a−c)2(δ+2λδ+1)
2(1−λ2δ)≥0i b >1−λ2δ
1+λ
,
WE−S
2=
(a−c)2(λ4δ2+(1−b2)((2λ−2b−2bλ+λ2+2λ2δ+2)−4λ2δ))
2((1−b2)−δλ2)2i b ≤1−δλ2
1+λ
(a−c)2(λ+1)2
2(1−δλ2)2i b >1−δλ2
1+λ
.
No es
1
Exclusi i y con ac s o e usal o deal a e o en iewed wi h suspicion by he An i us Au ho i ies. No e ha exclusion may
a ise in a ma ke h ough e ical in eg a ion oo.
2
Ma hewson and Win e (1987) [
5
] a gue ha manu ac u e s unde exclusi e dealing a angemen s compe e on he basis o
wholesale p ices o he igh o be selec ed by he e aile . Besanko and Pe y (1993) [
6
], in a di e en ia ed p oduc s oligopoly,
suppo ha exclusi e dealing can elimina e in e -b and ex e nali ies due o inc eased p omo ional in es men s. O’B ien and
Sha e (1993) [7] examine co ne solu ions in an oligopolis ic e ical se ing.
3
The e is also wo k on e ical con ac ing wi h in en o ies and dynamic conside a ions o enego ia ing con ac s, e.g., Jong-Say
Jong (1999) [19] o Taylo and Plambeck (2006) [20].
4
I he ba gaining powe was mo e e enly dis ibu ed ac oss he ups eam and downs eam i ms, addi ional ensions would a ise,
pa ly e lec ing esul s om ea lie esea ch on downs eam ‘bo leneck’ models. A sho discussion on such u he esea ch is
delega ed o he Conclusion.
Games 2024,15, 9 22 o 23
5
The ep esen a i e consume is cha ac e ized by he quad a ic and s ic ly conca e u ili y unc ion
U(qA
,
qB) = a(qA+qB)−
q2
A+q2
B+2bqAqB
2(see Singh and Vi es (1984)) [28].
6
An al e na i e way o o eclosu e an inpu supplie is he e aile o explici ly sign an exclusi e dealing con ac wi h he i al
manu ac u e .
7
We assume ha i he manu ac u e is indi e en be ween ob aining ze o p o i s om p oduc ion and no p oducing a all, hey
will choose o p oduce.
8A mo e de ailed discussion abou he consume s’ su plus and he o al wel a e is de e ed o Sec ion 4.
9
The analysis he e con ains he subcase (i) non-exclusi i y in he i s pe iod, i.e., no subsequen cos asymme y, and non-
exclusi i y in he second pe iod, and (ii) exclusi i y in he i s pe iod, i.e., subsequen cos asymme y, and non-exclusi i y in he
second pe iod. Ne e heless, a co ne solu ion may a ise in he la e case.
10
The analysis he e con ains he subcase (i) non-exclusi i y in he i s pe iod and exclusi i y in he second pe iod, and (ii)
exclusi i y in he i s pe iod and exclusi i y in he second pe iod. P oduc ion cos cA2is a unc ion o qA1.
11 Since a(2a−cA2−cB2)
4(1+b)+(a−cA2)(bcB2−cA2)+(a−cB2)(bcA2−cB2)
4(1−b2)≥(a−cA2)2
4.
12
The subcase non-exclusi i y in pe iod one, i.e., subsequen cos symme y, and exclusi i y in pe iod wo, canno be an equilib ium
ou come.
13
We ob ain posi i e quan i ies and he second-o de condi ions a e sa is ied when 4
(
1
+b)2−λ2δ>
0, 4
(1−b)2−λ2δ>
0, 41−b2−λ2δ>0.
14
All ele an exp essions ob ained unde non-exclusi i y a e in he Appendix A. This subcase co esponds o non-exclusi i y in
bo h pe iods.
