Bidimensional screening with intrinsically motivated workers

Ba igozzi, F ancesca; Bu ani, Nadia Wo king Pape Bidimensional sc eening wi h in insically mo i a ed wo ke s Quade ni - Wo king Pape DSE, No. 866 P o ided in Coope a ion wi h: Uni e si y o Bologna, Depa men o Economics Sugges ed Ci a ion: Ba igozzi, F ancesca; Bu ani, Nadia (2013) : Bidimensional sc eening wi h in insically mo i a ed wo ke s, Quade ni - Wo king Pape DSE, No. 866, Alma Ma e S udio um - Uni e si à di Bologna, Dipa imen o di Scienze Economiche (DSE), Bologna, h ps://doi.o g/10.6092/unibo/amsac a/3897 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/159705 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. 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I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by-nc/3.0/ ISSN 2282-6483 Bidimensional Sc eening wi h In insically Mo i a ed Wo ke s F ancesca Ba igozzi Nadia Bu ani Quade ni - Wo king Pape DSE N°866 Bidimensional Sc eening wi h In insically Mo i a ed Wo ke s∗ F ancesca Ba igozzi † and Nadia Bu ani ‡ Uni e si y o Bologna Abs ac We s udy he sc eening p oblem o a fi m ha needs o hi e a wo ke o p oduce ou pu and ha obse es nei he he p oduc i e abili y no he in insic mo i a ion o he job applican . We comple ely cha ac e ize he se o op imal con ac s acco ding o whe he mo i a ion o abili y is he main de e minan o he wo ke ’s pe o mance. We show ha i is always in he fi m’s in e es o hi e all ypes o wo ke and o offe diffe en con ac s o diffe en ypes o employees. In e es ingly, when mo i a ion is e y high, incen i es o ce he fi m o pay highe in o ma ional en s, o inc ease effo dis o sions o mo i a ed wo ke s, and o offe a s ic ly posi i e wage o wo ke s enjoying a posi i e u ili y om effo p o ision, who hus become paid olun ee s. These esul s sugges ha , om he p incipal’s iewpoin , e y high mo i a ion migh no be a desi able wo ke ’s cha ac e is ic. Jel classifica ion: D82, D86, J31, M55. Key-wo ds: bidimensional sc eening, sel -selec ion, in insic mo i a ion, skills. 1 In oduc ion A ecen li e a u e add esses he issue o he selec ion o applican s in a labo ma ke whe e po en ial wo ke s can be in insically mo i a ed o he job, as in he ma ke o ci il se an s, heal h p o essionals and eache s (Handy and Ka z 1998, Del gaauw and Du 2007, 2010 F ancois 2000, Heyes 2005). A sha ed iew om his li e a u e is ha high wages a e necessa y o a ac applican s wi h high skills, bu his comes a he cos o employing wo ke s who a e less mo i a ed o he ask o be pe o med. ∗ We a e g a e ul o Renaud Bou lès, Giacomo Calzola i, Vincenzo Denicolò, Robe Du , Mai eesh Ga hak, Tommaso Reggiani, Claudio Zoli and o semina pa icipan s a he Uni e si y o Bologna, Uni e si y o Ve ona and Bos on Uni e si y. The pape was p esen ed a he 11 h edi ion o he Jou nées d’Economie Publique Louis-And é Gé a d-Va e in Ma seille, a he Asse 2012 Mee ing in Cyp us, a he 54 h Mee ing o he Socie y o I alian Economis s SIE in Bologna and a he UECE 2013 Lisbon Mee ings. † Depa men o Economics, Uni e si y o Bologna and CHILD, P.zza Sca a illi 2, 40126 Bologna (I aly). E-mail: ancesca.ba [email protected] ‡ Co esponding au ho . Depa men o Economics, Uni e si y o Bologna, S ada Maggio e 45, 40125 Bologna (I aly). E-mail: nadia.bu [email protected] . Tel. +39 051 209 2642. Fax +39 051 209 2664. 1 Con e sely, low mone a y wages selec highly mo i a ed wo ke s, who migh no necessa ily be alen ed o skilled. This sugges s ha fi ms a e no able o design op imal compensa ion p ac ices in o de o sc een po en ial applican acco ding o bo h p oduc i e abili y and mo i a ion, despi e he ac ha he wo ke s’ o e all pe o mance depends on he in e play o bo h cha ac e is ics. We depa om his iew claiming ha , in a wo ld whe e wo ke s’ a ibu es a e no obse able, a fi m can succeed in sol ing his p oblem offe ing simple con ac s based on one sc eening ins umen only and a non-linea wage. Take he ma ke o nu ses, whe e hospi als ypically offe con ac s cha ac e ized by a diffe en numbe o wo king hou s: in he U.S., pa - ime con ac s equi e abou 24 hou s each week, ull- ime nu ses wo k an a e age o abou 43 hou s a week; mo eo e nu ses can choose paid olun a y o e ime up o a o al amoun ha canno exceed 60 hou s a week. 1 As o nu ses’ mone a y compensa ion, he o al sala y hey ecei e is gene ally ep esen ed by an hou ly wage ha depends on he numbe o hou s wo ked pe day: i encompasses pa - ime penal ies and/o o e ime p emia. We show ha such simple con ac s, defined only by he numbe o hou s wo ked pe week and by he o al sala y, a e likely o enable he hospi al o sc een applican s wi h espec o wo diffe en dimensions o p i a e in o ma ion, namely abili y and in insic mo i a ion. In pa icula , ou model p edic s ha high-abili y mo i a ed applican s choose he con ac wi h he la ges olun a y o e ime and low-abili y non-mo i a ed nu ses a e a ge ed o pa - ime con ac s. As is well known, also wo ke s’ ca ee conce ns can be used as a sc eening de ice. Typically, wo ke s sel -selec in o diffe en ca ee pa hs: some o hem accep asks in ol ing s ong pe o mance e alua ion in exchange o mo e likely and as e p omo ions; some o he s p e e a slowe p og ess up he job ladde oge he wi h lowe pay and almos no pe o mance e alua ion. In he academia, o ins ance, junio p o esso s can ei he choose enu e- ack posi ions, which equi e hem o demons a e, wi hin a sho ime span, a s ong eco d o published esea ch, g an unding, eaching and adminis a i e se ice o posi ions off he enu e ack (such as lec u e o adjunc p o esso ), which equi e hem o each ull- o pa - ime bu wi h ew o no esea ch esponsibili ies. He e an op imal con ac consis s o he ca ee pa h and he o e all compensa ion. In ui i ely, enu e- ack posi ions a e a ge ed o a ac he bes esea che s. We in es iga e he p oblem o he selec ion o wo ke s whose o e all pe o mance esul s om he in e play o skills and mo i a ion and we hus con ibu e o he exis ing li e a u e by explici ly accoun ing o he bidimensional na u e o wo ke s’ p i a e in o ma ion. Indeed, he li e a u e ei he deals wi h he 1 Bae (2012) p esen s a quan i a i e su ey da a collec ed om egis e ed nu ses who wo ked in hospi als as s aff nu ses in No h Ca olina and Wes Vi ginia in 2010-2011. Conce ning o e ime, he au ho shows ha 33.2% o nu ses wo king o e ime a e choosing o pe o m olun a y paid o e ime; among hem, 42% a e wo king o e ime mo e han 12 hou s a week. In e es ingly, he su ey also conside s he easons epo ed by nu ses as o why hey wo ked o e ime. Nea ly hal (46.3%) o nu ses choosing olun a y o e ime decla ed ha hey “like o wo k o e ime”. 2 selec ion o wo ke s who a e p i a ely in o med abou hei mo i a ion only (see Heyes 2005 and Del gaauw and Du 2007) o conside s asymme ic in o ma ion on bo h mo i a ion and skills bu does no explici ly sol e he bidimensional sc eening p oblem, because he p incipal is only asked o hi e a limi ed se o ypes (see Del gaauw and Du 2008). A de ailed desc ip ion o he ela ed li e a u e is p o ided in a sepa a e sec ion which ollows. We conside a p incipal-agen ela ionship whe e agen s’ skills (o p oduc i e abili y) and in insic mo i a ion a e independen ly and disc e ely dis ibu ed, and ake wo possible alues. P oduc i e abili y lowe s he wo ke ’s cos o p o iding effo whe eas mo i a ion is in e p e ed as he wo ke ’s enjoymen o he pe sonal con ibu ion o he fi m’s ou come o as a non-mone a y benefi acc uing o he wo ke when pe o ming a gi en ask. Since wo ke ’s cha ac e is ics can no be obse ed by he employe , hey can no be con ac ed upon. Ins ead, we assume ha he fi m can obse e and e i y he effo le els p o ided by diffe en ypes o wo ke s. Thus, he employe offe s a menu o con ac s consis ing in diffe en combina ions o wage a e and effo p o ision. Ou goal is hen o desc ibe he se o op imal sc eening con ac s and, in pa icula , o analyze which ypes o wo ke s a e hi ed and which a e he op imal compensa ion p ac ices ha he fi m adop s. 2 The comple e cha ac e iza ion o op imal con ac s allows us o deli e some no el and in e es ing insigh s. Despi e ha ing only one ins umen ( he obse able effo le el) a his disposal, he fi m always offe s con ac s ha en ail ull sepa a ion and ull pa icipa ion o ypes (whene e hey exis ) which domina e menus wi h pooling o exclusion. Thus, when ull sepa a ion and ull pa icipa ion o ypes is easible, sc eening is no oo cos ly o he fi m, nei he in e ms o in o ma ion en s ha he p incipal lea es o he mos mo i a ed and/o mos able ype, no in e ms o dis o ions o effo le els ha less mo i a ed and/o less able ypes a e equi ed o p o ide. F om his iewpoin , ou esul s s and in con as wi h he li e a u e on mul idimensional sc eening which p edic s ha bunching and/o exclusion a e ine i able. As o he op imal wage schemes, unde ull in o ma ion, high-abili y non-mo i a ed wo ke s a e always paid he highes wage, while mo i a ed low-abili y employees a e always paid he lowes sala y. Unde asymme ic in o ma ion, howe e , he e is a e e sal in he anking o ewa ds: high-skilled mo i a ed wo ke s ecei e he highes ans e , while low-skilled non mo i a ed wo ke ob ain he lowes ewa d. This is because mo i a ed wo ke s a e able o mimic non-mo i a ed ones and hus equi e an in o ma ion en o u h ul e ela ion, which inc eases hei compensa ion. Ou esul s a e d i en by he ela i e impo ance o he diffe ence in mo i a ion is à is he diffe ence in abili y, which influences he p incipal’s “p e e ence o de ing” o e he possible ypes. High-skilled mo i a ed wo ke s a e unambiguously he bes ypes, since hey p o ide he highes possible le el o 2 Wi h a sligh loss o gene ali y, ou analysis could be en i ely eph ased in e ms o a go e nmen al agency ( he p incipal) willing o hi e a manage ( he agen ) who migh be endowed wi h public se ice mo i a ion. 3 effo and con ibu e he mos o ou pu p oduc ion, low-skilled non-mo i a ed employees a e he wo s ypes, while he e is no na u al anking o in e media e ypes. Acco dingly, he e a e wo possible s a es o he wo ld o be s udied. The fi s one is cha ac e ized by he e ogenei y in mo i a ion p e ailing o e he e ogenei y in abili y (meaning ha mo i a ion has a la ge impac han abili y on he wo ke ’s effo p o ision and on he fi m’s ou pu ), in which case he low-abili y mo i a ed wo ke is asked o p o ide a highe effo han he high-abili y non-mo i a ed ype. The second is cha ac e ized by he e ogenei y in abili y being mo e significan han he e ogenei y in mo i a ion (meaning ha abili y con ibu es ela i ely mo e o effo p o ision) so ha high-skilled non-mo i a ed wo ke s a e induced o exe highe effo han low-skilled mo i a ed ones. When mo i a ion has a la ge impac han abili y on effo p o ision, we ob ain an in iguing