i n Compet i ng Fi rms
Pau l W .J. De Bijl 3
Mar ch 1995
Abstra ct
Thi s p aper investi gates th e strate gi c i mpact of organi zati onal desi gn
on pro duc t marke t competi tion. In a du opol y mo del of hori zontal
and verti cal p ro d uctdi erenti ati on , each rm's man ager can i mp ose
ap ro d uctl o cation,ordel egaterespon si bi l itytose lec tpro du ctl o cati on
tohi s subordi nate. The task of a subordi nate i sto devel op and
pro-duc ethego o d. Q uali tyi sdetermi nedbyhiseortl evel ,whi chd ep e nds
on h is private b en ets. The manage rs comp ete on a pro du ct market
byse ll in g the go o d s pro du ced by thei r subordin ates. Condi ti ons for
exi ste nce of equ il ib ria are deri ved, and imp li cati ons for man agement
strategyare di sc ussed.
Keywords: Firm Organ ization, Agen cy, Dele gation, Empowerme nt,
Ol i gop ol y,Pro ductDierentiation,Manage me nt S trategy;JEL
Clas-sication: D43,L13, L 20,M21.
3
CentERforEc onomicRes earch,TilburgUn iver sity,P.O.B ox90153,5000LETilbur g,
TheN ethe rlands . Themainres earchforthispap e rwasdoneatGREMAQ,Unive rsitede
Toulouse I,asapart ic ipantofthegraduate exchange programENTER.I amgrateful to
1 Introduction
To pmana gersofrmsdo notonlymake\stra teg ic"decisions ,fo rinstan ceo n
productchoiceandpricesetting ,buta ls odecideo no rga nizationa lissues like
delega ting resp o nsib ility to su b ordinates. Th ink, for insta nce, o f a product
manag erwh o isresponsible forhis rm'sma rket strategy, a nd has to decide
whichproduct va riety to sell insome marketsegment. A layer b elowhimin
thehiera rchy,thereisamiddlema nager,suchastheheado fthedevelo pment
and pro d uction depa rtment. In this paper, I s tudy the s trategic imp act of
org aniza tio nal structure, or more s p ecic, o f giving the middle mana ger a
say inthechoiceofaproductva rietythat hisd epa rtmentha stodevelo pa nd
produce.
Co nsider, a s an exa mple o f th e mo del, an olig opolistic market for so me
so ft drink, say cola, in which co nsumers h ave dierent preferences fo r
dif-ferent va rieties (such as reg ula rco la ,cherryco la , diet cola, an d caeine- free
cola). Fo r ea ch variety,con sumersa rewilling to pay mo reforhigherqu ality.
Suppose p rice co mpetition is erce: fo r given qua lities, a rm g ains more if
itpos itionsits bra ndinama rketniche(by dierentiatingits product), tha n
if it s ells a drink a imed at an \ averag e" taste. 1
Ea ch rmcons ists o f a product ma nag er and hissub o rdin ate (o rmid dle
manag er),whorepresentsthedevelopmenta ndproductiond epa rtment. 2
The
product mana ger has to ch o o se w hich cola type to s ell, and a t w hich price.
The subordinate perfo rms development an d production activities; qu ality is
determinedby his eo rtlevel. Wh ereas aproductma nag er ca resabout sales
or pro ts, his subordinate is motivated by priva te benets. For ins tance,
b ecauseo f ca reerconcerns he nds the acqu isitio nof pro fess io nal experien ce
important, or altern atively, he is ch allenged by techn ical innovativeness of
1
Cas ualempir icale vide ncesu gges tsthatpr o ductdi ere ntiationisanimp ortantsource
ofprots ins oft drinkmarkets . Co ca-Cola,forinstanc e,hasre ce ntlyintr o duce d, among
othervarie ties,ginsen g-base dandmilk-bas eddrinksinJ apan,andsugar-fr eec olorle ssc ola
inA me ric a( The E conomist,\Fizzing,"Sept ember4th1993,67-71).
2
Obvious ly,the remayals ob ec onic tsofint ere stb e twe enthemiddlemanage randthe
products. D eveloping a nd p ro ducing a certa in type o f cola requires sp ecic
technica lknowledg e(e.g . aboutchemicalsandproductio npro cesses),soth at
hisenthus ia smfor dierent typ es of colas will vary.
A product ma nag er does no t know how his subordinate's preferences.
The subordinate, however, ha s to invest cos tly time an d eort to nd o ut
hisp otentia lperso nal gains. A ma nagercaneither imp o sew hichva rietyha s
to b e produced (e.g . imp o se diet co la ), or give his s ub o rdina te a say inthe
cho iceofvariety (e.g. lethimcho o seb etweendietcola andca eine-free cola,
but notregularand cherryco la). Ifthe subordinate h assucient discretion,
he will want to acquire information a b ou t the poss ible drink types, so th at
he ca n reco mmend his preferred variety. If h e is allowed to d evelop a nd
produce his preferred variety, he will exert maxima l eort, and high quality
will result. 3
In the model,a product ma nag er fa cesthe following tra deo. Ifhe gives
his subordinate more discretion, it b ecomes more likely tha t he will get
in-formed in o rder to ma ke a pro p o sal wh ich , if a ccepted , w ill lead to a hig h
qualitydrink(apremiumbrand). Thesubordinate'sp ro p o sal,however,may
imply little di erentia tio n from other co la varieties, and th erefore res ult in
erce price co mpetitio n. Less discretion enables the mana gerbetter to p o
si-tio nadrinkinama rketniche,s otha tlo calmono p o lypro tsca nb eenjoyed.
The subordinate's incentivesto take initiative and exert e ort,however,
de-crea se, s otha t expected quality will be lower.
In the model, th e p o ssible cola varieties co rres p on d to locations on a n
interval representing co nsumers ' dierent tas tes . It is therefore convenient
to make a co mp aris on with the Ho telling model. In the sta ndard Hotellin g
3
Th e Econo mist disc usse se mpiricals upp ort fortheclaimthatrms\[:::]whichgive
middle manager s a s ay in forming s trate gy p e rfor m b e tter " and provide s examples of
delegationofres p onsibility. Forinst anc e,\Hondadeve lop edit sCiviccarbygivingagroup
ofyoungmiddlemanagersb roadguide lines( makeityou th-fr ie ndlyan dfuel-ecient)and
le ttingthe mge tonwiththejob." Also,\Motor ola'smiddle managers haveh ad asayin
des igningitsIr idiums at elliteproje ct." (\Thes alarymanridesagain,"p.70,Febr uar y4th,
1995.) O bviously, there may b e ac ombin ation of re asons(e .g. ince ntive s, infor mation,
mo delwithqua dratictrans p o rtatio n co sts,the deman d e ect (rmswantto
b e \w here the demand is"), outweigh s the stra teg ic e ect (rms want to b e
lo calmono p o lists) (seeD'Aspremonteta l.[3]). Co nsequ ently,rms
dieren-tia tetheirproductsas mucha sp os sib lein orderto s oftenprice comp etition.
