Yves R ichel le
University of Laval(Queb ec)
and
Paolo G. Garel la
Univ ersitadi Bologna
marc h1995
JEL Classication: L13, L41
Abstract
The pap er analyzes the question of whic h cost c haracteristicsare exhibited
bythe rmsthatexitanoligop olistic mark etwhencostsareasymmetricand
rmscan crediblyb eforced outbythe remainingcompetitors. Themain
re-sultsare: (i)ifreen tryisimpossible(duetothepresenceoflargesunkcosts),
then the rm with the highest marginal cost function dtays in; if reen try is
costlessthen therm with the highestaveragecost exits. Consequentysunk
costs not onlyaect the numb erof rms in anindustry,but they also en ter
the determination of the typ eof rms that resist predation.
Inmark etswherermsdierastotheircostfunctionsisitp ossibletopredict
what are the cost c haracteristicsof the rms that stayor that exit? In p
er-fectly competitivemark etsone canpredict thatthe rms exitingthe mark et
are those with highest average costs. This prediction has b een extended by
Ghema watand Nalebu (1985), (1990)and Fudenb ergand Tirole (1986)to
the case of declining industries with few competitors. Their analyses show
indeed that, in a war of attrition, the less ecien t rm will b e the rst to
exit 1
. However many recen t works (see c hapters 8 and 9 in Tirole (1988)
and Wilson (1992)) show that exit can o ccur in a wide variet y of
circum-stances. We are thereforeled to askif the ab ov epredictioncontinue tohold
in imperfectlycompetitiv emark etswhere rms are not engaged in a war of
attrition.
This pap er argues that the exiting rm may b e the one with the lowest
averagecost function. Toiden tifythe basic argumen tleadingtothis
conclu-sion, consider the follo wing example. Three rms decide,at a rst stage, to
stay in the mark et or to exit and, at a second stage, those that staydecide
how m uchto pro duce. All rms have iden ticalxed costs. They also have
constant marginal costs with rm i 's marginal cost b eing strictly smaller
than rm j's marginal cost whic h, in turn, is strictly smaller than rm k's.
Firms can therefore b eranked according to their av erage
cost function with
rm i having the lowestone.
Thensupp osethatifallrmsstayinthemark et,eac hofthemwillobtain
a strictly negativ eprot at the Cournot equilibrium while, if only tworms
stay in, their Cournot prot is p ositiv e and the third rm receiv es a zero
prot. It immediately follo ws that for eac h couple of rms to stay in the
mark et and to pro duce their Cournot quatit y is an equilibrium of the two
stage game. They are thereforethree equilibriaand aprediction onthe cost
c haracteristics of the exiting rm cannot b e based only on this two stage
game. Note,inciden tally,that theseequilibriaare allParetoecien tsothat
one cannot use coalitional pro ofness (see Bernheim, Peleg and Whinston
1
For instance, Ghema wat and Nalebuf (1985) show that in a war of attrition with
completeinformationwherermsdieraccordingtotheirpro ductioncapacity,thebiggest
rmisthersttoexit. Buttheseauthorsassumethatrmsincuronlya o wmain tenance
costwhichis prop ortionalto theircapacity. Accordingly,thebiggestrmistheone with
A p ossible route to follo w for obtaining a prediction is indicated by
the literature on endogenous coalition formation, as in Aumann and
My-erson(1988), Gul (1989), and esp ecially Bloc h (1990a) and (1990b). These
works use a non-coop erativ e sequen tial game to analyze the formation of
coalitionstructures. Inthesameway,onecanassumethatacoalition
forma-tiongame precedesthe playof agameofthe kindillustrated bythe example
ab ov e. Wehere adopt a sp ecication of the coalition formationgame where
eac hrminturnmak esadeclarationconsisting of(i)asetofrmsthat stay
in, (ii) a pay o v ector for the three rms that can b e obtained by the play
of a non-cooperativ e equilibrium of the two stage game. One can in terpret
these declarations as \oers", and we model the acceptance (refusal) of an
oerasthemakingofaniden tical(dieren t)declaration. Sinceadeclaration
corresp onds toone equilibrium, if tworms mak ethe same declarationthey
agree to play the same equilibrium. This determines whic h equilibrium is
played and the payos to all rms irrespectiv e of the declaration made by
the third rm.
It is importantto realizethat oncetworms hav eadopted their
equilib-rium strategies the third rm has no b etter alternativ ethan the play of its
ownb estreplytothosestrategies,whic hcoincideswiththestrategysp ecied
in the equilibrium c hosenby the other tworms. Thereforethe waypay os
are determinedhas nothing todo with the application of a majority rule in
collectiv edecision making.
For further reference we call this sequen tial game \the cartel formation
game". Theequilibriumofthisgamegiv esapredictionofthermthatexits.
Foreac horderinwhic hrmsdeclare,therewillb eauniquesubgamep erfect
equilibrium outcome inthe cartel formation game. But, as one can expect,
the equilibrium outcome will in general dep end on the order of declaration.
We are nev ertheless able to show that, as long as a rm exits the mark et
atthe equilibrium, the cost c haracteristicsof this rm can b e iden tiedand
are independen tof the order of declaration. The cartel formationgame will
therefore provide a strong prediction on the c haracteristics of the exiting
rm.
In the example the unique equilibrium is with rms j and k making the
same declaration of the form (i) fj;kg, and (ii) pay o zero for rm i , and
Cournot pay osforj andk. A pro ofof this statemen t istrivial. Indeedthe
mark et and rm k mak es the highest equilibrium prot when j stays on.
Hence the rm exiting the mark et is the one with the lowest av erage costs,
and not withthe highest, asitwouldb e predicted inawarof attrition orin
p erfect competition.
One is led to wonder if the dierence in prediction could disapp ear if
rms playa sup ergame insteadof a one-shotgame. Indeed, inasup ergame,
rmsare generallyable tomaximizejointprotsand,since jointprot
max-imization requires the minimization of variablecost, theywill b e inducedto
in ternalizethe gain made byhaving anecien tpartner.
In what follo wswe shall generalizethe examplegiv enab ov eby
consider-ing a general cost function and a pro duction game consisting of an innite
rep etition of the two-stage game of the example. In this game, we saythat
a two-rmcartel is feasible if there exists an equilibrium of the pro duction
game where these rms stay in and the third stays out along the
equilib-riumpath. Obviouslythe analysis isin terestingonlyif the pro duction game
displays atleast twodieren t feasiblecartels.
The main results are that (i) if reen try is impossible, then the rm with
the highestmarginal cost function stayson atall equilibriaof the cartel
for-mation game, for anyorder of declaration, while xed costs only determine
the set of feasible cartels. (ii) If reen try is p ossible then the rm with the
highest averagecost exits. These resultsimply that the dierence in
predic-tiondo es not dep endonthe p ossibilit yornot tocollude,but ratherdep ends
up on the existenceor not of sunk costs forreen try.
A nov elimplication of the presence of sunkcosts app ears here: not only,
asitisalready wellknownfromthe literatureonen trypreemption,theycan
determine the numb erof rms,but theyalso en terthe determination ofthe
typ e ofrms that stayina mark et.
The pap er is organised as follo ws: in the next Section we in tro duceour
assumptions relativ e to the cost and demand functions and we analyse the
equilibrium outcomes of the pro duction game. In Section 3, the cartel
for-mation game is formally presen ted. Our results are stated in Section 4 for
the unprotablereen trycase andinSection5for thecase ofcostlessreen try.
In Section6wetest the robustnessof the resultsfor the case ofunprotable
reen trytoc hangesinthecartelformationgame. Theresultsstatedinsection
4areshowntogothrough. Section7presen tssomeconcludingconsiderations
We consider a sup ergame in volving three rms. We shall denote this game
by 0
and the set of rms by N. 0
consists of the innite rep etition of
the two-stagegamewhere(i)atthe rststageeac hrm decidestostayinor
to stay out of the mark et and (ii) at the second stage the rms whic hhave
decided tostayin the mark et,hereafter referred to asthe activ erms,play
an usual Cournot game whilst an inactive rm pro duces nothing. A t eac h
stage decisionsare made sim ultaneouslyand actions taken atthe rst stage
arep erfectlyobservedbyallrmsb eforetheyc ho osetheirpro duction atthe
secondstage. The scalar,b elonging totheop en in terval(0;1), denotesthe
discount factor common to allrms.