esul : o in e media e ypes, wages migh no inc ease mono onically in effo p o ision, in which case low- skilled mo i a ed wo ke s become “paid olun ee s” a he op imal con ac , because hey always ea n a s ic ly posi i e sala y (gi en he in o ma ion en s ha hey ecei e) e en i hey enjoy a ne u ili y om effo p o ision. Con e sely, when abili y has a la ge impac han mo i a ion on effo p o ision, bu oca ion is s ill high, a ension ealizes: on he one hand, a any op imal con ac , he high-skilled non-mo i a ed wo ke is equi ed o p o ide a highe effo han he low-skilled mo i a ed one; on he o he hand, as mo i a ion inc eases, he mo i a ed wo ke aces a diminishing disu ili y o effo so ha i becomes mo e and mo e con enien o inc ease he effo and mo e and mo e difficul o mee he p e ious mono onici y condi ion. This ension gene a es a coun e -in ui i e pa e n o binding incen i e compa ibili y cons ain s and leads o he mos in e es ing classes o solu ions whe e: (i) he en s paid by he p incipal o he diffe en ypes o wo ke s a e lowe han in he case in which mo i a ion p e ails, (ii) he e is no dis o ion in he effo p o ided by low-skilled mo i a ed wo ke s, oge he wi h he s anda d esul o no dis o ion a he op. In his si ua ion, we find ha , al hough wages inc ease mono onically in effo le els, indi ec u ili ies (i.e. in o ma ion en s) a e mono onically dec easing in effo o in e media e ypes. Hence, low-skilled mo i a ed wo ke s exe a lowe effo , gain a lowe sala y bu a e mo e sa isfied han skilled, non mo i a ed ypes. To conclude, we ob ain an unexpec ed esul : when mo i a ion is e y high, all wo ke ypes ecei e highe in o ma ional en s and he low-skilled mo i a ed wo ke migh become a “paid olun ee ”, while she p o ides an effo le el which is no dis o ed a he solu ions ob ained when mo i a ion is lowe . This sugges s ha , om he p incipal’s iewpoin , e y high mo i a ion migh no be a desi able wo ke ’s cha ac e is ic. The es o he pape is o ganized as ollows. In he ollowing Subsec ion we desc ibe he ela ed li e a u e. In Sec ion 2, we se up he model. Sec ion 3 desc ibes he benchma k cases, s a ing wi h he fi s -bes (Sec ion 3.1) and conside ing hen asymme ic in o ma ion on one dimension only, be i 4 abili y (Sec ion 3.2) o in insic mo i a ion (Sec ion 3.3). In Sec ion 4, we conside he in e ac ion be ween he wo sou ces o incomple e in o ma ion. We dis inguish be ween he wo pola cases in which: (i)mo i a ion has a la ge impac han abili y on effo p o ision, o (ii)abili y has a la ge impac han mo i a ion on effo p o ision. We desc ibe he se o op imal con ac s ocusing on hei quali a i e ea u es. Sec ion 5 conside s mo e in de ail he wo mos significan op imal con ac s wi h ull sepa a ion and ull pa icipa ion o ypes and offe s some in e es ing compa isons be ween hem. All p oo s a e elega ed o he Appendix as well as he o mal analysis o bunching and/o exclusion. Sec ion 6 concludes. 1.1 Rela ed li e a u e Ou wo k con ibu es o wo diffe en s ands o li e a u e: om an economic poin o iew, i adds o he ecen and apidly g owing li e a u e on he selec ion o wo ke s wi h in insic mo i a ion; om a echnical poin o iew, i explici ly sol es he p incipal-agen p oblem in a labo ma ke whe e wo ke s a e cha ac e ized by wo diffe en dimensions o p i a e in o ma ion. The p oblem o he design o op imal incen i e schemes o in insically mo i a ed wo ke s has been ackled by Mu dock (2002), Besley and Ga hak (2005) and Gha ak and Muelle (2011), whose a en ion has p ima ily been de o ed o mo al haza d, while we conside he sc eening p oblem. Heyes (2005) and Del gaauw and Du (2007) a e he fi s pape s ha add ess he issue o he selec ion o wo ke s who a e p i a ely in o med abou hei oca ion. They show ha , as a wo ke ’s mo i a ion inc eases, he wo ke ’s ese a ion wage dec eases. The e o e, as he wage inc eases, he a e age mo i- a ion o he wo ke s who a e willing o accep he job de e io a es. Del gaauw and Du (2007) use a di ec ed sea ch amewo k à la Diamond, Mo ensen and Pissa ides and hey show ha op imal wage schemes en ail a ade-off be ween he p obabili y o filling a acancy, he en s le o he wo ke s and he expec ed mo i a ion o job applican s. Ou analysis depa s om his wo k because i includes a second sou ce o asymme ic in o ma ion (p oduc i e abili y) and, mos impo an ly, because i eso s o a di ec e ela ion mechanism allowing he p incipal o in e he wo ke s’ ue ypes. Del gaauw and Du (2010) conside a iche amewo k whe e wo ke s a e he e ogeneous wi h espec o bo h hei in insic mo i a ion and hei abili y. They ocus on he issue o manage ial sel -selec ion in o public s p i a e sec o s unde ull in o ma ion on he wo ke s’ cha ac e is ics: hey a gue ha he e u n o manage ial abili y is always lowe in he public han in he p i a e sec o , p o ided ha he demand o public sec o ou pu is no oo high and ha mo i a ion is un ela ed o ei he effo p o ision o o he fi m’s ou come. They conclude ha a ac ing a mo e able manage ial wo k o ce o he public sec o by inc easing emune a ion up o he p i a e sec o le els is no efficien . Finally, Ba igozzi e al. (2013) and Ba igozzi and Tu a i (2012) conside labo supply in a ma ke whe e wo ke s ha e p i a e 5 in o ma ion on bo h p oduc i e abili y and mo i a ion. They show ha he lemons’ p oblem migh be exace ba ed by he p esence o mul idimensional asymme ic in o ma ion because an inc ease in he ma ke wage can de e mine a simul aneous dec ease bo h in a e age oca ion and in a e age p oduc i i y o applican s. Ou pape is also closely ela ed o Handy and Ka z (1998) and Del gaauw and Du (2008). The fi s au ho s a gue ha non-p ofi s a ac mo i a ed manage s by offe ing hem compensa ion packages in ol ing lowe money wages and a la ge componen o ins i u ion-specific inge benefi s as compa ed o he p i a e sec o . Bu hei esul s a e d i en by an exogenously gi en anking o ese a ion wages o he diffe en ypes o manage s. Del gaauw and Du (2008) cha ac e ize he op imal incen i e schemes offe ed by a public agency when wo ke s diffe in laziness ( he opposi e o ou p oduc i e abili y) and public se ice mo i a ion. They show ha , when wo ke s’ effo is con ac ible and when he p oduc ion equi ed by he public ins i u ion is sufficien ly high, he ins i u ion a ac s dedica ed and p oduc i e wo ke s as well as he lazies wo ke s. 3 We depa om he las pape in wo main ways: we conside one sec o in isola ion and ou p incipal is no cons ained o hi e a mos wo ypes o agen s. The li e a u e on he analysis o op imal sc eening o agen s wi h unknown cha ac e is ics has flou - ished in he las wo decades o he wen ie h cen u y. None heless, his p oblem has mainly been examined unde he assump ion o unidimensional asymme ic in o ma ion. The in e es ing and pos- sibly mo e ealis ic cases whe e agen s ha e se e al unobse able cha ac e is ics ha e been s udied by ew impo an excep ions: A ms ong and Roche (1999), A ms ong (1996), Roche and Chonè (1998), A ms ong (1999), Baso (2001, 2005) and Denecke e and Se e ino (2011). They all show ha i is al- mos impossible o ex end o he mul idimensional en i onmen he quali a i e esul s and he egula i y condi ions o he unidimensional case. A ms ong and Roche (1999) p o ide a comple e cha ac e iza ion o he op imal con ac s when he dimensionali y o ac ions is he same as he dimensionali y o p i a e in o ma ion, he ype space is disc e e and he e a e common alues (namely he pa ame e s o p i a e in o ma ion en e di ec ly he objec i e unc ion o he p incipal). Ou model oo is cha ac e ized by a disc e e ype space, bu he e is one sc eening ins umen only a ailable o he p incipal (namely he con ac ible effo le el) and he e a e p i a e alues, hence he esul s in A ms ong and Roche (1999) canno be applied o ou se ing. When he dimensionali y o ac ions is smalle han he dimensionali y o p i a e in o ma ion and he ype space is con inuous, Laffon e al. (1987) explici ly sol e he p oblem o op imal nonlinea p icing by a egula ed monopoly. Again in he con inuous se up, A ms ong (1996), Roche and Chonè (1998), Baso (2001, 2005) and Denecke e and Se e ino (2011) p esen se e al use ul echniques o sol e he p oblem o mul idimensional sc eening, which canno be applied when he ypes space is disc e e. These 3 Dedica ed wo ke s exe highe effo han in he p i a e, pe ec ly compe i i e sec o whe eas lazy wo ke s’ effo is dis o ed downwa ds. 6 pape s show ha exclusion is gene ic and ull sepa a ion o ypes is impossible. In o he wo ds, i is ypically op imal o he p incipal no o se e he lowe pa o he agen ’s dis ibu ion and o offe he same con ac o diffe en (usually in e media e) ypes o agen s. Ou analysis owes much o A ms ong (1999), who conside s op imal p ice egula ion o a monopoly ha is p i a ely in o med abou bo h i s cos and demand unc ion. Like ou s, his model is disc e e and ea u es one ins umen and wo dimensions o p i a e in o ma ion, bu his analysis has common alues while ou s has p i a e alues. A ms ong (1999) dis inguishes wo main classes o p oblems. I cos unce ain y is ela i ely mo e impo an han demand unce ain y, hen op imal p ices a e always weakly abo e ma ginal cos s. Con e sely, i demand unce ain y is mo e significan han cos unce ain y, hen p icing below ma ginal cos could be op imal. In bo h classes o solu ions some pooling o ypes always exis s, a esul which is d i en by he ac ha “pa icipa ion cons ain s do no ie-in nicely wi h he na u al o de ing o ypes”. 4 Mo eo e , he au ho explici ly igno es he issue o exclusion by es ic ing he pa ame e space so ha i is ne e op imal o he egula o o shu down some ypes o fi m. These ac s ep esen a main diffe ence wi h espec o ou model whe e pa icipa ion cons ain s a e well-beha ed and whe e we find ha , om he p incipal’s iewpoin , ull sepa a ion o ypes always domina es pooling and ull pa icipa ion always domina es exclusion. Finally, some pape s analyze he issue o mul idimensional sc eening in insu ance ma ke s. C ocke and Snow (2011) conside a pe ec ly compe i i e ma ke whe e consume s possess hidden knowledge o hei p obabili y (high o low) o incu ing a loss. They show ha bundled co e age o diffe en pe ils and diffe en losses allows insu e s o sc een applican s in se e al dimensions he eby educing he cos s ha low- isk applican s bea o dis inguish hemsel es om high- isk ones. Bu hei efficien bundling o co e age in o a single insu ance policy essen ially educes he p oblem o a one-dimensional sc eening en i onmen . Oli ella and Sch oyen (2013) desc ibe a monopolis ic insu ance company acing a popula ion o indi iduals who diffe bo h in hei isks and in hei isk a e sion. They sol e a disc e e bidimensional sc eening p oblem ha bea s many simila i ies o ou s and comple ely cha ac e ize he se o op imal con ac s. 5 They find ha i is ne e op imal o ully sepa a e all he ypes and ha exclusion o some high- isk indi iduals may be op imal. I is also wo h men ioning ha in heal h economics, some esea ch has been conduc ed on al uis ic physicians, whose p e e ences include bo h mone a y ewa ds and hei pa ien s’ heal h s a us, which in u n depend on physicians’ se ices. Chonè and Ma (2011) and Liu and Ma (2013) s udy con inuous se ings whe e he physician is he agen cha ac e ized by bidimensional p i a e in o ma ion bo h on he 4 See oo no e 11 on page 207. 