In my mo del,a n ince ntiv e e ect a lso countera cts the strategice ect. Ifthis
eect b ecomes s tro nger, man agers will deleg ate more respons ibility to their
subordinates,a nd p ro ductswill belessdi erentia ted . In pa rticular,ahigher
impa ct of quality on prots favo rs more discretionin equilib rium.
Deleg ation d ecis io nsrela teto orga nizationa l structure and ma rket stra
t-egy. T hu s,s tudying the strategicna ture o fdeleg ationyield ssevera limplica
-tio nsintheeldofmanag ements trategy. 4
Theo ptimallevelofdiscretion,a s
afunctio no fthe discretion levelintheriva lrm,maybein creas ing(\s
trate-gicco mplements") as wella sdecreasing(\ strategicsubstitutes "),dep endin g
on the revenuefun ctions. Delega tio n o f res p on sibility ma kesa rm\ tough "
inth esensethatitreducesthepro tsoftheriva lrm;mo rediscretionresults
in a higher pro bability o f high quality, and a les s horizontally dierentiated
product. Mo reover, from the view p oint of an in cumb ent fa cing a potential
entra nt, a n optimal entry acco moda tio n strategy is to g ive the subordinate
littlediscretion (in the terminology of the ta xo nomy of mana gement s
trate-gies ofFud enb erg a ndT iro le[6 ]: ad opta\pup py dog "strategy). T hereaso n
isthat deleg atinglessresp o nsibility resultsin amore di erentia tedproduct,
which s often s price comp etition if entry occurs. By the sa me intu ition, the
optimalentrydeterrencestrategyisto emp owerthea gent(to beco mea\to p
dog ").
An interesting observa tion is th at in the mo del,an autho rita ria n
leader-shipstyle(thes ub o rdina te getslittledis cretio n)correspondsto aso ftstan ce
on the product market, a nd \hand s-o" mana gement corresponds to an ag
-gressive market s tance. Without claiming g enera lity, this result p o ints o ut
that lead ership styles may b e perceived quite d i erently inside and outs ide
a rm.
4
Man age me nts trategystudieshowamanage roptimallydes ignstherm'sorganiz at ion
In typical models o f industrial org anizatio n, rms a re viewed a s \ bla ck
b oxes." Althoug h this appro ach has led to imp o rtant insights, it ha s ma
-jor sho rtco mings . As Spulb er [10 ] arg ues: \For eco nomic mo dels to have
practica lvalueto ma nagers,theyneed toaddressthe choice ofboth
compet-itive ac tions and organ izational design " (p. 5 36, emphas is in o rig ina l). By
combiningo rganizationtheory andindus tria lo rg anizatio n,thispaperma kes
a prelimina ry attempt a t s hortening the ga p between eco nomic theory a nd
manag ement strategy.
The main litera tureo ncomp etitiona nd org aniza tio nal incentivesstudies
situationsinwhichmana gersplayamarketg ameo nb ehalfofowners(seefor
instanceVickers[14 ],Fershtma nandJ udd[5 ],Skliva s[9],andKa tz[8]). The
ques tio nintha tlitera tureiswhetherco ntra ctsbetweenow nersandmana gers
can serve a s precommitments. Having a n ag ent play the ma rket ga me may,
forinstance,res ultinlowerqua ntitieso rhig herprices. The fundamenta l
dif-ferencewiththa tliteraturesisthatIabs tractfroma gencyproblemsb etween
owners and ma nagers, and ins tea d lo o k at delegation inside rms. Delega
-tio no f res p o nsib ilityservesa norg anizatio nalpurpose{namely,itmotivates
as ub o rdina te to takeinitiative a nd exert eort (a ltho ugh co mmitmentmay
playarole). Als o,animp orta ntdierenceisthatinmymodel,theprin cipals
comp ete on th emarket, by selling g o ods produced by their ag ents.
InHo rneta l.[7 ],co ntra ctsbetweenownersandmana gersgiveamana ger
incentivestoreducethecostofproduction. Acommo nfeature ofth eirpa p er
and mine is that org aniza tio nal design ta kes pla ce b efore ma rket decis io ns
are taken. Their a nalys is su ggests a nega tive rela tio n b etween incentives
to reduce costs a nd the comp etitiveness o f product ma rket intera ction. In
my model,wh ich focus es on quited i erent issues, stro ngerincentives (more
resp o nsibility for asubordinate) result in more severe pricecomp etition.
The org aniza tio nal model is based on De Bijl [4]. In that pa p er, which
inturnwa sins pired byAghion andT irole [1],Iinves tiga teaprincip al- ag ent
rela tion shipinw hichtheprincipalappea lstotheag ent'sprivateb enetsfrom
formal a uthority to select a project, it is in his interes t to pick one th at
generates interes t fro m the agent. Thus, a ltho ugh the s up erio r has formal
auth ority (thedecisionright),the sub o rdin ate mayto so meextenthavereal
auth ority (seealso Tiro le [13 ]).
The model is presented in the next s ection. The formal results a re
de-rivedinsection3. Sectio n4 discuss es imp lica tio nsforma nag ementstra teg y.
Fina lly,s ection 5concludes.
2 The M odel
The modelco nsists of three building blocks: a Hotelling-typ e p ro duct ma
r-ket,th eo rganization ofa rm, and comp etition betweenvertical s tructures.
These will be taken u pin turn.
Product M arket Competition:
Therea re two rms ,called1and 2 . Firm1 ca nchoose a horizo ntalproduct
specica tio n (o r product loca tio n) x
1 2 [01 ;0], a nd rm 2 a p roduct sp ec-ica tio n x 2 2 [0;1]. 5
T he vertica l product qua lity of rm i is deno ted by
r
i .
Co nsumers are unifo rmlydistributed alo ng the interval [01;1 ]. The
will-ingnessto payo f acons umer\ located"a tz for rmi'sproductisdecrea sin g
in the distance b etween z a nd x
i
, a nd increa sin g in r
i
. A cons umer h as a n
inelastic demand for one unit; she purcha ses the good tha t gives her the
highest net su rplus .
