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Yves R ichel le

University of Laval(Queb ec)

and

Paolo G. Garel la

Univ ersitadi Bologna

marc h1995

JEL Classi cation: 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 the rm with the highestaveragecost exits. Consequentysunk

costs not onlya ect the numb erof rms in anindustry,but they also en ter

the determination of the typ eof rms that resist predation.

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Inmark etswhere rmsdi erastotheircostfunctionsisitp 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 tical xed 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 osethatifall rmsstayinthemark et,eac hofthemwillobtain

a strictly negativ epro t at the Cournot equilibrium while, if only two rms

stay in, their Cournot pro t is p ositiv e and the third rm receiv es a zero

pro t. 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

completeinformationwhere rmsdi eraccordingtotheirpro ductioncapacity,thebiggest

rmisthe rsttoexit. Buttheseauthorsassumethat rmsincuronlya o wmain tenance

costwhichis prop ortionalto theircapacity. Accordingly,thebiggest rmistheone with

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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 eci cation of the coalition formationgame where

eac h rminturnmak esadeclarationconsisting of(i)asetof rmsthat 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 \o ers", and we model the acceptance (refusal) of an

o erasthemakingofaniden tical(di eren t)declaration. Sinceadeclaration

corresp onds toone equilibrium, if two rms mak ethe same declarationthey

agree to play the same equilibrium. This determines whic h equilibrium is

played and the payo s to all rms irrespectiv e of the declaration made by

the third rm.

It is importantto realizethat oncetwo rms hav eadopted their

equilib-rium strategies the third rm has no b etter alternativ ethan the play of its

ownb estreplytothosestrategies,whic hcoincideswiththestrategysp eci ed

in the equilibrium c hosenby the other two rms. Thereforethe waypay o s

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 esapredictionofthe rmthatexits.

Foreac horderinwhic h rmsdeclare,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 ti edand

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 o sforj andk. A pro ofof this statemen t istrivial. Indeedthe

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mark et and rm k mak es the highest equilibrium pro t 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 di erence in prediction could disapp ear if

rms playa sup ergame insteadof a one-shotgame. Indeed, inasup ergame,

rmsare generallyable tomaximizejointpro tsand,since jointpro t

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 in nite

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 twodi eren 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 di erence 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 of rms 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 unpro tablereen trycase andinSection5for thecase ofcostlessreen try.

In Section6wetest the robustnessof the resultsfor the case ofunpro table

reen trytoc hangesinthecartelformationgame. Theresultsstatedinsection

4areshowntogothrough. Section7presen tssomeconcludingconsiderations

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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 in nite rep etition of

the two-stagegamewhere(i)atthe rststageeac h rm 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 e rms,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 erfectlyobservedbyall rmsb eforetheyc ho osetheirpro duction atthe

secondstage. The scalar ,b elonging totheop en in terval(0;1), denotesthe

discount factor common to all rms.

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

ofthein niterep 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 in nitely 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 de ne 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 e rmhas 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 eci cparametersand q

i

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confusion). If a rm decides to stay out, it pro duces nothingand incurs no

cost. Furthermore if a rm, 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 unpro table 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-speci c 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

de ned 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 de ned 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 variablecostfunctionofany rmarethe

follo wing:

Assumption 3 Letq

i

>0. Thevariable cost function is twice continuously

di erentiable with respect to q

i and  i on X i 2R ++ . In addition, c satis es

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 ),satis es:

Assumption 4 For all Q 2 [0; P 3 i=1  q i

] , f is twice continuously

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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 pro t 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 two rms are activ ethen,for anyquantityits opp onent can pro duce,a

rmcanachieveap ositiv epro t. Formally,letusde new

i

(s)withs=fi;jg

asthe minimal pay o rmi can guarantee toitself whenit faces rm 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, de ning 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 erfectequilibriumpayo v ectorsofthein nitelyrep eatedgame0

(s)

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The typical pay o for rm i in the game 0

(s) is giv enby: P i =(10 ) 1 X t =0 t  i (q i ;Q 0i )

In nitely 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 o v 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

pro t 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.

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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 payo s in 0

We now consider the game 0

, namely the in nite 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 o v 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

three rmsstayinthemark etateac hp eriod. Thesetof payo v ectorswhic h

canb eobtainedatequilibriaofthiskindisdenotedbyV

(N) anditiseasily

v eri edthat itcoincides with W

(N)\R 3

+

. Weshall furthermoredenoteby

e

k

(N)a subgame p erfectequilibrium of 0

wherethe three rms stayinat

eac hp eriodalongtheequilibriumpathandwhere rmk obtainsap er-p eriod

pro t equal to maxf0;w

k

(N)g and eac h other rm, sayh, obtains a pro t

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

. For sucien 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 by rm 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 yo stoi andj willb eas sp eci edine

k (N).

4

Weimplicitlyassumethat rmk'soutsideopp ortunitieshasap erp erio dvalueofzero.