5 They ha e a simila double-c ossing condi ion o in e media e ypes and wo possible o de ings o co-insu ance a es ha in u n gene a e wo classes o solu ions depending on how he diffe ence in expec ed losses compa es wi h he diffe ence in he deg ee o absolu e isk a e sion. 7 subjec o he pa icipa ion cons ain o he non-mo i a ed ype and o he incen i e compa ibili y cons ain o he mo i a ed wo ke . In ac , mo i a ed agen s ha e in e es in mimicking non-mo i a ed ones whene e he effo hey a e equi ed o p o ide is sufficien ly high so as o cause a disu ili y. Sol ing o effo le els we find e BM iH =1 + γ θ i =e F B iH and e BM iL =(1 −µ)−µγ (1 −µ)θ i whe e he esul s o no dis o ion a he op and downwa d dis o ion in he effo exe ed by he non- mo i a ed wo ke bo h hold. Also, e BM iL >0 o γ < 1−µ µ≡γ BM whe e γ < γ BM always holds i µ < 1 2 .In wo ds, when γis sufficien ly high, he in o ma ion en ha he p incipal mus pay o he mo i a ed ypes is so cos ly ha he p e e s o exclude non-mo i a ed wo ke s. Howe e , he necessa y condi ion o ull pa icipa ion, ha is e BM iL >0, is always sa isfied i he p opo ion µo mo i a ed wo ke s is sufficien ly low. Following he same p ocedu e as in Foo no e 16, i can be checked ha e BM iL >0is bo h necessa y and sufficien o ull pa icipa ion. As o wages, hey a e always inc easing in mo i a ion and w iH > w iL .Hence, when mo i a ion is wo ke s’ p i a e in o ma ion and abili y is obse able, he anking o sala ies o wo ke s who a e equally skilled bu ha e diffe en oca ion is e e sed wi h espec o he fi s -bes . This is because o he in o ma ion en s ha mo i a ed wo ke s ecei e. 4 Sc eening on bo h abili y and mo i a ion The benchma k cases wi h unidimensional hidden in o ma ion p o ide he ollowing p edic ions: (i) When he p incipal canno obse e wo ke s’ skills (bu is pe ec ly in o med abou hei mo i a ion), he migh ake ad an age o mo i a ed wo ke s and make hem wo k o ee; as we will see, his u ns ou o be impossible a he second-bes . (ii)When he p incipal canno obse e wo ke s’ mo i a ion (bu is pe ec ly in o med abou hei skills), he migh find in his in e es o exclude non-mo i a ed employees, no ma e whe he hey ha e high- o low-abili y; he exclusion o skilled, non-mo i a ed wo ke s will always be domina ed a he second-bes . Fu he mo e, gi en abili y, mo i a ed employees a e always offe ed a highe sala y han non-mo i a ed ones; his esul s ands in con as wi h he fi s -bes bu will be confi med a he second-bes . Suppose now ha bo h he wo ke s’ abili y θ i and mo i a ion γ j a e he agen s’ p i a e in o ma- ion,we call his si ua ion he second-bes . The p incipal offe s he wo ke a choice o ou effo -wage 14 combina ions. Fo i=L, H and j=L, H, he p incipal’s p og am is max (e ij ,w ij ) E(π) = νµ (e LH −w LH ) + ν(1 −µ) (e LL −w LL ) + (1 −ν)µ(e HH −w HH ) + (1 −ν) (1 −µ) (e HL −w HL )(SB) subjec o ou pa icipa ion cons ain s whose gene ic o m is w ij −1 2θ i e 2 ij +γ j e ij ≥0(PC ij ) and wel e incen i e compa ibili y cons ain s ha a e such ha w ij −1 2θ i e 2 ij +γ j e ij ≥w i ′ j ′ −1 2θ i e 2 i ′ j ′ +γ j e i ′ j ′ (IC ij si ′ j ′ ) wi h ij diffe en om i ′ j ′ . All cons ain s a e lis ed in Appendix B, whe e we show ha incen i e compa ibili y and pa icipa ion cons ain s sa is y some egula i y condi ions. When all pa icipa ion and incen i e compa ibili y cons ain s a e sa isfied, he solu ion o he p inci- pal’s p og am SB is cha ac e ized by ull sepa a ion and ull pa icipa ion o ypes. In such ins ances, we find ha i he pa icipa ion cons ain o he wo s ype HL holds hen all o he PCs a e also sa isfied. Mo eo e , he ollowing mono onici y (o implemen abili y) condi ion holds e LH ≥max {e LL ;e HH } ≥ min {e LL ;e HH } ≥ e HL .(6) Conce ning in e media e ypes HH and LL, one can add IC LL sHH and IC HH sLL and find ha ei he e SB HH > e SB LL and e SB LL +e SB HH ≤2γ ∆θ,(7) o e SB LL > e SB HH and e SB LL +e SB HH ≥2γ ∆θ(8) holds. O he wise e SB HH =e SB LL and bunching be ween in e media e ypes necessa ily occu s. Al hough condi ions (7) and (8) a e less anspa en han he co esponding fi s -bes condi ions (4) and (5), we can s ill obse e ha e HH > e LL holds a he second-bes when he quan i y γ ∆θ is high: pa alleling he fi s -bes , we will say ha condi ion (7) is sa isfied when he diffe ence in mo i a ion p e ails o e he diffe ence in abili y in de e mining effo and ou pu p o ision. On he con a y, i condi ion (8) and e LL > e HH hold a he second-bes , hen i is because he quan i y γ ∆θ is low and we will say ha he diffe ence in abili y p e ails o e mo i a ion in de e mining effo and ou pu p o ision. A diffe ence wi h espec o he fi s -bes is ha he e m γ ∆θ , eflec ing he ela i e impo ance o mo i a ion he e ogenei y is à is abili y he e ogenei y, is doubled in (7) and (8), since i mus now be compa ed wi h he sum o he wo effo le els exe ed by in e media e ypes. 17 Obse a ions 1 and 2, espec i ely, will show ha 17 No e ha condi ion (4) is pe se mo e es ic i e han (7) and ha condi ion (5) is again mo e es ic i e han condi ion (8). Hence, one can in p inciple expec some misalignmen be ween fi s - and second-bes effo le els as o in e media e ypes, bu his will no be he case as P oposi ion 2 poin s ou . 15 while mono onici y condi ion (7) implies ha he incen i e compa ibili y cons ain IC HH sLL is binding, condi ion (8) does no ule ou si ua ions in which IC HH sLL s ill binds o in which he alloca ions o ypes HH and LL a e en y- ee. When conside ing solu ions cha ac e ized by ull sepa a ion and ull pa icipa ion o ypes, we will sol e a elaxed p og am in which only PC HL and some (mos ly downwa d) incen i e cons ain s will bind; we will check ex pos ha he omi ed cons ain s a e e ified as well. None heless, i migh well be ha some cons ain s ail o hold and ha he op imal con ac calls o he exclusion o some wo ke s’ ypes o o pooling o bo h. I is possible o show ha he solu ion o p og am SB en ailing ull pa icipa ion and ull sepa a ion o ypes always yields he highes p ofi s o he p incipal, who will hen always implemen i when possible. 18 P oposi ion 1 Independen ly o whe he mo i a ion o abili y p e ail, i.e. independen ly o whe he con- di ion (7) o condi ion (8) holds, he p incipal’s p ofi s a e maximal a he solu ion wi h ull pa icipa ion and ull sepa a ion o ypes. P oo . The p oo o he si ua ion in which mo i a ion p e ails is illus a ed in Appendix C.3. I equi es all esul s de i ed om he ull cha ac e iza ion o he diffe en classes o op imal con ac s ( ull sc eening, pooling and con ac s wi h exclusion) and he condi ions o hei exis ence. The p ocedu e o he case in which abili y p e ails is equi alen and hen omi ed. The abo e P oposi ion is an icipa ed he e o exposi ional pu poses. In wha ollows, we will p esen he classes o solu ions cha ac e ized by ull pa icipa ion and ull sepa a ion o ypes, classi ying hem acco ding o whe he condi ion (7) o condi ion (8) holds. We a e going o explain which incen i e cons ain s a e binding in he wo al e na i e si ua ions. The aim o such discussion is mainly desc ip i e, since we would like o gi e he eade a comple e o e iew o he esul s we ob ained. The mo e de ailed analysis o he diffe en classes o op imal con ac s is gi en in Sec ions 5.1 and 5.2 and in he Appendix. A he end o his Sec ion, we will b iefly discuss he emaining classes o solu ions en ailing bunching o exclusion, bu hei o mal analysis can be ound in he Appendix. Obse a ion 1 Mo i a ion p e ails (Case M) When condi ion (7) holds, mo i a ion has a highe impac on effo p o ision han abili y, and a solu ion o p og am SB wi h ull pa icipa ion and ull sepa a ion o ypes such ha e HH > e LL migh be a ained. The binding downwa d incen i e cons ain s specific o his case a e hose o high-abili y ypes mimicking low-abili y ones, ha is IC LH sHH and IC LL sHL . The addi ional downwa d incen i e cons ain is IC HH sLL , connec ing he p e ious ones. I mo i a ion has a highe impac on effo and ou pu p o ision han abili y, hen om he p incipal’s iewpoin , ypes can be o de ed as LH ≻HH ≻LL ≻LH. Now we ha e o sol e a bidimensional 18 As men ioned in he Rela ed Li e a u e, he implemen abili y o ully sepa a ing and ully pa icipa ing solu ions is unusual in models o mul idimensional sc eening wi h disc e e ypes and impossible when he ypes space is con inuous. 16 sc eening p oblem which embeds and gene alizes he wo sub-p oblems wi h ad e se selec ion on he wo ke s’ abili y only (Benchma k BA in Subsec ion 3.2). The wo sub-p oblems BA a e now conside ed simul aneously and linked by incen i e cons ain IC HH sLL . Figu e 1 desc ibes his case. On he ho izon al axis, we ep esen effo cos o abili y, while on he e ical axis we ha e mo i a ion. Types a e loca ed a he co ne s o a ec angle whose wid h is he diffe ence in abili y ∆θ, and whose heigh is he diffe ence in mo i a ion ∆γ, o simply γ. 19 An a ow om one ype o ano he ep esen s ha he incen i e cons ain ha he o me ype does no choose he con ac designed o he la e ype is binding. Inse Figu e 1 and Figu e 2a a ound he e In ui i ely, when he diffe ence in mo i a ion is mo e ele an han he diffe ence in abili y, he ec angle on which ypes a e loca ed has heigh g ea e han wid h. Since ypes LH and HH on one hand and ypes LL and HL on he o he hand a e close o each o he , hen i is na u al ha he incen i e cons ain s ha bind fi s a e IC LH sHH and IC LL sHL . The emaining binding cons ain is he one ha conce ns in e media e ypes, namely IC HH sLL . Indeed, Case Moccu s when mo i a ion γis high enough, hen ype HH is asked o p o ide a ela i e high effo in exchange o a ela i ely low sala y and she migh find he con ac (e LL , w LL )po en ially con enien . Obse a ion 2 Abili y p e ails (Case A) When condi ion (8) holds, abili y has a highe impac on effo p o ision han mo i a ion, and a solu ion o p og am SB wi h ull pa icipa ion and ull sepa a ion o ypes such ha e LL > e HH migh be a ained. The binding downwa d incen i e cons ain specific o his case is ha o he high-abili y mo i a ed ype mimicking he high-abili y non-mo i a ed agen , ha is IC LH sLL . As o he o he ele an binding cons ain s, h ee sub-cases mus be conside ed: (1) Case A.1. The binding incen i e cons ain s a e he wo