Ishac, Richa d
A icle
Disc imina ion, egula ions and he op imal hi ing p ocess
Re iew o Economic Analysis (REA)
P o ided in Coope a ion wi h:
In e na ional Cen e o Economic Analysis (ICEA), Wa e loo, On a io
Sugges ed Ci a ion: Ishac, Richa d (2024) : Disc imina ion, egula ions and he op imal hi ing
p ocess, Re iew o Economic Analysis (REA), ISSN 1973-3909, In e na ional Cen e o Economic
Analysis (ICEA), Wa e loo (On a io), Vol. 16, Iss. 2, pp. 221-241,
h ps://doi.o g/10.15353/ ea. 16i2.5189
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/328165
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Re iew o Economic Analysis 16 (2024) 221–241 1973-3909/2024221
Disc imina ion, egula ions and he op imal hi ing p ocess
RICHARD ISHAC*
Queen’s Uni e si y
This a icle p o ides a heo e ical amewo k o compa ing wo di e en hi ing p ac ices:
an unpaid compe i i e in e nship ha is ollowed by a po en ial job o e e sus a s an-
da d se ies o in e iews. A e ully cha ac e ising he op imal hi ing p ocess, I show ha
high-abili y mino i ies can be ha med by labou egula ions ha cause employe s o shi
owa ds a hi ing p ocess in which hey a e mo e likely o disc imina e. Fu he mo e, p e-
en ing employe s om gi ing u h ul e e ences is shown o exace ba e he obs acles o
employmen o a communi y adi ionally acing disc imina ion.
Keywo ds: Disc imina ion, Regula ions, Labou
JEL Classi ica ions: J01, J15, D2, L51
1 In oduc ion
I is an un o una e ea u e o many mode n economies ha membe s o ce ain his o ically
ma ginalised g oups con inue o ace some o m o disc imina ion when seeking employmen ,
as e idenced by pape s like B addock and McPa land (1987), Goldin and Rouse (2000), Be and
and Mullaina han (2004), Ca lsson and Roo h (2007), Giuliano e al. (2009) and Heywood and
Pa en (2012). In he con ex o such employmen disc imina ion, his a icle seeks o show how
some labou egula ions aiming o p o ec job applican s and in e ns can ac ually ampli y he
p ejudices aced by membe s o hese communi ies.
An impo an pa o he li e a u e on economics has been on he ma ching p ocess be ween
employees and employe s. Howe e , ela i ely li le wo k has been done on he op imal hi ing
p ocess i sel . The s anda d hi ing p ac ice consis s o conduc ing a se ies o in e iews o
acqui e in o ma ion abou po en ial employees in o de o ge he igh ma ch. An al e na i e
o his me hod is he y be o e you buy (see Coco, 2000) app oach o in e nships: wo in e ns
compe e o a job and he company hi es he mos p oduc i e in e n. Gi en hese wo op ions,
his a icle seeks o model he op imal hi ing p ocess, as well as o model he undamen al
ade-o be ween hese wo p ocesses.
To achie e hese aims, a amewo k is de eloped wi h one p incipal and wo agen s. An
agen can be o ei he a low- o high-p oduc i i y ype, and an agen ’s ype is his p i a e in-
*Co esponding au ho : Phone: (613) 533-2250; Fax: (613) 533-6668
©2024 Richa d Ishac. Licenced unde he C ea i e Commons A ibu ion-Noncomme cial 3.0 Licence
(h p://c ea i ecommons.o g/licenses/by-nc/3.0/). A ailable a h p:// o ea.o g.
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Re iew o Economic Analysis 16 (2024) 221–241
o ma ion. The p incipal wishes o hi e one o he agen s, p e e ably a high-p oduc i i y ype,
o conduc a p ojec in which p o i s a e dis ibu ed among he hi ed agen and he p incipal
acco ding o wo exogenously gi en sha es. The p incipal has he choice be ween wo hi ing
p ocesses. Fi s , he can choose he in es iga ion p ocess, whe e he spends some esou ces in
o de o acqui e a se o signals abou he agen s’ ypes. This can be hough o as he adi ional
se ies o in e iews applican s go h ough when applying o a job. Second, he p incipal can
choose he in e nship p ocess, whe e he wo agen s compe e o he job h ough an in e nship
ou namen . Howe e , he p incipal has some p ejudice owa ds one o he agen s who belongs
o a ma ginalised g oup, which subsequen ly hu s he agen ’s chances o being hi ed. (An
agen ’s associa ion wi h a a ou ed o ma ginalised g oup is public in o ma ion.)
