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
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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.
In si ua ion (iii),one obse es he ollowing op imal con ac s: when 0< γ < γ
SBP b
he e a e he
same solu ions as in (i)and (ii),when γ
SBP b
≤γ < 1we 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.
Re e ences
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Gi ing”, Econome ica, 64, 51-75.
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52
[3] A ms ong M. and J-J. Roche (1999), “Mul i-dimensional Sc eening: A Use ’s Guide”, Eu opean
Economic Re iew, 43, 959-979.
[4] A ms ong M. (1999), “Op imal Regula ion wi h Unknown Demand and Cos Func ions”, Jou nal
o Economic Theo y 84, 196-215.
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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