adjacen ones IC LL sHH and IC HH sHL ; (2) Case A.2. The binding incen i e cons ain s a e IC LL sHL and IC HH sHL ; (3) Case A.3. The binding incen i e cons ain s a e IC LL sHL and he upwa d IC HH sLL . I he diffe ence in abili y has a highe impac on effo and ou pu p o ision han he diffe ence in mo i a ion, hen, om he p incipal’s iewpoin , ypes can be o de ed as LH ≻LL ≻HH ≻LH. Now we ha e a plu ali y o si ua ions a ising because he p incipal aces a ade-off be ween he need o sa is y condi ion e LL > e HH and he incen i e o inc ease e HH as mo i a ion g ows. Case A.1is he mos na u al one and is symme ic o Case M: i equi es o sol e a bidimensional sc eening p oblem ha consis s o he wo sub-p og ams ela ed o ad e se selec ion on wo ke s’ mo- i a ion only (as Benchma k BM in Subsec ion 3.3) oge he wi h incen i e cons ain IC LL sHH (see 19 Fo simplici y, Figu es om 1 o 2c a e all d awn le ing he diffe ence in mo i a ion ∆γ a y while keeping he diffe ence in effo cos ∆θcons an . Ac ually, i migh well be ha diffe en con ac s wi h ull sepa a ion and ull pa icipa ion exis o diffe en alues o bo h ∆γand ∆θ. 17 Figu e 2a). Now he ec angle on which ypes a e loca ed has heigh smalle han wid h, whe eby he ypes ha a e closes o each o he a e LH and LL on one hand and HH and HL on he o he hand. Then he incen i e cons ain s ha bind fi s a e hose o he closes pai s IC LH sLL and IC HH sHL . The emaining binding cons ain is he one ha conce ns in e media e ypes, namely IC LL sHH . He e, he mo i a ion le el γis sufficien ly low and hen ype LL migh be induced o mimic ype HH because he o me can benefi om a lowe effo e HH and s ill enjoy a sala y w HH which canno be oo low (gi en ha mo i a ion plays a mino ole). Be ween Case Mand Case A.1, ha is when mo i a ion is s ill less impo an han he diffe ence in abili y bu is no oo low, he wo less in ui i e si ua ions occu : Case A.2and Case A.3. He e a ension ealizes since, on he one hand, ype LL is asked o p o ide a highe effo han ype HH; on he o he hand, when mo ing om Case A.1 o Case M(i.e. when le ing mo i a ion γg ow), ype HH wo ke aces a diminishing disu ili y o effo so ha i becomes mo e and mo e con enien o he p incipal o ask he o p o ide a la ge effo and o pay he a lowe sala y. This ension leads anomalous incen i e cons ain s o bind. In pa icula , Cases A.2and A.3eme ge when ype LL p e e s o mimic HL a he han HH so ha he local downwa d incen i e cons ain IC LL sHH is no binding anymo e and he global IC LL sHL is binding ins ead. Then, oge he wi h he s anda d no-dis o ion a he op, he p incipal will ha e no in e es in dis o ing he effo equi ed o wo ke HH, because no o he ype is willing o mimic HH. Case A.2 ep esen s a bidimensional sc eening p oblem consis ing o he wo sub-p og ams ela ed o ad e se selec ion on wo ke s’ mo i a ion (as in Benchma k BM) which a e now connec ed by incen i e cons ain IC LL sHL (see Figu e 2b). In e es ingly, i is he unique case in which he e is no en y be ween ypes LL and HH so ha nei he IC LL sHH no IC HH sLL a e binding. In Case A.3, mo i a ion keeps inc easing and he disu ili y om he effo exe ed by ype HH is e en lowe han in Case A.2.Thus, i u ns ou ha ype HH mimics LL a he han HL. Now, no only is he local downwa d incen i e cons ain IC LL sHH slack, so ha no-dis o ion o ype HH occu s, bu he upwa d incen i e cons ain IC HH sLL is binding ins ead. This occu s since he mo i a ed ype HH alues a ela i ely high wage associa ed wi h a high effo ( ha she would ob ain by mimicking LL) mo e han he combina ion o low wage and low effo ( ha she would ge by mimicking HL). Case A.3 is ep esen ed in Figu e 2c. Inse Figu e 2b and Figu e 2c a ound he e Figu e 3 illus a es he classes o op imal con ac s jus p esen ed. Which class o solu ion ealizes depends on he ela i e posi ion o he e m 2γ ∆θ wi h espec o he sum o diffe en pai s o effo le els. The dis inc ion be ween Case Mand Case Adepends on condi ions (7) and (8). In addi ion, in o de o disc imina e among he diffe en si ua ions a ising when abili y p e ails, we conside which incen i e 18 cons ain s a e binding and which a e slack, as desc ibed in Obse a ion 2. Inse Figu e 3 a ound he e Finally, Figu e 4 desc ibes op imal con ac s as a unc ion o mo i a ion, ocusing no only on he exis ence egions o he ou ully sepa a ing and ully pa icipa ing equilib ia bu also conside ing equilib ia wi h pooling and/o exclusion ha a ise in-be ween. All equilib ia a e mu ually exclusi e, since, o any gi en ealiza ion o he pa ame e s γ∈(0,1] and θ∈(1,2], a diffe en solu ion o p og am SB is ob ained. In o he wo ds, he diffe en con ac s appea ing in Figu e 4 a e he ones ha assu e he highes p ofi s o he p incipal among all possible solu ions ha migh coexis in a gi en pa ame e egion. The domina ing con ac s o Case M a e lis ed in Appendix C.4, while he domina ing con ac s o Case A can be de i ed in he same way om he esul s con ained in Appendix D. 20 Inse Figu e 4 a ound he e As o exclusion, Figu e 4 shows ha he occu ence o equilib ia wi h exclusion is eally limi ed and essen ially elega ed o small egions lying in-be ween ully pa icipa ing and ully sepa a ing solu ions o Case A.1and Case A.2and in-be ween ully pa icipa ing and ully sepa a ing solu ions o Case A.2 and Case A.3. Conce ning pooling con ac s, no e ha , when mo i a ion akes he lowes possible alues ( ha is below Case A.1wi h ull sepa a ion and ull pa icipa ion), hen a pooling equilib ium whe e he low- abili y ypes HH and HL a e gi en he same con ac eme ges (see Appendix D.1.2). A he o he ex eme, o he highes possible alues o mo i a ion ( ha is abo e Case Mwi h ull sepa a ion and ull pa icipa ion), a pooling equilib ium whe e he non-mo i a ed ypes HL and LL a e gi en he same con ac is a ained (see Appendix C.2). Mo e impo an ly, he e is a wide ange o si ua ions in which a solu ion wi h bunching o in e media e ypes HH and LL is a ained, especially when mo i a ion and abili y ha e a simila impac on effo p o ision, ha is o alues o γclose o γ ∗ . In a pooling equilib ium wi h e LL =e HH =e p , we mus dis inguish wo sub-cases: he fi s one whe e he binding incen i e cons ain is IC HH sHL , which is ele an when e p +e HL ≥ 2γ ∆θ (see Appendix D.4.1) and he second one whe e he binding incen i e cons ain is IC LL sHL , occu ing when e p +e HL ≤ 2γ ∆θ (see Appendix D.4.2). When sepa a ion o ypes LL and HH becomes impossible, Case A.1 u ns o he pooling equilib ium wi h IC HH sHL binding, whe eas Case M, Case A.2and Case A.3all u n o he pooling equilib ium wi h IC LL sHL binding. In e es ingly, Figu e 4 clea ly shows ha all op imal con ac s in ol ing sepa a ion o in e media e ypes HH and LL (including no only ull pa icipa ion and ull sepa a ion bu possibly exclusion o 20 Mo eo e , no e ha o some pa ame e configu a ions, some op imal con ac s migh no exis . Appendix E conside s he possible equilib ia a ising in he pa icula case in which he p obabili y dis ibu ion o ypes is uni o m. 19 bunching o o he ypes) unde Case Mhold when γ > γ ∗ while all op imal con ac s in ol ing sepa a ion o in e media e ypes HH and LL (again, including no only ull pa icipa ion and ull sepa a ion bu possibly exclusion o bunching o o he ypes) unde Case Aa ain when γ < γ ∗ . The e o e, when effo le els a e aligned in a gi en way a he fi s -bes , hen he same o de ing o effo le els a ises a he second-bes . P oposi ion 2 The anking o second-bes effo le els is always he same as he fi s -bes anking. P oo . Looking a Figu e 4, in o de o p o e he abo e P oposi ion i is sufficien o show ha : (i) he in e al o mo i a ion le els o which he e exis s a solu ion wi h ull pa icipa ion and ull sepa a ion unde Case Mlies o he igh o γ ∗ ( his is done in Appendix C.1); (ii) he in e al o mo i a ion le els o which he e exis s a solu ion wi h ull pa icipa ion and ull sepa a ion unde Case A.3lies o he le o γ ∗ (and his is done in Appendix D.3.1). 5 Op imal con ac s wi h ull sepa a ion and ull pa icipa ion In ligh o P oposi ion 1, in his Sec ion we will ocus on he cha ac e iza ion o con ac s wi h ull pa icipa ion and ull sepa a ion o ypes, elega ing o he Appendix he analysis o all si ua ions wi h pooling and/o exclusion o ypes. Ins ead o p esen ing all ou possible classes o solu ions, we will es ic a en ion o he ones ha a ise when mo i a ion is sufficien ly high, namely Cases Mand A.3. 21 The a ionale behind such a choice is many old: (i)since ou main con ibu ion is in s udying he impac o in insic mo i a ion on op imal sc eening con ac s ha a p incipal migh offe , i is easonable o ocus on si ua ions whe e mo i a ion is significan ; (ii)Cases Mand A.3a e almos exhaus i e because hey belong o he wo diffe en s a es o he wo ld whe e mo i a ion p e ails o abili y p e ails, espec i ely: (iii)Cases Mand A.3a e close o each o he in e ms o exis ence condi ions (see Figu es 3 and 4) and he e o e hey a e mo e easily compa able han dis an cases; (i )finally and mo e in e es ingly, con as ing he wo si ua ions, one can show ha e y high mo i a ion migh no be a desi able wo ke ’s cha ac e is ic o he p incipal. To imp o e eadabili y, in he ex we mainly p o ide a quali a i e desc ip ion o he wo diffe en solu ions wi h economic in ui ions; we e e he eade o Appendices C.1 and D.3.1 o quan i a i e esul s, echnical s a emen s conce ning exis ence condi ions and p oo s conce ning Cases Mand A.3, espec i ely. 21 The analysis o Cases A.1and A.2will be ca ied ou in Appendices D.1.1 and D.2.1, espec i ely. 20 5.1 The solu ion when mo i a ion p e ails (Case M) In Case M, a sepa a ing equilib ium occu s i condi ion (7) holds, ha is i e HH > e LL and e LL +e HH ≤ 2γ ∆θ bo h hold. The implemen abili y condi ion (6) hus becomes e LH > e HH > e LL > e HL .The cons ain s ha a e expec ed o bind a he op imum wi h ull pa icipa ion a e he downwa d local incen i e cons ain s IC LH sHH ,IC HH sLL ,IC LL sHL and PC HL (see Obse a ion 1 and Figu e 1). In his si ua ion, mo i a ion γis high enough o ype HH o be asked o p o ide a ela i e high effo in exchange o a sala y ha is qui e low (in ac w HH may also be lowe han he sala y offe ed o wo ke LL, as he inequali y in Rema k 6 which ollows poin s ou ). Thus, ype HH migh find he con ac (e LL , w LL )appealing. No e ha Case M( oge he wi h Case A.1) co esponds o he si ua ion whe e ou bidimensional sc eening p oblem is equi alen o a unidimensional sc eening one wi h ou ypes, he unidimensional pa ame e o p i a e in o ma ion being he wo ke s’ o e all cos o effo exe ion. 22 Gi en he binding cons ain s, we can sol e o he wage schedules, which allow us o isola e he in o ma ion en s ecei ed by each ype o wo ke w HL =1 2θe 2 HL ,(9) w LL =1 2e 2 LL +1 2∆θe 2 HL    In o en wo ke LL ,(10) w HH =1 2θe 2 HH −γe HH −1 2∆θe 2 LL +γe LL +1 2∆θe 2 HL    In o en wo ke HH (11) and finally w LH =1 2e 2 LH −γe LH +1 2∆θe 2 HH −1 2∆θe 2 LL +γe LL +1 2∆θe 2 HL    In o en wo ke LH (12) All ypes excep HL ecei e an in o ma ion en and in o ma ion en s cumula e when mo ing om he wo s ype HL up o he bes ype LH. Since in o ma ion en s a e always inc easing in he effo exe ed by he ypes ha can be mimicked, we ob ain he amilia esul s o no dis o ion a he op and downwa d dis o ions in effo le els o all agen s’ ypes o he han LH. Mo eo e , all in o ma ion en s include a leas one exp ession o he o m 1 2 ∆θe 2 ij ,as in Benchma k BA. Only mo i a ed ypes HH and LH ecei e an in o ma ion en depending also on mo i a ion γ, which comes om he incen i e cons ain IC HH sLL linking he wo unidimensional sc eening p oblems o Benchma k BA. In pa icula , 22 No ice ha , when mo i a ion p e ails o e abili y, we ob ain a sc eening p oblem which is equi alen o he one o benchma k BA, whe e he unidimensional p i a e in o ma ion is wo ke s’ abili y. The ac ha he e he ele an incen i e cons ain s a e he ones o benchma k BA migh seem coun e in ui e. None heless, when mo i a ion p e ails, he cons ain s ha bind fi s a e p ecisely he ones connec ing ypes cha ac e ized by he same mo i a ion bu diffe en abili y (see Figu e 1a). 