Job quality, search, and optimal unemployment contracts

da Cos a, Ca los E.; Maes i, Lucas Jó e ; San os, Céza
Wo king Pape
Job quali y, sea ch, and op imal unemploymen con ac s
IDB Wo king Pape Se ies, No. IDB-WP-1667
P o ided in Coope a ion wi h:
In e -Ame ican De elopmen Bank (IDB), Washing on, DC
Sugges ed Ci a ion: da Cos a, Ca los E.; Maes i, Lucas Jó e ; San os, Céza (2025) : Job quali y,
sea ch, and op imal unemploymen con ac s, IDB Wo king Pape Se ies, No. IDB-WP-1667, In e -
Ame ican De elopmen Bank (IDB), Washing on, DC,
h ps://doi.o g/10.18235/0013396
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J
ob Quali y, Sea ch, and Op imal
Unemploymen Con ac s
Ca los da Cos a
Lucas Maes i
Ceza San os
WORKING PAPER No IDB-WP-1667
In e -
A
me ican De elopmen Bank
Depa men o Resea ch and Chie Economis
Janua y 2025
* FGV EPGE
** In e -Ame ican De elopmen Bank and CEPR
J
ob Quali y, Sea ch, and Op imal
Unemploymen Con ac s
Ca los da Cos a*
Lucas Maes i*
Ceza San os**
In e -
A
me ican De elopmen Bank
Depa men o Resea ch and Chie Economis
Janua y 2025
Ca aloging-in-Publica ion da a p o ided by he
In e -Ame ican De elopmen Bank
Felipe He e a Lib a y
Da Cos a, Ca los.
Job quali y, sea ch, and op imal unemploymen con ac s/ Ca los da Cos a,
Lucas Maes i, Ceza San os.
p. cm. — (IDB Wo king Pape Se ies ; 1667)
Includes bibliog aphical e e ences.
1. Unemploymen insu ance-Econome ic models-Uni ed S a es. 2. Job
hun ing-Econome ic models-Uni ed S a es. 3. Labo ma ke -Econome ic
models-Uni ed S a es. I. Maes i, Lucas. II. San os, Ceza . III. In e -Ame ican
De elopmen Bank. Depa men o Resea ch and Chie Economis . IV. Ti le.
V. Se ies.
IDB-WP-1667
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Abs ac
When sea ching o employmen , wo ke s conside non-wage job cha ac e is ics, such
as e o equi emen s o ameni ies. We s udy an en i onmen whe e unemployed
wo ke s sea ch o jobs o di e en quali y in a labo ma ke cha ac e ized by di ec ed
sea ch. In equilib ium, i ms a e mo e likely o pos acancies o low-quali y jobs,
as hese a e mo e p o i able. Hence, high-quali y jobs a e ha d o come a c oss. The
non-obse abili y o hese employmen con ac s in luences he o p imal unemploy-
men insu ance (UI) p og am, leading o dis o iona y axa ion. Calib a ing he model
o he U.S. economy, we ind ha non-obse abili y o employmen con ac s esul s in
as e -declining UI bene i s, s eepe axes upon e-employmen , dis o iona y axa ion,
and a 10.5% cos lie p og am han an obse able con ac scena io p o iding equal
wel a e.
JEL classi ica ions: H21, J64
Keywo ds: Unemploymen insu ance, Di ec ed sea ch, In ensi e ma gin, Ameni ies,
Hidden sa ings
Filipe Fiedle , Ped o Gue a, A u Rod igues and Alejand a To es p o ided excellen esea ch assis-
ance. The iews exp essed in his a icle a e hose o he au ho s and do no necessa ily ep esen hose o
he In e -Ame ican De elopmen Bank. da Cos a hanks CNPq p ojec 304955/2022-1 o inancial suppo .
This s udy was inanced in pa by he Coo denac¸˜ao de Ape eic¸oamen o de Pessoal de N´ı el Supe io - B asil
(CAPES) - Finance Code 001.

1 In oduc ion
The labo ma ke encompasses mo e han jus wage compensa ion, as jobs a y signi i-
can ly in e ms o ameni ies, pe ks, wo k hou s, and e o equi emen s. When sea ching
o jobs, wo ke s conside hese non-pecunia y ac o s ha de e mine job quali y, and i ms
ailo posi ions acco dingly. Recognizing and unde s anding hese complexi ies o he labo
ma ke is c ucial o c a ing e ec i e economic policies, pa icula ly in he con ex o un-
employmen insu ance. Unemploymen insu ance p og ams mus s ike a balance be ween
p o iding adequa e insu ance and a oiding disincen i es o job sea ch and eemploymen .
By conside ing di e en aspec s o job quali y, policymake s can de elop unemploymen
insu ance policies ha be e align wi h he complexi ies o he mode n job ma ke .