Once product cha ra cteristics are xed (seeb elow ),the rms co mpeteo n
the product market by s imu ltan eo usly setting prices. Ma rginal costs are
equa la nd no rma lized to zero. Befo re the price comp etition stag e, the rms
obs erve ea ch others' product cha racteristics. Tokeepthe a nalys is tractable,
pricecomp etitio nisno tmo deledexplicitly. I willa ssumetha t givenproduct
lo cationsx 1 andx 2 ,andqualitiesr 1 an dr 2
,thereexistsauniqueequilibrium
5
inthe price sub game. Also,qua litiesa resuciently hig hso tha tthe market
is always covered.
Given the unique equilibriu m o utcome in the price s ubga me, rm i's
revenu e (or pro t) function is denoted by R
i (x 1 ;x 2 ;r 1 ;r 2 ), wh ich is twice
continuo uslydierentiableinx
1 andx 2 (i=1;2 ). Also,R i (x 1 ;x 2 ;r 1 ;r 2 )0.
Assu mption 1 (Rev en ue func tions)
(i) R i (x 1 ;x 2 ;r 1 ;r 2 ) is strict ly increasing in r i
, an d st ric tly decreasing in r
j , for all x 1 ;x 2 , i=1;2 . (ii) R i (x 1 ;x 2 ;r 1 ;r 2
)is strictl ydec reasin gin x
1
, and strictl yinc reasing in x
2 , for all r 1 ;r 2 , i=1 ;2.
The interpretation of as sumptio n 1 is d irect. A rm's pro t level is
in-crea sin ginitsownverticalp ro ductquality,a nd decreasinginitsriva l's
qual-ity. Furth ermore, givenqu ality levels ,the rms wouldliketo di erentia tea s
much a s poss ible to soften price co mpetition. So implicitly, on the interval
[01 ;1 ]thestra teg iceect(rmswantto beloca lmono p o lists)do minatesthe
dema nd e ect (rms want to b e\ where the d emand is"). 6
Thus , the model
app liestoma rketsinw hichitisprotab leforrmstop o sitionbra ndsinma
r-ketniches. Moreover,theas sumptio nwillallowfo rea syco mparison w iththe
maximumdi erentia tio nresult o f the Hotelling model w ith qu adratictrans
-p o rtationcos ts.
Organization of a Firm:
The way a rm is o rganized is a dapted from De Bijl [4]. Firm i cons ists of
a principal P
i
(the mana ger) a nd an ag ent A
i
(the man ager's s ub o rdina te),
i = 1;2. The role of a principal in a rm is either to impose a ho rizontal
products p ecicationor to d eleg ate theproduct locationto hisag ent. Given
6
Cf. t he Hotelling lo c at ion mo de l with quad ratic transp ortation costs , some nite
re ser vation value for consume rs, and p oss ibly die re nt ve rtical pro duct qu alit ies . The
willingnes stop ayofaconsu me rlo catedatzforgo o diinthatmo delisr
i 0p i 0d(z0x i ) 2 , wher ep i
isth ep ric eofthego od ,anddame as ureofthetr ans p ortationcost . Forr
1 =r
2 ,
pro duc tlo c at ionsinequilibriumarex 3
=0 1andx 3
productloca tio n,thes ub o rdina teta kesca reo fd evelopmenta nd pro d uction,
and vertica l product quality isd etermined by h is e ort level. Once locatio n
andqua litya redetermined ,thema nag erch o o sesapriceinordertomaximize
expected pro ts.
Anag entismotivatedtoexerteortbyprivatebenets,whicha rerelated
to horizo ntal product cha racteristics . P rivate b ene ts may include jo b sa
t-isfactio n, a sense of achievementa nd acco mplis hment, p erks on th e job,the
acquis ition of pro fes sio nal exp erience, career concerns , a nd s o o n. Fo r
sim-plicity, theagentdoesnot respondto pecuniaryincentives . For insta nce,the
ag entisinnitelyriskaversewithresp ecttoinco me. Acco rding ly,eachag ent
receivesa cons tantsa la ryequa lto his reserva tio nwa ge,which is normalized
to zero. 7
A
1
's priva te b ene ts are determined by Nature as follow s. Exactly one
p o int in [01 ;0 ] yields the a gent benets b; all the o ther pro du ct loca tio ns
yield b <b (w here b >0 ). The location o f the high private-benet p o int is
uniformly dis tributed on [01 ;0]. 8
T heprivate b ene ts of A
2
are d etermined
inasimilarfa shion onthe interval[0;1 ],a nd areindep endentof A
1 's private b enets. L et 1b0b: IfA i
isno tallowedto pro d ucethe highprivate-benetsg o odthenh ewill
exertlowe ort,which resultsin low verticalquality r
i
=`>0. Conversely,
producing a goo d which yield high priva te b ene ts results in high product
quality r
i
= h > `. 9
Note th at by ab stracting fro m pecuniary incentives,
punishments based o nloweo rtare ruled out.
The realization of A
i
's p rivate benets ca n only b e obs erved by A
i , but
7
InDe B ijl[4] I showthat abstr ac tin gfrompayment s do es notharmgene ralityifan
age ntisrelativelymoreres p onsive topr ivateb e nets thant omoney.
8
Thedisc ont inuityint hedistr ibutionsimp liesthee xp osition;itisnotc rucialforthe
in sights .
9
Onecane xplicitlymo delanagent'sb e havior. S upp oseanage nthasautilityfunct ion
U(b;e ),whe rebdenoteprivateb ene tsandehise ortle ve l. As sumeU(b;e)isinc reas ing
in b for alle ,s tric tlyc oncave ine forall b,and satis es@ 2
U(b;e)=(@b@e)>0. It follows
hehas to incur aprivate co stF 0 (forinsta nce, time andeo rt) to do so.
The principa lca nnot verify whetherhis ag ent g ets in fo rmed.
P
1
's delegation decision is expressed by a function p
1
: [01 ;0] ! [0 ;1 ],
suchthatifA
1
recommendsproductlocatio nx
1
,heisallowedto pro d ucethe
good located a t x
1
with pro bability p
1 (x
1
), but has to produce th e good at
01w ithprobab ility10p
1 (x
1
). Simila rly,P
2
'sdeleg ations chemeisd escrib ed
by a function p
2
: [0;1]! [0;1 ] (A
2
has to p roduce good 1 with proba bility
10p 2 (x 2 )givenp ro p o sal x 2 ). So p i (x i )=P r(A i is allowedto pro du ceg o odx i j A i p rop o sed x i ):
Whetheranag entwilllea rnhisp riva tebenetsdepends onthedis cretio n
hehas . A
1
g etsinformed if a nd o nly if 10 Z 0 01 [p 1 (x 1 )b+(10p 1 (x 1 ))b]dx 1 0F b; or equivalently, Z 0 01 p 1 (x 1 )dx 1 F 1 : (1)
One can write dow n a simila r inequa lity for A
2
. To make the model
inter-esting , the following assu mption is ma de:
Assu mption 2 F <1,implyingt hatifanage nthascompl etere sponsib ility
conce rnin g produc t l ocation (p
i (x
i
)=1 for all x
i
) the n he w ill g et informed.