15 This subcase co esponds o exclusi i y in pe iod one and non-exclusi i y in pe iod wo.
16 This subcase co esponds o exclusi i y in bo h pe iods.
17
We need 4
1−b2−δλ2>
0 o he second-o de condi ions o be sa is ied and o ob ain posi i e quan i y in pe iod one.
O he wise, no p oduc ion occu s. All emaining exp essions o his Lemma a e in he Appendix A.
18
(N)E s ands o (non-)exclusi i y. We ha e al eady p o ed ha NE in pe iod one and E in pe iod wo, canno be an equilib ium
ou come.
19 We ha e 1 >b=√λ4δ4−2λ2δ3(λ2+2)+δ2(λ(24λ+λ3+32)+16)+δ(4(λ(8−λ)+8))+16−λδ(λ(1+δ)+4)
4(δ+λδ+1)≥4−λ2δ
2(λ+2)>0
20 Fo b≤4−δλ2
2(λ+2), we ob ain ∆Π =ΠE
R1+δΠE
R2−ΠNE
R1+δΠNE
R2<0.
21
Fo
b∈(4−λ2δ
2(λ+2)
,
b)
we ob ain
∆Π <
0 while o
b∈(b
, 1
)
we ob ain
∆Π >
0. The e o e, i is ne e an equilib ium ou come o
pu chase p oduc A in he i s pe iod and bo h p oduc s in he second pe iod.
22 See Singh and Vi es, 1984 [28]
23
We p o e ha
d(CSE
−CSNE
)
db =qNE
i −qNE
i −2(1+b)dqNE
i
db >
0, wi h
dqNE
i
db <
0,
=
1, 2, ha is,
∆CS
is inc easing in
b
.
Fu he mo e, we ob ain ha ∆CS(b=0)<0 and ∆CS(b=1)>0.
24
To ob ain posi i e quan i ies and o sa is y he second-o de condi ions, we should ha e
(1+b)2−δλ2≥
0,
(1−b)2−δλ2>
0
and 1−b2−δλ2>0.All ele an exp essions ob ained in his case a e in he Appendix A.
25
We need
1−b2−δλ2>
0 o he second-o de condi ions o be sa is ied and o ob ain posi i e quan i y in pe iod one. All
ele an exp essions ob ained in his case a e in he Appendix A.
26 Fo he second pe iod, we ha e qC
i2=a−c
2+b+λ(a−c)((2−b)(2+b)2+4δλ)
(2+b)((2−b)(2+b)3−4δλ2).
27 Wi h he p esence o he e aile , we also ob ain q∗
B1=
(a−c)(2(1+b)+δλ)
4(1+b)2−δλ2i b ∈−1, b
−i b ∈b, 1.
Re e ences
1.
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[C ossRe ]
2.
Rey, P.; Ti ole, J. A P ime on Fo eclosu e. In Handbook o Indus ial O ganiza ion III; A ms ong, M., Po e , R.H., Eds.; No h
Holland: Ams e dam, The Ne he lands, 2007. A ailable online: h ps://econpape s. epec.o g/bookchap/eeeindchp/3-33.h m
(accessed on 7 Feb ua y 2024).
3.
K a enmake , T.; Salop, S. Compe i ion and coope a ion in he ma ke o exclusiona y igh s. Am. Econ. Re . 1986,76, 109–113.
4. Aghion, P.; Bol on, P. Con ac s as a ba ie o en y. Am. Econ. Re . 1987,77, 388–401.
5. Ma hewson, F.; Win e , R. The compe i i e e ec s o e ical ag eemen s: Commen . Am. Econ. Re . 1987,77, 1057–1062.
6.
Besanko, D.; Pe y, M.K. Equilib ium incen i es o exclusi e dealing in a di e en ia ed p oduc s oligopoly. Rand J. Econ. 1993,24,
646–667. [C ossRe ]
7.
O’B ien, D.P.; Sha e , G. On he dampening-o -compe i ion e ec o exclusi e dealing. J. Ind. Econ. 1993,41, 215–221. [C ossRe ]
Games 2024,15, 9 23 o 23
8.
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