In his pape , we s udy he p oblem o a go e nmen ha o e s unemploymen insu -
ance in a dynamic en i onmen ea u ing di ec ed sea ch. We inno a e by conside ing
non-wage aspec s o job quali y, which i ms can p o ide a di e en le els. Fi ms may
expand he supply o acancies o jobs o lowe quali y, such as jobs ha equi e mo e
e o and/o p o ide ewe ameni ies o he same le el o ea nings. As a p ime exam-
ple, many people p e e home o ice jobs due o he lexibili y hey p o ide in balancing
pe sonal and p o essional li es, such as elimina ing long commu es and accommoda ing
amily esponsibili ies. Fu he mo e, emo e wo k educes expenses on commu ing, dining
ou , and p o essional a i e, making i a mo e economical choice, which mo i a es indi id-
uals o seek ou such oles. A g owing body o empi ical esea ch, discussed in he nex
sec ion, s ongly emphasizes he signi icance o job quali y.
These non-pecunia y dimensions o job quali y a e impo an o ou analysis. F om
he wo ke s’ pe spec i e, hey can educe he expec ed unemploymen spell i hey look
o lowe -quali y jobs. Fo he design o op imal policies hey a e impo an because hese
adjus men s in job quali y a e ypically no con olled by he planne .
We cha ac e ize he op imum o gene al sepa able p e e ences when he planne con-
ols he agen ’s sa ings. A he op imum, unemploymen bene i s and ne ea nings decline
wi h he leng h o he unemploymen spell. The epea ed mo al haza d na u e o he p ob-
lem implies ha , a he op imum, he s ochas ic p ocess go e ning consump ion sa is ies
he in e se Eule Equa ion. In he long un, unemploymen bene i s con e ge o ze o. The
op imal con ac also p esc ibes a posi i e wedge on he ma ginal a e o subs i u ion be-
ween consump ion and job quali y, i.e., dis o iona y axa ion. This esul ma e ializes
e en hough he planne can use non-dis o iona y ins umen s and he e is no dis ibu i e
mo i e.
2
The logic is as ollows. Conside a i m ha inc eases job quali y (lowe wo k e-
qui emen s o inc eased ameni ies) o a ixed le el o ea nings o a ac wo ke s. This
inc ease in he alue o he job, which goes unde he planne ’s ada since only ea nings
a e obse ed, leads o a highe p obabili y o hi ing. Bu , om a wo ke ’s pe spec i e,
high-quali y jobs a e ha de o ind. Because agen s sea ching o a job a e en i led o
unemploymen bene i s, high-quali y jobs a e expensi e o he unemploymen insu ance
p og am. The ques ion is how he planne can discou age i ms om o e ing hese high-
quali y jobs. Now, a wo ke who manages o ge such a job has a highe u ili y o job
quali y han hose who ge lowe -quali y jobs. In an economy wi hou dis o ions a he
ma gin, hese wo ke s would like he i m o p o ide less job quali y in exchange o a
p opo ional inc ease in ea nings. By axing ea nings a he ma gin, he planne discou -
ages such a change and makes hese high-quali y jobs less a ac i e. This esul elies on
h ee ealis ic assump ions embedded in ou amewo k: di ec ed sea ch, in ensi e ma gin
adjus men s, and unobse abili y o he de ails o he employmen con ac .
We calib a e ou model o he U.S. economy. The non-obse abili y o employmen
con ac s has a signi ican quan i a i e impac on he op imal unemploymen insu ance con-
ac designed by he social planne . When con ac s a e unobse able, he unemploymen
insu ance bene i s decline as e , axes upon e-employmen inc ease mo e apidly wi h un-
employmen du a ion, and dis o iona y ax a es a e in oduced. These ac o s lead o an
unemploymen insu ance p og am ha is 10.5% mo e expensi e han one ha p o ides he
same le el o wel a e in a wo ld in which con ac s a e obse able.
To implemen he op imal alloca ion desc ibed abo e, he planne mus con ol he
agen ’s sa ings, which may no be possible in p ac ice. We ake he possibili y o hidden
sa ings and bo owing in pe ec capi al ma ke s in o accoun . Fo his case, we es ic
ou analysis o p e e ences o he G eenwood e al. [1988] ype specialized o he case o
Cons an Absolu e Risk A e sion (GHH-CARA p e e ences). The op imal alloca ion can
be implemen ed by a simple s a iona y con ac : an up on unemploymen ins allmen ,
cons an g oss ea nings, and axes when he agen inds a job. The pa e n o declining con-
sump ion in bo h employmen and unemploymen is achie ed by he wo ke ’s (dis)sa ings
along he unemploymen spell. In his hidden-sa ings case oo, a posi i e wedge a he
ma gin cha ac e izes he op imum.
Go e nmen agencies deploy a ious s a egies o o e see unemployed indi iduals du -
ing hei job sea ch o alida e eligibili y o unemploymen bene i s. Despi e widesp ead
epo ing manda es, such as eco ding job applica ions and in e iews, ensu ing job seek-
e s pu sue sui able employmen p o es challenging. Con i ming indi iduals do no solely
3
a ge highly compe i i e ye appealing posi ions ha equen ly d aw nume ous applican s
p esen s a signi ican hu dle. Ou esea ch demons a es ha he bene i s o es ablishing an
e ec i e unemploymen insu ance moni o ing agency a e subs an ial.