An uninformed agent is in di erent b etween the p o ssible loca tio ns. For
simplicity, hewill then pro p os e the principal's preferredlo cation.
I a ssumetha taprincipal ca ncommithimself to adelegation scheme;the
focusofthep ap erisondeleg atio nasamean s tomotivateasubordinate. 11
A
justica tio nisthat a ma nag er caresa b o uthis repu tatio nto keepapromis e.
Since s elling ah ig h-quality goodlo ca tedat x
i
may yieldhigher pro ts tha n
sellingamoredierentiatedlow- qua litygoodforallx
i thatsatisfyp i (x i )>0, 10
Tob e pre cise, b andb r epre se ntthe privateb ene ts obtainedbyth eagentgive n his
optimaleortle vel;e.g.,usingnotationintro duc edinfo otnote9,br epre se ntsU(b;e 3
(b)).
11
Itwillb eshownthatp
i (x
i
)2f0;1gforallx
i
delega tio n schemes may be o ptimal ex p os t; the a ssumption is not crucia l.
This istypically the ca seif high qualityhas arela tivelylarg e impacton
rev-enues,comparedto dierentiation.
Competing Organiza tional Structures:
The p rincip als comp ete with ea ch other; they face each other on th e
prod-uct ma rket. Th ere is no intera ction between the ag ents, a nd they cann ot
communica te with ea ch other. The course o f events is as follows :
t =0: Na ture selects the a gents 'private benets, uno bserved a tthissta ge.
t =1: The prin cipals simultaneous ly choos e deleg ation schemes, un
observ-able o utside ea ch rm. Each principa l co mmunicates the deleg atio n
sch eme to his a gent, who then decides whether to learn h is private
b enets. The latter decision is private info rma tio n for a n ag ent. The
ag entsthensimultaneous lyrecommendp ro ductloca tio nstotheir
prin-cipa ls. Product locatio ns a resimultaneou sly selected acco rding to the
delega tio n schemes. An ag ent's proposa la nd the selected locationare
unob servable o utside ea ch rma tthis sta ge.
t =2: Eacha gentpicksapro du ctione ortlevel,andverticalproductqualities
are realized.
t =3: Product locations a nd qualities are obs erved . Th e principa ls simulta
-neously setprices and the g oo dsare so ld on th emarket.
Itisimp o rtanttono ticetha ton cep ro ductionhastakenplace,deleg atio n
schemes no lo nger matter;on lyproduct lo cationsand qualities in uen cethe
prices th at a re charg ed in th e ma rket.
In the a nalysis that follows , subg ame p erfect equilibria in pure s
trate-gies a re derived. Since the price stag e is not mo deled explicitly, ess entially
the principals comp eteby simulta neously selecting delega tion schemes . The
3 Analysis
Therstpro p os itionallowsustorepresentdeleg ationschemesbywell-dened
\discretion levels." In p articular, in any equilibrium p 3 i (x i )= 1 fo r all x i in
so me intervalconta ining rm i's maximally dierentiated pro du ct lo cation,
and p 3 i (x i )=0 o therw ise. 12
A discretion level for rmi's ag ent, deno ted by
X
i
, is a cco rding ly dened as the length of the interval on which p
i (x i ) =1. A higherlevelof X i
co rres p on dsto more respons ibility foragentA
i
. In pa
r-ticular, if X
i
= 0 then rm i's ma nag er imp o ses his a gent to p ro duce the
maximally di erentia ted product. If X
i
=1,a gent A
i
has fullres p on sibility.
Proposition 3.1 Inanyequ ilibrium,thereex istdiscre tionle ve lsX 3 i 2[0;1], i = 1;2, such that A 1 's recomme ndation x 1
is follow ed up if and onl y if
x 1 01 +X 3 1 , an d A 2 's recommendation x 2
is followed up if and onl y if
x 2 10X 3 2 .
Proof: See the appen dix.
Intu itively,g iventhelevelofresponsibilitythe rivalrm'sa genth as,each
principalfa cesthefollow ingtradeo . Givinghisa gentlittlediscretio nresults
inala ck o finitia tive: thea gentha sno incentiveto lea rnhisprivate b enets
and makea recommenda tio n. The ma ximally di erentiated pro du ct will b e
produced, but qua lity will be low. Much discretion results in initia tive: the
ag entw illg et info rmeda nd reco mmend hispreferredp ro duct loca tio n. The
product will b e less di erentia ted,but quality willb e high if the prop o sa l is
followed up.
Usin g(1),adirectco nsequenceofpro p os ition3 .1istha tA
i
g etsinfo rmed
if and only if heha s enou ghdiscretion.
Corollary 3.1 Agen t A
i
get s informed if and onl y if X
i F 1 . 12
A similar res ultis obtainedinDe B ijl[4], with adiscr etenumb e r of projec tsandin
So me additio nal no tation is intro du ced. Let
i
:[0;1]2[0;1]!<denote
P
i
's exp ected revenue as a functio n of (X
1 ;X
2
), given that both a gents get
informed, i=1 ;2. Acco rd ing ly,
i (X 1 ;X 2 ) = Z 01+X 1 0 1 Z 1 10 X 2 R i (x 1 ;x 2 ;h;h)dx 2 +(10X 2 )R i (x 1 ;1;h;`) dx 1 +(10X 1 ) Z 1 10 X2 R i (01 ;x 2 ;`;h)dx 2 +(10X 2 )R i (01;1 ;` ;`) :
Firm i's expected pro ts, a function
i :[0;1 ]2[0 ;1 ] ! <, ca n now b e dened as follow s. i (X 1 ;X 2 )= 8 > > > > > > < > > > > > > : i (X 1 ;X 2 ) if X 1 F 1 andX 2 F 1 , i (X 1 ;0) if X 1 F 1 andX 2 < F 1 , i (0;X 2 ) if X 1 < F 1 and X 2 F 1 , i (0;0) otherwis e.
With expected pro ts written a s functio ns o f levels of discretion, we are
ready to derive the main results. The following lemma will b e invoked
re-p eatedlyin the ana lysis below.