I ind ha he op imal hi ing p ocess is based on a undamen al ade-o be ween he accu-
acy o he signals and hei cos s. I also show ha i he in es iga ion’s signals a e su icien ly
p ecise, hen he in es iga ion p ocess is mo e likely o be op imal when he di e ence be ween
a low-p oduc i i y wo ke and a high-p oduc i i y wo ke is high. This sugges s ha highly
p oduc i e employees a e mo e likely o be hi ed h ough he in es iga ion p ocess i he i m
has access o a su icien ly compe en ec ui ing depa men . Addi ionally, I show ha banning
unpaid in e nships and discou aging nega i e e e ences om p e ious employe s can ampli y
he disc imina ion aced by ma ginalised wo ke s, e en hough bo h policies aim o p o ec
labou e s. I conclude by discussing how hese esul s ela e o he ‘ban- he-box’ deba e (see
Agan and S a , 2018; Doleac and Hansen, 2020).
An al e na i e applica ion o his amewo k would be o compa e a se ing whe e en y-
le el wo ke s ace an up-o -ou scheme o a se ing whe e a i m ills a posi ion wi h ex e nal
hi es. Howe e , a consequence o his al e na i e applica ion would be ha one o he agen s
would ace disc imina ion a he ‘p omo ion s age’ ins ead o he ‘candida e s age’. Gi en ha
he labou egula ions on in e nships and p e ious employe s’ e e ences s udied in his a icle
e ol e a ound he ‘candida e s age’, I will ocus on applying his amewo k o he op imal
hi ing p ocess ins ead o he op imal p omo ion p ocess. Fo simila easons, disc imina ion
ha would occu a an ea lie hi ing s age ( il e ing applican s o in e iews and in e nships) is
no he a ea o ocus o his pape (see Knouse e al. (1999) o e idence o acial di e ences in
in e nship a ainmen ).
Some pape s in o ganiza ional economics ha e app oached he hi ing p oblem by ocusing
on he idiosync a ic ea u es o he i m. Fo ins ance, Lazea (2009) showed ha wo ke s a e
mo e aluable a some i ms han o he s based on employe alua ion. Lazea (2000) showed
ha incen i e pay a ac s mo e skilled applican s, as po en ial wo ke s sel -so . O he pape s
such as Ande son e al. (2009), Hayes e al. (2006) and Woodcock (2015) ha e ocused on he
impo ance o ge ing he bes ma ch be ween he employee and wo ke s. Ano he app oach has
been o ocus on he signals agen s can send abou hei ype o po en ial employe s (see Spence,
1973; Lazea , 1986). Au o (2001) and Au o and Sca bo ough (2008) concen a e on he ole
222
ISHAC Disc imina ion, egula ions and he op imal hi ing p ocess
ha labou ma ke in e media ies can ha e in smoo hing he hi ing p ocess, while pape s like
Fe nandez and Weinbe g (1997), Holze (1987), Mon gome y (1991) and Salone (1985) a -
gue o he impo ance o e e als and ne wo ks o cu en employees o sc een applican s.
Mac oeconomic condi ions can also a ec hi ing decisions (see Russo, Go e and Sche ka ,
2001), and he choice be ween hi ing employed o unemployed wo ke s is s udied by Fallick
and Fleischman (2004) and T anaes (2001). This a icle build upon amewo ks wi h one p in-
cipal and wo agen s wi h limi ed signalling abili ies, in o de o ocus on how he op imal hi ing
p ocess can gene a e he g ea es amoun o in o ma ion o employe s1.
Mul iple pape s sough o model (see Phelps, 1972) o p o ide a a ionale o disc imina-
ion wi h o wi hou using p ejudice as an assump ion. Milg om and Os e (1987) a gue ha
he skills o disad an aged wo ke s a e ha de o unco e o i ms, which hen keep high-
skilled disad an aged wo ke s in low-paying jobs o hide hei ype om o he employe s.
Ti ole (1996) models disc imina ion as a consequence o he noisy signals i ms ecei e on
po en ial wo ke s’ ypes. Fi ms hen eac o his noisy signal by complemen ing i wi h he
s e eo ypes su ounding he applican ’s g oup. Be nha d (1995) de elops a amewo k whe e
classes o wo ke s di e in hei skill dis ibu ion and, e en wi h he absence o disc imina-
ion, shows ha less able wo ke s om he dominan popula ion ecei e oppo uni ies be o e
mo e able wo ke s om he domina ed g oup. Focusing ins ead on i ings, Oye and Schae e
(2000) show e idence ha me hods o i ing changed di e en ly by ace ollowing he ci il
igh s ac s o 1999, while Oye and Schae e (2002) demons a e ha he e u ns o expe ience
a ied ac oss g oups du ing he same ime pe iod. Casella and Hanaki (2006) and Casella and
Hanaki (2008) s udy he op imal hi ing p ocess by compa ing e e al-based e sus ce i ica ion
hi ing s uc u es and highligh he dominance o he e e al-based p ocess and he subsequen
consequences o disc imina ion.