The purp ose of this preparatory section istwofold. On the one hand, we
giv eaprecisecontenttothe conceptof afeasible cartel. Onthe other hand,
for eac h feasible cartel s, we c haracterize the set of all payo v ectors that
rms can obtain at a subgame p erfect Nash equilibrium of 0
where along
the equilibrium path only the rms in cartel s stay in the mark et at eac h
p eriod.
Tosimplify the exposition we shall pro ceed in three steps. First, in
sub-section 2.1., we shall in tro duce the assumptions on the cost and demand
functions. In the second step, in subsection 2.2., we shall ignore the rst
stageof theconstituen tgameandconcen trateonthegame 0
(s) consisting
oftheinniterep etitionoftheCournotgamewherethesetofplayersisgiv en
bys(i.e. rmsinsdecidetostayinthe mark etatev eryp eriodandthe rm
outside s, if s 6=N, decides tostay out of the mark et atev ery p eriod). We
can thereb y use the results fromthe literature on innitely rep eated games
to bring forth a c haracterization of the set of equilibrium pay o v ectors of
0
(s). Inthe last step,subsection 2.3.,we in tro ducethe p ossibilit yforeac h
rm to exit the mark et. This will allow us to dene what we mean by a
feasible cartel and c haracterize,foreac h feasible cartel,the setof attainable
payo v ectorsV
(s).
2.1 Assumptions
Anactiv ermhas topaya(time-in variant)xedcost F
i
aswellasavariable
costgiv enbythefunctionc(q
i ; i ;q i )where i andq i
are(time-in variant)
rm-sp ecicparametersand q
i
confusion). If a rm decides to stay out, it pro duces nothingand incurs no
cost. Furthermore if arm, sayi , has decidedto stayout atp eriod t01,it
m ust pay a reen try cost, R
i
, if it decides to stay in the mark et at p eriod t .
Wesimplify the analysis byconsidering in turn twop olar cases namely:
Assumption 1 Reentry is unprotable i.e. R
i
is as large as we want, for
i=1;2;3.
Assumption 2 Reentryis costless i.e. R
i
=0, for i=1;2;3.
The variable cost function c dep ends up on two rm-specic parameters,
i and q i . q i
stands for the rm i 's capacit y constraint whic h means that,
for a giv en
i
, rm i cannot pro duce more than q
i
and accordingly cis only
dened for 0 q
i q
i
. On the other hand,
i
is a convenient wayto rank
rmsaccordingtotheirmarginalcostfunction. Weshallindeedsupp osethat
for any quantity q such that the marginal cost to pro duce this quantit y is
well dened for rms i and j, rm j's marginal cost is strictly greater than
the one of rm i if and only if
j >
i
. Moreprecisely way, let X
i
= [0;q
i ],
thenour assumptionsregardingthe variablecostfunctionofanyrmarethe
follo wing:
Assumption 3 Letq
i
>0. Thevariable cost function is twice continuously
dierentiable with respect to q
i and i on X i 2R ++ . In addition, c satises
the following properties:
1. c(0; i ;q i )=0, 8 i 2R ++ , 2. 0@c(q i ; i ;q i )=@q i i , i 2]0;1 [ ,and@c(q i ; i ;q i )=@ i >0,8( i ;q i )2 R ++ 2X i , 3. @ 2 c(q i ; ;q i )=@q i @ i >0, 8( i ;q i )2R ++ 2X i .
NowletQstandforaggregateoutput. A teac hp eriod,thein versedemand
function for the homogeneousgo o d, denoted f(Q ),satises:
Assumption 4 For all Q 2 [0; P 3 i=1 q i
] , f is twice continuously
b e able to concen trateourselv es only up on the in uence of the cost c
harac-teristics onmark etstructure.
For an activ e rm the prot function (gross of the reen try cost) in the
Cournot game will b e written as:
i (q i ;Q 0i )=f(q i +Q 0i )q i 0c(q i ; i ;q i )0F i (1) where Q 0i =Q0q i . We shall assume:
Assumption 5 Foralli2N,
i isstrictlyquasi-concaveonX i 2 [0; P j6=i q j ] .
Furthermore there exists (q
1 ;q 2 ;q 3 ) 2 X 1 2 X 2 2X 3
such that, for all i ,
i (q i ;Q 0i )>0.
Obviouslytheseassumptions,togetherwiththerestrictionthatanyactiv e
rm i m ust c ho ose a quantit y in [0;q
i
], are sucient for the existence of a
Cournot equilibrium. The second part of Assumption 5 will ensure, as we
shall see later on, that there exists some \agreements" b et ween the three
rms with all of them remaining on the mark et. This could b e assumed
away, infact simplifying the analysis without c hanging the results, but it is
k eptfor the sakeof generalit y.
The last restriction on the cost and demand functions is that whenev er
only tworms are activ ethen,for anyquantityits opp onent can pro duce,a
rmcanachieveap ositiv eprot. Formally,letusdenew
i
(s)withs=fi;jg
asthe minimal pay ormi can guarantee toitself whenit facesrm j, i.e.:
w i (i;j)= min q j 2 X j max q i 2 X i i (q i ;q j ) Under assumption 4 i
is a strictly decreasing function of q
j . Therefore, dening q R i (q j )=arg max q i 2 X i i (q i ;q j ), we have: w i (i;j)= i (q R i (q j );q j ) Weshall require: Assumption 6 For all i;j 2N, w i (i;j)0and q R i (q j )<q i .
This will guarantee on the one hand that the mark et cannot b e
mo-nop olized and on the other hand that there exist couples (q
i ;q j ) such that i (q i ;q j ) > w i
(i;j). We now turn to the c haracterization of the set of
sub-gamep erfectequilibriumpayov ectorsoftheinnitelyrep eatedgame0
(s)
The typical pay ofor rm i in the game 0
(s) is giv enby: P i =(10) 1 X t =0 t i (q i ;Q 0i )
Innitely rep eated games with discounting have b een extensiv ely analysed
in the literature. It has b een established (see, for instance, Theorem 3.2
in Sorin (1992)) that the set of subgame p erfect Nash equilibrium payo
v ectors of 0
(s) converges (with resp ect tothe Haussdorf top ology) to the
setofindividuallyrationalandfeasiblepay ov ectorsoftheconstituen tgame
asthediscountfactortendstoone 2
. Thisisoneofthev ersionoftheso-called
Folk Theorem. Accordingly, for our purp oses we only need to c haracterize
the set of individually rational and feasible payo v ectors of the Cournot
one-shot game where the set of players is giv en by s. This set is denoted
W(s).
To b egin with letus denote the set of feasible payo v ectors with three
activ e rms by F(N) and the one with two activ e rms by F(i;j). Let
X =X i 2X j 2X k
,F(N) and F(i;j) are giv enby:
F(N)=convexhull f(P 1 ;P 2 ;P 3 ) j 9 (q 1 ;q 2 ;q 3 )2X suchthat P i = i (q i ;Q 0i )for i=1;2;3g:
F(i;j)=conv exhull f(P
i ;P j ) j 9 (q i ;q j ;0)2X such that P i = i (q i ;Q 0i ) and P j = j (q j ;Q 0j )g:
When the three rms are activ e, eac h activ e rm can guarantee to itself a
prot giv enby:
w i (N)= min qj2 Xj;q k 2 X k max qi2 Xi i (q i ;q j +q k )
Accordinglythe set of individuallyrationaland feasible payo v ectorswhen
rms ins b eing activ eand # s2is simply:
W(s)=f(P 1 ;P 2 ;P 3 )j (P i ) i2 s 2F(s) ; P k =0 for k62s and P i w i (s) 8i2sg: (2) 2
Providedthesetofindividuallyrationalandfeasiblep oin tshasa non-empt yin terior.
byW
(s) ,we hav efromthe FolkTheoremthat W
(s) convergestoW(s)
when tends to one. We shall furthermore denote by w
i
(s) the minimal
payo rm ican obtain inW
(s).
2.3 Equilibrium payos in 0
We now consider the game 0
, namely the innite rep etition of the
two-stage constituen t game where, at the rst stage, rms decide to stay in or
to stay out of the mark et and, at the second stage, activ e rms decide the
quantitytheypro duce. LetV
(s) denotesthesetofattainablepay ov ectors
for cartels. A tthe end of this sectionit willp ossible toc haracterizeV
(s) .
Note that this set will dep end on whic h of the two assumptions, 1 or 2, is
taken tohold.
