Op imal Coexis ence o Long- e m and
Sho - e m con ac s in Labo Ma ke s∗
Inés Macho-S adle †Da id Pé ez-Cas illo‡Nicolás Po ei o§
May 5, 2011
Abs ac
We conside a ma ke whe e i ms hi e wo ke s o un hei p ojec s and such
p ojec s diffe in p o i abili y. A any pe iod, each i m needs wo wo ke s o suc-
cess ully un i s p ojec : a junio agen , wi h no speci ic skills, and a senio wo ke ,
whose effo is no e i iable. Senio wo ke s diffe in abili y and hei compe ence
is e ealed a e hey ha e wo ked as junio s in he ma ke . We s udy he leng h o
he con ac ual ela ionships be ween i msandwo ke sinanen i onmen whe e
he ma ching be ween i ms and wo ke s is he esul o ma ke in e ac ion. We
show ha , despi e in a one- i m-one-wo ke se -up long- e m con ac s a e he op-
imal choice o i ms, ma ke o ces o en induce i ms o use sho - e m con ac s.
Unless he ma ke only consis s o i ms wi h e y p o i able p ojec s, i ms ope -
a ing highly p o i able p ojec s offe sho - e m con ac s o ensu e he se ice o
high-abili y wo ke s and hose wi h less luc a i e p ojec s also use sho - e m con-
∗We a e g a e ul o he pa icipan s a he semina a CREST (Pa is, 2011) o hei insigh ul com-
men s. Financial suppo om Minis e io de Ciencia y Tecnología (ECO2008-04321 and ECO2009—
07616), Gene ali a de Ca alunya (2009SGR-169), Jun a de Andalucía (SEJ-02936 and SEJ-04992),
Ba celona G adua e School o Economics and ICREA Academia is g a e ully acknowledged. The i s
wo au ho s a e ellows o MOVE.
†Uni e si a Au onoma de Ba celona - Ba celona GSE; Dep . Economía e His . Económica; Edi icio
B; 08193 Bella e a - Ba celona; Spain. Email: [email p o ec ed].
‡Co esponding au ho . Uni e si a Au onoma de Ba celona - Ba celona GSE; Dep . Economía e
His . Económica; Edi icio B; 08193 Bella e a - Ba celona; Spain. Email: da id.p[email p o ec ed].
§Depa men o Economics (Uni e sidad Pablo Ola ide). Email: npo ei [email protected].
1
ac s o sa e on he junio wo ke s’ wage. In e media e i ms may (o may no )
hi e wo ke s h ough long- e m con ac s.
JEL numbe s: D86, C78
2
1 In oduc ion
Pa ne s may es ablish ela ionships ha las o se e al pe iods. In job con ac s, o
ins ance, some i ms hi e he same wo ke o se e al yea s wi h a long- e m con ac ,
while o he s may p e e o sign con ac s pe iod by pe iod, some imes wi h he same
wo ke , some imes wi h diffe en wo ke s o e ime. One ques ion ha a ises is when is
i be e o he employe o w i e a long- e m con ac ha co e s he whole leng h o
he ela ionship and when is a sho - e m con ac signed o a ce ain pe iod o ime
supe io , knowing ha once ha pe iod is o e he i m has o offe ano he sho - e m
con ac .
The con ibu ions by Lambe (1983), Roge son (1985), Malcomson and Spinnewyn
(1988), and Chiappo i e al. (1994), among o he s, add ess he p e ious ques ion in
se ings cha ac e ized by mo al haza d whe e one i m ( he p incipal) en e s in o a long-
las ing ela ionship wi h a wo ke ( he agen ). In hese se ings, i he i m and he agen
can commi o a long- e m con ac , he i m can design a long- e m ag eemen ha
domina es he sequence o op imal sho - e m con ac s. In ac , long- e m con ac s can
always eplica e he sequence o he op imal sho - e m con ac s while he e e se is, in
gene al, no possible. This esul is obus o a numbe o diffe en speci ica ions and
implies ha , i commi men on he pa o he pa icipan s is possible, we should expec
i ms o use mainly long- e m con ac s in p ac ice.1Howe e , his is no he case and
he ac ha diffe en i ms in simila ma ke s ollow diffe en ime-du a ion con ac s
sugges s ha he e may be o he explana ions beyond lack o commi men .
In his pape we a gue ha no all he cha ac e is ics o he op imal con ac be ween
an employe and a wo ke can be deduced om he analysis o his ela ionship in a
one-wo ke , one- i m se ing. As he ecen con ibu ions by Dam and Pé ez-Cas illo
(2006), Se es (2008), Te iö (2008) and Alonso-Paulí and Pé ez-Cas illo ( o hcoming)
ha e shown, when he analysis is enla ged o ake in o conside a ion ma ke in e ac ion,
hen he o m o he op imal con ac s may subs an ially diffe om he ag eemen s ha
one ob ains o a gi en ela ionship s udied in isola ion. When he e ogenous p incipals
1When commi men on he agen side is difficul , con ac s may include, o example, non-compe e
clauses unde which he agen ag ees no o pu sue a simila p o ession o ade in a i m in he same
indus y i he b eaks he con ac . Con ac s may also include o he clauses ha will educe mobili y by
inc easing he cos o hi ing he wo ke by ano he i m.
3
compe e o he e ogenous agen s, he iden i y o he pa ne s in each ela ionship (in
addi ion o he con ac signed) is endogenous, as i is he le el o u ili y ob ained by
he agen s. Hence, e en i sho - e m con ac s a e no op imal when we conside an
isola ed ela ionship, hey may a ise as pa o he ma ke equilib ium whe e some i ms
may choose long- e m con ac s while o he hi e wo ke s only on sho - e m ag eemen s.
We conside a ma ke whe e he e ogenous i ms hi e wo ke s o un hei p ojec s.
Fi ms diffe in he p o i abili y o hei p ojec . A any pe iod, each i m needs wo ypes
o wo ke s: a junio agen , wi h no speci ic skills, and a senio expe ienced, wo ke whose
expe ise is c ucial o he good de elopmen o he p ojec . E e y wo ke s a s as a
junio agen in he i s pe iod he wo ks in a i m. A his poin in hei li es, all wo ke s
a e iden ical. A e he pe iod as an app en ice, he wo ke becomes senio o he second,
and inal, pe iod o his job ca ee . In ou model, he i s pe iod has a aining componen :
wo ke s acqui e he knowledge and expe ience needed o un a p ojec when senio s. We
assume ha his pe iod gi es human capi al speci ic o he indus y. This allows any
wo ke ha was hi ed (and ained) by a i masajunio o unap ojec asasenio in
any i m o he ma ke .2
The model is dynamic no only because he ela ionships (may) in ol e se e al pe i-
ods, bu also because in o ma ion abou wo ke s’ cha ac e is ics changes o e ime: a e
he agen has wo ked o a i m as a junio , he i ms and he wo ke himsel lea n his
compe ence as senio , which was ex-an e unknown o all ma ke pa icipan s. The e o e,
while all junio wo ke s a e indis inguishable, his is no he case o senio wo ke s as no
only hei esponsibili ies ( he p ojec hey wo k on), bu also hei abili ies may be qui e
diffe en . Consequen ly, we model a e y simple echnology o aining ha combines wo
dimensions o lea ning. Fi s , he e is lea ning because he inna e abili y o he wo ke is
e ealed h ough his aining as a junio and his in o ma ion becomes common knowl-
edge o he indus y. Second, he e is a lea ning-by-doing componen since wo king as
junio is a p e equisi e o la e unning a p ojec as a senio . In his espec , he pape is
ela ed, hough diffe en , o he li e a u e on on- he-job alen disco e y. Te iö (2009)
also p esen s a si ua ion whe e wo ke s’ inna e abili y is unknown o he ma ke (and
2In o he wo ds, we do no deal wi h i m-speci ic aining in he model. I he e is some i m-speci ic
abili y, wo ke s who change jobs in he second pe iod o hei li es lose hei i m-speci ic human capi al
and jus keep hei gene al aining in he indus y. This will end o dec ease he p o i abili y o sho -
e m con ac s.
4
he wo ke s hemsel es) un il hey ac ually wo k o a i m. Howe e , in Te iö’s pape
heobjec i eisdiffe en asheisconce nedby hepossibli y ha ma ke impe ec ions
hampe he p ocess o disco e ing alen .
In he model, a mo al haza d p oblem is p esen as senio wo ke s’ effo o decision is
no con ac ible. On he con a y, and jus o simplici y, we conside junio s’ effo o be
con ac ible. All he pa icipan s a e isk neu al and hey all ha e he capaci y o commi
o a long- e m con ac .3Howe e , wo ke s a e p o ec ed by limi ed liabili y: hei sala y
when junio and hei sala y when senio canno be lowe han ce ain h esholds.
We cha ac e ize equilib ia in his ma ke , which accoun s o he ype o con ac
offe ed by each i m and he cha ac e is ics o hese con ac s. Ou equilib ium concep is
close o he idea o “s abili y” used in he ma ching li e a u e ha has analyzed con ac s
in en i onmen s whe e he ma ching be ween i ms and wo ke s is endogenous.4To be
an equilib ium, an ou come ( ha is, a ma ching and a se o con ac s) mus be immune
o de ia ions. In ou en i onmen , a equilib ium, i mus be he case ha a i m canno
make mo e p o i by changing i s s a egy, ha is, by offe ing con ac s o wo ke s ha
make bo h he i m and he wo ke s be e -off han be o e.
We i s show ha i ms signing equilib ium long- e m con ac s offe low sala ies o
junio wo ke s oge he wi h he p omise o high ewa d when senio . This allows he i ms
o alle ia e he incen i e p oblem hey ace wi h senio agen s, imp o ing he efficiency o
he ela ionship and also hei p o i s. Since hey commi o do so, hese i ms will keep
he agen s when senio , i espec i e o hei abili y. Fi ms ha sign sho - e m con ac s
hi e junio agen s wi h no p omise o con inua ion. They also sign sho - e m con ac s
wi h senio wo ke s (who may o may no be he same hey hi ed he p e ious pe iod as
junio s); he e ms o he ag eemen may depend on he wo ke s’ abili y le el.
We ha e al eady a gued ha he op imal long- e m con ac always (a leas weakly)
3I no pa icipan can commi o a long- e m con ac , hen all mus be sho - e m con ac s. I he
pa icipan s in one o he sides o he ma ke , say he i ms, can commi while he o he s canno , hen
he e can s ill be oom o long- e m con ac s, bu hey a e ypically less efficien han in he en i onmen
wi h ull commi men . In e ms o he commi men possibili ies, we place ou sel es in he bes scena io
o he p e alence o long- e m con ac s.
4S abili y and compe i i e equilib ium a e e y close concep s. Any s able ou come is also a compe -
i i e equilib ium and ice- e sa. Fo (ea ly) ma ching models whe e he pa ies decide on money ins ead
o con ac s see, o ins ance, he o iginal con ibu ion by Shapley and Shubik (1972), and he excellen
e iew o he li e a u e by Ro h and So omayo (1990).
5