21 he en ecei ed by ype HH when mimicking LL is gi en by − 1 2 ∆θe 2 LL +γe LL which is always posi i e and inc easing in e LL when mo i a ion p e ails. Subs i u ing he wage schedules in o he p incipal’s objec i e unc ion and maximizing yields he op imal effo le els e SBM ij , whose exp essions a e p o ided in Appendix C.1. In e es ingly, e SBM HH has he same exp ession as in Benchma k BA. None heless, effo le els equi ed om wo ke s LL and HL a e mo e dis o ed han hose in p og am BA, because o he cumula i e effec o in o ma ion en s s emming om he bidimensional na u e o ad e se selec ion. Gi en he op imal effo s, he op imal wage le els and he in o ma ional en s (i.e. indi ec u ili ies) can be anked as ollows. Rema k 6 When mo i a ion p e ails (Case M), a he op imal con ac wi h ull sepa a ion and ull pa icipa ion, he anking o wages is w SBM LH >max w SBM HH , w SBM LL ≥min w SBM HH , w SBM LL > w SBM HL >0 and he o de ing o in o ma ion en s (indi ec u ili ies) is u SBM LH > u SBM HH > u SBM LL > u SBM HL = 0. In o ma ion en s ha e he same o de ing as effo le els, while he e can be a e e sal in he anking o sala ies o in e media e ypes. In o he wo ds, he p incipal could offe he mo i a ed bu high-cos ype HH a con ac in which effo p o ision is highe and emune a ion is lowe han in he con ac p oposed o ype LL. None heless, ype HH always enjoys a highe u ili y han ype LL, because he high mo i a ion gene a es highe in o ma ional en s. This esul is no i ial and depends on he peculia i y o mo i a ed wo ke s’ u ili y unc ion, which admi s olun a y wo k. 23 Mo eo e , when w SBM HH < w SBM LL holds, hen i is always he case ha e SBM HH < 2γ θ ,implying ha mo i a ed high-cos ypes HH enjoy a ne posi i e u ili y om effo p o ision. This has a nega i e impac on o al sala y w HH (see he fi s wo e ms in exp ession 11), which is mo e han compensa ed by he posi i e effec o in o ma ion en s. Co olla y 1 When mo i a ion p e ails (Case M), a he op imal con ac wi h ull sepa a ion and ull pa icipa ion, wo ke HH migh be a “paid olun ee ”: she is offe ed a posi i e wage, bu she enjoys a posi i e u ili y om effo exe ion. Ac ually, i is also possible (al hough e y a e) ha e SBM HH < γ θ ,in which case effo equi ed o ype HH is in he ange in which he u ili y inc eases in effo and he indiffe ence cu e is downwa d sloping in 23 In pa icula , w SBM HH < w SBM LL holds when he p obabili y o high mo i a ion µis low ela i e o he p obabili y o low effo cos ν, when he diffe ence in abili y ∆θis high and when he le el o mo i a ion is high oo. 22 he space (e, w). Then ype HH would be willing o exe mo e effo han he op imal con ac specifies and i would be necessa y o he p incipal o o bid his ype o engage in olun a y o e ime. 24 , 25 Finally, he in e al o mo i a ion le els o which ull pa icipa ion and ull sepa a ion is a ained in Case Malways lies o he igh o γ ∗ .Hence Case M ealizes a he second-bes when condi ion (2) holds a he fi s -bes : his confi ms he esul an icipa ed in P oposi ion 2, namely ha effo le els ha e he same anking bo h a fi s - and a he second-bes . 5.2 The solu ion when abili y p e ails (Case A) When abili y p e ails, ull sepa a ion occu s unde condi ion (8), ha is i e LL > e HH and e LL +e HH ≥ 2γ ∆θ bo h hold. The implemen abili y condi ion (6) hus becomes e LH > e LL > e HH > e HL . Th ee diffe en solu ions cha ac e ized by ull sepa a ion and ull pa icipa ion o ypes migh be ound acco ding o which incen i e compa ibili y cons ain s a e binding oge he wi h cons ain s IC LH sLL and PC HL (see Figu e 3). In wha ollows, we concen a e on Case A.3. 5.2.1 Case A.3 Suppose ha he binding incen i e cons ain s a e IC LH sLL ,IC LL sHL and he upwa d incen i e con- s ain IC HH sLL , oge he wi h pa icipa ion cons ain PC HL (see Figu e 2c). This esul s in inequali y e HL +e LL ≤ 2γ ∆θ ≤e HH +e LL . This p og am b idges Cases Aand Case M. Indeed, he incen i e con- s ain ha is sha ed wi h he o he cases in which abili y p e ails is IC LH sLL ,whe eas he o he wo binding cons ain s a e IC HH sLL and IC LL sHL as in Case M. The ele an wage le els and in o ma ional en s a e now w HL =1 2θe 2 HL ,(13) w HH =1 2θe 2 HH −γe HH −1 2∆θe 2 LL +γe LL +1 2∆θe 2 HL    In o en wo ke HH ,(14) w LL =1 2e 2 LL +1 2∆θe 2 HL    In o en wo ke LL (15) and w LH =1 2e 2 LH −γe LH +γe LL +1 2∆θe 2 HL    In o en wo ke LH .(16) 24 The same conclusion also holds when he op imal con ac equi es pooling be ween ypes HH and LL (see Appendix D.4.2). Con e sely, no such ins ance occu s when abili y p e ails. 25 None heless, being effo pe ec ly obse able and con ac ible, o bidding olun a y o e ime does no in alida e ou esul s. 23 whe e γ SBM <1is always he case o (3µν −ν−µ)≥0, ha is o ν > 1 3 and µ≥ ν (3ν−1) ,whe eas, o (3µν −ν−µ)<0,inequali y γ SBM <1is ue when θ < µ+ν−3µν + (1 −ν) (1 −(1 −ν) (1 −µ)) µ+ν−3µν =θ M 2 wi h θ M 2 > θ M 1 i and only i µ > µ 0 (wi h µ 0 < 1 2 ).Mo eo e , e SBM HL < e SBM LL holds o γ < (1 −µ) (1 −(1 −ν) (1 −µ)) ∆θ µ(∆θ+ (1 −µ) (1 −ν)) =γ SBM , wi h γ SBM <1being always he case o µ≥µ 0 . Recall ha condi ion (7), which amoun s o e SBM LL +e SBM HH ≤ 2γ ∆θ ,mus be sa isfied and his is equi alen o γ≥∆θ(2ν(1 −µ) (1 −ν) + (ν−µ) ∆θ) 2ν(1 −ν) (1 −µ) + ∆θν (2 −µ(θ+ 1)) −∆θ(1 −ν) (1 −(1 −ν) (1 −µ)) =γ SBM 1 , whe e γ SBM 1 < γ SBM i and only i θ < θ M 1 . Finally, no e ha he chain o inequali ies γ SBM 1 < γ ∗ < γ SBM < γ SBM < γ 0 holds p o ided ha he denomina o o e SBM LL is posi i e, ha is p o ided ha θ < θ M 1 . Resul 1 summa izes wha we ha e ound so a . Resul 1 Full pa icipa ion and ull sepa a ion when mo i a ion p e ails. A solu ion o he p incipal’s p og am SB which en ails ull pa icipa ion and ull sepa a ion o ypes, which sa isfies he mono onici y condi ion e LH > e HH > e LL > e HL >0,and which is such ha effo le els a e gi en by exp essions om (17) o (20) exis s i and only i θ < min θ M 1 , θ M 2 and γ SBM < γ < γ SBM wi h γ SBM ≡ (µ(1−ν)+ν(1−µ))∆θ (νµ∆θ+(1−ν)(1−(1−ν)(1−µ))) γ SBM ≡ (1−µ)(1−(1−ν)(1−µ))∆θ µ(θ−(1−(1−ν)(1−µ))) θ M 1 ≡ (1−(1−ν)(1−µ)) µ θ M 2 ≡ ((µ+ν−3µν)+(1−ν)(1−(1−ν)(1−µ))) (µ+ν−3µν) . In e es ingly, bo h γ ∗ < γ SBM and min θ M 1 , θ M 2 <2hold, so ha he alignmen o second-bes effo le els wi h he anking ob ained a fi s -bes unde condi ion (4) necessa ily holds. C.2 Pooling and exclusion The p incipal can also eso o con ac s in ol ing pooling o ypes and e en ually exclusion o some wo ke s’ ypes. Obse e ha he ully sepa a ing and ully pa icipa ing solu ion is cha ac e ized by he lowe bound γ SBM ,which comes om he condi ion e SBM HH > e SBM LL .The e o e, i γ≤γ SBM , he p incipal is o ced o offe he same con ac o bo h ypes HH and LL. Likewise, he ully sepa a ing and ully pa icipa ing 30 solu ion is cha ac e ized by he uppe bound γ SBM ,which co esponds o e SBM LL > e SBM HL . And i γ≥γ SBM we always expec a pooling con ac whe e ypes HL and LL ecei e he same offe . We e e he eade o Appendix D.4.2 o he de ailed analysis o he fi s si ua ion, while we conside he second one in wha ollows. Suppose ha he e’s pooling be ween non mo i a ed ypes and ha e LL =e HL =e p .Then he o de ing o effo le els is e LH > e HH > e LL =e HL =e p and he ele an downwa d incen i e cons ain s ha one expec s o be binding a e IC LH sHH and IC HH sLL (o IC HH sHL , which is equi alen ) oge he wi h pa icipa ion cons ain PC HL . Since he e wo ke ypes LL and HL ecei e he same wage and p o ide he same effo , u LL > u HL necessa ily holds. The wages a e w LL =w HL =w p =1 2θe 2 p ,(22) w HH =1 2θe 2 HH −γe HH +γe p  In o en wo ke HH and w LH =1 2e 2 LH −γe LH +1 2∆θe 2 HH +γe p    In o- en wo ke LH . Subs i u ing he wages in o he objec i e unc ion o he p incipal and maximizing yields e LH =e SBM LH =e F B LH = 1 + γ, e HH =e SBM HH =(1 −ν) (1 + γ) (θ−ν) and e LL =e HL =e SBM p =(1 −µ)−µγ (1 −µ)θ=e BM HL (23) No e ha he exp essions o e LH and e HH a e he same as in Case M, meaning ha no dis o ion a he op is e ified and ha he effo o indi idual HH is lowe han he co esponding fi s -bes le el. Mo eo e , e LH > e HH s ill holds. Conce ning e SBM p ,i is he same as in Benchma k BM, i is s ic ly posi i e o γ < γ BM and such ha e HH > e SBM p holds i and only i γ > ν(1 −µ) ∆θ θ(1 −ν) + µν∆θ=γ p whe e γ p < γ SBM always holds. We a e hen able o s a e he ollowing esul . Resul 2 Full pa icipa ion and pooling be ween ypes LL and HL when mo i a ion p e ails. The solu ion o he p incipal’s p og am SB which en ails ull pa icipa ion and pooling be ween ypes LL and HL, which sa isfies he mono onici y condi ion e LH > e HH > e LL =e HL >0,and which is such 31 ha effo le els a e gi en by exp essions (17), (18) and (23) exis s i and only i γ p < γ < min γ BM ,1 wi h γ p ≡ ν(1−µ)∆θ θ(1−ν)+µν∆θ γ BM ≡ (1−µ) µ being γ BM <1 o µ > 1 2 . The condi ions o exis ence o an equilib ium wi h ull pa icipa ion and pooling o wo ke s HL and LL a e less s ingen han he ones we ob ained in Resul 1 because he equi emen e SBM HL < e SBM LL is no longe ele an . Also, he pooled effo e SBM p is always in-be ween exp essions (19) and (20): in pa icula , e SBM HL > e SBM p > e SBM LL holds i and only i γ > γ SBM . No e ha γ BM ≥1i and only i µ≤ 1 2 , he e o e he p incipal always p oposes a pooling con ac o ypes LL and HL when mo i a ion is sufficien ly high (i.e. o γ≥γ SBM ) and he p obabili y o being mo i a ed is sufficien ly low (i.e. o µ≤ 1 2 ). Con e sely, when µ > 1 2 and γ BM <1, hen o γ≥γ BM he p incipal is expec ed o exclude