To b egin with, letus remark that, from assumption 5, there will always
existsubgame p erfectequilibriain0
wherealong theequilibrium paththe
threermsstayinthemark etateac hp eriod. Thesetof payov ectorswhic h
canb eobtainedatequilibriaofthiskindisdenotedbyV
(N) anditiseasily
v eriedthat itcoincides with W
(N)\R 3
+
. Weshall furthermoredenoteby
e
k
(N)a subgame p erfectequilibrium of 0
wherethe three rms stayinat
eac hp eriodalongtheequilibriumpathandwherermk obtainsap er-p eriod
prot equal to maxf0;w
k
(N)g and eac h other rm, sayh, obtains a prot
strictly greater than max f0;w
h (N)g
3
.
Then let us supp ose that there is a rm, say k, such that w
k
(N) < 0.
Consider rst the existence of subgame p erfect equilibria of 0
where rm
k staysout ofthe mark etateac hp eriod 4
. Forsucien tlyclosetoone,the
follo wingstrategycom binationisasubgame p erfectNashequilibrium ofthe
subgame starting after the three rms hav e decided to stay in the mark et
at a giv en p eriod: pro duce (q
i ; q j ; q R k (q i +q j
)) follo wed by, wathever the
quantitypro duced byrm k, either e
k
(N) if rms i and j hav epro duced or
3
Note that there are many output com binationsthat giv e rm k a pa yo equal to
w
k
(N). Forsomeofthemthepa yostoi andj willb eas sp eciedine
k (N).
4
Weimplicitlyassumethatrmk'soutsideopp ortunitieshasap erp erio dvalueofzero.
Howeverthe readerwill easily verifythat all ourresults gothroughif we ha veassumed
thatthep erp erio dvalueofrmi'soutsideopp ortunities,i=1 ;2 ;3,isgiv enbyafunction
O(
i
)satisfying@O=@
i
0. Thep oin tistodenew^
i
(s)=w
i
(s)0O(
i
)forallp ossible
cartels andworkwithw^
i
(s)insteadofw
i (s) .
i j i
q
i
while rm j has pro duced q
j , or e
j
(N) if rm j did not pro duce q
j while
rm i has pro duced q
i
. This is an equilibrium, in the subgame starting at
theno de wherethethreermshaveen teredthemark et,withrmkhavinga
negativ e pay o. Thereforeusing this equilibrium as a punishmen ttriggered
bythe rm k'sdecision tostayin the mark et,rms i and j can forcerm k
to stayout of the mark etat eac hp eriod. Weshall referto this situation by
saying that the cartel fi;jg is feasible. It m ust b e clear that the feasibilit y
of a particular cartel, say fi;jg, neither excludes nor implies the feasibilit y
of anothercartel lik efi;kg forinstance.
Tosumup: weconsiderthat thegrandcartel,N,isalwaysfeasiblesince,
byassumption5,therealwaysexistsubgame p erfectequilibriaof0
where,
along the equilibrium path the three rms decide to stayin at eac h p eriod.
Onthe otherhandwesaythatatwo-rmscartel,sayfi;jg,isfeasibleifand
only if rms i and j can credibly predaterm k i.e. w
k
(N) <0 5
. We shall
denote the setof feasible cartels by S.
Before examiningthe setofequilibrium payov ectors ofafeasible cartel
inthegame0
,wemustrecognizethatthere isathirdkindofmark et
struc-ture whic hcould emergenamelythe monopolyone. Howeverthis p ossibilit y
is ruled out, whether reen tryis costless orunprotable, byassumption 6.
We nowturn tothe c haracterizationof the setof equilibrium payov
ec-tors that a feasible cartel, sayfi;jg, can obtain ina subgame startingafter
the decision of rm k to stay out of the mark etand the decisionsof rms i
and j tostayinthemark et. Considerrst the casewhere thereen trycost is
so high that it is nev erprotable for rm k todecide to stay in the mark et
at any p eriod of this subgame (i.e. assumption 1 holds). In this situation
assumption 6will ensurethat rms iand j will decidetostayinthe mark et
at eac h p eriod of the subgame. This immediately implies that the set of
equilibrium pay ov ectors of the subgame coincides with the set of
equilib-rium pay ov ectors of the game 0
(s) , with s = fi;jg. Therefore we have
V
(s)=W
(s) for allfeasible two-rmscartel s.
Consider nowthe case wherereen try is costless,i.e. assumption 2 holds.
Thentheworstp er-p eriodpay ormiwillobtainisequaltomax f0;w
i (N) g.
Indeed consider the subgame whic h starts after rms i and j have decided
5
Theuseofw
k
(N)0insteadofw
k
(N)<0inthedenitionofafeasiblecarteldo es
rm i or rm j has deviated from their resp ectiv e quantit y sp ecied in a
giv en strategy com bination. Since reen try is costless, e
i
(N) and e
j
(N) are
p erfect equilibrium of this subgame. In other words, with costless reen try,
rm k can b e used topunish a deviation by a memb er of the cartel sothat
the worst payoa rm, sayrm i ,will obtainif it deviates froma sp ecied
quantity is simply the maxim um b et ween 0 and w
i
(N). Note that if
car-tel fj;kg is feasible, if rm i deviates it could b e punished by its exclusion
of the mark et leading to a zero pay o. But since cartel fj;kg is feasible if
and only if w
i
(N) < 0, the worst payo rm i will obtain is still equal to
max f0;w
i
(N)g. Itfollo wsthatthesetofequilibriumpayov ectorafeasible
two-rm cartel s can obtain under assumption 2, V
(s), is simply equal to
the follo wingset:
V (s)=f(P 1 ;P 2 ;P 3 ) j (P i ) i2 s 2F(s) ;P k =0for k 62s; and P i max f0;w i (N)g8i2sg: (3)
Note that, since w
i
(N) is strictly smaller than w
i
(s) for any s 6= N to
whic hi b elongs, then V
(s) islarger than W
(s).
3 The cartel formation game
Due to the multiplicity of equilibria in the pro duction game, rms must
co ordinate up on the play of a particular equilibrium b efore the pro duction
game starts. The wayrms solvethis co ordination problem is describedby
a three-stage game where at eac h stage one rm mak es a declaration. The
orderinwhic hdeclarationsaremadeisexogeneouslygiv enand itiscommon
knowledge.
A rm i 's declaration, d
i
, consists of a feasible cartel, s
i
, to whic h rm
i m ust b elong and of a payo v ector p
i = (p i1 ;p i2 ;p i3
) whic h mustb elong
to V
(s
i
). The set of rm i 's declarations is thus D
i = f(s;p) ji 2 s;s 2 S and p2V (s)g. FurthermoreletH i
denotethe setof declarations
preced-ing that of rm i with H
i
= ; if rm i is the rst to mak e a declaration.
Then a strategy forrm i is a mapping
i :H i !D i .
Let d denote a declaration v ector, rm i will receive the pay o g
i giv en
g i (d)= 8 > > > > > > < > > > > > > : p ii i 8 > < > : either d i =d j ; ord i =d k ; ord i =d j =d k ; p ji i d i 6=d j and d j =d k ; g i otherwise :
Accordingly rm i will receiv e its prop osed payop
ii
if itsdeclaration is
compatible with (iden tical to) either that of rm j (i.e. d
i = d j ) or that of rmk (i.e. d i =d k
)orb oth. Ifthedeclarationsofrmsj andkare
compati-blethenrmiwillobtainthepayoassignedtoitinthermj'sdeclaration.
This payo assignmen t follo ws from the fact that, once two rms agree to
playaccordingtoapartiacularequilibriumof thepro ductiongame,thethird
one's b estresp onse inthe pro duction game istoplayaccordingto this
equi-librium to o. Consequently all rms will receiv e the pay ocorresp onding to
the equilibriumup on whic htworms havedecidedtoco ordinate. Finally,if
all the declarations are pairwise incompatible no agreemen t is p ossible and
rm i will receive some predetermined payo g
i
. We supp ose that rm i 's
reservation payo, g
i
, corresp onds to the worst payo it can obtain in the
pro duction game 0
or, inother words,the b est payorm i can guarantee
toitselfinthis game. Accordingly 6 ,g i =maxf0;w i (N)g8i2N. Toendup
note that since any strategy com bination (
i ; j ; k ) leads to a declaration
v ector the pay oofrm iin the cartel formationgame will b egiv enby g
i .