Since he causes o employmen disc imina ion a e beyond he scope o his a icle, I assume
a o m o p ejudice ha econciles he p emise o a ional p o i -maximising employe s wi h
he exis ence o disc imina ion o link ce ain labou policies o a po en ial ampli ica ion o
employmen disc imina ion.
2 P elimina ies
2.1 Model F amewo k
P e e ences. This amewo k consis s o one p incipal and wo agen s. All h ee playe s a e
isk-neu al. An agen ’s ype is de ined by his cos pa ame e , which can only ake wo possible
alues θi∈ {θL, θH}, wi h θL< θH. The p incipal needs o hi e one o he agen s o conduc
a p ojec and p e e s o hi e a ype-L agen . The p incipal wan s o choose he op imal hi ing
p ocess in o de o maximise his chances o hi ing a ype-L agen and maximise his p o i s
1See Oye and Schae e (2011) o a mo e comple e li e a u e e iew on he subjec o hi ing in economics.
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Re iew o Economic Analysis 16 (2024) 221–241
while p o iding an expec ed u ili y o each agen o a leas hei ese a ion u ili y, which is
assumed o be 0.
In o ma ion S uc u e and G oup A ilia ion. Each agen lea ns his ype a e en e ing ei he
o he hi ing p ocesses ou lined below, bu he p incipal and he o he agen ne e ully do so.
Fu he mo e, one o he agen s is a ilia ed wi h a ma ginalised g oup, meaning he p incipal
is p ejudiced agains his agen , who migh subsequen ly be disc imina ed agains du ing he
hi ing decision ( he exac o m o disc imina ion is desc ibed below), while he o he agen is
om a a ou ed g oup ha is sa egua ded om his disc imina ion. The sha e o H- ype agen s
is 1
2in each g oup, and he agen s’ ypes a e d awn independen ly om one ano he . These wo
ac s and he agen s’ g oup a ilia ion a e public in o ma ion.
Hi ing P ocesses. The p incipal can hi e he agen s h ough an in e nship p ocess o an
in es iga ion p ocess. Du ing he in e nship p ocess, each agen compe es in a ou namen
and he winne o he in e nship ou namen is po en ially e ealed o he p incipal. Du ing
an in es iga ion p ocess, he p incipal conduc s his own esea ch abou he agen s’ ypes and
po en ially un eils he agen s’ ypes. Each hi ing p ocess is desc ibed in de ail in sec ion 2.2.
In ei he case, he p incipal ge s some signals conce ning he agen s’ ypes and hi es he agen
who he belie es is he mos likely o be an L- ype agen 2.
Disc imina ion. In bo h hi ing p ocesses, he p incipal can ecei e ei he an unambiguous
signal (which sends a clea , unmis akable message abou he agen s’ ypes) o an ambiguous
signal (which is open o in e p e a ion). Du ing he in e nship p ocess, he p incipal ecei es
an unambiguous signal wi h p obabili y (1-p) indica ing who won he in e nship ou namen
and ecei es, du ing he in es iga ion p ocess, an unambiguous signal wi h p obabili y (1-q)
indica ing he esul o his esea ch. Howe e , when ecei ing an ambiguous signal, which
happens wi h p obabili ies pand qin he in e nship and in es iga ion p ocesses, espec i ely,
he signals a e open o in e p e a ion, in which case he p incipal disc imina es agains he agen
om he ma ginalised g oup by hi ing he agen om he a ou ed g oup. I should be no ed
ha he p incipal’s p ejudice is consequen ial only a e ecei ing an ambiguous signal: i is
i ele an ollowing a clea unambiguous signal abou he agen s’ ype3. This kind o p ejudice
econciles he p emise o employe s being a ional p o i -maximise s ( his p ejudice does no
dec ease expec ed p o i s) while s ill allowing o disc imina ion.
P oduc ion Technology and Payo s. Once an agen is hi ed, he will exe an e o eiwi h
i∈ {L,H}a a cos θie2
i
2in o de o in luence he ou come o he p ojec . The payo s o his
p ojec a e d awn om a dis ibu ion wi h a mean o ηei, whe e ηis a e enue componen ha is
independen o he agen ’s ype. The expec ed e enue ηeiis dis ibu ed be ween he p incipal,
2This se up excludes he possibili y o he p incipal spending esou ces o acqui e a mo e p ecise signal
abou he agen s’ ype, akin o Au o (2001).
