Optimal coexistence of long-term and short-term contracts in labor Markets

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. 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