domina es any sequence o sho - e m con ac s when he iden i y o he pa ies ma ched
in a job con ac is p ede e mined. Howe e , sho - e m con ac s can be bene icial o
i ms when he i m-wo ke ma ching is endogenous. Sho - e m con ac s allow he
i ms o sc een wo ke s be o e pu ing hem in cha ge o leading a p ojec . This non-
commi men s a egy gi es i ms he eedom o ocus on he pa icula ype o senio
wo ke ha i s hei needs. The e o e, i ms acea ade-offbe ween choosing he op-
imal con ac o a gi en ma ch (long- e m a e supe io o sho - e m ag eemen s) and
selec ing a con ac ha allows a be e selec ion (hi ing high-abili y senio wo ke s is
mo e impo an o some i ms han o o he s).
The ma ke equilib ium depends on he cha ac e is ics o he se o i ms and he se
o wo ke s. We sol e he model o ma ke s whe e he e is a la ge p opo ion o low-
p oduc i i y (no mal) senio wo ke s and a small p opo ion o high-p oduc i i y senio
wo ke s (s a s) and whe e hese highly- alen ed wo ke s eally make a diffe ence in he
i ms heywo k o .
When only i ms wi h e y p o i able p ojec s exis in he ma ke , all o hem sign
long- e m con ac s a equilib ium. Each i m offe s he same ag eemen ha i would
offe i no ma ke would ha e exis ed. Mo e in e es ingly, we show ha , excep in his
case whe e he ma ke only consis s o i ms wi h e y p o i able p ojec s, he e is always
ase o i ms ha sign sho - e m con ac s wi h hei junio wo ke s and specialize in
a pa icula ype o senio s. Depending on he alue hey a ach o hei p ojec s, some
i ms always look o high-abili y while o he s hi e low-abili y senio wo ke s. Fi ms wi h
highly p o i able p ojec s gi e a g ea deal o ele ance o hi ing high-abili y senio agen s
o un hei p ojec s and, hence, hey a e willing o offe high wages o a ac hem.
As a esul , he expec ed u ili y o junio wo ke s when hey accep sho - e m con ac s
becomes highe because, i hey u n ou o be o high abili y, hey will ob ain a high
ewa d when senio . The expec a ion o his po en ial ewa d leads wo ke s o accep ,
when junio , a low wage. Fi ms wi h ela i ely poo p ojec s ake ad an age o his
educ ion in he wage o junio wo ke s. These i ms pu mo e weigh on he sa ings on
junio s’ wages han o he ac ha hey end up con ac ing wi h a low-abili y senio
wo ke . The e o e, a equilib ium, he i ms wi h he mos p o i able p ojec s use sho -
e m con ac s o ensu e he se ices o high-abili y wo ke s while he i ms wi h he leas
p o i able p ojec s use sho - e m con ac s o sa e in he cos o hi ing junio wo ke s.
6
In his sense, he ma ching be ween i ms and senio agen s is posi i e asso a i e, o he
se s ha choose sho - e m con ac s.5
The ade-offbe ween he ad an ages o long- e m and sho - e m con ac s is o en
sol ed in a o o he use o long- e m con ac s o i ms wi h in e media e p ojec s. The
likelihood o he coexis ence o he wo ypes o con ac s is highe as he dis ibu ion
o i ms is mo e biased owa d good p ojec s, he discoun a e is lowe , he diffe ence
be ween he ese a ion u ili y o junio wo ke s and he minimum sala y is lowe , he
diffe ence in pe o mance be ween high and low-abili y senio wo ke s is highe , and he
cos o he wo ke s’ effo is lowe .
Wo ke s ecei e pa o he inc eased su plus c ea ed by he op imal so ing o senio
wo ke s p omo ed by he sho - e m ag eemen s. Indeed, al hough all junio wo ke s a e
iden ical and hey pe o m iden ical asks, hose who sign sho - e m con ac s expec a
highe u ili y han hose signing long- e m con ac s. Long- e m ag eemen s allow he
i ms o a oid he compe i ion o he bes wo ke s, who ob ain high sala ies unde sho -
e m equilib ium con ac s.
In ou analysis, we ocus on ma ke s wi h a small p opo ion o e y alen ed senio
wo ke s ha makeadiffe ence o he i ms hey wo k o and whose le el o abili y is pub-
lic o all he i ms inside he indus y. Mo eo e , he human capi al acqui ed by senio s is
indus y-speci ic and no jus i m-speci ic. This model can p o ide a schema ic e sion o
he uni e si y job ma ke . The pe o mance o esea che s du ing he i s yea s a e he
comple ion o hei Ph.D., ha we can associa e o hei “abili y”, is public in o ma ion
since i can be measu ed, o ins ance, by hei publica ion eco d. In his job-ma ke
some uni e si ies offe Ph.D. g adua es a enu e ack posi ion ha gua an ees enu e i ,
a e he p oba iona y pe iod, he candida e sa is ies some p ede e mined pe o mance
c i e ia (in e ms o publica ions and o he measu es). The enu e- ack sys em co e-
sponds in ou model o sho - e m con ac s. O he uni e si ies sign enu e con ac s
om he e y beginning and ake he commi men o keeping he esea che independen
o he ou come o u he e alua ion (e en i con ac condi ions may indeed depend on
pe o mance). This co esponds o a long- e m con ac .
A s and spo s a e also examples o ma ke s whe e he abili y o senio s is well-known,
5See Leg os and Newman (2007) o condi ions unde which mono one ma chings eme ge in en i on-
men s whe e u ili y is no ully ans e able.
7
as i is subjec o public sc u iny h ough hei pe o mance and whe e his human capi al
is mainly indus y-speci ic. Singe s o socce playe s, o ins ance, may sign exclusi e
con ac s (wi h a s udio, a eco d company, o a club) o a long pe iod in which hey a e
p e en ed om eco ding an album o ano he company o playing wi h ano he club.
O he companies, howe e , choose o offe sho e con ac s, pa icula ly o young singe s
o playe s. The s a s o hese ma ke s, a ew indi iduals, a ain p ominence and success
and hei alue and ea nings a e signi ican ly g ea e han he ea nings o he s anda d
wo ke in he ma ke s. The same can be said abou su geons o c ea i es in ad e ising.
Finally, he ma ke o uppe execu i es also sha es some simila ea u es: hese high
execu i es a e well-known wi hin hei indus y and hei con ac s may (o may no )
include special clauses aimed a p e en ing hem om mo ing o ano he i m.
To he bes o ou knowledge, ou s is he i s pape o s udy how he choice o he
con ac ual leng h may be de e mined by ma ke in e ac ion. The e a e o he pape s ha
ha e s udied he implica ions o diffe en con ac ual a angemen s bu in widely diffe en
se -ups. Rice and Sen (2008) show how a educ ion in he leng h o a con ac can help
o alle ia e he mo al haza d p oblem when explici incen i es canno be included in he
e ms o he con ac . In hei pape , he op imal choice o he p incipal depends on
he balance be ween mo e incen i es o effo (sho - e m con ac s) and lowe wages
(long- e m con ac s).
The con ibu ion by Ghosh and Waldman (2010) compa es wo con ac ual a ange-
men s: up-o -s ay s. up-o -ou con ac s in a se ing wi h mul iple i ms compe ing o
a wo ke . The pape does no add ess he issue o endogenous ma ching since i s udies
he i ms’ Be and compe i ion in wages o a ac he single wo ke a ailable in he
ma ke . They show ha up-o -ou p e ails when i m-speci ic human capi al is low and
when high- and low-le el jobs a e simila . O he wise, s anda d (up-o -s ay) p ac ices a e
op imal. Simila o ou pape , Gha ak e al. (2001) s udy an o e lapping gene a ions e -
sion o a p incipal-agen p oblem whe e con ac s a e de e mined in gene al equilib ium.
In hei model, all young wo ke s a e iden ical bu ha e diffe en in es men possibili ies
when senio , depending on hei pe o mance. They do no allow o long e m con ac s
because hei he au ho s’ conce n is o explain he senio s’ decision be ween becoming
en ep eneu s o emaining wo ke s.
In ou pape , sho - e m con ac s ac as a o m o p oba iona y pe iod ha allows
8
i ms and wo ke s o achie e a be e ma ching. I is, he e o e, no a way in which i ms
y o es i he wo ke is good enough o he job, bu a he i allows senio wo ke s
o be ma ched wi h hose i ms whe e hey a e mo e p oduc i e. In his sense, he sho -
e m con ac se es as a so ing de ice. A ela ed, bu diffe en , a gumen can be ound
in Loh (1994) whe e i is a gued ha in oducing an employmen p oba ion can se e as
a so ing de ice as i will induce sel -selec ion by wo ke s. Fi ms offe ing p oba iona y
employmen will end o a ac wo ke s who a e mo e con iden abou hei capabili ies.
Finally, he coexis ence o ixed paymen schemes and incen i e-based paymen schemes
ela ed o he cha ac e is ic o he wo ke s is also p esen in a s a ic ad e se selec ion
amewo k whe e i ms compe e o agen s. Ma u es e al. (1994) s udy he choice o
compensa ion schemes by wo i ms ha compe e in a labo ma ke whe e agen s a e
he e ogenous and hey ha e p i a e in o ma ion abou hei ype. They show ha , in
equilib ium, i i ms a e no oo diffe en in he eyes o wo ke s, one i m offe s a wage
a e and he o he offe s a piece a e. By p oposing diffe en compensa ion schemes, i ms
induce sel -selec ion among wo ke s, which he eby dec eases he in ensi y o compe i ion
in he labo ma ke .
The emainde o he pape is o ganized as ollows. Sec ion 2 p esen s he model.
Sec ions 3, 4 and 5 analyze he candida e long- e m and sho - e m con ac s o equi-
lib ium. Sec ion 6 cha ac e izes he iden i y o he i ms and wo ke s ha en e in o he
ela ionship, he equilib ium sala ies, as well as he con ac s ha eme ge as a esul o
he ma ke in e ac ion. Finally, Sec ion 7 concludes.
2Model
We model he economy as an o e lapping gene a ion model whe e a each pe iod ,wi h
=12 , i ms con ac wi h wo ke s o de elop p ojec s. Fi ms a e in ini e-li ed playe s
and he se o i ms is cons an o all pe iods. On he o he hand, wo ke s (agen s) li e
o wo pe iods. Bo h, i ms and wo ke s discoun he u u e acco ding o he discoun
ac o ,whe e∈(01).
Allpa icipan sa eassumed obe iskneu al.Wealsoassume ha awo ke ,a any
age, enjoys limi ed liabili y o e income. This cons ain implies ha his wage in any
pe iod and con ingency canno be lowe han a ce ain h eshold .
9
…wJ
wJ
wJ
wJ
…
(wS, S)(wS, S)(wS, S)(wS, S)
– 1 + 1 + 2 …
wJw’Jw’Jw’J…
(wS, S)(wS, S)(w’S, ’S)(w’S, ’S)…
…wJ
wJ
wJ
wJ
…
(wS, S)(wS, S)(wS, S)(wS, S)
– 1 + 1 + 2 …
wJw’Jw’Jw’J…
(wS, S)(wS, S)(w’S, ’S)(w’S, ’S)…
Figu e 1: A i m conside s changing he con ac
The si ua ion is simila i a i m which is cu en ly offe ing he con ac  decides
o swi ch o a se ies o ST con ac s: i i s changes hecon ac i offe s o he junio
agen o be able o ully implemen he new s a egy in he subsequen pe iod. Also, we
ace he same si ua ion i a i m is cu en ly offe ing ST con ac s and plans o swi ch
o LT con ac s: i needs o change he ag eemen wi h he junio agen oday bu s ill
needs o hi e a senio agen h ough an ST con ac o be able o ully implemen he
change omo ow. Finally, when a i m swi ches om ST con ac s o ano he s eam o
ST con ac s in a pe iod, i can do i immedia ely, wi hou wai ing ill he subsequen
pe iod. Indeed, i can keep hi ing junio agen s unde he same condi ions as be o e ( ha
is, unde he lowes sala y ha he agen is eady o accep ). Whe he we compu e he
cos o he junio agen as 1
o isno ele an o hecompa isono p o i s in
he wo s a egies, since he i m pays he same cos unde bo h, he old and he new
con ac s.
The e o e, we can de elop he analysis o he (s a iona y) equilib ia o ou model by
ocusing on he p o i s i ms make in one pe iod, p o ided ha we conside he cos o he
junio agen as being gene a ed he p e ious pe iod, ha is, as long as we associa e a cos
o 1
, ins ead o , o he junio agen . F om now on, we will e e o his le el o p o i s
as “a i m’s one-pe iod p o i s” and we will deno e e=−1
+(−∆)−.A i m
hasincen i es oswi ch omcon ac  o con ac 0i and only i e()e(0).
16