Lemma 3.1 (i) i (X 1 ;X 2 )isstrictl ydecreasing inX j , forall X i , i;j =1 ;2, i6=j (ii) i (X 1 ;X 2 ) isstrict ly concav e in X i , for all X j , i6=j, an d (iii) @ 1 (0 ;X 2 )=@X 1 >0 , for all X 2 ; an d @ 2 (X 1 ;0 )=@X 2 >0 , for all X 1 . Proof: Dierentiate i (X 1 ;X 2
)pa rtially(twicetoprovepa rt(ii))andap ply
as sumptio n 1. 2
If wesuppose thata gentscan co stless ly o bservetheir private b enets,s o
that i (X 1 ;X 2 ) = i (X 1 ;X 2
), i = 1;2, then lemma 3.1 has stra ig htfo rwa rd
interpretatio ns. Acco rdingto part(i),aprincip alwa ntstheag entoftherival
rm to have as little discretion a s p os sible. No tice the simila rity with the
as sumptio ntha t a rmwantstherivalrmto locatea sfar away as possible.
thattheriva lproductwillbeo fhig hqua lity. Usingtermin olo gyofFudenberg
andTirole[6],d eleg ationofresp o nsibility makesarm\to ugh," inthe sense
of reducingthe rivalrm's prots .
A stra ightforwa rdimplication oflemma3 .1(iii) is the follow ing :
Corollary 3.2 If F = 0 the n in any equil ibriu m eac h prin cipal giv es his
age nt some re sponsib ility,i.e ., X 3 1 >0 an d X 3 2 >0.
The next propos ition gives necessa ry a nd s ucient co ndition s fo r exis
-tenceo fan equilibriuminw hichb o thag entshavefulldiscretion. Info rma lly,
proposition 3.2 states that b o th a gents h ave full discretion in a n
equilib-riu m wh en s elling a hig h-quality product is more prota ble than s elling a
maximally dierentiated pro d uct. Expected p ro duct locations a re 0 1
2 a nd
1
2
. Since the agents have complete freedom to pick product loca tio n, b o th
products will be of high qua lity.
Proposition 3.2 There exists an equ ilibrium in which eac h princ ipal giv es
his ag ent complet e re spon sib ility,i.e ., X 3 1 =X 3 2 =1 , if an d on ly if Z 1 0 R 1 (0;x 2 ;h;h )dx 2 Z 1 0 R 1 (01 ;x 2 ;`;h)dx 2 : (2)
Proof: By lemma 3.1 we h ave
1 (X 1 ;1 ) is strictly concave in X 1 , and a ls o @ 1 (X 1 ;1 )=@X 1 j X 1 = 0 >0. Therefo re, X 3 1 = 1 is a bes t response to X 3 2 = 1 if and only if @ 1 (X 1 ;1 ) @X 1 X 1 =1 0 ;
equivalent to inequa lity (2 ). The result follows by symmetry. 2
Inequa lity(2) canbeinterpreteddirectly intermsof productchara cteris
-tics: g iventha t the riva l rm's ag ent has fulldiscretion (which implies hig h
vertica lpro du ctqua lity),aprincipalprefersto sellahigh- qua lity pro d uctlo
-catedat the center (tha tis,a t0 )to alow-qua lity productthat isma ximally
dierentiated.
to emp ower his subordinate to s elect product lo cation beca use it w ill result
in high product qua lity. Und er conditio n(2 ), a nd a ls ounder the conditio ns
for equilibria w ith intermediate discretion that are g iven in propos ition 3.4
b elow,theincentiveeectissucientlystrongsotha twedonolon gero bserve
the maximal dierentiation result o f the Ho telling model.
Bycoro llary3.2,a nequilibriuminwhicheachprincipalimp o seshisag ent
to pro du cethe maxima lly di erentia ted product exists o nly if F >0.
Proposition 3.3 Suppose that F >0 . There e xists an equil ibrium in w hich
each prin cipal giv es his ag en t no responsibil ity, i.e., X 3 1 = X 3 2 = 0, if and onl y if Z F 1 01 01 R 1 (x 1 ;1;h;`)dx 1 < F 1 R 1 (01;1;` ;`): (3)
Proof: Let F > 0. By lemma 3.1,
1 (X 1 ;0) is strictly concave in X 1 , a nd @ 1 (X 1 ;0 )=@X 1 j X 1 =0 > 0 . Therefore, X 3 1 = 0 is a b est response to X 3 2 = 0 if a nd only if 1 ( F 1 ;0) < 1
(0 ;0), equivalent to inequality (3). The result
follow s by symmetry. 2
A necess aryconditio nfor (3 )is
R
1
(0 ;1;h;`) <R
1
(01 ;1;`;` ): (4)
Toseethis,no ticetha tbylemma3.1,inequa lity(3)(equivalentto
1 ( F 1 ;0)< 1 (0;0)) implies R 1 (01 ;1;`;` )= 1 (0;0 )> ma x X12[ F 1 ;1] 1 (X 1 ;0 ) 1 (1 ;0) = Z 0 01 R 1 (x 1 ;1 ;h;` )dx 1 >R 1 (0;1;h;` ):
Inequa lity (4 ) ca n b e interpreted more directly tha n co ndition (3). It
says th at a principal prefers to s ell a low-quality, ma ximally dierentiated
product to a hig h-quality, min imally dierentiated pro du ct, given tha t the
rival rm produces a low -quality product that is maximally di erentia ted.
As proposition 3.3 demonstra tes , the mo d el is a ble to g enera te the
well-known maximumd i erentia tionresultofthe Ho telling modelwithquadra tic
trans p orta tio nco sts. T hisoccurswhenth eincentiveeectisrelativelywea k,
so tha t the s tra teg ice ectdomina tes both th edema nd eecta nd the
incen-tivee ect.
Theremaya ls oexistequilib riainw hichagentshavea nintermediatelevel
of discretio n,eno ugh to motivate themto g et info rmed.
Proposition 3.4 There exists an equ ilibrium in which eac h princ ipal giv es
his ag ent l imit ed re sponsibil ity, i.e., X 3 1 = X 3 2 2 [ F 1 ;1), if an d on ly if there ex ists an X 3 1 2[ F 1 ;1) su ch t hat Z 1 10 X 3 1 [R 1 (01+X 3 1 ;x 2 ;h;h)0R 1 (01 ;x 2 ;`;h)]dx 2 8 > > > < > > > : =(10X 3 1 )[R 1 (01 ;1;`;` )0R 1 (0X 3 1 +1 ;1 ;h;`)] if X 3 1 2( F 1 ;1 ); (10X 3 1 )[R 1 (01 ;1 ;`;`)0R 1 (0X 3 1 +1;1 ;h ;` )] if X 3 1 = F 1 ; (5) and 1 (X 3 1 ;X 3 2 ) 1 (0;X 3 2 ) if X 3 1 = F 1 .