3This is simila o a simpli ied o m o F ye e al. (2019), in which an agen ’s bias would disappea in he
limi o a pe ec ly in o ma i e signal.
224
ISHAC Disc imina ion, egula ions and he op imal hi ing p ocess
who ge s αηei, and he hi ed agen , who ge s (1−α)ηei. The p o i sha e αis aken as exogenous
by all playe s. The hi ed agen ’s p oblem is
max
ei
(1 −α)ηei−θie2
i
2,(1)
wi h a esul ing i s -o de condi ion o e∗
i=(1−α)η
θi. The e o e, du ing he p ojec s age, he
hi ed agen o ype i∈ {L,H} ecei es an expec ed payo o (1−α)2η2
2θi, while he p incipal ge s
α(1−α)η2
θi. The agen s a e p o ec ed by a limi ed liabili y cons ain p e en ing he p incipal om
asking he agen s o up on paymen s, side con ac s be ween he agen s a e no easible, and
he p incipal is con ac ually obliga ed o pay he hi ed agen ’s sha e o he o e all p o i s.
Timing. The iming o he game is as ollows. Fi s , he p incipal chooses be ween he
in es iga ion p ocess o he in e nship p ocess. I he in e nship p ocess is chosen, hen he
agen s lea n hei ypes and exe some e o in he hopes o winning he in e nship ou na-
men . I he p incipal ecei es an unambiguous signal indica ing he winne o he in e nship
ou namen , hen he hi es ha agen . I he ecei es an ambiguous signal, hen he hi es he agen
om he a ou ed g oup. I he in es iga ion p ocess is chosen and he p incipal ecei es an
unambiguous se o signals, hen he ecei es wo signals, one o each agen ’s ype, upda es his
p io belie s abou he agen s’ ypes and hi es he agen he belie es is he mos likely o be a
ype-L agen . I he ecei es an ambiguous se o signals, he once again hi es he agen om
he a ou ed g oup. Finally, he hi ed agen now has o exe some e o in a p ojec in which
e u ns a e dis ibu ed among he p incipal and himsel .
2.2 The Expec ed P o i s o Bo h P ocesses
2.2.1 In e nships
I an in e nship p ocess is chosen a a cos S o he p incipal, hen each agen is asked wi h
p oducing he highes possible alue yi=˜ei+ϵi. Agen idi ec ly con ols his e o ˜ei∈E⊂R
bu has no con ol o e he noise ϵi. Fo ac abili y pu poses, I assume ϵ1−ϵ2∼U[−L,L]. A
his s age, i agen iis om he ma ginalised g oup, his p oblem is
max
˜ei∈E
(1 −α)2η2
2θi
P ob(winning) −θi˜e2
i
2
=max
˜ei∈E
(1 −α)2η2
2θi
[(1 −p)P ob(˜ei+ϵi>˜e−i+ϵ−i)] −θi˜e2
i
2
=max
˜ei∈E
(1 −α)2η2
2θi
[(1 −p)( ˜ei−˜e−i+L
2L)] −θi˜e2
i
2,(2)
which esul s in a i s -o de condi ion o
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Re iew o Economic Analysis 16 (2024) 221–241
˜e∗
i=
(1−α)2η2(1−p)
4Lθ2
i
i L>(1−α)η[(1−p)(θ2
i−1
2θ2
−i)] 1
2
2θiθ−i
0 o he wise.
The condi ion on he pa ame e s ensu es ha agen i has a posi i e expec ed u ili y om
en e ing he in e nship p ocess4. I agen iis om he a ou ed g oup, his p oblem is
max
˜ei∈E
(1 −α)2η2
2θi
[(1 −p)( ˜ei−˜e−i+L
2L)+p]−θi˜e2
i
2,(3)
which also yields a i s -o de condi ion o
˜e∗
i=
(1−α)2η2(1−p)
4Lθ2
i
i L>(1−α)η[(1−p)(θ2
i−1
2θ2
−i)] 1
2
2θiθ−i
0 o he wise.