ype HL since he p obabili y o mo i a ed ypes is high and he p oduc i i y loss om ype HL is low. As o exclusion, he necessa y and sufficien condi ion o ull pa icipa ion equi es in gene al ha , o any ype ij, he expec ed p ofi om employing ype ij be highe han he expec ed in o ma ion en s ha ha e o be paid o he mimicke s; his condi ion is sa isfied as long as ype ij’s effo is s ic ly posi i e. Howe e , he condi ion e ij >0migh call o some es ic ions on he pa ame e space, as occu ed in Benchma k BM (see oo no e 16). In o de o de i e he condi ions o exis ence and o cha ac e ize he con ac wi h exclusion o ype HL, we p oceed as in he case wi h ull pa icipa ion, bu we ob iously d op wo ke HL om he p incipal’s maximiza ion p og am SB and omi he mono onici y condi ion e LL > e HL .Since he uppe bound γ SBM o he exis ence ange o an equilib ium wi h ull pa icipa ion comes p ecisely om he cons ain e SBM LL > e SBM HL , he ange o he exis ence o a sepa a ing equilib ium wi h exclusion o HL is b oade on he igh side wi h espec o he in e al γ SBM , γ SBM .Mo eo e , he op imal effo le els o he emaining ypes a e gi en by he same exp essions om (17) o (19), e en wi h exclusion. Ins ead, he op imal wages o he emaining ypes will be lowe han exp essions om (10) o (12), since he po ions o he h ee in o ma ion en s ha depend on e HL disappea . Resul 3 Exclusion o ype HL when mo i a ion p e ails. The solu ion o he p incipal’s p og am SB, which en ails sepa a ion and exclusion o ype HL, which sa isfies he mono onici y condi ion e LH > e HH > e LL > e HL = 0 and which is such ha effo le els a e gi en by exp essions om (17) o (19), exis s i and only i θ < min θ M 1 , θ M 2 and γ SBM < γ < γ 0 ≡ ν(1−µ) µ . Up o now we ha e iden ified all possible classes o solu ions o he p incipal’s p og am SB, when mo i a ion p e ails, and hei co esponding (possibly o e lapping) exis ence egions. Ac ually, he e s ill 32 emains o cha ac e ize he solu ion when he e is bunching be ween in e media e ypes HH and LL and we e e he eade o Appendix D.4.2 o he analysis o such a si ua ion. In o de o single ou he op imal con ac chosen by he p incipal, we s ill ha e one s ep o go: when diffe en solu ions coexis , we mus pick he one ha yields he highes p ofi s o he fi m and disca d he o he s. This is p ecisely wha we do nex . C.3 P oo o P oposi ion 1 We wan o show ha he solu ion en ailing ull pa icipa ion and ull sepa a ion o ypes domina es (meaning ha i p o ides highe p ofi s o he p incipal) bo h ull sepa a ion bu exclusion o a leas wo ke HL and ull pa icipa ion bu pooling o wo wo ke s’ ype. Mo eo e , we p o e ha ull pa ici- pa ion and pooling o wo diffe en ypes domina es ull sepa a ion and exclusion o (a leas ) wo ke HL, whene e he wo solu ions coexis . We conside he si ua ion in which mo i a ion p e ails o e abili y (Case M); he same line o easoning applies o Case Aas well, which is he e o e omi ed. S a wi h he compa ison be ween ull pa icipa ion and ull sepa a ion o ypes and exclusion o a leas wo ke HL. We mus e alua e he cos s and benefi s om pa icipa ion o he wo s wo ke ype HL. The p incipal’s benefi om employing wo ke HL is he expec ed p ofi (1 −µ) (1 −ν) (e HL −w HL ),(24) whe eas he cos om pa icipa ion o HL is ep esen ed by he in o ma ion en s paid o he h ee emaining wo ke s’ ypes, which add up o 1 2(1 −(1 −µ) (1 −ν)) ∆θe 2 HL (25) Thus, he p incipal p e e s ull pa icipa ion o exclusion o ype HL i and only i (24) is s ic ly g ea e han (25). Taking in o accoun exp ession (9) o he wage w HL in Case M, he inequali y educes o 2e SBM HL > e SBM HL , which is ob iously sa isfied as long as e SBM HL >0.Al e na i ely, subs i u ing bo h exp ession (9) o he wage and exp ession (20) o e HL in (24) and (25), we ob ain ha ull sepa a ion and ull pa icipa ion domina es ull sepa a ion and exclusion o ype HL i and only i θ−(1 −(1 −µ) (1 −ν)) >0.This is he denomina o o e HL in exp ession (20). The p e ious inequali y says ha he effo cos o a low abili y ype θmus be la ge han he join p obabili y o hi ing ypes ha a e o e all mo e efficien han HL. This condi ion always holds in ou se ing, gi en ha θ > 1. Simila conclusions can be d awn conside ing exclusion o bo h wo ke s HL and LL. Conside now he compa ison be ween ull sepa a ion and ull pa icipa ion o ypes and ull pa ic- ipa ion bu pooling o wo ke s HH and LL (as said, his solu ion is conside ed in de ail in Appendix D.4.2 bu we an icipa e some findings he e o exposi ional con enience). Now he ade-off be ween cos s and benefi s om ull sepa a ion becomes less clea , so le us eso di ec ly o he compa ison be ween 33 he p incipal’s p ofi s unde he wo solu ions. The p incipal’s payoffs unde ull sepa a ion and ull pa icipa ion o ypes a e π SBM F S,F P = 1 2 νµ (1 + γ) 2 +µ (1−ν) 2 (1+γ) 2 θ−ν + (ν(1−µ)−µγ) 2 (1−(1−ν)(1−µ))−µθ + (1−ν) 2 (1−µ) 2 θ−(1−(1−ν)(1−µ))  while, unde ull pa icipa ion bu pooling o wo ke s HH and LL, p ofi s amoun o π SBM F P,HH=LL = 1 2 νµ (1 + γ) 2 + (ν(1−µ)+µ(1−ν)−γµν) 2 ν(1−µ)+µ(1−ν) + (1−ν) 2 (1−µ) 2 θ−(1−(1−ν)(1−µ))  I is immedia e o check ha π SBM F S,F P > π SBM F P,HH=LL always holds. Conside now he compa ison be ween ull sepa a ion and ull pa icipa ion o ypes and ull pa - icipa ion bu pooling o wo ke s HL and LL (see Appendix C.2). The p incipal’s payoffs unde ull pa icipa ion bu pooling o wo ke s HL and LL a e gi en by π SBM F P,HL=LL = 1 2 νµ (1 + γ) 2 + µ(1−ν) 2 (1+γ) 2 (θ−ν) + ((1−µ)−µγ) 2 θ(1−µ)  and, again, i is s aigh o wa d o check ha π SBM F S,F P > π SBM F P,HL=LL always holds. Finally, conside he compa ison be ween ull pa icipa ion bu pooling o wo ke s HL and LL and ull sepa a ion bu exclusion o wo ke HL. Since bo h solu ions a e domina ed by ull sepa a ion and ull pa icipa ion, hey can be candida e op imal con ac s only abo e γ SBM . The p incipal’s p ofi s a he la e solu ion a e π SBM F S,HL=0 = 1 2 νµ (1 + γ) 2 + µ(1−ν) 2 (1+γ) 2 (θ−ν) + (ν(1−µ)−µγ) 2 ν(1−µ)−µ(θ−1)  and π SBM F P,HL=LL > π SBM F S,HL=0 i and only i ((1 −µ)−µγ)e SBM p >(ν(1 −µ)−µγ)e SBM LL . The abo e inequali y is always e ified since ((1 −µ)−µγ)>(ν(1 −µ)−µγ)always holds and e SBM p > e SBM LL is ue abo e γ SBM . No e ha he compa ison be ween ull pa icipa ion bu pooling o wo ke s HH and LL and ull sepa a ion bu exclusion o wo ke HL is meaningless because, below γ SBM , i is ne e easible o sepa a e ypes HH and LL. So we a e done. C.4 Op imal con ac s when mo i a ion p e ails Conside ing P oposi ion 1, Resul s om 1 o 3 and Resul 13 in Appendix D.4.2, i is now possible o cha ac e ize he op imal con ac s when mo i a ion p e ails. Resul 4 When mo i a ion p e ails, he op imal con ac s p oposed by he p incipal a e as ollows: (i)Full pa icipa ion and pooling be ween ypes HH and LL and IC LL sLH binding (cha ac e ized in 34 Resul 13) is implemen ed i and only i θ < θ M 1 and γ ∗ ≤γ < γ SBM . (ii)Full pa icipa ion and ull sepa a ion o ypes (cha ac e ized in Resul 1) is implemen ed i and only i θ < min θ M 1 , θ M 2 and γ SBM ≤γ≤γ SBM . (iii)Full pa icipa ion and pooling be ween ypes LL and HL (cha ac e ized in Resul 2) is implemen ed i and only i θ < θ M 1 and γ SBM ≤γ≤min γ BM ,1. (i )Full sepa a ion and exclusion o ype HL (cha ac e ized in Resul 3) is implemen ed i and only i µ > 1 2 , θ < θ M 1 and γ BM < γ ≤1. D Abili y p e ails (Case A) When abili y p e ails, condi ion (8) holds and e LL > e HH oge he wi h e LL +e HH ≥ 2γ ∆θ mus be sa isfied. In line wi h Obse a ion 2, in o de o find a ully sepa a ing and ully pa icipa ing solu- ion o he p incipal’s p oblem SB ake he pa icipa ion cons ain PC HL and he ollowing incen i e cons ain s: IC LH sLL ,IC LL sHH o e en ually IC LL sHL (whiche e one binds fi s ), IC HH sHL o IC HH sLL (again whiche e one binds fi s ). No e ha all incen i e compa ibili y cons ain s conside ed a e downwa d cons ain s excep o IC HH sLL which poin s upwa ds. Since IC LL sHH and IC HH sLL canno be simul aneously binding a a sepa a ing equilib ium, hen he possible si ua ions a e he ol- lowing: (A.1) all downwa d local ICs a e binding and hus IC LH sLL ,IC LL sHH and IC HH sHL hold wi h equali y, as shown in Figu e 2a; (A.2) he downwa d local cons ain s IC LH sLL and IC HH sHL and he global downwa d cons ain IC LL sHL a e all binding, as shown in Figu e 2b; (A.3) cons ain s IC LH sLL ,IC LL sHL and he upwa d IC HH sLL hold wi h equali y, as shown in Figu e 2c. Such h ee possible cases will be analyzed in de ail in wha ollows. D.1 Case A.1 D.1.1 Full sepa a ion and ull pa icipa ion This ep esen s he mos in ui i e case whe e all downwa d local ICs a e binding and occu s when γis sufficien ly low so ha wo ke HH ecei es a ela i ely high sala y in exchange o a ela i ely low effo , and such a con ac is a ac ing o ype LL. Sol ing he binding cons ain s o sala ies, one ob ains he ollowing wage schedules and in o ma ional en s w HL =1 2θe 2 HL ,(26) w HH =1 2θe 2 HH −γe HH +γe HL    In o en wo ke HH ,(27) 35 w LL =1 2e 2 LL +1 2∆θe 2 HH −γe HH +γe HL    In o en wo ke LL (28) and w LH =1 2e 2 LH −γe LH +γe LL +1 2∆θe 2 HH −γe HH +γe HL    In o en wo ke LH .(29) All in o ma ion en s, excep he one o ype HL, a e posi i e and ha e he usual cumula i e s uc- u e. They all include a leas one exp ession o he o m γe ij as in Benchma k BM whe e asymme ic in o ma ion conce ns mo i a ion only. Only ype LL ecei es an in o ma ion en which also depends on he diffe ence in abili y ∆θ: his comes om he ac ha his p og am embeds he wo subcases in Benchma k BM and links hem h ough cons ain IC LL sHH .Type LH cumula es his en oo when ying o mimic LL. This case is peculia because an addi ional cons ain needs o be sa isfied, which comes om ex- p ession (28). The en acc uing o ype LL when mimicking HH mus be posi i e and his occu s i and only i e HH > 2γ ∆θ (which is mo e es ic i e han condi ion 8). 28 In diffe en wo ds, only when γis sufficien ly low, does ype LL benefi om mimicking ype HH. O he wise, ype LL will a he p e e o mimic ype HL as in Case A.2and Case A.3 ha ollow. Subs i u ing he wage schedules in o he objec i e unc ion and de i ing wi h espec o effo le els we ob ain e SBA1 LH = 1 + γ(30) e SBA1 LL =(1 −µ)−µγ (1 −µ)=e BM LL ,(31) e SBA1 HH =(1 −ν)µ+ (1 −(1 −ν) (1 −µ)) γ (1 −(1 −ν) (1 −µ)) θ−ν(32) and e SBA1 HL =(1 −ν) (1 −µ)−(1 −(1 −ν) (1 −µ)) γ (1 −ν) (1 −µ)θ.(33) The s anda d esul o no dis o ion a he op and downwa d dis o ion in effo le els o all o he agen ’s ypes is ob ained. No e ha e SBA1 LL is equal o he effo le el we ob ained o ype LL in Benchma k BM. Ins ead, he effo le els equi ed om he less p oduc i e wo ke s ( ypes HH and HL)a e cha ac e ized by a la ge downwa d dis o ion han in p og am BM because o he cumula i e effec o in o ma ional en s. Also obse e ha e SBA1 LH and e SBA1 HH a e s ic ly posi i e, while e SBA1 LL >0i and only i γ < γ BM , and e SBA1 HL >0i and only i γ < (1 −ν) (1 −µ) (1 −(1 −ν) (1 −µ)) =γ SBA1 1 . 