4 Equilibria in the cartel formation game
with unprotable reentry
Weturnnow tothe c haracterizationofthe equilibriaof the cartelformation
game when, in the pro duction game, reen try is unprotable i.e.
assump-tion 1 holds. To simplify the exposition we shall in tro duce some pieces of
terminology: Weshall saythat, fora giv enorder of declaration,say (i;j;k),
cartels forms ifand onlyif, for this order ofdeclaration, thereexists a
sub-gamep erfectNashequilibriumofthe cartelformationgame, 3 ,leadingtoa declaration v ector d 3 dened by 3 i = d 3 i ; 3 j (d 3 i )= d 3 j ; 3 k (d 3 i ;d 3 j ) =d 3 k such 6
Recall that outside opp ortunities, if there exist, are tak en in to account in the stay
d 3 h =d 3 l forh;l 2s
On the other hand, we shall say that cartel s wil l not form for a giv en
order of declaration if and only if there do es not exist at least one subgame
p erfect Nash equilibrium of the cartel formation game satisfying the ab ov e
requiremen t.
As one could expect subgame p erfect equilibria of the cartel formation
game dier according to the order of declaration. Howeverwe can iden tify
the cartelswhic hdonotformwhateverthe orderofdeclaration. LetW
i (s)
b e the highest one-p eriod prot thatrm i can obtainin W
(s). Then,
Proposition 1 Supposeallourassumptionsexceptassumption2hold. Then
there exists < 1 such that for all 2 (; 1) and whatever the order of
declarationwe have: 1. A cartelforms, 2. if S =ff i;jg;fi;kg;Ngand j < k
, then cartelfi;jg does not form,
3. if S =ff i;jg;fi;kg;fj;kg;Ng and i < j < k
, then (a) cartel fi;jg
does not form and (b) if in addition q
j q i and W j (i;j) W j (N)
then cartels fi;jg and fi;kg do not form.
Fixed costs, marginal cost functions and capacit y constraints determine
alltogetherwhic hcartelsarefeasiblei.e. whic hrmscanb eforcedtostayout
bythe twoothers. Howeveronthe basis ofmarginal cost functionsalone we
can conclude that if the rm with the highest marginal cost function belongs
to a feasible two-rms cartel then it wil l stay in the market. This means
thata sucientconditionforthe highestmarginalcost rmtosurviv eisthe
existenceof apartnerwithwhic hitcanpredatethe thirdrm. Furthermore
if inaddition wesupp ose that q
j q i 7 and W j (i;j)W j (N) then we can
b emorepreciseab outthesurvivalofthelowestmarginalcostrm. Indeedin
this case the resultsstate that if arm isforced tostayout of the mark etit
7
Asthereaderwillsee in thepro of ofLemma 4theresultwillhold undertheweaker
butlesstransparent(sucient)condition:
i (q R j (q i ); q i ) i (q R i (q j ); q j ).
cartel to whic h this rm do es not b elong. Accordingly a low marginal cost
functionand,sincetheresultsdonotdep endonthelev elofxedcosts, alow
averagecostfunctionisnot anadvantagetosurviv einamark et,ifweexcept
itsrole inthedeterminationofthesetoffeasible cartels. Putinanotherway,
our results suggest that toface predation the use of a tec hnologyleadingto
a small xed cost and a (relativ ely) high marginal cost function do provide
strong advantages with resp ect to the use of a tec hnologyleadingto a large
xed cost and alow marginal cost function.
The cornerstone underlying our results is:
Lemma 1 Suppose all our assumptions except 2hold andlet W
i
(s) bethe
highest payo rm i can obtain in V
(s). Then there exists <1 such that
for all 2(; 1), W
i
(i;j)<W
i
(i;k) if and only if
j < k . Proof: LetW i
(s) b e the highest payorm ican obtain inW(s). If we
areable toprov ethatW
i
(i;j)<W
i
(i;k) ifand onlyif
j <
k
then theresult
will follo w immediately bythe application of the FolkTheorem.
It is obviousthat: W i (i;j)= max q i 2 X i q j 2 X j i (q i ;q j ) subject to j (q j ;q i )=w j (i;j); W i (i;k)= max q i 2 X i q k 2 X k i (q i ;q k )subject to k (q k ;q i )=w k (i;k):
Remark immediatelythat theconstraintsinthese optimizationprograms do
notdep endonthe xedcost andhencethe constraintsdieronlyinthelev el
of .
Forq
i =q
i
,theconstraintsinthesemaximizationprogramsaresatisedif
and only if q j =q R j (q i )and q k =q R k (q i
)resp ectively. Under our assumptions
we obviouslyhav eq R j (q i )>q R k (q i )if and onlyif j < k
. Futhermorefor any
giv enq
i
2[0;q
i
)letusdieren tiate
j (q j ;q i )=w j
(i;j) withresp ecttoq
j and j . We obtain: @ j (q j ;q i ) @q j dq j = " @c(q j ; j ;q j ) @ j 0 @c(q R (q i ); j ;q j ) @ j # d j :
where all derivatives are evaluated at (q
j ;q i ) such that j (q j ;q i ) =w j (i;j). Since j
is a strictly quasi-concav e function we hav e that @
j (q j ;q i )=@q j is
strictly negativ e for any q j 2 (q (q i );q j
] and strictly p ositiv e for any q
j 2 [0;q R (q i
)). Moreover, by assumption 3, marginal cost is increasing in so
that @c(q j ; j ;q j )=@ j 0@c(q R (q i ); j ;q j )=@ j
is strictly p ositiv efor any q
j 2 (q R (q i );q j
] and strictly negativ e for any q
j 2 [0;q R (q i )). It then follo ws that dq j =d j
is strictly negativ e for any giv en q
i 2 [0;q i ) and q j satisfying j (q j ;q i ) = w j
(i;j). This establishes that the isoprot curv e
j (q j ;q i ) = w j
(i;j) shiftsrightwardsin the co ordinate (q
j ;q i ) as j decreases. As shown
in Figure1the result then follo ws. 2
As shown by the pro of, the xed cost do es not matter since it aects
b oth sides of the constraint in the maximization program in the same way.
Hence, for anyquantity pro duced by rm i , the quantity requiredto satisfy
the constraint isindependen tof thexed cost. On the other hand,the lev el
of ,i.e. the levelofthe marginal cost fora giv enquantitypro duced, aects
the constraint in two ways. First, if the right-hand side of the constraint
wereindependen tof then, inthe co ordinates of Figure1,rm k'sisoprot
curv e will b e en tirely b elo w the rm j's one as long as
j <
k
. This eect
re ectsthe advantagetoformacartelwithalowmarginalcostrm. Second,
however, the minimal pay o required by a rm to participate in a cartel
with rm i clearly decreases with . This translates the in tuitionthat alow
marginal cost rm will b e more greedy than a rm with a higher marginal
cost. What the Lemma states is that the second eect dominates the rst
one.
According to this result, for
k
larger than
j
, rm k can always giv eto
rm i agreater payothan the highest pay orm ican obtain with rm j.
On the otherhand ifcartelfi;jgformsthen rmk will receiv eazero payo
whileitwill obtainatleastw
k
(i;k)> 0ifcartel fi;kgforms. Hence,lo osely
sp eaking, rm k has always the opp ortunity and the willingness to prev ent
the formation of cartel fi;jgso that this cartel cannot form.
Remark that we cannot exclude the formation of the grand cartel, N,
for all orders of declaration. Indeed consider, for instance, the case where
S = ff i;jg;fi;kg;Ng with
j <
k
and rm i is the rst rm to declare.
If w
k
(i;k) is sucien tly large it could happ en that W
i
(i;k) is strictly
smaller than the greatest pay orm i can obtain in V
(N). Consequently,
rm iwill prop osethe formationofthe grand cartelandthe b esteither rm
of rm i . This shows that the availability of a predatory strategy is not
sucien tforpredation too ccur.
Finally,Part 3of Prop osition 1 corresp onds tothe example giv eninthe
In tro duction ab ov e, except for the feasibilit y of the grand cartel. However,
tohav ethesame prediction, i.e. that the low marginal cost rm isexcluded
for all orders of declarations, two additional requirements are needed. The
rst is that the maximal capacit y of rm j is smaller than that of rm i ;
the second is that the maximal payo that j can obtain in the cartel fi;jg
is larger than the one it can obtain in cartel (N). These conditions seem
fairly unrestrictiv e: as rm i has a lower marginal cost function than j it
is reasonable to assume that it has installed a higher capacit y; while it is
quite plausible that a rm can obtain more in a two-rm than in a
three-rm cartel. obviously, if cartel (N) was not feasible, as in the in tro ductory
example, then this second condition istrivially met.