I make he pa ame e es ic ions gi en below:
L>(1 −α)η[(1 −p)(θ2
H−1
2θ2
L)]1
2
2θHθL
.(4)
Assump ion 4 ensu es ha ype-H agen s, and he e o e all agen s, ega dless o ype o
g oup a ilia ion, exe a posi i e le el o e o du ing he in e nship p ocess, e en i he o he
agen is o ype L. The p incipal only obse es who c ea ed he highes yi, so he can ecei e wo
possible unambiguous signals:
si∈ {Agen 1 wins,Agen 2 wins}≡{s1,s2}.(5)
I in oduce he ollowing no a ion o con enience: deno e he ype o bo h agen s by θ=
(θ1, θ2) and w i e θLL =(θ1
L, θ2
L), θLH =(θ1
L, θ2
H), θHL =(θ1
H, θ2
L) and θHH =(θ1
H, θ2
H). Gi en
his p io belie s, he p incipal belie es ha P(θ=θLL)=P(θ=θHH)=P(θ=θLH)=P(θ=
θHL)=1
4. Assuming ha he p incipal will ecei e an unambiguous signal, he p obabili ies o
he p incipal ecei ing signal sicondi ional on he agen s’ ype a e
P(s1|θLL)=P((1 −α)2η2(1 −p)
4Lθ2
L
−(1 −α)2η2(1 −p)
4Lθ2
L
≥ϵ2−ϵ1)
=P(0 ≥ϵ2−ϵ1)=1
2,(6)
4The inequali y can be ob ained by en e ing ˜e∗
iin o agen i’s expec ed u ili y (see (2)) and sol ing o he
se o pa ame e s ha ensu es ha i is s ic ly posi i e.
226
ISHAC Disc imina ion, egula ions and he op imal hi ing p ocess
P(s1|θHL)=P((1 −α)2η2(1 −p)
4Lθ2
H
−(1 −α)2η2(1 −p)
4Lθ2
L
≥ϵ2−ϵ1)=
1
2−(1 −α)2η2(1 −p)(θ2
H−θ2
L)
8L2θ2
Hθ2
L
(7)
and
P(s1|θLH)=P((1 −α)2η2(1 −p)
4Lθ2
L
−(1 −α)2η2(1 −p)
4Lθ2
H
≥ϵ2−ϵ1)=
1
2+(1 −α)2η2(1 −p)(θ2
H−θ2
L)
8L2θ2
Lθ2
H
.(8)
Fo simila easons, i can be seen ha P(s1|θHH)=P(s2|θLL)=P(s2|θHH)=1
2,P(s2|θLH)=1
2−
(1−α)2η2(θ2
H−θ2
L)
8L2θ2
Hθ2
L
and P(s2|θHL)=1
2+(1−α)2η2(θ2
H−θ2
L)
8L2θ2
Lθ2
H
.The e o e, a e ecei ing some unambiguous
signal si, he p incipal’s pos e io s a e
P(θLL|s1)=P(θHH|s1)=P(θLL|s2)=P(θHH|s2)=1
4,(9)
P(θLH|s1)=P(θHL|s2)=1
4+(1 −α)2η2(θ2
H−θ2
L)
16L2θ2
Lθ2
H
(10)
and
P(θHL|s1)=P(θLH|s2)=1
4−(1 −α)2η2(θ2
H−θ2
L)
16L2θ2
Lθ2
H
.(11)
F om hese pos e io s, i can be seen ha i he p incipal ecei es he unambiguous signal si, he
will hi e agen i, belie ing ha he is he mos likely o ha e he smalles cos pa ame e :
P(θ1=θL|s1)=P(θ2=θL|s2)=1
2+(1 −α)2η2(θ2
H−θ2
L)
16L2θ2
Hθ2
L
(12)
and
P(θ1=θL|s2)=P(θ2=θL|s1)=1
2−(1 −α)2η2(θ2
H−θ2
L)
16L2θ2
Hθ2
L
.(13)
I he p incipal ecei es an ambiguous signal, he hi es he agen a ilia ed wi h he a ou ed
g oup. In he e en o an ambiguous signal, he p incipal expec s he same p o i s om hi ing
ei he agen , so applying he p incipal’s p ejudice does no dec ease his expec ed p o i . The
227
Re iew o Economic Analysis 16 (2024) 221–241
p incipal’s expec ed u ili y is hen
p[(1
2)α(1 −α)η2
θL
+(1
2)α(1 −α)η2
θH
]
+(1 −p)(1
2{[1
2+(1 −α)2η2(θ2
H−θ2
L)
16L2θ2
Lθ2
H
]α(1 −α)η2
θL
+[1
2−(1 −α)2η2(θ2
H−θ2
L)
16L2θ2
Lθ2
H
]α(1 −α)η2
θH
}
+1
2{[1
2+(1 −α)2η2(θ2
H−θ2
L)
16L2θ2
Lθ2
H
]α(1 −α)η2
θL
+[1
2−(1 −α)2η2(θ2
H−θ2
L)
16L2θ2
Lθ2
H
]α(1 −α)η2
θH
})−S
=[1
2+(1 −α)2η2(1 −p)(θ2
H−θ2
L)
16L2θ2
Lθ2
H
]α(1 −α)η2
θL
+[1
2−(1 −α)2η2(1 −p)(θ2
H−θ2
L)
16L2θ2
Lθ2
H
]α(1 −α)η2
θH
−S,(14)
whe e he i s line ep esen s he p incipal’s expec ed p o i s, should he ecei e an ambiguous
signal, and he second o i h lines ep esen wha would happen i he p incipal ecei ed an
unambiguous signal ha agen 1 won and he hi ed agen was o ype θLo o ype θH(second
and hi d lines) o ecei ed an unambiguous signal ha agen 2 won and he hi ed agen was
o ype θLo o ype θH( ou h and i h lines). Once simpli ied, he six h line ep esen s he
expec ed u ili y o hi ing he in e nship winne who u ns ou o be ype L, and he se en h line
ep esen s he expec ed u ili y o hi ing a ype-H winne .