28 No e ha condi ion e HH > 2γ ∆θ implies condi ion e HH > 2γ θ .Hence i LL ecei es a posi i e in o ma ion en when mimicking HH, hen i mus be ha ype HH is no a olun ee and ha she is expe iencing a ne cos om p o iding effo . 36 Ac ually, e SBA1 LL >0always holds when µ≤ 1 2 o when e SBA1 HL is s ic ly posi i e, since e SBA1 HL >0implies e SBA1 LL >0(being γ BM > γ SBA1 1 ). As o he mono onici y condi ions, i can be checked ha e SBA1 LH > e SBA1 LL always holds, ha e SBA1 LL > e SBA1 HH is ue o γ < (1−µ)(1−(1−ν)(1−µ))∆θ µ(1−(1−ν)(1−µ))∆θ+(ν(1−µ)+µ(1−ν)) =γ SBA1 2 and ha inequali ies e SBA1 LH > e SBA1 HH , e SBA1 LL > e SBA1 HL and e SBA1 HH +e SBA1 LL > 2γ ∆θ all hold when γ < γ SBA1 2 . No e ha γ SBA1 2 < γ SBA1 1 i and only i θ < µ(1 −ν(1 −ν)) + ν(1 −µ) ν(1 −(1 −ν) (1 −µ)) ≡θ 3 . Finally, e SBA1 HH > e SBA1 HL o γ > ν(1 −ν) (1 −µ) ∆θ (1 −(1 −ν) (1 −µ)) (θ−ν)=γ SBA1 whe e i is always he case ha γ SBA1 <min γ SBA1 1 , γ SBA1 2 . In addi ion, i mus be ue ha e SBA1 HH > 2γ ∆θ .Such condi ion is equi alen o γ < µ(1 −ν) ∆θ ν∆θ+µ(1 −ν) (θ+ 1) =γ SBA1 3 , whe e γ SBA1 3 > γ SBA1 holds i and only i µ > ν 1+ν =µ 0 .Thus, µ > µ 0 is a necessa y condi ion ensu ing ha he wo equi emen s e SBA1 HH > e SBA1 HL and e SBA1 HH > 2γ ∆θ can bo h be me . 29 Finally, γ SBA1 3 < γ SBA1 1 i and only i θ < µ(1 + ν)−ν (2µ−1) (1 −(1 −µ) (1 −ν)) ≡θ 4 , which is always he case o µ≤ 1 2 .Obse e ha γ SBA1 3 < γ SBA1 2 i and only i µ < (1−2ν)+ √ 1+4ν(1−ν) 4(1−ν) ≡ µ 2 ,wi h µ 2 > 1 2 and ha θ 3 < θ 4 i and only i µ < µ 2 . We a e hen able o s a e he ollowing esul . Resul 5 Full pa icipa ion and ull sepa a ion when abili y p e ails and IC LL sHH and IC HH sHL a e binding. The solu ion o he p incipal’s p og am SB, which en ails ull pa icipa ion, ull sepa a ion o ypes and cons ain s IC LL sHH and IC HH sHL binding, which sa isfies he mono onici y condi ion e LH > e LL > e HH > e HL >0and which is such ha effo le els a e gi en by exp essions om (30) o (33) exis s and ep esen s he op imal con ac i and only i µ > ν 1+ν ≡µ 0 and γ SBA1 < γ < γ SBA1 wi h γ SBA1 ≡ ν(1−ν)(1−µ)∆θ (1−(1−ν)(1−µ))(θ−ν) γ SBA1 = min γ SBA1 1 , γ SBA1 2 , γ SBA1 3  29 Full pa icipa ion and ull sepa a ion in Case A.1is possible only i µ > µ 0 ,o i he p obabili y o mo i a ed wo ke s is sufficien ly high, implying ha in o ma ion en s a e no oo cos ly. 37 and γ SBA1 1 ≡ (1−ν)(1−µ) (1−(1−ν)(1−µ)) γ SBA1 2 ≡ (1−µ)(1−(1−ν)(1−µ))∆θ µ(1−(1−ν)(1−µ))∆θ+(ν(1−µ)+µ(1−ν)) γ SBA1 3 ≡ µ(1−ν)∆θ (ν∆θ+µ(1−ν)(θ+1)) . Finally no e ha γ ∗ >max γ SBA1 1 , γ SBA1 2 , γ SBA1 3 is always ue, he e o e Case A.1wi h ull pa icipa ion and ull sepa a ion is always a subse o he fi s -bes s a e o he wo ld in which condi ion (3) holds. To be e cha ac e ize he op imal con ac s in Case A.1,we add he ollowing. Rema k 10 A he op imal con ac wi h ull pa icipa ion and ull sepa a ion unde Case A.1, he o de ing o wages is w SBA1 LH > w SBA1 LL > w SBA1 HH > w SBA1 HL >0 and he o de ing o in o ma ion en s (indi ec u ili ies) is u SBA1 LH > u SBA1 LL > u SBA1 HH > u SBA1 HL = 0. Case A.1 ep esen s he unique ins ance in which wages and in o ma ion en s always ha e he same o de ing as effo le els. As in Case M, he e he bidimensional sc eening p oblem is equi alen o he unidimensional sc eening one wi h ou ypes, he unidimensional pa ame e o p i a e in o ma ion being he wo ke s’ o e all cos o effo exe ion. D.1.2 Pooling and exclusion In ligh o P oposi ion 1, he op imal con ac will always be cha ac e ized by ull pa icipa ion and ull sepa a ion o ypes, excep when he o me solu ion is no iable, in which case pooling and possibly exclusion will also be pa o he op imal con ac . Fi s o all conside pooling. Obse e ha he lowe bound γ SBA1 co esponds o condi ion e SBA1 HH > e SBA1 HL .Thus, i γ≤γ SBA1 , hen we expec a pooling equilib ium whe e ypes HH and HL ecei e he same con ac . Suppose ha he e’s pooling be ween he less p oduc i e ypes and ha e HH = e HL =e p holds.Then he o de ing o effo le els is e LH > e LL > e p >0and he ele an downwa d incen i e cons ain s ha one assumes o be binding a e IC LH sLL and IC LL sHL (o IC LL sHH , which is equi alen ) wi h pa icipa ion cons ain PC HL . Since he e he incen i e cons ain s IC LL sHH and IC LL sHL a e bo h binding by cons uc ion, we do no need any condi ion on he sum o e HH and e HL . Mo eo e , since he wo ypes o wo ke s ecei e he same wage and p o ide he same effo , u HH > u HL necessa ily holds. The wages a e w HH =w HL =w p =1 2θe 2 p , 38 w LL =1 2e 2 LL +1 2∆θe 2 p    In o en wo ke LL and w LH =1 2e 2 LH −γe LH +γe LL +1 2∆θe 2 p    In o en wo ke LH . Subs i u ing he sala ies in o he objec i e unc ion o he p incipal and maximizing wi h espec o effo le els yields e SBA1 LH = 1 + γ, e SBA1 LL =(1 −µ)−µγ (1 −µ)=e BM LL and e HH =e HL =e SBA1 p =(1 −ν) (θ−ν)=e BA HL (34) No e ha he exp essions o e LH and e LL a e he same as in Case A.1(and A.2 ha ollows) wi h ull sepa a ion, meaning ha no dis o ion a he op is e ified and ha he effo o indi idual LL is lowe han he co esponding fi s -bes le el. Mo eo e , e LH > e LL s ill holds. Conce ning e SBA1 p ,which is s ic ly posi i e, we expec ha his effo lies in-be ween he effo exe ed by ypes HH and HL in Case A.1wi h ull sepa a ion. One can easily check ha e SBA1 HH < e SBA1 p < e SBA1 HL i and only i γ < γ SBA1 . Finally, e SBA1 LL > e SBA1 p i and only i γ < (1 −µ) ∆θ µ(θ−ν)=γ p whe e γ p > γ SBA1 always holds. Now conside he uppe bounds ( ecall ha condi ion γ < γ SBA1 1 is equi alen o e SBA1 HL >0, ha inequali y γ < γ SBA1 2 is equi alen o e SBA1 LL > e SBA1 HH and finally ha γ < γ SBA1 3 ensu es ha equi emen e SBA1 HH > 2γ ∆θ holds): i γ≥γ SBA1 ,we expec an equilib ium in which ei he ypes HH and LL a e pooled oge he o exclusion occu s o bo h. 30 Resul 6 (i)Full pa icipa ion and pooling be ween ypes HH and HL when abili y p e ails. The solu ion o he p incipal’s p og am SB which en ails ull pa icipa ion, pooling be ween ypes HH and HL, which sa isfies he mono onici y condi ion e LH > e LL > e HH =e HL =e p >0and which is such ha effo le els a e gi en by exp essions (30), (31) and (34) exis s i and only i 0< γ ≤γ p ≡ (1−µ)∆θ µ(θ−ν) and ep esen s he op imal con ac when 0< γ ≤γ SBA1 . (ii)Full pa icipa ion and pooling be ween ypes HH and LL when abili y p e ails. The solu ion o he p incipal’s p og am SB which en ails ull pa icipa ion, pooling be ween ypes HH and LL 30 We e e he eade o Appendix D.4.1 o he de ailed analysis o his si ua ion. 39 by exp essions (40), (43) and (39) ep esen s he op imal con ac when γ SBA3 ≤γ≤γ ∗ and when γ SBP b ≤γ≤γ SBA3 . Below γ SBA3 one also finds pooling be ween ypes HH and LL and exclusion o ype HL and (pos- sibly) a solu ion wi h sepa a ion bu exclusion o ype HL. In e es ingly, in he la e case, i is possible o ha e an upwa d dis o ion o he effo equi ed o ype LL, bu no so impo an as o allow o a pooling equilib ium whe e ypes LH and LL a e gi en he same con ac . Suppose ha ype HL is le ou . In his ci cums ance, he op imal le els o effo a e he same as unde ull pa icipa ion, excep o e HL = 0,and all ele an cons ain s a e sa isfied whene e he chain o inequali ies e LL ≤ 2γ ∆θ ≤e LL +e HH holds. Now, 2γ ∆θ ≤e SBA3 LL +e SBA3 HH is always sa isfied when γ < γ SBA3 ,whe eas e SBA3 LL ≤ 2γ ∆θ is ue i and only i γ≥ν(1 −µ) ∆θ (2ν(1 −µ)−µ∆θ(1 −2ν)) =γ SBA3 whe e γ SBA3 < γ SBA3 always holds and whe e γ SBA3 > γ SBA3 1 i and only i ν > 1 2 .Hence a solu ion wi h exclusion o ype HL unde Case A.3exis s o γ SBA3 ≤γ < γ SBA3 and θ < θ A3 .Obse e ha , when ν < 1 2 and γ SBA3 ≤γ < γ SBA3 1 , he solu ion en ails an upwa d dis o ion in he le el o effo p o ided by ype LL. Resul 11 (i)Pooling be ween HH and LL and exclusion o ype HL when abili y p e ails and PC LL is binding. A solu ion o he p incipal’s p og am SB wi h pooling be ween ypes LL and HH and exclusion o ype HL, wi h PC LL binding and wi h effo le els desc ibed by exp essions (40) and (39) ep esen s he op imal con ac when γ SBP b < γ < min γ SBA3 , γ SBP b ,whe e γ SBA3 ≡ ν(1−µ)∆θ (2ν(1−µ)−µ∆θ(1−2ν)) γ SBP b ≡ ∆θ(ν(1−µ)+µ(1−ν)) (νµ∆θ+2(ν(1−µ)+µ(1−ν))) . (ii)Sepa a ion and exclusion o ype HL when abili y p e ails and IC HH sLL and PC LL a e binding. A solu ion o he p incipal’s p og am SB wi h exclusion o ype HL and IC HH sLL and PC LL binding and wi h effo le els desc ibed by exp essions om (40) o (42) ep esen s he op imal con ac only i γ SBA3 < γ SBP b and γ SBA3 ≤γ < γ SBP b . Resul 11(i)desc ibes p ecisely he same pooling equilib ium ob ained in Case Mand Case A.2. 46 D.3.3 P oo o Rema k 9 Conside he con ac s wi h ull sepa a ion and ull pa icipa ion in Cases Mand A.3and le ∆θbe he same in he wo si ua ions. 32 Expec ed in o ma ional en s paid by he p incipal in Case Ma e highe han in Case A.3i and only i µνu M LH +µ(1 −ν)u M HH +ν(1 −µ)u M LL > µνu A3 LH +µ(1 −ν)u A3 HH +ν(1 −µ)u A3 LL , whe e u HL is omi ed om bo h sides because i is equal o ze o. A sufficien condi ion o he abo e inequali y o hold is ha u M ij ≥u A3 ij o e e y ype o wo ke ij, wi h a leas one s ic inequali y, whe e he ac ual exp essions o in o ma ional en s appea in he wages gi en by (10) o (12) o Case Mand by (14) o (16) o Case A.3. Now, u M LH > u A3 LH holds i and only i 1 2 (θ−1) e M HH  2 − 1 2 (θ−1) e M LL  2 +γe M LL + 1 2 (θ−1) e M HL  2 > γe A3 LL + 1 2 (θ−1) e A3 HL  2 . Gi en ha 1 2 (θ−1) e M HH  2 − 1 2 (θ−1) e M LL  2 is always posi i e in Case Mand ha e M HL =e A3 HL , a sufficien condi ion o he abo e inequali y o hold is simply ha e M LL > e A3 LL which is indeed he case. Mo eo e , u M HH > u A3 HH i and only i − 1 2 (θ−1) e 2 LL +γe LL + 1 2 (θ−1) e 2 HL >− 1 2 (θ−1) e 2 LL +γe LL + 1 2 (θ−1) e 2 HL . Since e M HL =e A3 HL one can simpli y he abo e inequali y as − 1 2 (θ−1) e M LL +e A3 LL e M LL −e A3 LL +γe M LL −e A3 LL >0 and, being e M LL > e A3 LL ,one can u he simply i as e M LL +e A3 LL <2γ ∆θ. Subs i u ing o he exp essions o e M LL and e A3 LL , he la e condi ion is equi alen o γ > (θ−1)(1−µ)(2µ+2ν−2θµ−3µν+θµν) (2µ+2ν−2θµ−6µν+2θµν−3µ 2 +4θµ 2 +4µ 2 ν−2θµ 2 ν−θ 2 µ 2 ) ≡γ LL . No e ha γ LL < γ SBA3 always holds so e M LL +e A3 LL < 2γ ∆θ is always sa isfied when bo h Cases Mand A.3 a e ele an . Finally, u M LL =u A3 LL because u LL = 1 2 (θ−1) e 2 HL and again e M HL =e A3 HL . 32 I is always he case ha min θ M 1 , θ M 2 < θ A3 bu he e is a wide ange o p obabili ies µ < 1 2 and νsuch ha θ A3 <min θ M 1 , θ M 2 ,meaning ha he wo subse s o θa e a leas pa ially o e lapping. 