Proof: (i) Supp o se th at X 3 2 2 ( F 1 ;1 ). By lemma 3.1, 1 (X 1 ;X 3 2 ) is strictly concave in X 1 , a nd @ 1 (X 1 ;X 3 2 )=@X 1 j X 1 =0 > 0. Therefo re, X 3 1 = X 3 2 is a b est respons e to X 3 2 if a nd on ly if @ 1 (X 1 ;X 3 2 ) @X 1 X 1 = X 3 2 =0;
equivalent to the equality in (5). T he resu ltfo llows by symmetry.
(ii) Supp o se tha t X 3 1 = F 1 . By lemma 3 .1 , X 3 1 = F 1 is a bes t response to X 3 2 = F 1 if an d only if @ 1 (X 1 ; F 1 ) @X 1 X 1 = F 1 0
(equivalent to the inequa lity in (5)) a nd
1 ( F 1 ; F 1 ) 1 (0; F 1 ). Th e result follow s by symmetry. 2
Co ndition (5) in pro p o sition 3.4 sta tes tha t X 3 1 is a bes t response to X 3 2 = X 3 1 . Fo r X 3 2 2 ( F 1
;1), we have a s tanda rd rst- order condition. For
X 3 2 = F 1
, the discontinuity o f rm 1's pro t function implies that we must
require that a marg ina l increase in A
1 's dis cretio n (a t X 3 1 = F 1 ) does n ot
increas e rm1 's expected pro ts. This explain s the inequa lity in (5 ).
It is straightforward to derive existence co nditions for as ymmetric
equi-libria,bu tthis invo lvestedio us nota tio n w itho ut g etting additional insights.
For simplicity, s upp o se that F = 1. Then there exis ts an equilibrium in
which one principal g ives his ag ent res p ons ibility an d the other d o es not,
that is, eith erX 3 1 =1 and X 3 2 =0 or X 3 1 =0 and X 3 2 =1,if a nd o nly if 1 (1;0 ) 1 (0;0 ) and 2 (1 ;1)< 2 (1;0): (6)
These inequa lities are sta ndard Nas h equilibrium conditio ns. T he seco nd
conditio n in (6 ) can a ls o b e written a s
1 (1;1) < 1 (0;1). Since 1 (1;1 ) < 1 (1;0) and 1 (0 ;1) < 1
(0 ;0),asymmetricequilibria may indeedexist.
4 M anagement Strat eg y
Mana gement stra teg y studies how a mana ger o ptimally chooses org aniza
-tio nalstructure a nd marketstra teg y, givenany p o liticaland regu lato ryco
n-stra ints. Inthis pa p er,o rga nizationa ldesign isdetermin edwh ileta king into
accounttheo utco meofmarketcomp etition{themana ger'sdecis io np roblem
is so lved by abackwa rdinductio nprocess(see alsoS pulber [1 1]).
Inthemodel,ama nag erselectsadiscretionlevelforhissubordinatewhile
reecting on res ulting product lo cations ,qualities,an dprices. In particular,
amana ger'sdecis iono fd eleg ationofresp o nsibilityca ptureshismarketstra
t-egy concerning p ro duct characteristics and price, an d therefore repres ents,
in the context o f the model, the rm's overa ll stra teg y. In this section, I
Strategic Complements or Subs titutes?
Fromamana ger'sview p oint,itis interestingto knowhowthe riva lrmwill
react if he g ives his subordinate more or less d iscretio n. Applying no tio ns
develo p ed by Bulow et al. [2] a nd Fudenb erg and Tirole [6], I will a nalyze
whetheranincreaseo fthelevelo fdis cretioninarivalrmin ducesamana ger
to deleg atemo reor lessres p on sibilitytohissubordinate. Intheformercas e,
reactio n functions are u pwa rd sloping , and discretion levels a re sa id to b e
stra teg ic complements. In the latter case, reactio n functio ns a re d ownwa rd
sloping,an d discretion levels are stra teg icsub stitutes. 13
Givena unique equilibrium o utco me o f th eprice su bgame, we can focus
on co mpetitio nindelega tio nschemes , represented by thelevelso f dis cretio n
X
1
and X
2
. Firm i's b est resp o nse (o r reactio n fun ction) to X
j (j 6= i) is dened as X 3 i (X j )arg max X i 2[0;1] i (X 1 ;X 2 ):
The fo llowingexampleillustra tes one o f ma ny p os sib le situations.
Figure 1 Reactio n functio nsa nd equilib ria -6 0 X 2 X 1 1 1 X 3 2 (X 1 ) X 3 1 (X 2 ) X 3 1 (X 2 ) X 3 2 (X 1 ) 8 8 8 8 8 8 1 1 1 1 1 1 X 3 X 3 F 1 F 1 13
Exa mple (see g ure 1): Fo r an intermediate value of F, su pp os e that
in-equa lities (3) an d (5 ) ho ld. By propositio ns3.3 and 3.4, there are two
sym-metric equilibra, namely (0;0) a nd (X 3 ;X 3 ) for some X 3 2 [ F 1 ;1). For a n
expositional purpose, rea ction fun ctions are assu med to b e increasing inthe
regions wherea gents acquireinformation.
Suppos enowtha tF =0 ,sotha tweneednot worrya b outdisco ntinuities
in the reaction functio ns. Denefor allX
1 , 1 (X 1 )R 1 (01+X 1 ;1 ;h ;` )0R 1 (01 ;1 ;`;` ); and for a llX 1 and X 2 , 1 (X 1 ;X 2 )R 1 (01+X 1 ;10X 2 ;h;h)0R 1 (01;10X 2 ;` ;h): The va lue o f 1 (X 1
) is rm 1's g ain from selling a high-qua lity product
lo cated at 01+ X
1
co mpa red to selling a ma ximally di erentia ted , low
-quality product,g iven that rm2produces alow-qua lity product loca tedat
the extreme. The value of
1 (X
1 ;X
2
) repres ents a s imilar g ain g iven th at
rm2 sells a hig h-quality product lo cateda t 10X
2 . Proposition 4.1 Suppose F =0 . (i) If 1 (X 1 ) > 1 (X 1 ;X 2 ) for all X 1 ;X 2
, lev el s of discre tion are strateg ic
compleme nts. (ii) If 1 (X 1 ) < 1 (X 1 ;X 2 ) for all X 1 ;X 2
, le ve ls of discre tion are strateg ic
subst itutes.