4 Long- e m con ac s in equilib ium
Conside a i m ha owns a p ojec whose addi ional alue in case o success is ∈£ ¤
and ha signs LT con ac s wi h junio wo ke s. A each pe iod , he i m uns he p ojec
wi h hejunio agen ha i hi esa and wi h he senio wo ke ha i hi ed a pe iod
−1. The senio agen has abili y wi h p obabili y and abili y wi h p obabili y
1−, as his abili y was unknown a −1. As p e iously said, he abili y o he agen
is publicly known be o e he s a s wo king as a senio ; hence, he LT con ac signed a
−1may ha e paymen s con ingen on he abili y o he agen when senio .12
All wo ke s a e ex-an e iden ical and he e a e mo e junio wo ke s han posi ions
o ill. The e o e, a any pe iod he e a e unemployed junio agen s eady o accep
any LT con ac ha p o ides hem wi h an expec ed u ili y equal o hei ( wo-pe iod)
ou side u ili y +. Hence, he pa icipa ion cons ain (PC ) speci ies ha he o al
expec ed u ili y he wo ke ob ains in he ela ionship be a leas equal o +.
Following he discussion o he p e ious sec ion, a candida e LT con ac o equi-
lib ium (
∆
∆)maximizes he i m’s one-pe iod p o i s, also aking in o
accoun he ICCs and he limi ed liabili y cons ain s (LLC), ha is, i sol es
max
(∆∆)−1
+((−∆)−)+(1−)((−∆)−)
s. . +£(∆+−()2)+(1−)(∆+−()2)¤≥+
=1
22∆,=1
22∆
≥
≥
≥.
I he con ac does no sa is y he p e ious p og am, hen he i m can de ia e by offe ing
adiffe en accep able LT ag eemen o junio agen s and ob ain la ge discoun ed p o i s.
We s a e he cha ac e is ics o he candida e LT con ac in P oposi ion 2, whe e we
deno e
e≡q2
+(1−)2
,