As itcan b e seenfromthe Pro of ofProp osition 1inthe App endix, these
twoconditions are sup er uous for all ordersof declaration except when k is
the rst to declare. In this case k in order to induce j to en ter the cartel
fj;kg mustgiv e toj a pay oat least as great as the maxim um payothat
j could obtain inthe cartel fi;jg, W
j
(i;j). The same is true if k wantsto
inducei toen ter thecartel fi;kg,that isk m ustgiv eW
i
(i;j). Thus,k will
prefer the cartel fj;kg if it gets a higher payoin it rather than in fi;kg ,
giv enthe constraints imposed by what hemustoer toj and i . This isthe
case if the conditions in Part 3 of Prop osition 1 are met, as it is shown by
Lemma 4in the App endix.
5 Equilibria of the cartel formation game
with costless reentry
It has b een shown in section 2.3, that the case with costless reen try diers
from the one with unprotable reen try only by the fact that in the former
case the minimal payo required by a rm, say j, to participate to a
two-rms cartel, say fi;jg, is equal to max f0;w
j
(N)g while in the latter case
8
Notethatifthecartelf j;kgwerealsofeasiblethenthegrandcartelcouldform with
rm i b eing the rst rmto declareand rm j (resp. rm k) the second one provided
that W
j
(j;k) (resp. W
k
(j;k))is strictly smallerthanthe highestpa yo rmj (resp.
rmk)canobtaininV
(N) .
j
is,if w
j
(N)and w
k
(N) are negativ e,then the minimal payoobtained by
rmsj andk inthese cartelsareequal tozero. Therefore,thehighestpayo
rmican obtaininacartel,W
i
(s),dep endsonb oth themarginalandxed
costs of its partner. More precisely,let q 0
h
b e the rival's output whic h leads
to zero prot for rm h when it plays its b est reply, q R h (q o h ), that is, q o h is such that h (q R h (q 0 h );q 0 h
)=0. Furthermore, for allq 2X 0 h with X 0 h =[0;q 0 h ), letq^ h
(q) b e thesmallest quantitypro duced byhwhic hgiv esitazero prot
whenev er its rivals pro duce q, that is, q^
h (q) is such that: h (q^ h (q);q) 0 and @ h (q^ h (q);q)=@q h >0. We hav e:
Lemma 2 Suppose all our assumptions except assumption 1 hold. There
exists < 1 such that, if cartels fi;jg and fi;kg are feasible and (
j ;F j ) and ( k ;F k
)are suchthat q^
k (q)>q^ j (q) forall q 2X 0 j \X 0 k , then W i (i;j)> W i (i;k)> W i (N) for all 2(;1).
Considering Figure 2, the pro of of this result is clearly quite obvious
and is thus omitted. It m ust b e noticed that a necessary and sucien t
condition for W
i
(i;j)>W
i
(i;k) to hold wouldin volveacomparison ofthe
coststructureofthethreerms. Wethusc ho osetostateourresultsinterms
ofa sucien tcondition whic hactually requiresonlythe comparisonof rms
j and k av eragecost function.
Clearly, Lemma 2 here will play the role of Lemma 1 in the case of
no-reen try. It thereforefollo ws:
Proposition 2 Supposeallourassumptionsexceptassumption1hold. There
exists <1 such that for all 2(; 1) and for any order of declaration we
have:
1. A cartelforms,
2. let S = ff i;jg;fi;kg;Ng and q^
j (q)<^q k (q) for all q 2 X 0 j \X 0 k , then
cartelsfi;kg and N do not form,
3. let S =ff i;jg;fi;kg;fj;kg;Ng and (i ) q^
i (q) <q^ j (q) for all q 2 X 0 i \ X 0 j , (ii ) q^ i (q) < q^ k (q) for all q 2 X 0 i \X 0 k , (iii ) q^ j (q) < q^ k (q) for all q2X 0 j \X 0 k
This Prop osition contrastswith our previous results intwoways: First,
the grand cartel, N, do es not form, so that if seeing that there exists a
predatorystrategyitwillb eplayedi.e. predationoccurs. Thiscomesfromthe
fact that, aslong as cartels fi;jg and fi;kgare feasible, the minimal payo
rms j andk will obtainin b oth atwo-rmcartel and inthe grandcartel is
equal to zero. It then follo ws that rm i can always obtain a larger payo
in a two-rm cartel than in the grand cartel (see Lemma 3). Consequently
if rm i isthe rst rm to declareitwill nev erprop ose the formation ofthe
grand cartel. On the other hand if it is rm j (resp. rm k) whic h is the
rst to declare then it will nev erprop ose the formation of the grand cartel.
Indeed if it do es so then b oth rm i and rm k (resp. rm j) can obtain a
higher payo than the one prop osed in rm j's (resp. rm k's) declaration
bymakingcompatibledeclarationswhic hprop osetheformationofthecartel
fi;kg (resp. fi;jg).
The second dierence b et weenthe results with costless reen try and the
ones with unprotable reen try can b e illustrated if we supp ose that rms
have iden tical xed costs 10 . In this case q^ j (q) < q^ k (q) for all q 2 X 0 j \X 0 k
will hold if and only if
j <
k
. Then Prop osition 2 states simply that the
rm with the highest marginal cost function will b e predated. Therefore
with costlessreen try,contrarytowhat happ ens inthe unproptable reen try
case, a low marginal cost constitues a strong
advantage to face predation.
On the other hand, if we supp ose that
i =
j =
k
then the conditions
used inProp osition 2 will b e satisedif and only ifF
i <F j <F k . Hence we
ndbackaresultstatedrst byGhema watandNalebu(1985)fordeclining
industriesaccordingtowhic hthermwiththelargestcapacitiesi.e. withthe
highest xed cost lev elis the rst rm to exit the mark et. Such conclusion
has also b e drawn by Fudenb erg and Tirole (1986) from the analysis of an
incomplete information game.
To conclude with, if we are able to rank the rms with resp ect to their
average cost function then Prop osition 2 states that the exiting rm is the
one with the highest averagecost function.
9
Thepro ofofthisProp ositionfollo wssocloselythatofProp osition1thatitsomitted.
10
Recallthatourresultsin thecaseofunprotablereentrydonotdep endontherms
reentry
One sp ecic feature of the cartel formation game presen ted ab ov e is that
eac h rm in its declaration prop oses simultaneously a particular cartel and
the payos that eac h mem b er of the cartel will receiv e. As a consequence,
the cartel formation game giv es to all rms a strong in uence on the way
payos are allo catedamong cartel mem b ers.
This seems reasonable when reen try costs are negligible. In this case
indeedthepro ductiongameremainsathreeplayersgameev enifarmexits
the mark et. However when reen try costs are large, the pro duction game
b ecomes a two playersgame once a rm decides to stay out of the mark et.
Inthiscaseonecanaskthequestionifthecartelformationgamedo esnotgiv e
totheexitingrmanunrealisticallyexcessivein uence ontheequilibrium of
the resulting tworms pro duction game whic h shall b e played. In order to
provideananswer,weshallanalyzethe sensitivit yof theinecienciesstated
in Prop osition 1 tothe way rms are supp osed toco ordinate.
Toin vestigatethisissuewelo okatatwostepco ordinationpro cesswhere
theexitingrmhasnoin uenceonthewaytheremainingrmswillsharethe
gains from co op eration in the pro duction game. This co ordination pro cess
constitutesagame: itsrststepisasubstituteforthecartelformationgame
presen ted b efore. The only dierenceis that it isnow supp osed that a rm
declaration only consists of a feasible cartel, s. If all declarations dierthe
game ends and eac h rm receiv es its reservation payo g
i
. Otherwise one
mo vesto the second step.
The second step consists of a negotiation b et ween the members of the
cartelgiven in the identical declarationsof the rst step, says, todetermine
apay ov ector,p,b elonging toV
(s). If armdo es notb elong tos thenits
action setinthis step is simply fdonothingg.
We shall not sp ecify explicitely the bargaining game pro cedure. We
as-sume instead that, the gains from co op eration (i.e. the actual payo min us
thesum ofappropriatelydiscountedCournotprotsoftheone-shotquantity
game)aresharedaccordingtoabargainingsolution. Thebargainingsolution
solutionsas axiomatized byKalai (1977) and Kalai and Samet(1985) .