Howeverthe readerwill easily verifythat all ourresults gothroughif we ha veassumed

thatthep erp erio dvalueof rmi'soutsideopp ortunities,i=1 ;2 ;3,isgiv enbyafunction

O(

i

)satisfying@O=@

i

0. Thep oin tistode new^

i

(s)=w

i

(s)0O(

i

)forallp ossible

cartels andworkwithw^

i

(s)insteadofw

i (s) .

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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 wherethethree rmshaveen teredthemark et,with rmkhavinga

negativ e pay o . Thereforeusing this equilibrium as a punishmen ttriggered

bythe rm k'sdecision tostayin the mark et, rms i and j can force rm 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 predate rm k i.e. w

k

(N) <0 5

. We shall

denote the setof feasible cartels by S.

Before examiningthe setofequilibrium payo v ectors ofafeasible cartel

inthegame0

,wemustrecognizethatthere isathirdkindofmark et

struc-ture whic hcould emergenamelythe monopolyone. Howeverthis p ossibilit y

is ruled out, whether reen tryis costless orunpro table, byassumption 6.

We nowturn tothe c haracterizationof the setof equilibrium payo v

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. Consider rst the casewhere thereen trycost is

so high that it is nev erpro table 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 o v ectors of the subgame coincides with the set of

equilib-rium pay o v 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 o rmiwillobtainisequaltomax 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)<0inthede nitionofafeasiblecarteldo es

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rm i or rm j has deviated from their resp ectiv e quantit y sp eci ed 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 payo a rm, say rm i ,will obtainif it deviates froma sp eci ed

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 wsthatthesetofequilibriumpayo v 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 way rms 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 for rm 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

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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 payo p

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. Ifthedeclarationsof rmsj andkare

compati-blethen rmiwillobtainthepayo assignedtoitinthe rmj'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 o corresp onding to

the equilibriumup on whic htwo rms 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 payo rm 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 o of rm iin the cartel formationgame will b egiv enby g

i .

4 Equilibria in the cartel formation game

with unpro table reentry

Weturnnow tothe c haracterizationofthe equilibriaof the cartelformation

game when, in the pro duction game, reen try is unpro table 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 de ned 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

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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 di er according to the order of declaration. Howeverwe can iden tify

the cartelswhic hdonotformwhateverthe orderofdeclaration. LetW

i (s)

b e the highest one-p eriod pro t that rm 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 h rmscanb 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 third rm. Furthermore

if inaddition wesupp ose that q

j q i 7 and W j (i;j)W j (N) then we can

b emorepreciseab outthesurvivalofthelowestmarginalcost rm. Indeedin

this case the resultsstate that if a rm 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 ).

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cartel to whic h this rm do es not b elong. Accordingly a low marginal cost

functionand,sincetheresultsdonotdep endonthelev elof xedcosts, 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 payo rm 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 constraintsdi eronlyinthelev el

of  .

Forq

i =q

i

,theconstraintsinthesemaximizationprogramsaresatis edif

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

)letusdi eren 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

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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 isopro t 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 a ects

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 the xed cost. On the other hand,the lev el

of  ,i.e. the levelofthe marginal cost fora giv enquantitypro duced, a ects

the constraint in two ways. First, if the right-hand side of the constraint

wereindependen tof  then, inthe co ordinates of Figure1, rm k'sisopro t

curv e will b e en tirely b elo w the rm j's one as long as 

j < 

k

. This e ect

re ectsthe advantagetoformacartelwithalowmarginalcost rm. 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 e ect dominates the rst

one.

According to this result, for 

k

larger than 

j

, rm k can always giv eto

rm i agreater payo than the highest pay o rm 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 o rm i can obtain in V

(N). Consequently,

rm iwill prop osethe formationofthe grand cartelandthe b esteither rm

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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 o at least as great as the maxim um payo that

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 payo in it rather than in fi;kg ,

giv enthe constraints imposed by what hemusto er 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 di ers

from the one with unpro table 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) .

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j

is,if w

j

(N)and w

k

(N) are negativ e,then the minimal payo obtained by

rmsj andk inthese cartelsareequal tozero. Therefore,thehighestpayo

rmican obtaininacartel,W

i

(s),dep endsonb oth themarginaland xed

costs of its partner. More precisely,let q 0

h

b e the rival's output whic h leads

to zero pro t 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 pro t

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

coststructureofthethree rms. 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

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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 di erence b et weenthe results with costless reen try and the

ones with unpro table 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 unprop table 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 satis edif and only ifF

i <F j <F k . Hence we

ndbackaresultstated rst byGhema watandNalebu (1985)fordeclining

industriesaccordingtowhic hthe rmwiththelargestcapacitiesi.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 thecaseofunpro tablereentrydonotdep endonthe rms

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reentry

One sp eci c 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 payo s 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

payo s are allo catedamong cartel mem b ers.