1≡2
e 1
(−)+−and 
2≡4
e 1
(−)+−.
12As will be clea la e , his lexibili y has no effec on he op imal con ac . The e o e, a he candida e
equilib ium con ac , no hi d pa y needs o e i y he abili y o he agen .
17
P oposi ion 2 I i m is in he se R , heni offe s he ollowing LT con ac :
Region  :I 

1, hen
13
 ()=µ
=

=
=1
(−)+−1
422e2∆
=∆
=¶
Region  :I ∈£
1

2¤ hen
 ()=Ã
=

=
= ∆
=∆
=2
e 1
(−)+−!
Region  :I 

2 hen
 ()=µ
=

=
= ∆
=∆
=
2¶
We now explain he main cha ac e is ics o he LT con ac  (). Despi e he ab-
sence o isk a e sion, he mo al haza d p oblem o he senio agen induces an inefficiency
due o he p esence o limi ed liabili y ha es ic s he capaci y o he i m o induce
hesenio wo ke oexe ahigheffo . The e o e, he i m is in e es ed in elaxing he
senio agen ’s limi ed liabili y cons ain , which explains why i concen a es as much as
possible he agen ’s paymen s in his second pe iod o li e (i.e., he i mpays oayoung
wo ke he minimum possible wage: 
=.) Young agen s accep con ac s wi h a
low payoffbecause o he c edible p omise o be “well” paid when hey a e senio . The
limi ed liabili y cons ain s also explain why, unless is e y low, wo ke s a e paid he
minimum sala y i he ou come u ns ou o be a ailu e: 
=
=.
The impac o limi ed liabili y on bonuses and on payoffs ob ained by agen s and i ms
diffe s depending on he p o i abili y o he p ojec (as well as on he le el o agen s’
ese a ion u ili y +,cos o effo , and “a e age” p obabili y o success e).
Some cha ac e is ics a e shown in Figu e 2.
Fo high alues o (Region  ), heop imalbonusdependsonlyon he alueo
he p ojec . The i m sha es hal o he alue in he e en o success because i maximizes
p o i s when he senio agen supplies effo 
=1
42 o = . Gi en his bonus,
he wo ke ends up wi h a u ili y la ge han +(i.e., he ob ains in o ma ional
en s).
13In his egion, he e a e o he con ac s ha a e also candida es o equilib ium. Any combina ion
o ,and  ha sa i ies +(+(1−))+1
422e2=+andsuch ha each
a iable is highe han , is also a candida e as i would gi e he same p o i s o he i m.
18
R1LT R
(PC) does no bind
(LLC) binds
1/2 o FB e o s
(PC) binds
(LLC) binds
e o s inc ease in
UJ and US
(PC) binds
(LLC) does no bind
FB e o s
R
R
R/2






R2LT
Figu e 2: Incen i es in he op imal LT con ac s
Fo in e media e alues o (Region  ), he equilib ium paymen scheme also
depends on 
,and , as he pa icipa ion cons ain ( oge he wi h he lim-
i ed liabili y cons ain ) binds. Gi en ha he i m needs o p o ide a le el o u ili y
o +, i gi es i in e ms o bonuses, which lead o a senio agen ’s effo o

=
q1
(−)+− o = .
Finally, i mswi hlow- aluedp ojec s(Region ) gi e all he p ojec ’s e u ns o
he wo ke ( hey se ∆=)inexchange o a ixed paymen (a anchise- ype con ac ).
The e o e, agen s ob ain hei o al ou side u ili y +and hey p o ide, when
senio , he i s -bes le el o effo 
=1
22 o = .
Nex co olla y p o ides he exp ession o he i m’s one-pe iod p o i s o  ().
Co olla y 1 The i m’s one-pe iod p o i s unde  ()a e:
Region  :I 

1, hen
e ()=−1
−+1
422e2
Region  :I ∈£
1

2¤ hen
e ()=+1
e 1
(−)+−−1
[2−(1 + )]−2
Region  :I 

2 hen
e ()=+2e2
82−(1 + )

19
The p o i unc ion e ()is con inuously diffe en iable and con ex in .
5 Sho - e m con ac s in equilib ium
All i ms signing ST con ac s hi e simila young wo ke s, as hey a e indis inguishable
ex-an e. Conce ning senio wo ke s, hey can decide o hi e high-abili y o low-abili y
wo ke s.
Conside an equilib ium whe e some i ms sign ST con ac s. A ac ion o hose
i ms offe con ac s o high-abili y senio agen s. Deno e by  he (minimum) le el o
u ili y ha his ype o agen ob ains a he equilib ium.14 Simila ly, deno e by  he
(minimum) le el o u ili y ecei ed by low-abili y senio wo ke s. Bo h and need o
be highe han o equal o Addi ionally, gi en he limi ed liabili y cons ain and he
compe i ion among i ms, and, possibly, can be s ic ly highe han The e o e,
a junio agen is eady o sign an ST con ac ha p o ides a u ili y le el lowe han 
as long as he educ ion is no highe han he expec ed ex a u ili y he will ob ain when
senio . Fo mally, he sala y  ha he junio agen is eady o accep mus sa is y:
+[+(1−)]≥+,
whe e we deno e and  he expec ed u ili y o a high- and a low-abili y wo ke . Fo
example, i all he low-abili y wo ke s ob ain he same in all he possible jobs, hen
=.
The candida e equilib ium con ac o i m in R(R) o a high- (low-) abili y
senio agen mus be he op imal one-pe iod con ac o his agen , aking in o accoun
ha i mus g an him a le el o u ili y o a leas (); ha is, i sol es
max
(∆)+(−∆)−
s. . ∆+−()2≥
=1
22∆
≥
14Gi en he limi ed liabili y cons ain , simila senio agen s migh ob ain diffe en u ili y le els a
equilib ium. A i m wi h a e y high ends up p o iding i s senio agen a u ili y le el highe han as
i s pa icipa ion cons ain will no be binding (see also, Alonso-Pauli and Pé ez-Cas illo, o hcoming).
20
o = . Nex p oposi ion p o ides he candida e equilib ium con ac o hose
i ms, whe e we use he no a ion