Proof: Bydi erentia tingtherst-ord erconditio n@
i (X 3 i (X j );X j )= @X i =0
withres p ecttoX
j
(as suminga ninterio rs olutio n),an da pplyinglemma3.1(ii),
it follows that the sign of dX 3 i (X j )=dX j
(determining the s lope o f reactio n
functio n X 3 i (X j ))is equ alto the s ig nof @ 2 1 (X 1 ;X 2 ) @X 1 @X 2 = R 1 (01+X 1 ;1 ;h;`)0R 1 (01+X 1 ;10X 2 ;h;h)+ R (01;10X ;` ;h)0R (01 ;1 ;`;`): (7)
By rewriting (7 )a s @ 2 1 (X 1 ;X 2 )=(@X 1 @X 2 )= 1 (X 1 )0 1 (X 1 ;X 2 ), the re-sult follows. 2
The interpretatio n is d irect. Su pp o se revenues of selling a hig h-quality
productco mpa redto ma ximally dierentiatingitspro d uct(wh ichwo uld
im-plylowqua lity)arehig herifitsrivalsellsalow-qua lityproductloca tedatthe
extreme,than ifitsrivalsellsa high -qualityproduct(no t necessarily located
at the extreme). Then P
j
's b est respons e to more discretion fo r ag ent A
i is
to g ivehisagentA
j
mo rediscretio naswell. Thereisasimila rinterpretatio n
of th esucient con dition fo r stra teg icsubs titutes.
Top Dog or P uppy Dog?
Suppose tha t only one rm, say rm 1 , is active in the ma rket, and th at
rm2isap otentia lentrant. One cand istinguishtwocas es : the incumbent's
manag erwantstodeter entry,o rhewantstoa ccomodateentry(forinstan ce
b ecause entry deterrence is no t p rotable). In ea ch ca se, the incumbent's
manag erha s to formula te a n appro pria testra teg y. In case of accomo d atio n
for ins tance, he will want to ch o o se a stra teg y that so ftens p o st-entry price
comp etition. In wha t fo llows, I a ssume tha t rm 2's ma nager decides o n
entry (a nd if he enters, o n how much respons ibility h e w ill d eleg ate) after
havingob servedinwhichmarketnicherm1 'spro du ctisloca ted,and which
quality rm1 is selling .
The taxono my ofma nag ement stra teg iesproposed byFudenberg and
Ti-role [6 ] is used to cha racterize emp owerment a s a strateg y to a ccomodate
or deter entry. C onsider the level of discretion of an agent a s the stra teg ic
\investment" va riab le. A di erence w ith Fudenberg and Tirole's set-up is
thatin mymodel,the pro d uctch aracteristicsres ulting from\investment" is
obs erva ble,whereasintheira nalys is,investmentitselfcan b eobs erved. This
dierence,however,doesnotma tter. The reas onisthatalthoug h deleg atio n
schemes a re u nobservab le, each mana ger can o bserve the oth er's product
-matter;o nly the p roduct characteris tics a rethen relevant.
In the product market subg ame, prices are strategic co mplements for
given pro du ct cha ra cteristics. 14
Mo reover, by lemma 3.1 (i), deleg ation of
resp o nsibility ma kes a rm to ugh in the sense of reducin g the riva l rm's
pro ts .
Suppos etha t,fo r axedlevelo f discretio n forA
2
,the principalof rm1
delega tes more resp o nsibility to A
1
. T hetotal eect,whichis P
1
's incentive
to deleg ate respons ibility, is given by @
1 (X 1 ;X 2 )= @X 1 . T his e ect ca n b e
deco mpos ed into two e ects. First, a direct (or pro t maximizing) e ect
of giving A
1
mo re resp o nsibility is th at fo r g iven prices, rm 1 's expected
market sha re and product quality, an d therefore pro ts, increa se. Second,
there is a s trategic eect, res ulting fro m rm 2's price reactio n. If A
1 gets
more discretion, the pro bability that rm 1 's product w ill b e located closer
to the center increa ses . T herefore,in exp ectations the products w ill b e les s
dierentiated,sothatpriceco mpetitionb eco mesmoreintens e. Inparticular,
itwillb e expected rm2will reactby lowering its price,th ereby decrea sin g
rm1'smarket share a nd pro ts.
Given that rm 1 wants to accomo da te entry, the fa ct that deleg atio n
makes armtoug h impliesthat P
1
sho uld \ underinvest" in delega tio n. 15
In
the terminolog y of Fudenb erg and T iro le, P
1
shou ld adop t a \puppy do g
ploy," that is, it sho uld b e nice and s ma ll in order to avo id to trigger a n
ag gressive response fro m rm 2 . The op timal entry deterrence strategy for
rm1isto \overinvest" indelega tio n,thatis,adop t a\to pdog " stra teg yin
order to b e a toug h rival. Sucha s trategy will reduce prots ofa n entra nt.
Dierent Perception s of a Man agement Style
The p revious dis cus sio n points a t a n interesting link between a ma nager's
sta nceins idearmand hisp o stureo nthe productma rket. In particula r,in
14
SeeTirole [12],chapte r 7,foradis cus sion.
15
Mor eprec is ely,X
1
willb elower thantheop e n-lo ops olution,wh ich isde ned asthe
optimalvalue ofX
1 if P
2
cannotobs er ve t he pro duc tcharacte ris tic s ofrm 1's pro duc t
the mo delthere are dierent perceptio nso f a sing le lea dership style.
Being nice to the rival rm co rres p on ds to ado pting a to ugher posture
vis- a-vis his sub o rdina te, b ecause there is u nderinvestment in d eleg ation of
resp o nsibility. Mo re g enera l, the mo d el demo nstrates tha t motivating the
subordinate to take in itiative by delega ting resp o nsibility corresponds to a
more agg res sives tance onthe pro d uctmarket. According ly,aproduct ma
n-ag er may give h is su b ordinates a loto f freedom (\han ds-o " ma nag ement);
not beca use he is s ucha niceand frien dly p erso n, bu t b eca use heis atoug h
comp etito r. Vicevers a,a na uthoritarianma nag er(i.e.,a mana gerw ho gives
his subordinate littleo r no discretion) is a so ft rival in the pro du ct market.
Summa rizing: a toug he r posture of a manag er in side a rm (i.e ., with r
e-gardt o hissubordinate )corre spondstoasoft erpostureontheproductmarke t
(i.e., with reg ard to the rival rm), an d vice v ersa.
Without claiming g enerality of this dichoto my, the result tells us that it
is imp orta nt to reco gnize th e strategic cons equences of dierent leadership
styles. Moreover, sta tements like \Mr. X is a toug h manag er" may have
littlemeaningif o ne does not s p ecifywith regard to w hom.
5 Conclusion
A mana ger o f a rm in a comp etitive enviro nment ha s to ta ke decis io ns
concern ing org aniza tio nal design a nd comp etitive a ction s. In this pa p er, a
mo delis develo p edtha t integra tes b o thmanag ementaspects.