Tob e precise, letus rst assumethat:
Assumption 7 For any feasible cartel, s, the Cournot equilibrium in the
quantity game is unique.
Then,let c
i
(i;j)denotethermi 'sCournotequilibriumprotwhenonly
rmsiandjareactiv eonthemark et. Furthermoredenoteby(q e i (i;j);q e j (i;j))
thequantityv ectorwhic hmaximizesP
i subjecttoP i 0 c i (i;j)=P j 0 c j (i;j) andletP e i (i;j)(resp. P e j (i;j))b egiv enby(10) P 1 t =0 t i (q e i (i;j);q e j (i;j)) (resp. (10) P 1 t =0 t j (q e j (i;j);q e i (i;j)) ). Obviously (P e i (i;j);P e j (i;j)) is
the symmetric egalitariansolution 13
tothe co op erativ e bargaining game
de-ned by a set of outcomes giv en by F(i;j) and a statu-quo p oint giv enby
( c i (i;j); c j (i;j)).
We can immediately state:
Lemma 3 Let all our assumptions except 2 be satised. Furthermore, for
anyfeasibletwo-rmscartel,sayfh;lg,supposethat(q e h (h;l);q e l (h;l))belongs to ]0;q h [ 2 ]0;q l
[ and that there exits (q
h ;q l ) 0 which maximizes P h +P l .
Then there exists <1 such that, for all >, P e i (i;k)> P e i (i;j) if and only if k > j . 11
Thiskindofstructurehasalreadyb eenusedintheliterature. Forinstance,in
Gross-man and Hart (1986), two agents rst cho ose non-co op eratively and sim ultaneously a
levelof in vestment and then, giv enthese in vestments, tak e actions such that the gains
from renegotiation, which correspond to the gains from co op eration in our framew ork,
is shared equally. In their context, this corresponds also to the Nash bargaining
solu-tion. The Groosmanand Hart's analysis hasb een extended by Hart and Mo ore(1990)
toman yagentsandthebargainingsolutionadoptedtheretosharethegainfromtradeis
theShapleyvalue.
Weadopthereanegalitariansolutiononetheonehandb ecauseitismuchmoretractable
thanthe otherones (inparticular theNashbargaining solution),andon theotherhand
b ecause theegalitarian solutions are theonly ones which, in thepresence of other
stan-dardrequirements,satisfythemonotonicit yprop erty(seeKalaiandSamet(1985)). This
conditionsimply statesthat ifthe feasibleset of one coalitionincreases and thefeasible
sets of all other coalitions remain thesame, then none of the memb ersof this coalition
should b ecomeworseob ecause ofthischange.
12
Notethat similarresultscould b eobtained byusing thesymmetricNsahbargaining
solution.
13
It will b e ob viousto verify that theresults presented b elo wwill hold if we tak ean
asymmetric egalitarian solution provided the weight of rm i in the solution dep ends
negativ elyon
i
guarantee that there exists a feasible payov ector strictly greater than the
Cournot equilibrium prots v ector.
Againanincreasein
j
willhavetwoeectsontheco op erativ ebargaining
game in volving rms iand j: On the one hand,it leads toamodication in
thesetoffeasibleoutcomeswhic haectsnegativ elythepayoofrmiatthe
egalitarian solution while, on the other hand, it increases (resp. decreases)
rm i 's(resp. rm j's)statu-quo pay owhic hwill rise therm i 's payoat
the egalitarian solution. The Lemma 14
states simply that the p ositiv eeect
arising from the movein the statu-quo pay odominates the negativ eeect
coming fromthe reduction in the setof feasibleoutcomes.
This resultwillplay,forProp osition3b elo w,theroleplayedbyLemma1
and 2 for Prop osition 1 and 2 resp ectively. To see this it suces to realize
that the setof subgame p erfectequilibria of the gamederiving fromthe two
steppro cedure hereconsideredcoincideswiththe oneofthecartelformation
gamewherearmi 'sdeclarationconsistsofafeasiblecartel,s
i
,towhic hrm
i b elongsand ofa pay ov ectorwhic hgiv estoeac hrm ins
i
the symmetric
egalitarian payodenedab ov e 15
and a zero payoto a rm (if any) whic h
do es not b elong to s
i
. Formallythe set of rm i 's declarations is now D
i = f(s;p) j i 2 s; s 2 S; for allh 2 s p h = p e h (s) and, for l 62 s; p l = 0g. Therefore we have:
Proposition 3 Let all assumptions in Lemma 3 hold. Then there exists
< 1 such that for all 2 (; 1) and whatever the order of declaration we
have: 1. A cartelforms, 2. if S =ff i;jg;fi;kg;Ngand j < k
, then cartelfi;jg does not form,
3. if S = ff i;jg;fi;kg;fj;kg;Ng and i < j < k
, then cartels fi;jg
and fi;kg do not form.
14
The pro of of this result comes quite straightforw ardly from the application of the
envelop etheoremaswellas theFolktheorem. Henceitwillb e omitted.
15
To savespace we do notdene formally theegalitarian pa yowhen thethree rms
areactiv e. Howeverthiscaneasilyb edoneevenifonewantstoconsideracoalitionform
game instead of a co op erative bargaining game. Anyway this do es not matter for our
of the exiting rm on the waypay osare allo cated inthe pro duction game.
7 Concluding remarks
Wehaveconsideredinthis pap eradynamicpro ductiongamein volvingthree
rmswhic haredieren tiatedaccordingtotheircostfunction. Moreprecisely
wehaveassumedthatrms canb erankedunam biguouslyaccordingtotheir
marginal cost function and that their xed cost ma ydier. Furthermorewe
supp ose that one rm can b e credibly forced to stay out of the mark et by
the two others and that at least two rms can b e put under such a threat.
We then in vestigate the cost c haracteristics of the exiting rm under two
alternativ ehyp othesis concerning the p ossibilit y of reen trynamely the case
where reen try is unprotable in any circumstances due to the presence of
large sunk costs, and the one where reen tryis costless.
We have obtained two predictions (whic h app ears quite robust to the
sp ecicationof the cartelformationgame). Firstif reen tryisalways
unprof-itablethentheexitingrmhasthelowestmarginalcostfunctionascompared
with the marginalcost functionof the rmswhic hcan crediblyb e predated.
Furthermore this result do es not dep end on the lev el of xed costs 16
.
Ac-cordingly, in this case, cost ineciencies will arise since the exiting rm is
the one whic h uses the most ecien ttec hnology.
A second result is that when reen try is costless and when we can rank
rms according to their averagecost function then the exiting rm has the
larger av eragecost functionas compared to the averagecost function ofthe
rms whic hcan b eput underthe threat ofpredation. Therefore inthis case
cost ineciencies donot app ear.
The result obtained in the no-reentry case lo oks strange since it go es
against the common b elief that the most ecien t rm will remain on the
mark et. But this b elief has b een dev elopp edinthe contextof \neo-classical
economics". Ifinsteadwelo okatthisresultfromthep ointofviewof
\trans-action cost economics" (as dev elopp ed in Williamson (1985) for instance)
then they app ear rather unsurprising. Indeed in this context such kind of
inecienciesare frequen tlyobtained. It isworthwhileemphasizingthe deep
16
Howeverthesetofrmswhichcanb epredateddep endob viouslyonthelevelofxed
deed although the latter approach fo cuses mainly on the in ternal
organisa-tion of the rm the presen tstudy shows that the basic p oints whic h
distin-guishtransactioncosteconomicsfromothereconomicapproc hesarealsowell
suited to study the composition of an industry and more generally to mak e
substantialprogressesintheunderstandingoftheformationandcomposition
of groups orcoalitions on amark et.
Roughly sp eaking transaction cost economics seeks toanalyse situations
in volvingagentsc haracterizedbyopportunismandboundedrationalitywhere
(i ) agents will meet frequently, (ii ) agents do not rely on courts for settling
disputes among them i.e. private ordering prevails, (iii ) agents hav e the
opp ortunity to mak easset specic investments and (iv) agentsev olvein an
uncertainen vironment.