1()≡2
p−and 
2()≡4
p−.
P oposi ion 3 I i m is in he se Rwi h ∈{ }, heni offe s he ollowing
ST con ac o a senio agen :
Region 
():I 

1(), hen

( )=µ
=−1
422
2∆
=¶
Region 
():I ∈£
1()

2()¤ hen

( )=µ
= ∆
=2
p−¶
Region 
():I 

2() hen

( )=µ
= ∆
=
2¶
In Region 
(), senio agen ’s effo is he i s -bes le el 
=1
22 while in
Region 
()his effo is lowe han he i s -bes le el: 
=1
√− In hese wo
egions, he agen ’s expec ed u ili y is Finally, in Region 
()whe e he p ojec is
e y aluable, he senio agen ’s effo is 
=1
42 o =  and he ecei es an
in o ma ional en . His expec ed u ili y in his egion is +1
1622
2

Co olla y 2 p o ides he exp ession o he i m’s one-pe iod p o i s unde 
( ),
deno ing 
 he equilib ium sala y paid o junio agen s.
Co olla y 2 A i m in he se Rwi h ∈{ }ob ains he ollowing one-pe iod
p o i s wi h 
( )
Region 
():I 

1(), hene
¡ 

¢=+1
422
2−−1


Region 
():I ∈£
1()

2()¤ hen
e
¡ 

¢=−2++1
√−−1


Region 
():I 

2() hen e
( 
)=−+1
822
2−1


The p o i unc ion e
¡ 

¢is con inuously diffe en iable and con ex in .
21

6 Equilib ium ma ching and equilib ium con ac s
The p e ious sec ions iden i y he equilib ium con ac s once we know he ype o ag ee-
men s i ms offe ( ha is, once he se s R ,Rand Ra e de e mined) and he le els
o u ili y and  ha hey mus gua an ee o low- and high-abili y agen s. In he
p esen sec ion, we cha ac e ize equilib ia whe e a leas some i ms offe ST con ac s.
The e o e, we iden i y he dis ibu ion o i ms in R ,Rand R, he le els and
and he minimum sala y  ha i ms mus offe o junio s unde ST con ac s.
We look o equilib ia whe e =. Low-abili y wo ke s do no ha e special skills
and he i ms will no compe e o hem.15 On he o he hand, he le el o will be
de e mined by he equilib ium condi ions, ha is, by he (ma ginal) i m’s willingness
o pay o a ac a high-abili y wo ke ins ead o ei he a ac ing a low-abili y one, o
signing an LT con ac .
We de elop he analysis o ma ke s whe e high-abili y wo ke s a e no abundan bu
hey make a diffe ence o he i m hey wo k o . Tha is, we conside en i onmen s wi h
many “no mal” wo ke s and some “s a s”. Assump ion 1 e lec s his idea, oge he wi h
he easonable hypo hesis ha he ou side ese a ion u ili y o a senio agen is la ge o
equal o ha o a junio wo ke (pa (i)). Assump ion 1 (ii) s a es ha he p opo ion o
high-abili y agen s is small enough. Finally, Assump ion 1 (iii) ep oduces he idea ha
he diffe ence among he wo ypes o agen is la ge enough.
Assump ion 1 The pa ame e s sa is y he ollowing condi ions:
(i) ≥,
(ii)  
1+2,
(iii) ³
´21+ 1
 
Why may some i ms be in e es ed in LT ela ionships while o he s p e e o secu e
high-abili y agen s h ough ST con ac s? E en mo e, why would a i m choose a s a egy
ha implies con ac ing low-abili y agen s h ough ST con ac s, ins ead o offe ing LT
con ac s and, some imes, bene i ing om high-abili y senio agen s? The wo main
equilib ium a iables ha make i ms p e e one o ano he ype o con ac a e he
15Howe e , a equilib ium he measu e o senio wo ke s wi h low abili y is he same as he measu e o
i ms looking o hem. The e o e, o he equilib ia may exis whe e 
 o all low-abili y playe s.
22
sala y o a young wo ke 
(o a he , he compa ison be ween 
and )and he
diffe ence be ween he cos o a high- e sus a low-abili y senio agen , ha is, −
The i ms ha ob ain la ge p o i s in he e en o success, ha is, i ms wi h a high ,
a e eady o pay a high p ice o always hi e a good senio agen gi en his added alue in
e ms o inc eased p obabili y o success. The e o e, i ms a he igh end o he in e al
£ ¤mus be hose mos in e es ed in signing ST con ac s o hi e high-abili y senio
agen s. Simila ly, i ms ha do no ca e much abou agen s’ effo , i.e., i ms wi h a low
, pay mo e a en ion o he po en ial sa ings hey can make in a junio ’s con ac i
hey offe him an ST con ac han o he gains ob ained h ough an LT con ac , o
by secu ing a high-abili y agen . The e o e, i ms a he le end o £ ¤a e he likely
candida es o sign ST con ac s o hi e low-abili y senio agen s.
Lemma1p o idesa i s con i ma ion o he p e ious in ui ions. I compa es he
slopes, in e ms o , o he p o i s ob ained om he diffe en ypes o con ac .
Lemma 1 Unde Assump ion 1, he slopes o he p o i unc ions sa is y he ollowing
ela ions:16
(a) 

 ( )
 (), o all ;
(b) 

 ( )

 ( 
), o all ,and o all≥;and
(c) 
 ()

 ( 
), o all ,and o all≥.
A i m’s ST p o i s inc ease wi h he alue o success when i hi es a low-abili y
wo ke . Howe e , his inc ease is smalle han ha o a i m’s p o i s unde he op imal LT
con ac (pa (a)). I is also smalle han he a e a which i s p o i s inc ease i i hi es
high-abili y wo ke s h ough ST con ac s (pa (b)). A highe implies a la ge in e es
in secu ing he se ices o a high-abili y wo ke , which explains he p e ious ela ions. A
simila a gumen gi es he in ui ion o pa (c) in he lemma.
Le usdeno eby he alue ha would “balance” he se o i ms i all he i ms
wi h 
would hi e low-abili y wo ke s while all he i ms wi h ≥would hi e
high-abili y wo ke s, ha is, is cha ac e ized by
()
1−()≡1−

16Lemma 1 (a) and 1 (b) do no depend on Assump ion 1. Howe e , i Assump ion 1 does no hold,
hen Lemma 1 (c) may ail i 
≡h2
+(1−)2

2
i£1
(−)+−¤+.
23
Also, we deno e b
 he alue ha makes he i m indiffe en be ween using LT con ac s
and hi ing low-abili y senio wo ke s h ough ST con ac s, when he junio sala y is

=, ha is, b
is cha ac e ized by
e (b
)=e
(b
 ).
As we check in Claim 1in he p oo o Theo em 1, unde Assump ion 1 i m b
lies in
egions  and 
()The e o e, we can easily calcula e b
:b
≡2
(2
−2
)p−.
We i s conside he case whe e b
∈[
) ha is, some o he i ms in he ma ke
ha e a low- alued p ojec , bu he e is a ela i ely high numbe o i ms wi h aluable
p ojec s.
Theo em 1 Suppose ≤b