2.2.2 In es iga ions
In his se ing, he p incipal can acqui e a se o signals (ϕ1, ϕ2) a some cos E. The signals
indica e ϕi∈ {θL, θH}, wi h
P(ϕi=θL|θi=θL)=P(ϕi=θH|θi=θH)=σ∀i∈ {1,2},(15)
P(ϕi=θL|θi=θH)=P(ϕi=θH|θi=θL)=1−σ∀i∈ {1,2},(16)
whe e σ > 1
2can be seen as he p ecision o he signal. Mo e speci ically, wi h p obabili y
σ(1 −σ), he p incipal will ecei e a co ec signal abou one o he agen s, bu an inco ec
signal abou he o he agen . Wi h p obabili ies σ2and (1 −σ)2, he p incipal ecei es wo
co ec and inco ec signals, espec i ely. This implies ha when he p incipal ecei es some
unambiguous signal ϕi, his pos e io belie s become
P(θi=θH|ϕi=θH)=P(θi=θL|ϕi=θL)=σ∀i∈ {1,2},(17)
228
ISHAC Disc imina ion, egula ions and he op imal hi ing p ocess
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6 Appendix
P oo o P oposi ion 1:
Pa i)
I mus i s be no iced ha Πin −Πin can be ew i en as
=α(1 −α)η2
θH
[1
2−(1 −q)(2σ−1)
4]+α(1 −α)η2
θL
[1
2+(1 −q)(2σ−1)
4]−E
−α(1 −α)η2
θH
[1
2−(1 −α)2η2(1 −p)(θ2
H−θ2
L)
16L2θ2
Lθ2
H
]
−α(1 −α)η2
θL
[1
2+(1 −α)2η2(1 −p)(θ2
H−θ2
L)
16L2θ2
Lθ2
H
]+S.(22)
Fo pa i), when E=S=0, (22) simpli ies o Vin −Vin .
Pa ii)
Fo pa ii), Πin −Πin >0 can be ew i en as
(1 −q)(2σ−1)
4[α(1 −α)η2(θH−θL)
θHθL
]
−(1 −α)2η2(1 −p)(θ2
H−θ2
L)
16L2θ2
Lθ2
H
[α(1 −α)η2(θH−θL)
θHθL
]−(E−S)>0 (23)
⇔σ > 1
2+α(1 −α)3η4(1 −p)(θ2
H−θ2
L)(θH−θL)+16(E−S)L2θ3
Hθ3
L
8(1 −q)L2θ2
Lθ2
Hα(1 −α)η2(θH−θL).(24)
Pa iii)
236
ISHAC Disc imina ion, egula ions and he op imal hi ing p ocess
Fo pa iii), (23) can be ew i en as
α(1 −α)η2θH(1 −q)(2σ−1)
4θLθH
−α(1 −α)η2θL(1 −q)(2σ−1)
4θLθH
−α(1 −α)3η4(1 −p)(θ3
H−θ2
LθH−θLθ2
H+θ3
L)
16L2θ3
Lθ3
H
=α(1 −α)η2(1 −q)(2σ−1)
4θL
−α(1 −α)η2(1 −q)(2σ−1)
4θH
−α(1 −α)3η4(1 −p)
16L2θ3
L
+α(1 −α)3η4(1 −p)
16L2θLθ2
H
+α(1 −α)3η4(1 −p)
16L2θ2
LθH
−α(1 −α)3η4(1 −p)
16L2θ3
H
.(25)
The de i a i e o (25) wi h espec o −θLyields
−[−α(1 −α)η2(1 −q)(2σ−1)
4θ2
L
+3α(1 −α)3η4(1 −p)
16L2θ4
L
−α(1 −α)3η4(1 −p)
16L2θ2
Lθ2
H
−α(1 −α)3η4(1 −p)
8L2θ3
LθH
].(26)
The equa ion in (26) is posi i e when
0<α(1 −α)η2(1 −q)(2σ−1)
4θ2
L
−3α(1 −α)3η4(1 −p)
16L2θ4
L
α(1 −α)3η4(1 −p)
16L2θ2
Lθ2
H
+α(1 −α)3η4(1 −p)
8L2θ3
LθH
⇔e>1
2+[(1 −α)2η2(1 −p)
4(1 −q)L2](3θ2
H−θ2
L−2θHθL
2θ2
Hθ2
L
),
which p o es he i s pa o he esul . The de i a i e o (25) wi h espec o θHyields
α(1 −α)η2(1 −q)(2e−1)
4θ2
H
−α(1 −α)3η4(1 −p)
8L2θLθ3
H
−α(1 −α)3η4(1 −p)
16L2θ2
Lθ2
H
+3α(1 −α)3η4(1 −p)
16L2θ4
H
.(27)