In the presen tanalysis we hav eruled out b oth uncertain ty and b ounded
rationalit ysincethesec haracteristicsapp earunessentialforourresults. Note
furthermore that frequency will not b e relevant here as the example giv en
in the In tro duction p oints out. The dierence b et ween opp ortunism and
self-in terested b ehavior do es not matter here b ecause the set of subgame
p erfectNashequilibriaandthe setofNashequilibriaofthepro duction game
coincides for a discount factor sucien tly close to zero. We shall however
argue that if we mak e abstraction of the presence of either
private ordering
orassetsp ecicin vestmentsthenthecostinecienciesobtainedinthepap er
disapp ear.
Letusb eginwithprivateordering. Manyexchangeanalysissupp osethat
ecacious rules of law are in place so that any disagreemen t regarding the
execution of a contract is settled bycourts in a fully informed and low-cost
way. This assumption of court ordering is v ery convenientsince it allowsto
disregard the ex-p ost side of a contract. In our context, rms cannot rely
on court since the kind of contract they are willing to do is simply illegal.
An immediate consequence of private ordering is that we cannot disregard
the execution phase of the contract since the latter mustb e self enforcing.
Thisen tailsthatrms,asissupp osedinthecartelformationgame,willonly
considerpay ov ectorswhic hcanb eassociated withasubgame p erfectNash
equilibrium of the pro duction game.
But supp ose to the contrary that rms can rely costlessly on court to
enforce anagreemen t. This implies that the setof payo v ectors that must
cor-toexist can committoob ey anagreemen tinwhic hit receiv esazero pay o.
Without this p ossibilit y of commitmen t, such an agreemen t is not credible
in the no-reentry case while it is in the case of costless reen try.
Conse-quen tly,theresultstatedinLemma2will holdev enifreen tryisunprotable
and the exiting rm is the one with the highest av erage cost function (see
Prop osition 2foramore precisestatemen t). Thereforethe costineciencies
disapp ear once courtordering isallowedfor. Remarkthat this clearlyshows
that considering tacit co op eration b et weenrms asillegal is p ossibly costly.
Let us now turn to the asset sp ecic c haracter of in vestments. In
vest-men ts are said wholly asset sp ecic if they are unredeplo yable. Accordingly
in vestment costs are sunk for wholly asset sp ecic in vestments while they
are xedwhen in vestmentlo oses its asset sp ecic c haracter. The main
con-sequence of the presence of asset sp ecic in vestmentis the o ccurence of the
fundamental transformation . Thelatterconceptreferstothetransformation
in the nature of the competition prevailing b efore and after the adoption of
the contract.
In our context, the sunk reen try cost we hav e in tro duced can simply
b e in terpreted as the cost of unredeplo yable in vestments. More precisely,
the unprotable reen trycase corresp onds to the situation where large asset
sp ecic in vestments m ust b e achieved b efore b eing activ e on the mark et
while in the costless reen try case such in vestments are negligible. When
reen try is unprotable the fundamental transformation o ccurs since, once
a rm exits, the pro duction game b ecomes a two players game. If instead
reen try is costless this transformation do es not o ccur. Indeed, in this case
ev enif a rm exits itcan participate tothe punishmen tof a deviationfrom
the equilibrium path by one of the two rms whic h remain on the mark et.
Inother wordsthepro duction gamestill in volvesthree playersev enif arm
exits the mark et. As we hav eshown, cost ineciencies app ear only in the
case of unprotable reen try whic h means that the presence of large asset
sp ecic in vestments is a necessary condition for such cost ineciencies to
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b et ween players and coalitions: An application of the Shapley value". In:
The Shapley Value: Essays in Honor of Lloyd Shapley, edited by A. Roth.
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Bernheim, D., D. Peleg and M. Whinston 1987: "Coalition-proof
Nash equilibria. I: Concepts". Journal of Economic Theory ,v ol 42
pages1-12.
Bloch, F. 1990a: "Sequen tialformationof coalitionsand stablecoalition
structures insymmetricgames". Mimeo,Departmen tofEconomics, Univ
er-sit yof Pennsylvania.
Bloch, F. 1990b: "Endogenousstructures of associations in oligop olies".
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Fudenberg, D. and J. Tirole 1986: "A theory of exit in duop oly".
Econometricav ol54 pages 943- 960.
Ghema wat, P. and G. Nalebu1985: "Exit". Rand Journal of
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Ghema wat, P. and G. Nalebu 1990: "The ev olution of declining
industries". Quaterly Journal of Economicsv ol105 pages 167-186.
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A theory of v ertical and lateral in tegration". Journal of Political Economy
v ol94, pages691-719.
Gul, F. 1989: "Bargaining foundations of Shapley value", Econometrica
v ol57 pages 81-95.
Hart, O. and J. Moore1990: "Propert y rights and the nature of the
rm". Journal of Political Economyv ol 98pages 1119-1158.
Kalai, E. 1977: "Proportional solutions to bargaining situations: In
ter-p ersonal utilit ycomparisons". Econometricav ol45pages 1623-1630.
Kalai, E. and D. Samet 1985: "Monotonic solutions to generalco
op-erativ e games". Econometricav ol53 pages 307-327.
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Free Press New-York.
Wilson, R.1992: "Strategicmodels of en trydeterrence". In: Handbook
of Game Theory ed. by Aumann, R. and S. Hart, North-Holland/ Elsevier
8 Proof of Proposition 1
8.1 A cartel forms.
The pro of that a cartelforms is straightforwardand it istherefore omitted.
8.2 Let S = ff i;jg;fi;kg;Ng and
j <
k
, then cartel fi;jg does not
form.
Remark immediately that since fi;jg and fi;kg are b oth feasible we have
g
j = g
k
= 0. Obviously,it is useless to consider strategies that lead to the
formationof cartel N since weare onlyin terestedto provethat cartel fi;jg
do es not form.
8.2.1 Suppose that rm i is the rst rm to declare:
1. (a) Letrmkb ethelastrmtodeclare. Ifd
i 62D
k
(i.e. theprop osed
cartelinthermi 'sdeclarationisthecartelfi;jg)then,whatever
is rm j's declaration and whatever is its own declaration rm k
will obtain a zero pay o (either d
i = d
j
and in this case rm
k obtains p ik = p jk = 0 since p i 2 V (i;j) implies p ik = 0 or d i 6=d j
in whic h case rm k will obtain g
k
=0). Nowif d
i 2D
k
we m ust distinguish two cases: On the one hand, if d
i 62 D
j (i.e.
rm i prop oses the formation of cartel fi;kg) then rm k will
obtain a zero payo by declaring d
k 6= d
i
while it will receiv e
p ik w k (i;k) > 0 by declaring d k = d i
. On the other hand if
d i 2D j \D k
thenrm k will obtainp
ik 0 (sincew k (N)<0)if either it declaresd k = d i or d i = d j
while it will receiv e g
k =0if d i 6=d j 6= d k
. It follo ws that all rm k's b est-resp onse to (d
i ;d
j )
is c haracterized by: (i) d
k
can b e anything if either d
i 62 D k or d i =d j orp ik =0;(ii) d k =d i if d i 2D k and p ik >0 and d j 6=d i .
soning will lead to the conclusion that all rm j's b est-resp onse
to (d
i ;d
k
) will satisfy (i) d
j
can b e anything if either d
i 62D j , or d i =d k , orp ij =0; (ii) d j =d i if d i 2D j and p ij >0and d k 6=d i .
2. (a) Let us considerthe problem faced by the second rm to declare,
say rm j. If d
i 62 D
j
then, whatever its own declaration is, j
will receiv e azero pay o. Giv enp oint 1.(b) ab ov e, if d
i 2D
j nD
k
(i.e. rm i prop oses the formation of cartel fi;jg) then it will
obtain a payo equal to p
ij w j (i;j) if it declares d j = d i or g j =0 if it declares d j 6=d i . If d i 2 D j \D k
(i.e. rm i prop oses
the formation of cartel N) then, since rm k b est-resp onse to
(d i ;d j ), with d j 6= d i is d k = d i , j will obtain p ij 0 either by declaring d j = d i
or by declaring anything else. On the other
hand if d i 2 D j \D k and d i
is such that rm k's b est-resp onse
is d k 6= d i for d j 6= d i
then rm j will obtain a zero payo if it
declares d j 6=d i while it receiv esp ij 0 ifit declares d j =d i .
Therefore all rm j's b est-resp onses to d
i
will satisfy: (i) d
j can b e anythingifeither d i 62D j orp ij =0, ord i is suchthat rmk's b est-resp onse to (d i ;d j ), with d j 6= d i , is d k = d i ; (ii) d j = d i if d i 2D j , and p ij >0,and d i
is suchthat rm k's b est-resp onse to
(d i ;d j ), with d j 6=d i , isd k 6=d i .