, and deno e  he i m such ha ³b
´=
(1 −)(). Then, unde Assump ion 1, an equilib ium exis s whe e
(i) i ms wi h ≤b
offe ST con ac s:  o junio wo ke s and 
( ) o low-
abili y senio wo ke s,
(ii) i ms wi h ∈³b
 ´offe he LT con ac s  (),
(iii) i ms wi h ≥ offe ST con ac s:  o junio wo ke s and 
( 
) o
high-abili y senio wo ke s, whe e 
is such ha e ()=e
(
),
(i ) junio wo ke s accep bo h LT con ac s ha gua an ee hem +and ST con-
ac s wi h 
=,and
( ) senio wo ke s accep con ac s ha gua an ee hem .17
When is high enough, ha is, he popula ion o i ms is no concen a ed on low
le els o  hen, a equilib ium, i ms a e di ided acco ding o h ee hi ing s a egies.
Fi ms wi h low- alued p ojec s use ST con ac s and only hi e low-abili y senio s; i ms
wi h a high also use ST con ac s bu hey only hi e high-abili y senio s; and i ms wi h
in e media y s use LT con ac s.
The a ionale behind Theo em 1 is he ollowing. Fi ms wi h mo e p o i able p ojec s
gi e mo e impo ance o hi ing he high-abili y wo ke , and hey offe mo e o a ac
hem. This inc eases he expec ed u ili y o a junio wo ke when he accep s he ST
con ac : i he u ns ou o be o high abili y he will ob ain a la ge u ili y le el. The
17A equilib ium, high-abili y wo ke s ecei e a le el o u ili y o , a leas , 

.Howe e ,ou o
equilib ium, hey should be eady o accep lowe offe s, as long as hey gua an ee .
24
R
ˆR Ro
o
)(
~RE LT


S
ST
LUwRE ,,
~

R
o
R


oo
H
ST
HHwRE ,,
~

Sho Te
m
low-abili y
wo ke s
LongTe
m
Sho Te
m
high-abili y
wo ke s
Figu e 3: P o i unc ions a equilib ium
expec a ion o his po en ial ewa d leads wo ke s o accep a wage =when junio
which is unde hei ese a ion u ili y because hey will be compensa ed in he u u e
(in expec ed e ms) o his sac i ice. Fi ms wi h low  ake ad an age o his educ ion
in hewage ha canbeoffe ed o junio wo ke s who sign ST con ac s: hei alue o
he p ojec is low enough so ha he educ ion in he wage o junio agen s mo e han
compensa es he ac ha hey always end up hi ing low-abili y senio wo ke s.
Gi en he diffe ence in equilib ium sala ies be ween high- and low-abili y senio wo k-
e s, i ms wi h in e media y do no pe cei e a la ge diffe ence be ween hi ing one ype
o ano he . The e o e, i is be e o hem o p o i om he addi ional imp o emen in
efficiency due o he commi men hey make h ough LT con ac s.
Figu e 3d aws he LT and ST p o i s, as a unc ion o , o he equilib ium alues
o sala ies and u ili y . As shown in Lemma 1, he slope o e
( 
)is always
highe han ha o e ()which in u n is highe han he slope o e
( ).
A equilib ium, he ma ke p ice ha a i m has o pay in o de o a ac a high-abili y
wo ke (
), is such ha he h ee p o i unc ions c oss as shown in Figu e 3.
I iswo hno ing ha e en houghalljunio wo ke sa eiden icalwhen heysign hei
equilib ium con ac s and hey pe o m iden ical jobs, hei expec ed u ili y is diffe en
25
up ecei ing high emune a ion when ST con ac a e in place which, in u n, allows he
educ ion o he paymen o junio s, who o esee he p ospec s o a e y high wage when
senio s. Consequen ly, some i ms ha ing less luc a i e en u es20 may no be able o
e ain he high- alen ed wo ke s, bu hey indi ec ly p o i om he exis ence o such
wo ke sasi allows hem ohi ejunio sa amuchlowe cos .
A equilib ium, we o en ind ha wo ypes o i ms use sho - e m con ac s: i ms
in which he success o he p ojec depends e y much on he senio ’s effo , which always
end up hi ing high-abili y senio wo ke s; and i ms whose p o i s do no depend oo
much on he effo , which hi e low-abili y senio wo ke s. In e media e i ms may use
long- e m o sho - e m con ac s, depending on se e al ma ke cha ac e is ics. We show
ha coexis ence o bo h ypes o con ac is mo e likely when he e is a ele an ac ion
o i ms wi h p o i able p ojec s, when he ese a ion u ili y o young wo ke s is low and
he minimum wage is high, when he discoun a e is small, when he e is a la ge diffe ence
be ween he p oduc i i y o high- and low-abili y wo ke s, and when he agen s’ effo is
no oo cos ly.
In addi ion o he equilib ium wi h sho - e m con ac s ha o en exis s, he e always
exis s an equilib ium whe e all i ms choose a long- e m con ac s (see P oposi ion 1).
Howe e , we a gue ha , in ou en i onmen , whene e he equilib ium wi h sho - e m
con ac s and he one wi h only LT con ac s coexis , he o me is mo e “ obus ” o
“sensible” as he la e is a “kni e-edge” esul . The ull long- e m ou come is sus ained
by he ac ha , since no o he i m is choosing a sho - e m con ac , no i m can p o i
om he enhanced lexibili y ha sho - e m con ac s offe . A small amoun o i ms
wi h low- alued p ojec s and ano he wi h high- alued p ojec s ha e incen i es o swi ch
om LT o ST ag eemen s o ob ain highe p o i s.
Appendix
A P oo o P oposi ion 1
P oo . We i s no e ha ,inasi ua ionwhe eall i ms sign LT con ac s, i a i m
ollows he s a egy o offe ing ST con ac s o i s wo ke s, i necessa ily hi es as senio
20Fi ms whe e he ole o he senio is less impo an o he ou come.
32