47 D.4 Pooling be ween in e media e ypes HH and LL Suppose ha he p incipal offe s a single con ac o bo h agen s LL and HH. Then one has e LL = e HH =e p and w LL =w HH =w p .The ele an cons ain s a e w LH −1 2e 2 LH +γe LH ≥w p −1 2e 2 p +γe p o ype LH, w p −1 2e 2 p ≥w HL −1 2e 2 HL (44) o ype LL o w p −1 2θe 2 p +γe p ≥w HL −1 2θe 2 HL +γe HL (45) o ype HH. Finally, o ype HL w HL −1 2θe 2 HL ≥0. The binding pa icipa ion cons ain is he one o ype HL abo e, while all o he pa icipa ion cons ain s a e sa isfied p o ided ha PC HL is. The mono onici y condi ion e LH ≥e p ≥e HL holds; bu which incen i e compa ibili y cons ain be ween (44), ha is IC LL sHL ,and (45), o else IC HH sHL , binds fi s ? Taking in o accoun he binding pa icipa ion cons ain o ype HL, i mus be ha w p ≥max 1 2θe 2 p −γe p +γe HL ;1 2e 2 p +1 2∆θe 2 HL . Thus, IC HH sHL is binding fi s when 1 2θe 2 p −γe p +γe HL ≥1 2e 2 p +1 2∆θe 2 HL ⇐⇒ e p +e HL ≥2γ ∆θ, whe eas IC LL sHL is binding when 1 2θe 2 p −γe p +γe HL ≤1 2e 2 p +1 2∆θe 2 HL ⇐⇒ e p +e HL ≤2γ ∆θ In wha ollows we s udy he wo sub-cases sepa a ely. D.4.1 Pooling be ween in e media e ypes wi h IC HH sHL binding Suppose ha when pooling occu s, IC HH sHL is binding while IC LL sHL is slack. We call his si ua ion Case P(a). Then one has e p +e HL ≥ 2γ ∆θ .Wages mus sa is y w HL =1 2θe 2 HL ,(46) w p =1 2θe 2 p −γe p +γe HL    In o en wo ke HH ,(47) 48 and w LH =1 2e 2 LH −γe LH +1 2∆θe 2 p +γe HL    In o en wo ke LH .(48) The wage w p has he same exp ession as w HH in Cases A.1and A.2(see equa ion 27). Since IC HH sHL is binding while IC LL sHL is no , we expec ha he in o ma ion en o wo ke LL is highe han he one o wo ke HH and his occu s o e p > 2γ ∆θ .This equi emen is mo e es ic i e han e p +e HL ≥ 2γ ∆θ and i mus be imposed ex-pos , as was done in Case A.1. Subs i u ing again he wage schedules in o he p incipal’s p og am we find e SBP a LH = 1 + γ, e SBP a p ≡e SBA1 p =(ν(1 −µ) + µ(1 −ν)) (1 + γ) νµ∆θ+ (ν(1 −µ) + µ(1 −ν)) θ(49) and e SBP a HL =(1 −ν) (1 −µ)−γ(1 −(1 −ν) (1 −µ)) (1 −ν) (1 −µ)θ=e SBA1 HL . No e ha e SBP a LH > e SBP a p and e SBP a LH > e SBP a HL always hold. Mo eo e e SBP a HL is he same as e SBA1 HL since in bo h cases pa icipa ion cons ain o wo ke HL is binding. Also obse e ha e SBP a HL is s ic ly posi i e i and only i γ < γ SBA1 1 ,and e SBP a p > e SBP a HL i and only i γ > νµ(1−ν)(1−µ)∆θ νµ(1−(1−ν)(1−µ))∆θ+θ(µ(1−ν)+ν(1−µ)) =γ SBP a , whe e γ SBP a < γ SBA1 always holds. Mo eo e , e SBA1 LL < e SBP a p < e SBA1 HH i and only i γ > γ SBA1 2 and he condi ion e p > 2γ ∆θ holds i and only i γ < (ν(1−µ)+µ(1−ν))∆θ 2νµ∆θ+(ν(1−µ)+µ(1−ν))(θ+1) =γ SBP a whe e γ SBP a > γ SBP a is always ue, γ SBP a < γ SBA1 1 i and only i θ < (ν(1−µ)(1−µ(1−ν))+µ(1−ν)(1−ν(1−µ))) ( (2ν−1)(ν(1−µ)+µ(1−ν))+2(1−ν) 2 µ 2 )=θ 5 (always o ν < 1 2 and µ < (1−2ν) 2 + √ (1−2ν)(1+2ν−4ν 2 ) 4(1−ν) 2 ≡µ 3 < 1 2 ), whe e θ 3 < θ 5 < θ 4 i and only i µ < µ 2 ,and γ SBA1 3 < γ SBP a < γ SBA1 2 i and only i µ < µ 2 . I is now possible o s a e he ollowing esul . Resul 12 (i)Full pa icipa ion and Pooling be ween ypes HH and LL wi h IC HH sHL bind- ing. A solu ion o he p incipal’s p og am SB which en ails ull pa icipa ion and pooling be ween ypes HH and LL and IC HH sHL binding, which sa isfies he mono onici y condi ion e LH > e HH =e LL > e HL >0,and which is such ha effo le els a e gi en by exp essions (30), (33) and (49) exis s i and 49 only i γ SBP a < γ < min γ SBA1 1 , γ SBP a wi h γ SBP a ≡ νµ(1−ν)(1−µ)∆θ νµ(1−(1−ν)(1−µ))∆θ+θ(µ(1−ν)+ν(1−µ)) γ SBP a ≡ (ν(1−µ)+µ(1−ν))∆θ 2νµ∆θ+(ν(1−µ)+µ(1−ν))(θ+1) γ SBA1 1 ≡ (1−ν)(1−µ) (1−(1−ν)(1−µ)) (ii)Pooling be ween ypes HH and LL wi h IC HH sHL binding and exclusion o ype HL. A solu ion o he p incipal’s p og am SB which en ails pooling be ween ypes HH and LL and PC HH binding, exclusion o ype HL and which sa isfies he mono onici y condi ion e LH > e HH =e LL >0,and which is such ha effo le els a e gi en by exp essions (30) and (49) exis s i and only i γ < γ SBP a . No e ha in his Case P(a)i ne e happens ha ype HH is asked o p o ide an effo which alls in he ange whe e he u ili y is inc easing in effo , namely i is ne e he case ha e HH =e LL =e SBP a p < γ θ . This migh occu in he subsequen Case P(b). D.4.2 Pooling be ween in e media e ypes wi h IC LL sHL binding Suppose now ha when pooling occu s, IC LL sHL is binding while IC HH sHL is slack. We call his si ua ion Case P(b), in which e p +e HL ≤ 2γ ∆θ .Wages mus sa is y w HL =1 2θe 2 HL , w p =1 2e 2 p +1 2∆θe 2 HL    In o en wo ke LL (50) and w LH =1 2e 2 LH −γe LH +γe p +1 2∆θe 2 HL    In o en wo ke LH . No e ha he wage w p now has he same exp ession as w LL in Case M(see equa ion 10), Case A.3(see equa ion 15) and Case A.2. Subs i u ing he wage schedules in o he p og am and de i ing yields e SBP b LH = 1 + γ, e SBP b p ≡e SBM p =e SBA2 p =(ν(1 −µ) + µ(1 −ν)) −γµν (ν(1 −µ) + µ(1 −ν)) (51) and e SBP b HL =(1 −ν) (1 −µ) θ−(1 −(1 −ν) (1 −µ)) =e SBM HL =e SBA3 HL , whe e e SBP b HL is equal o e SBM HL and e SBA3 HL since in all cases he incen i e cons ain IC LL sHL is binding. No e ha e SBP b p >0i and only i γ < ν(1 −µ) + µ(1 −ν) µν =γ SBP b , 50 which is always he case o µ < ν 3ν−1 ,and which is such ha γ SBP b > γ ∗ i and only i θ < θ M 1 and such ha γ SBP b > γ SBM and γ SBP b > γ SBA2 always hold. Fu he mo e, obse e ha e SBP b LH > e SBP b p and e SBP b LH > e SBP b HL always hold, while e SBP b p > e SBP b HL holds whene e e SBP b p >0is ue. Finally, he condi ion e SBP b p +e SBP b HL ≤ 2γ ∆θ holds i and only i γ≥ (ν(1−µ)+µ(1−ν))∆θ(∆θ+2(1−ν)(1−µ)) (θ−(1−(1−ν)(1−µ)))(2(ν(1−µ)+µ(1−ν))+µν∆θ) =γ SBP b whe e γ SBP b <min γ ∗ , γ SBP b is always ue and whe e γ SBA2 < γ SBP b and γ SBA3 < γ SBP b < γ SBA3 a e also ue. Resul 13 Full pa icipa ion and Pooling be ween ypes HH and LL wi h IC LL sHL binding. A solu ion o he p incipal’s p og am SB which en ails ull pa icipa ion and pooling be ween ypes HH and LL and IC LL sHL binding, which sa isfies he mono onici y condi ion e LH > e HH =e LL > e HL >0, and which is such ha effo le els a e gi en by exp essions (17), (20) and (51) exis s i and only i γ SBP b ≤γ < γ SBP b wi h γ SBP b ≡ (ν(1−µ)+µ(1−ν))∆θ(∆θ+2(1−ν)(1−µ)) (θ−(1−(1−ν)(1−µ)))(2(ν(1−µ)+µ(1−ν))+µν∆θ) γ SBP b ≡ ν(1−µ)+µ(1−ν) µν Conce ning exclusion o he wo s ype, we need o conside a simila p og am whe e, ins ead o ha ing IC LL sHL binding and IC HH sHL slack, we need PC LL o be binding and PC HH o be slack. In his case, he equi emen e SBP b p +e HL ≤ 2γ ∆θ educes o he mo e gene al condi ion e SBP b p ≤ 2γ ∆θ ,which is sa isfied i and only i γ≥∆θ(ν(1 −µ) + µ(1 −ν)) (νµ∆θ+ 2 (ν(1 −µ) + µ(1 −ν))) =γ SBP b whe e γ SBP b < γ SBP b . Resul 14 Pooling be ween ypes HH and LL wi h PC LL binding and exclusion o ype HL. A solu ion o he p incipal’s p og am SB which en ails pooling be ween ypes HH and LL wi h PC LL binding and exclusion o ype HL, which sa isfies he mono onici y condi ion e LH > e HH =e LL >0, and which is such ha effo le els a e gi en by exp essions (17) and (51) exis s i and only i γ SBP b ≤ γ < γ SBP b wi h γ SBP b ≡ ∆θ(ν(1−µ)+µ(1−ν)) (νµ∆θ+2(ν(1−µ)+µ(1−ν))) . Fo u he e e ence no e ha γ SBP b is smalle han γ SBA2 p o ided ha θ≤2,namely p o ided ha Assump ion 2 holds. Also no e ha i migh e en ually be he case ha e HH =e LL =e SBP b p < γ θ in which si ua ion ype HH would ha e incen i e o p o ide mo e effo han he one equi ed by he op imal con ac since he equi ed effo alls in he ange in which he indiffe ence cu e is downwa d sloping in he space (e, w). 51 E Example Le γ L = 0 and γ H =γ∈(0,1] and le θ L = 1 and θ H =θ∈(1,2]. Assume ha mo i a ion and skills a e uni o mly dis ibu ed ac oss wo ke s, so ha µ=ν= 1 2 .Case Mis a ained o 1< θ < 3 2 ,Case A.2does no exis , while Case A.3holds o 5 3 < θ ≤2.Hence one can ha e h ee classes o p oblems: (i) he diffe ence in abili y is low and 1< θ < 3 2 ,and ei he mo i a ion p e ails and Case Mis a ained o abili y p e ails and Case A.1holds; (ii) he diffe ence in abili y is high and 5 3 < θ ≤2,abili y always p e ails and ei he Case A.1o Case A.3hold depending on he alue aken by γ; (iii) he diffe ence in abili y is in e media e so ha 3 2 ≤θ≤ 5 3 ,abili y p e ails and only Case A.1holds. In si ua ion (i),one obse es he ollowing op imal con ac s: when 0< γ ≤ ∆θ 3(2θ−1) =γ SBA1 he p incipal offe s a pooling con ac o low-skilled ypes HH and HL, when γ SBA1 < γ < γ SBA1 = γ SBA1 3 = ∆θ 3θ−1 ull pa icipa ion and ull sepa a ion unde Case A.1is implemen ed, when γ SBA1 ≤γ < γ SBP a = ∆θ 2θ he p incipal offe s a pooling con ac o in e media e ypes HH and LL, which is such ha IC HH sHL is binding, when γ SBP a ≤γ < γ SBP b = 2∆θ θ+3 he e is exclusion o bo h ypes HH and HL, when γ SBP b < γ < γ SBP b = 4(2θ−1)∆θ (4θ−3)(θ+3) he e is pooling be ween in e media e ypes HH and LL wi h he cons ain IC LL sHL binding and exclusion o ype HL. No e ha γ SBP b < γ ∗ so ha we s ill a e in he domain in which abili y p e ails and e LL > e HH .When γ SBP b ≤γ≤γ SBM = 4∆θ 2θ+1 we ha e pooling be ween in e media e ypes HH and LL wi h he cons ain IC LL sHL binding bu ull pa icipa ion is a ained, and we c oss γ ∗ so ha mo i a ion p e ails and e HH > e LL .When γ SBM < γ < 3∆θ 4θ−3 = γ SBM < 1 2 , ull sepa a ion and ull pa icipa ion is a ained unde Case M. When γ SBM ≤γ < 1 he p incipal offe s a pooling con ac o non-mo i a ed ypes LL and HL. In si ua ion (ii),one obse es he ollowing: when 0< γ < γ SBP b he e a e he same op imal con ac s as in (i),when γ SBP b ≤γ < γ SBA3 = (3θ−1)∆θ 2(4θ−3) we ha e pooling be ween in e media e ypes HH and LL wi h he cons ain IC LL sHL binding and ull pa icipa ion, when γ SBA3 < γ < γ SBA3 = 2∆θ θ+2 he e is ull pa icipa ion and ull sepa a ion unde Case A.3, when γ SBA3 ≤γ≤1,we ha e ull pa icipa ion and pooling be ween in e media e ypes HH and LL wi h he cons ain IC LL sHL binding. 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(2007), “The Mo i a ion and Bias o Bu eauc a s”, Ame ican Economic Re iew 97(1), 180-196. [27] P ende gas C. (2008), “In insic Mo i a ion and Incen i es”, Ame ican Economic Re iew: Pape s & P oceedings 98(2), 201-205. [28] Roche J.-C. and P. Chonè (1998), “I oning, Sweeping, and Mul idimensional Sc eening” Econome - ica, 66, 783-826. [29] Van den S een (2006), “Too Mo i a ed?”, Sloan School o Managemen Wo king Pape Se ies, No. 4547—05. 54 γ H LH HH HL LL θ H θ θ L =1 1 γ Figu e 2c. Case A.3 (e HH <e LL ): eLL +eHL < 2γ/∆θ < eLL +eHH γ γ H LH HH HL LL θ H θ θ L =1 1 Figu e 2b. Case A.2 (e HH <e LL ): eHH +eHL < 2γ/∆θ < eLL +eHL. γ γ H LH HH HL LL θH θ θL=1 1 Figu e 2a. Case A.1 (e HH <e LL ): 2γ/∆θ < eHH +eHL. γ H LH HH HL LL θH θ θ L =1 1 γ Figu e 1. Case M (e HH >e LL ): 2γ/∆θ > eHH +eLL. e LL < e HH eLL > eHH Figu e 3. Exis ing classes o equilib ia as a unc ion o 2γ/∆θ. 2γ/∆θ e HH +e HL e LL +e HL e LL +e HH Case A.1 Case A.2 Case A.3 Case M