(b) If rm k were the second rm to declare a similar reasoning will
lead to the same c haracterization of all rm k's b est-resp onse to
d
i .
3. Nowconsiderthe problem facedbyrm i .
First, if it declares d j i (fi;jg;(W i (i;j);w j
(i;j);0)), rm j will
de-clare d j = d j i (since d j i 62 D k
) and consequen tly rm i will obtain
W
i
(i;j). Second if it declares d k i (fi;kg;(W i (i;k);0;w k (i;k))) rm k will declare d k = d k i
whatever the decalration of rm j (since
d k i 62 D j and p ik
> 0) and rm i will obtain W
i (i;k). Third if rm i declares d N i = (N;(P i
( );;0)), for some > 0 and P
i
( ) b eing the
maximal pay orm i can obtain in V
(N) when P j = and P k =0,
then rm j will declare d
j =d
N
i
and it follo ws that rm i will receiv e
P
i
( ). Remark immediately that P
i
( ) tends toW
i
(N) when tends
to zero. Finally if rm i mak es a declaration, d O
i
declare d j 6=d i and rm k to declare d k 6= d i
then rm i will receiv e
g =w
i (N).
Firm i will nev er mak e a declaration lik e d O
i
since doing so it will
receiv e w
i
(N) whic h is strictly smaller than the pay oitwill obtain
if it decalares d j
i
for instance. Furthermore since
j < k we know by Lemma 1 that W i (i;k) > W i
(i;j) for sucien tlyclose to one.
Therefore rm i will nev er declare d j
i
and the cartel fi;jg do es not
form.
8.2.2 Suppose that rm i is the second rm to declare:
1. Let rm k b e the last rm todeclare. An y b est-resp onse for this rm
satises the prop erties giv en in p oint B.2. 1a . It follo ws that by
declaringd k
i
,denedinp ointB.2.13,rmiwill obtainW
i
(i;k).
Con-sequen tly for rm i to maximize its pay o by declaring d
i = d j it is necessary that p ji W i
(i;k). Therefore all rm i 's b est-resp onses
to d
j
are such that: (i) d
i = d k i if p ji < W i (i;k); (ii) d i = d j if p ji >W i
(i;k). Now,byLemma1, for sucien tlyclose toone there
do es not exist d j 2 D j such that d j 62 D k and p ji W i (i;k). This
impliesthat cartelfi;jgdo es not form.
2. Let rm j b e the last rm to declare. Reasoning as ab ov e, it follo ws
that all rm i 's b est-resp onses to d
k
are such that: (i) d
i = d j i if p k i < W i (i;j); (ii) d i = d k if p k i > W i
(i;j). But for sucien tly
close to one, Lemma 1 ensures the existence of d
k 2 D k such that p k i > W i
(i;j). Moreover, by making a declaration d
k 62 D j , whic h induces rm i to declare d i = d k
, rm k will obtain a strictly greater
payo than the one it will obtain by declaring any other d
k 2 D
k .
Therefore cartelfi;jg do es not form.
8.2.3 Suppose that rm i is the last rm to declare:
1. If d
j =d
k
(i.e. cartel N is prop osed) then rm i will obtain p
ji =p
k i
whateverisitsowndeclaration. Ifd
j 6=d
k
thenrmiwillobtaineither
g i = w i (N) by declaring d i
whic h diers from b oth d
j and d k , or p ji by declaring d i = d j , or p k i by declaring d i = d k . But p ji w i (N) and p k i w i
(N) . It follo ws that all rm i 's b est-resp onses to (d
j ;d
k )
i j k i j ji k i d i =d k if p k i >p ji .
2. Let rm k b e the second rm to declare. If d
j 62 D
k
, i.e. rm j
prop oses the formation of cartel fi;jg, then rm k will receiv e a zero
payo if it mak es a declaration such that the b est-resp onse to (d
j ;d k ) is d i = d j
. However, for sucien tly close to one, Lemma 1 ensures
that, for any d
j 62 D k , there exists a d k 2 D k with p k i > p ji and p k k >w k (i;k)>p jk
=0. Thereforecartel fi;jg do es not form.
3. Let rm j b e the second rm to declare. Firm k receiv es zero if fi;jg
forms. This, however, can b e prev ented by rm k: from Lemma 1, it
can mak ea declaration with p
k i >W
i
(i;j) and s=fi;kg. Therefore,
since a cartel formsand using subgame p erfection, fi;jg cannotform.
1.3 Let S = ff i;jg;fi;kg;fj;kg;Ng and
i < j < k , then cartel
fi;jgdoesnotform. Furthermoreifin additionq
j q i andW j (i;j) W j
(N) then cartels fi;jg and fi;kg do not form.
Remark immediately since fi;jg, fi;kg and fj;kg are sim ultaneously
feasible we hav eg i =g j =g k =0.
To b egin with let us c haracterize the b est-resp onses of the last rm to
declare. For the generalit y of the argumen t we shall index the last rm by
o and the two others by m and n. First of all, if d
m = d
n
then whatever
its declaration rm o will receiv e p
mo . Second, if d m 6= d n and d m 62 D o and d n 62 D o
, then whatever itsdeclaration rm o will receiv e a zero pay o.
Finally,supp ose that d
m 6= d n and either d m 2D o ord n 2 D o or b oth, then
rm o will obtain either p
mo if d m 2 D o and it declares d o = d m , or p no if d n 2D o anditdeclaresd o =d n
,orzeroifitmak esadeclarationwhic hdiers
sim ultaneously from d
m
and d
n
. Accordingly, all rm o's b est-resp onses to
(d m ;d n )satisfy: (i)d o
canb eanythingifeitherd
m =d n ord m 6=d n ,d m 62D o and d n 62 D o or p mo =p no = 0; (ii) d o = d m if d m 2 D o and p mo > p no ; (iii) d o =d n if d n 2D o and p no >p mo .
1. Let rm j b e the second rm to declare. Forthe cartel fi;jg to form
it m ust b e the case that d
i = d j , d i 2 D j nD k . But if rm i mak es a declaration b elonging to D j nD k
, rm j will obtain at most W
j (i;j) by declaring d j = d i
while it will receiv e W
j (j;k) if it declares d k j = (fj;kg;(0;W j (j;k);w k
(j;k))),sinceforsuchdeclarations(d
i ;d
k
j )rm
k will maximizeitspayobydeclaring d
k =d
k
j
. ByLemma1we know
that,for sucien tlyclose toone, W
j
(j;k)>W
j
(i;j)and therefore
cartel fi;jg do es not form.
Forcartel fi;kg toform itm ust b e the case that d
i 62D i \D j . If rm
j mak es a declaration whic h induces rm k to declare d
k = d
i
then it
will receiv e a zero pay o. However for any p
ik 2 [ w k (i;k);W k (i;k)]
we know byLemma 1 that, for sucien tlyclose to one, there exists
d j 2 D j \D k such that p jk > p ik and p jj > w j (j;k) > 0. Therefore
cartel fi;kg do es not form.
2. Let rm i b e the second rm to declare. For the cartel fi;jg to
form it m ust b e the case that d
i = d j , d j 2 D i nD k . But if rm
j mak es such a declaration rm i will receiv e at most W
i (i;j) by declaring d i = d j
while it will receiv e W
i (i;k) if it declares d k i = (fi;kg;(W i (i;k);0;w k
(i;k))) since for such (d
j ;d
k
i
) rm k will
max-imize its pay oby declaring d
k = d
k
i
. By Lemma 1, for sucien tly
close to one, W
i
(i;k) is stricly greater than W
i
(i;j) and therefore
cartel fi;jg do es not form.
For cartel fi;kg to form it m ust b e the case that d
i 2 D k nD j and d i 62D j \D k andd j issuchthatp ik p jk
(theequalityb et weenp
ik and
p
jk
isallowedonlyif rm k,facing twoindieren talternativ es,c ho oses
to declare d
k = d
j
). In this situation rm j will obtain a zero payo
while rm k will obtain atmost W
k
(i;k). However,by Lemma 1, we
know that, for suciently close to one, there exists d
j 2 D j \D k suchthat p jk >W k (i;k)and p jj >w j (j;k)>0. Consequently cartel
fi;kgdo es not form.
Summing up, if k is the last rm to declare, Lemma 1 is sucien t to