agen a pe iod he same agen ha i hi ed as a junio a pe iod −1.Also, heonly
al e na i e occupa ion o he senio agen is o ge ou o he ma ke , since no o he
i m is in e es ed in hi ing him, independen on his abili y. Then, any sequence o ST
con ac s can be eplica ed as an LT con ac . The e o e, he op imal ST con ac s canno
gi e highe p o i s han he op imal LT con ac s.
B P oo o P oposi ion 2
P oo . Subs i u ing and by hei alue and mul iplying he objec i e unc ion by
, he i m’s p og am can be ew i en as:
max
(∆∆) + µ1
222
∆(−∆)−¶+(1 −)µ1
222
∆(−∆)−¶−
s. . + µ1
422
∆2
+¶+(1 −)µ1
222
∆2
+¶≥+(1)
≥
≥
≥.
Le  ,and be he Lag ange mul iplie s co esponding o he cons ain s. The
Kuhn-Tucke ( i s -o de ) condi ions o he abo e maximiza ion p oblem include he con-
s ain s, and he non-nega i i y o he mul iplie s: ≥0,≥0,≥0,≥0The
de i a i es o he Lag angian wi h espec o ∆and ∆a e:
 1
222
(−2∆)+ 1
222
∆=0 (2)
(1 −)1
222
(−2∆)+ (1 −)1
222
∆=0(3)
which imply ha ∆=∆, which we deno e ∆in he es o he p oo .
The de i a i es o he Lag angian wi h espec o ,and a e:
−1++=0
− + +=0(4)
−(1 −)+(1 −)+=0
which imply =1−,= (1 −)and =(1 −)(1−); he e o e, ei he he
h ee cons ain s a e binding o none is. The las Kuhn-Tucke condi ions a e:
∙+∙µ+1
422
∆2
¶+(1−)µ+1
422
∆2
¶¸−(+)¸=0
33
(−)=0
(−)=0
(−)=0
F om (2) and (4) we can deduce ha :
=2−
∆and = µ
∆−1¶
We s udy he diffe en egions whe e he Kuhn-Tucke condi ions may be sa is ied:
Case /1:0,0,0,0Paymen s when young and in case o ailu e a e
===and he bonus in case o success is ∆=2
q1
[(+)−(1 + )].
Finally, his is a candida e only i ∈[01], i.e.,
∈"2
e 1
[(+)−(1 + )]4
e 1
[(+)−(1 + )]#
Case 2 :=00
0,0.Then===,and∆=
2.In
his case he pa icipa ion cons ain holds only i ≥4
q1
[(+)−(1 + )]
(The candida e a he lowe bound o his case coincides wi h he candida e a he highe
bound o Case 1.)
Case 3:===0.Then=1and ∆=. We w i e he pa icipa ion cons ain
as
+(+(1−))+1
422e2=+
Any combina ion o ,and  ha sa is ies he p e ious cons ain and such ha he
h ee alues a e la ge o equal o cons i u es an op imal solu ion (in pa icula , he al-
ues p oposed in he p oposi ion). This can be he case only i +(+(1−))≥
(1 + ), ha is≤2
q1
[(+)−(1 + )].
Theuniquecandida e o each alueo is he op imal solu ion o he i m’s maxi-
miza ion p og am. F om he op imal con ac in each case, i is immedia e o compu e
agen ’s effo (s) and u ili y, and i m’s p o i s. Addi ionally, easy calcula ions show ha
he unc ion  ()is con inuously diffe en iable in .
34
C P oo o P oposi ion 3
P oo . Subs i u ing by i s alue in he i m’s p og am, we can ew i e i as
max
(∆)+1
222
∆(−∆)−
s. . 1
422
∆2
+≥
≥.
Le   be he Lag ange mul iplie s co esponding o he cons ain s. The Kuhn-Tucke
( i s -o de ) condi ions o he abo e maximiza ion p oblem include he cons ain s, and
he non-nega i i y o he mul iplie s: ≥0,≥0The de i a i es o he Lag angian
wi h espec o and ∆a e
−1++=0(5)
1
222
(−2∆)+1
222
∆=0(6)
F om (5) and (6) we can deduce ha :
=2−
∆
and =
∆−1
We s udy he diffe en egions.
Case /1:0,0Paymen a e =and ∆=2
√−. This is a candida e
only i ≥0and ≥0,i.e.,∈h2
√− 4
√−i
Case 2 :=00.Then=,and∆=
2. In his case he pa icipa ion cons ain
holds only i ≥4
√−
Case 3:=0.Then∆=, which implies =10. The pa icipa ion cons ain is
1
422
2+=The e o e, =−1
422
2≥i and only i ≤2
√−
Theuniquecandida e o each alueo is he op imal solu ion o he i m’s maximiza ion
p og am. F om he op imal con ac in each case, i is immedia e o compu e agen ’s
effo (s) and u ili y.
D P oo o Lemma 1
P oo . We highligh ha he h ee de i a i es ha we conside in he lemma, 

 ( ),

 (),and

 ( 
), ha e a simila shape: hey a e i s linea in un il hey
35
each some 1(ei he 
1(),o 
1,o 
1()), hen hey a e cons an un il hey
each a second h eshold 2and, om 2on, hey a e linea in again. The p oo o
he h ee pa s in he lemma is simila . We w i e a comple e p oo o pa (a) and we
poin ou he main elemen s o pa s (b) and (c).
(a) Fi s , no ice ha i lies in bo h egions 
()and  ,

 =1
222

1
22e2=
 The same compa ison holds i lies in bo h egions 
()and
 . Addi ionally, i lies in bo h egions 
()and  ,

 =1
p− 
1
eq1
(−)+−=
 
Second, i 
1≥
1()(and 
2≥
2()), hen 
 is inc easing in a
la ge egion o pa ame e s han 

 be o e becoming cons an (a a highe le el han


 in egion 
()). Finally, e en i 

 s a s inc easing again (i.e., i eaches
egion 
())be o e
 (because 
2()≤
2), i is always lowe han he
la e , since i is lowe e en when =
2, gi en ha we ha e seen ha 

 

o any which lies in bo h egions 
()and  .
Thi d, suppose 
1

1()(and 
2

2()). Gi en ha 

 is smalle
han 
 when 

 eaches he egion whe e i becomes cons an , and ha i is
ce ainly also smalle when i s a s inc easing again (because 
 has eached his
egion be o e), i is no possible ha he wo de i a i es c oss. The e o e, 

 

o any 0.
(b) I lies in bo h egions 
()and 
(),

 =1
222
 1
222
=


 The same compa ison holds in egions 
()and 
(). Also, i lies in
bo h egions 
()and 
(),

 =1
p−  1
√−=

 
The es o hep oo isiden ical o heoneinpa (a).
(c) Fo in bo h egions  and 
()(and simila ly in  and 
()),

 =1
22e2 1
222
=

 .I lies in bo h egions  and 
(),
hen 
 =1
eq1
(−)+−  1
√−=

 i and only i 
2
2
£1
(−)+−¤+. I his inequali y holds, he es o he p oo o Lemma
(c) is iden ical o he one in pa (a). A sufficien condi ion is
≥e2
2
∙1
(−)+−¸+(7)
which, gi en Assump ion 1 (i), is implied by (2
−e2)e2,i.e.,(1 −)(2
−2
)
2
+(1−)2
,o ,((1 −)−)2

2

(1+)(1−)Assump ion 1 (ii) implies ha (1−
36
)−0The e o e, gi en Assump ion 1 (iii), he inequali y holds i ((1 −)−)³1+ 1
 ´≥
(1 + )(1 −),i.e.,(1 −)−≥0which closes he p oo s.
E P oo o Theo em 1
P oo . We do he p oo h ough a se ies o claims.
Claim 1:I 
=, hen b


1()and b


1.
P oo o Claim1:I he alueb
 ha sa is ies e (b
)=e
(b
 )lies in bo h e-
gions  and 
()(i.e., b


1()and b


1), hen b
=2
(2
−2
)p−
Mo eo e , i is easy o check ha each o he inequali ies b


1()and b


1is
equi alen o he ollowing:
2
(−)¡2
−2
¢(−)(8)
Gi en Assump ion 1 (i), (8) is implied by Assump ion 1 (ii).
Claim 2: Conside he alue b
such ha e
³b
 
´=e
³b
 

=b
´.
I junio wo ke s an icipa e ha hey will ob ain a leas b
when senio i hey u n ou
o be high-abili y, hen hey a e eady o accep 
=.
P oo o Claim2: We p oceed as ollows. We conjec u e ha b
is such ha b


1³b
´=2
qb
− we will compu e he co esponding b
in his egion, and
hen we will show ha i is indeed he case ha b


1³b
´. The e o e, b
is de ined
by
−1

−+³
2´2b
2=−1

−b
+³
2´2b
2
i.e., b
=+¡1
2¢2(2
−2
)b
2o b
=+1
 (−)Fo his alue, 
1³b
´=
2
q+¡1
2¢2(2
−2
)b
2−.The e o e,b


1³b
´holds i and only i b
2
³2
´2h+¡1
2¢2(2
−2
)b
2−i, i.e., 2
b
2(2)2(−), which is equi alen
o (8). Finally, gi en b
, and aking in o accoun ha ≥b
and ≥,
a junio wo ke is eady o accep an ST con ac wi h 
whene e 
++
 ¡1
2¢2(2
−2
)b
2≥+, ha is, when 
≥.
Claim 3:e
¡ 
=¢e
¡ 
=
¢ o any ≥b
and
o any  b
.
37