In the mo delthere is a tension b etween p os itioning a bra nd ina market
nicheandpro du cing apremiumbrand. A pro du ctmana gercanmotivatehis
subordinate (which is important for qu ality) by g iving him a say in which
variety heha s to d evelop a nd produce. Giving th es ub o rdina te eno ugh
free-dom to s elect pro d uct location mo tiva tes him to get informed and make a
proposa l. In turn, follow ing up the a gent's recommend atio n induces himto
exerthig he ort,becaus e the a gentw ill workh arder o ndeveloping and pro
his a gent asay in p ro duct loca tio n(the incentive e ect). In the model, the
presence of incentive eects may result in less pro d uct di erentia tio n tha n
in the Ho telling mo del with quadra tic trans p o rta tio n costs .
A more genera l point of this pa p er is that when incentive e ects exist,
they may be imp o rtant. When manag erstake orga nizationa l incentives into
account, pro du ct dierentiation, and therefore als o comp etition, may be a
f-fected. In di erent models, thes e type of eects may inuence co mpetitio n
inva riou sways. Further wo rkin this direction isneededto enhan ceo uru
n-derstand ing o ftheinuenceofincentives inside orga niza tion son comp etitive
b ehavior.
In reality, there may b e a combination of reaso ns o f why to p mana gers
delega te res p ons ibility to mid dle mana gers { not only incentive issues, but
for instance als o work overlo ad, exibility (versus commitment) to ada pt to
cha nging ma rket ch aracteristics, or the co llection o f information a b o ut the
market. T he inves tiga tio n o f the strategic na ture of thos e and o th er is sues
rela tedto orga nizationa lstru cture seemsto b ea fruitful andimp o rta nta rea
for furth er research inin dustria lorga niza tion and ma nag ement s trategy.
Appendix
Proof of P rop osition 3.1:
Firs t, the fo llowing cla im w ill be proved:
Claim 1 Inan yequ ilibrium,the re exist say
1 2[01 ;0]an day 2 2[0;1]su ch that p 3 i (x i )= 8 < : 1 if jx i jj y i j; 0 ot herwise, for i=1;2 , x 1 2[01;0], an d x 2 2[0;1].
Proof of C la im 1: Let deleg ation s chemesp 3
i
(1), i=1 ;2, b eg iven.
(i)Su pp o se tha t A
2
is uninfo rmed, so tha t P
2
will s elect product location1.
If P
1
's b est res p o nse is to imp o se p ro duct location 01, then the prop o
that (1 ) holds. Acco rding ly, A
1
will get info rmed. S ince R
1 (x 1 ;1;h;`) is decreas ing in x 1
, there existsa y~2(01 ;0 ]such that
R 1 (x 1 ;1;h;` )0R 1 (01;1 ;` ;`)0 , x 1 y:~
Two ca ses can b e disting uished. First,
Z ~ y 01 p 3 1 (x 1 )dx 1 F 1 : (8) P 1
's exp ected returns are equa l to
Z 0 01 [p 3 1 (x 1 )R 1 (x 1 ;1 ;h ;` )+(10p 3 1 (x 1 ))R 1 (01 ;1;`;` )]dx 1 = Z ~ y 0 1 p 3 1 (x 1 )[R 1 (x 1 ;1 ;h;` )0R 1 (01 ;1 ;`;`)]dx 1 + Z 0 ~ y p 3 1 (x 1 )[R 1 (x 1 ;1;h;` )0R 1 (01;1 ;` ;`)]dx 1 +R 1 (01;1 ;` ;`)
(by mo noto nicity of R
1 ) Z ~ y 01 [R 1 (x 1 ;1;h;` )0R 1 (01;1 ;`;`)]dx 1 +R 1 (01;1 ;` ;`)= Z ~ y 0 1 R 1 (x 1 ;1;h;`)dx 1 0y R~ 1 (01;1 ;` ;`): It follows that P 1
ca n (weakly) increase his exp ected prots by selecting for
y 1 =y,~ p 1 (x 1 )= 8 < : 1 if x 1 y 1 ; 0 otherwise.
Seco nd, it may be the ca sethat
Z ~ y 0 1 p 3 1 (x 1 )d x 1 < F 1 : (9) If Z ~ y 01 1d x 1 F 1 ; (1 0)
then,bymono tonicityofR
1 ,P
1
can increaseh isexpectedpro tsbyselectin g
for y 1 =y ,~ p 1 (x 1 )= 8 < : 1 if x 1 y 1 ;
Nows upp o se tha t(1 0) does no tho ld. L et y^2(~y ;0]b e implicitly denedby Z ^ y 0 1 p 3 1 (x 1 )dx 1 = F 1 :
Note that by (9 ),y^is welldened. P
1
's exp ected returns are equalto
Z 0 01 [p 3 1 (x 1 )R 1 (x 1 ;1 ;h ;` )+(10p 3 1 (x 1 ))R 1 (01 ;1;`;` )]dx 1 = Z ~ y 0 1 p 3 1 (x 1 )[R 1 (x 1 ;1 ;h;` )0R 1 (01 ;1 ;`;`)]dx 1 + Z ^ y ~ y p 3 1 (x 1 )[R 1 (x 1 ;1 ;h;` )0R 1 (01 ;1 ;`;`)]dx 1 + Z 0 ^ y p 3 1 (x 1 )[R 1 (x 1 ;1;h;` )0R 1 (01;1 ;` ;`)]dx 1 +R 1 (01;1 ;` ;`)
(by mo noto nicity of R
1 ) Z ~ y 0 1 [R 1 (x 1 ;1 ;h;`)0R 1 (01 ;1;`;` )]dx 1 + Z ^ y ~ y p 3 1 (x 1 )[R 1 (x 1 ;1 ;h;`)0R 1 (01 ;1;`;` )]dx 1 +R 1 (01 ;1;`;` )
(by mo noto nicity of R
1 ) Z y1 01 [R 1 (x 1 ;1;h;` )0R 1 (01 ;1 ;`;`)]dx 1 +R 1 (01;1 ;`;`)= Z y 1 0 1 R 1 (x 1 ;1;h;` )d x 1 0y 1 R 1 (01;1 ;` ;`); where y 1 2(y;~ y ]^ is denedby Z y 1 01 1dx 1 = F 1 : It follows that P 1
can (weakly) increasehis exp ected prots by s electing
p 1 (x 1 )= 8 < : 1 if x 1 y 1 ; 0 otherwise.
(ii)Th eproo f of the caseinwhichA
2
lea rnshisprivate benets iss imilarto
case (i), andis o mitted. 2
Claim 1 allows us to dene the level of discretion o f a gent A
1
as the
mea sure of interva l [01;y
1 ], that is, X 1 y 1 +1 ,a nd s imilarly, A 2 's level of
discretion a s the meas ure o f [y
2
;1], tha t is , X
2
10y
2
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