The equa ion in (27) is posi i e when he inequali y below holds:
(1 −q)(2e−1) −(1 −α)2η2(1 −p)
2L2(1
θHθL
+1
2θ2
L
−3
2θ2
H
)>0
237
Re iew o Economic Analysis 16 (2024) 221–241
⇔e>1
2+[(1 −α)2η2(1 −p)
4(1 −q)L2](2θLθH+θ2
H−3θ2
L
2θ2
Lθ2
H
),
which shows he second pa o he esul .
QED
P oo o P oposi ion 2:
Pa i)
In an in es iga ion p ocess, an agen o ype θLwho is om a ma ginalised g oup has an
expec ed u ili y o
1
2{(1 −q)0 +q[(1 −α)2η2
2θL
][ee +e(1 −e)
2+(1 −e)e
2]}
+1
2{(1 −q)0 +q[(1 −α)2η2
2θL
][ee
2+e(1 −e)+(1 −e)2
2]}(28)
=(1 −q)(1 −α)2η2
2θL
(1+2e
4).(29)
The i s line o (28) ep esen s wha would happen i he o he agen was o ype θH(which he
agen belie es would happen wi h p obabili y 1
2) and he second ep esen s wha would happen
i he agen is o ype θL. Wi hin each se o swi ly b acke s, he second se o b acke s ep esen s
he p obabili y o being hi ed h ough he in es iga ion p ocess gi en he o he agen ’s ype. The
i s se o b acke s ep esen s he expec ed payo s in he e en o an ambiguous signal and an
unambiguous signal, espec i ely.
A eminde is in o de ha he p incipal’s bias is assumed o be o any consequence only
in he e en o an ambiguous signal being ecei ed by he p incipal. I is he e o e i ele an
ollowing a clea unambiguous signal abou he agen s’ ype. Subsequen ly, i he p incipal
ecei es an unambiguous signal e ealing ha bo h agen s a e o ype θL, hen he simply lips a
coin o de e mine he winne , which amoun s o each agen ha ing a p obabili y o winning o
1
2, o which he exp ession in he b acke s in he second pa o (28) simpli ies.
Simila ly, a ma ginalised wo ke o ype L expec s a payo in he in es iga ion p ocess o
1
2{(q)0 +(1 −q)[(1 −α)2η2
2θH
][e(1 −e)+(1 +e)2]}
+1
2{(q)0 +(1 −q)[(1 −q)2η2
2θH
][e2
2+1
2(1 −e)2+e(1 −e)]}
=(1 −q)(1 −α)2η2(3 −2e)
8θH
.(30)
238
ISHAC Disc imina ion, egula ions and he op imal hi ing p ocess
The e o e, a ma ginalised wo ke who has ye o lea n his ype has an expec ed u ili y in he
in es iga ion p ocess o
1
2[(1 −q)(1 −α)2η2(3 −2e)
8θH
]+1
2
(1 −q)(1 −α)2η2(1 +2e)
8θL
=(1 −q)(1 −α)2η2[θL(3 −2e)+θL(1 +2e)]
16θHθL
.(31)
In an in e nship p ocess, an agen o ype θLwho is om a ma ginalised g oup has an expec ed
u ili y o
(1 −p)[(1 −α)2η2(1 −p)(θ2
H−θ2
L)
16L2θ2
Lθ2
H
+1
2](1 −α)2η2
2θL
−θL
2[(1 −α)2η2(1 −p)
4Lθ2
L
]2(32)
=(1 −α)2η2(1 −p)
4θL
−(1 −α)4η4(1 −p)2
32L2θLθ2
H
.(33)
Simila ly, a ma ginalised wo ke o ype H has an expec ed u ili y o
(1 −α)2η2(1 −p)
4θH
−(1 −α)4η4(1 −p)2
32L2θ2
LθH