P oo o Claim3:Gi en hede ini ion o b
in Claim 3,e
³b
 
=´≥
e
³b
 
=
´ o any ≥b
. Then, he claim ollows a e Lemma 1 (b).
Claim 4:e
¡ 
=¢max ©e
¡ 
=
¢e ()ª o
any  b
.
P oo o Claim4:The i s inequali y ollows a e Claim 3,also akingin oaccoun ha
 b
implies 
b
. The second inequali y ollows he de ini ion o b
and Lemma
1(a).
Claim 5:e ()≥max ©e
¡ 
=
¢e
¡ 
=¢ª o
any ∈hb
 i.
P oo o Claim5:The i s pa o he inequali y ollows a e he cha ac e iza ion o 

in pa ( i) o he heo em, by he p ope y ha 
b
and Lemma 1 (c). The second
pa ollows he de ini ion o b
and Lemma 1 (a).
Claim 6:e
¡ 
=
¢max ©e ()e
¡ 
=¢ª o
any 
.
P oo o Claim6: By he same a gumen as in Claim 5, he maximum o he wo e ms
inside he maximiza ion is e (). Then, he inequali y is implied by he cha ac e i-
za ion o 
in pa ( i) o he heo em, by he p ope y ha 
b
and Lemma 1
(c).
FP oo o Theo em2
P oo . Gi en ha he beha io o he wo ke s is op imal by cons uc ion, we p o e he
heo em i we show ha i ms’ s a egies a e op imal. We do i h ough a se ies o claims.
Claim 1:
≤− ¡
2¢2(2
−2
)
P oo o Claim1.Gi en ha ≥
and ≥,−(+(1−)−)≤
− ¡
2¢2(2
−2
)Mo eo e ≤− ¡
2¢2(2
−2
)because his inequali y
is equi alen o ≤b
.
Claim 2:≤
1(
)
P oo o Claim2:≤2
q+¡
2¢2(2
−2
)−i and only i ≤2
p−=

1()which is implied by he ac ha ≤b
and b
≤
1()(by Claim 1in
he p oo o Theo em 1).
Claim 3:
¡


¢=
¡

¢≥ ()
38
P oo o Claim3.Gi en ha ≤
1()and ≤
1(
) he i s equali y comes
di ec ly om he de ini ion o 
To p o e he inequali y, we no ice ha ≤
1
because ≤b
and b
≤
1(by Claim 1in he p oo o Theo em 1). Gi en ha
≤
1and ≤
1() he inequali y can be w i en as −
+¡
2¢2≥−−
1
(−)+³
2´2
By Claim 1,asufficien condi ion is −³− ¡
2¢2(2
−2
)´+
¡
2¢2≥−−1
(−)+³
2´2This inequali y holds because i is equi alen o
≤b

Claim 4:e
¡ 
¢max ©e
( 

)e ()ª o any 

P oo o Claim4: I ollows om Claim 3and Lemma 1 (a) and (b).
Claim 5:e
( 

)≥max ©e
¡ 
¢e ()ª o any ≥
P oo o Claim5: I ollows om Claim 3and Lemma 1 (b) and (c).
G P oo o Theo em 3
P oo . We ecall ha b
is cha ac e ized by e (b
)=e
(b
 ).I b
, hen
b
 o all ∈£ ¤. The e o e, Lemma 1 (b) implies e ()e
( )
o all ∈£ ¤. I easily ollows ha e ()e
( 
) o all ∈£ ¤,
≥and ≥. The e o e, a equilib ium, no ST con ac can be signed, since
i would imply ha some i ms choose he s a egy o keeping low-abili y senio wo ke s,
which is domina ed by he s a egy o always offe ing LT con ac s.
H P oo o Theo em 4
P oo . The p oo s o heo ems 1 and 2 and ha o Lemma 1 only use Assump ion 1 o
show ha he inequali ies (7) and (8) hold. The e o e, we p o e heo em 4 i we show
ha Assump ion 2 also imply (7) and (8). We w i e Assump ion 2 as
 ¡2
−2
¢(−)
2
(−)
and
(1 −)¡2
−2
¢(−)e2(−).
The i s inequali y co esponds o (8). Mo eo e , i is easy o check ha he second
inequali y also co esponds o (7) (wi h s ic inequali y).
39
Re e ences
[1] Alonso-Paulí, E., Pé ez-Cas illo, D. ( o hcoming), Codes o Bes P ac ice in Com-
pe i i e Ma ke s o Manage s. Economic Theo y.
[2] Chiappo i, P.A., Macho, I., Rey, P., Salanie, B. (1994), Repea ed Mo al Haza d: The
Role o Memo y, Commi men , and Access o C edi Ma ke s. Eu opean Economic
Re iew 38, pp. 1527—1553.
[3] Dam, K., Pé ez-Cas illo, D. (2006), The P incipal Agen Ma ke . F on ie s in Eco-
nomic Theo y, Be keley Elec onics 1.
[4] Gha ak, M., Mo elli, M. and Sjös öm, T. (2001), Occupa ional Choice and Dynamic
Incen i es. Re iew o Economic S udies 68, pp. 781—810.
[5] Ghosh, S., Waldman, M. (2010), S anda d P omo ion P ac ices e sus Up-o -ou
Con ac s. The Rand Jou nal o Economics 41, pp. 301-325.
[6] Holms öm, B. (1999), Manage ial Inncen i es: A Dynamic Pe spec i e. Re iew o
Economic S udies 66, pp. 169-182.
[7] Loh, E.S. (1994), Employmen P oba ion as a So ing Mechanism. Indus ial and
Labo Rela ions Re iew 47, pp. 471-486.
[8] Lambe , R. (1983), Long-Te m Con ac s and Mo al Haza d. Bell Jou nal o Eco-
nomics 14, pp. 441—452.
[9] Leg os P., Newman, A. (2007), Beau y is a Beas , F og is a P ince: Asso a i e
Ma ching wi h Non ans e abili ies. Econome ica 75, pp. 1073-102.
[10] Malcomson J. Spinnewyn, F. (1988), The Mul ipe iod P incipal-Agen P oblem. Re-
iew o Economic S udies 55, pp. 391—408.
[11] Ma u es, C., Regibeau P., Rocke K. (1994), Compensa ion Schemes and Labo
Ma ke Compe i ion: Piece Ra e e sus Wage Ra e. Jou nal o Economics & Man-
agemen S a egy 3, pp. 325—353.
[12] Rice, J.B., Sen, A. (2008), Mo al-Haza d in Long- e m Gua an eed Con ac s.
Mimeo, Bos on Uni e si y.
40
[13] Roge son, W. (1985), Repea ed Mo al Haza d. Econome ica 53, pp. 69—76.
[14] Ro h, A., So omayo , M. (1990), Two-Sided Ma ching: A S udy in Game-Theo e ic
Modeling and Analysis. Camb idge Uni e si y P ess, New Yo k and Melbou ne,
Econome ic Socie y Monog aphs.
[15] Se es, K. (2008), Endogenous Ma ching in a Ma ke wi h He e ogeneous P incipals
and Agen s. In e na ional Jou nal o Game Theo y 36, pp. 587-619.
[16] Shapley, Ll.S., Shubik, M. (1972), The Assignmen Game I: The Co e. In e na ional
Jou nal o Game Theo y 1, pp. 111-130.
[17] Te iö, M. (2008), The Diffe ence ha CEOs Make: An Assignmen Model App oach.
Ame ican Economic Re iew 98, pp. 642-668.
[18] Te iö, M. (2009), Supe s a s and Medioc i ies: Ma ke Failu e in he Disco e y o
Talen . The Re iew o Economic S udies 76, pp. 829—850.
41