This seems reasonable when reen try costs are negligible. In this case

indeedthepro ductiongameremainsathreeplayersgameev enifa rmexits

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

totheexiting rmanunrealisticallyexcessivein uence ontheequilibrium of

the resulting two rms 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

theexiting rmhasnoin uenceonthewaytheremaining rmswillsharethe

gains from co op eration in the pro duction game. This co ordination pro cess

constitutesagame: its rststepisasubstituteforthecartelformationgame

presen ted b efore. The only di erenceis that it isnow supp osed that a rm

declaration only consists of a feasible cartel, s. If all declarations di erthe

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 o v ector,p,b elonging toV

(s). If a rmdo 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 ofappropriatelydiscountedCournotpro tsoftheone-shotquantity

game)aresharedaccordingtoabargainingsolution. Thebargainingsolution

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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)denotethe rmi 'sCournotequilibriumpro twhenonly

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 satis ed. 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 ecomeworseo b 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

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guarantee that there exists a feasible payo v ector strictly greater than the

Cournot equilibrium pro ts v ector.

Againanincreasein

j

willhavetwoe ectsontheco op erativ ebargaining

game in volving rms iand j: On the one hand,it leads toamodi cation in

thesetoffeasibleoutcomeswhic ha ectsnegativ elythepayo of rmiatthe

egalitarian solution while, on the other hand, it increases (resp. decreases)

rm i 's(resp. rm j's)statu-quo pay o whic hwill rise the rm i 's payo at

the egalitarian solution. The Lemma 14

states simply that the p ositiv ee ect

arising from the movein the statu-quo pay o dominates the negativ ee ect

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

gamewherea rmi 'sdeclarationconsistsofafeasiblecartel,s

i

,towhic h rm

i b elongsand ofa pay o v ectorwhic hgiv estoeac h rm ins

i

the symmetric

egalitarian payo de nedab ov e 15

and a zero payo to 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 notde ne formally theegalitarian pa yo when thethree rms

areactiv e. Howeverthiscaneasilyb edoneevenifonewantstoconsideracoalitionform

game instead of a co op erative bargaining game. Anyway this do es not matter for our

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of the exiting rm on the waypay o sare allo cated inthe pro duction game.

7 Concluding remarks

Wehaveconsideredinthis pap eradynamicpro ductiongamein volvingthree

rmswhic haredi eren tiatedaccordingtotheircostfunction. Moreprecisely

wehaveassumedthat rms canb erankedunam biguouslyaccordingtotheir

marginal cost function and that their xed cost ma ydi er. 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 unpro table 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 eci cationof the cartelformationgame). Firstif reen tryisalways

unprof-itablethentheexiting rmhasthelowestmarginalcostfunctionascompared

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

Howeverthesetof rmswhichcanb epredateddep endob viouslyonthelevelof xed

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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 speci c 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 di erence 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 eci cin 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 tailsthat rms,asissupp osedinthecartelformationgame,willonly

considerpay o v 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

(24)

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 tryisunpro table

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 ween rms asillegal is p ossibly costly.

Let us now turn to the asset sp eci c c haracter of in vestments. In

vest-men ts are said wholly asset sp eci c if they are unredeplo yable. Accordingly

in vestment costs are sunk for wholly asset sp eci c in vestments while they

are xedwhen in vestmentlo oses its asset sp eci c c haracter. The main

con-sequence of the presence of asset sp eci c 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 unpro table reen trycase corresp onds to the situation where large asset

sp eci c 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 unpro table 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 a rm

exits the mark et. As we hav eshown, cost ineciencies app ear only in the

case of unpro table reen try whic h means that the presence of large asset

sp eci c in vestments is a necessary condition for such cost ineciencies to

(25)

Aumann, R. and R. Myerson 1988: "Endogenousformation of links

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.

Cam bridgeUniv ersityPress, Cam bridge,U.K.

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".

Mimeo, Departmen tof Economics, Universityof Pennsylvania.

Fudenberg, D. and J. Tirole 1986: "A theory of exit in duop oly".

Econometricav ol54 pages 943- 960.

Ghema wat, P. and G. Nalebu 1985: "Exit". Rand Journal of

Eco-nomics v ol16 pages 184-194.

Ghema wat, P. and G. Nalebu 1990: "The ev olution of declining

industries". Quaterly Journal of Economicsv ol105 pages 167-186.

Grossman,S.andO.Hart1986: "ThecostsandBene tsofownership:

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.

Tirole J. 1988: The Theory of Industrial Organization. MIT Press,

(26)

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

(27)

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) Let rmkb ethelast rmtodeclare. Ifd

i 62D

k

(i.e. theprop osed

cartelinthe rmi '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

then rm 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 .

(28)

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 facedby rm 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 o rm 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

(29)

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 o itwill 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

satis es the prop erties giv en in p oint B.2. 1a . It follo ws that by

declaringd k

i

,de nedinp 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

then rmiwillobtaineither

 g i = w i (N) by declaring d i

whic h di ers 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 )

(30)

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 hdi ers

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 .

(31)

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 maximizeitspayo bydeclaring 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 o by 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 twoindi eren 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

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