.(34)
The e o e, a ma ginalised wo ke who has ye o lea n his ype has an expec ed u ili y in an
in e nship p ocess o
1
2[(1 −α)2η2(1 −p)
4θH
]−1
2[(1 −α)4η4(1 −p)2
32L2θ2
LθH
]
+1
2[(1 −α)2η2(1 −p)
4θL
]−1
2[(1 −α)4η4(1 −p)2
32L2θLθ2
H
]
=(1 −α)2η2(1 −p)(θL+θH)
8θLθH
−(1 −α)4η4(1 −p)2(θH+θL)
64L2θ2
Lθ2
H
.(35)
By compa ing bo h u ili ies, i can be seen ha he ma ginalised wo ke p e e s he in e nship
p ocess i and only i (35) >(31) holds, which is equi alen o
(1 −p)(θL+θH)[8L2θLθH−(1 −α)2η2(1 −p)]
4L2θLθH[θL(3 −2σ)+θH(1 +2σ)] ≥(1 −q),(36)
which is one o he condi ions o p oposi ion 4.2-i. I should also be no ed ha i (36) holds,
hen es ablishing a minimum wage o in e ns would simply ein o ce a ma ginalised wo ke o
ype L’s p e e ence o he in e nship p ocess.
239
Re iew o Economic Analysis 16 (2024) 221–241
Fu he mo e, i should also be no ed ha Πin >Πin when w=0 holding is he equi alen
o
α(1 −α)η2
θH
[−(1 −α)2η2(1 −p)(θ2
H−θ2
L)
16L2θ2
Lθ2
H
]
+α(1 −α)η2
θL
[(1 −α)2η2(1 −p)(θ2
Hθ2
L)
16L2θ2
Lθ2
H
]−S
>α(1 −α)η2
θL
[(1 −q)(2σ−1)
4]−α(1 −α)η2
θH
[(1 −q)(2σ−1)
4]−E
holding. Fo he p incipal o ind i op imal o swi ch o he in es iga ion p ocess, he minimum
wage es ablished by he go e nmen would ha e o be high enough ha
w>w′≡E−S+α(1 −α)η2
θH
[(1 −q)(2σ−1)
4−(1 −α)2η2(1 −p)(θ2
H−θ2
L)
16L2θ2
Lθ2
H
]
−α(1 −α)η2
θL
[(1 −q)(2σ−1)
4+(1 −α)2η2(1 −p)(θ2
H−θ2
L)
16L2θ2
Lθ2
H
] (37)
would hold.
Pa ii)
Di e en ia ing (35) wi h espec o p esul s in
(1 −α)4η4(1 −p)(θH+θL)
32L2θ2
Lθ2
H
−(1 −α)2η2(θL+θH)
8θLθH
,
which is posi i e i and only i
(1 −p)>4L2θLθH
(1 −α)2η2,
which p o es esul 2-ii. I should be no ed ha he inequali y (1 −p)>4L2θLθH
(1−α)2η2does no
p eclude he in e nship p ocess being op imal. Since he inequali y
2σ−1
2>α(1 −α)3η4(1 −p)(θ2
H−θ2
L)(θH−θL)+16(E−S)L2θ3
Hθ3
L
8(1 −q)L2θ2
Hθ2
Lα(1 −α)η2(θH−θL)
⇔8L2θ2
Lθ2
H[(1 −q)α(1 −α)η2(θH−θL)(2σ−1) −2(E−S)θHθL]
2α(1 −α)3η4(θ2
H−θ2
L)(θH−θL)>(1 −p)
gua an ees ha he in es iga ion p ocess is op imal, i can be seen ha
8L2θ2
Lθ2
H[(1 −q)α(1 −α)η2(θH−θL)(2σ−1) −2(E−S)θHθL]
2α(1 −α)3η4(θ2
H−θ2
L)(θH−θL)>4L2θLθH
(1 −α)2η2
240
ISHAC Disc imina ion, egula ions and he op imal hi ing p ocess
⇔θLθH[(1 −q)α(1 −α)η2(θH−θL)(2σ−1) −2(E−S)θHθL]
> α(1 −α)η2(θ2
H−θ2
L)(θH−θL)
does no necessa ily hold, meaning ha he inequali y (1 −p)>4L2θLθH
(1−α)2η2does no p eclude he
in e nship p ocess being op imal.
Pa iii)
As o esul 2-iii, i ollows di ec ly om equa ion (31).
QED
241