The es o he pape is o ganized as ollows. A e a b ie li e a u e e iew, in Sec ion
2, we mo i a e empi ically he in e ac ion be ween eceip o unemploymen insu ance, he
likelihood o inding employmen , and ce ain job cha ac e is ics. In Sec ion 3, we desc ibe
he en i onmen and o e a one-pe iod accoun o he o ces explaining ou indings. We
de i e he p ope ies o an op imal sys em unde he assump ion ha he planne con ols
agen s’ sa ings in Sec ion 4and analyze he op imal con ac quan i a i ely. Sec ion 5
desc ibes he op imal con ac o he case o hidden sa ings. Sec ion 6concludes.
Li e a u e Re iew
The mode n ea men o unemploymen insu ance p og am design has i s oo s in Sha el
and Weiss [1979] and ound i s i s canonical ea men in Hopenhayn and Nicolini [1997].
We con ibu e by ocusing on di ec ed sea ch and by in oducing he possibili y o selec -
ing jobs acco ding o hei e o equi emen s. Acemoglu and Shime [1999] conside a
gene al equilib ium model o di ec ed sea ch wi h isk a e sion. The s a ic e sion o ou
model gene alizes hei s by conside ing he possibili y o adjus ing he e o equi emen s
o di e en jobs. Mo eo e , while hei ocus is on he gene al equilib ium aspec s o
unemploymen insu ance, we concen a e on he planne ’s solu ion o he op imal policy.
Shime and We ning [2007,2008] e alua e he consequences o allowing agen s o bo -
ow and sa e in pe ec capi al ma ke s using McCall’s (1970) model o sequen ial job
sea ch. Unde CARA p e e ences, a policy comp ised o a cons an bene i du ing unem-
ploymen , a cons an ax du ing employmen , and ee access o a iskless asse is op imal.
In ou di ec ed sea ch en i onmen wi h he possibili y o in ensi e ma gin adjus men s
once employed, simple s a iona y policies a e also op imal unde CARA. We add o he
p esc ip ion by p o ing he op imali y o in oducing dis o iona y axa ion o incen i ize
sea ch owa ds easie - o- ind jobs.2
A s and o he li e a u e in es iga es edis ibu i e policies in he p esence o labo ma -
ke ic ions. Goloso e al. [2013] conside he edis ibu ion o esidual income. Unde
di ec ed sea ch, he op imal edis ibu ion o esidual income can be a ained wi h posi-
i e unemploymen bene i s and a posi i e, inc easing, and eg essi e income ax schedule.
They do no conside an in ensi e ma gin o non-wage job quali y as we do.
2We also con ibu e o he li e a u e ha s udies he op imal pa h o UI bene i s [e.g., Kols ud e al.,2018,
Lindne and Reize ,2020]. We add by conside ing non-obse able aspec s o job quali y.
4
The p esen a ion o ou heo y below is cen e ed on a ia ions o e o as he ele an
in ensi e ma gin adjus men . In p ac ice, wo ke s may adjus hei sea ch no only by
becoming mo e selec i e abou wages and how much e o hey mus exe once employed
bu also abou he quali y o hei p ospec i e wo k en i onmen , nei he o which is wi hin
he each o policy.3We show ha he same logic leading o he wedge in e o implies a
wedge in he supply o ameni ies. Recen esea ch shows ha job ameni ies a e impo an
o wo ke s [e.g., Sockin,2022]. Fo ins ance, Mo chio and Mose [2024] demons a e
ha ameni ies play an impo an ole in explaining he gende pay gap. Bagga e al. [2024]
show ha inc eased p e e ences o elewo k, a key job ameni y, help explain he pos -
pandemic labo ma ke expe ience in he Uni ed S a es. We con ibu e o his li e a u e
by showing ha hese non-wage cha ac e is ics o job quali y in luence he design o he
op imal unemploymen insu ance p og am.
K o e al. [2020] ind su icien s a is ics o he op imal combina ion o income axes
and unemploymen bene i s bu do no conside in ensi e ma gin adjus men s as we do. da
Cos a e al. [2022] s udy op imal dis ibu i e policies in he p esence o labo ma ke ic-
ions. While hey emphasize in ensi e ma gin choices, hei model is s a ic and ocused on
he in e ac ion be ween dis ibu i e mo i es and unemploymen insu ance design. He e, we
abs ac om edis ibu ion while highligh ing he dynamics o insu ance when con ac s
a e no obse ed and he e is scope o adjus men s in he in ensi e ma gin.
2 Empi ical Mo i a ion
This sec ion explo es da a om he Uni ed S a es o check whe he he e a e disce nible
di e ences in labo ma ke ou comes be ween indi iduals ecei ing unemploymen insu -
ance (UI) and hose wi hou such co e age. We ely on da a ex ac ed om he Ma ch
supplemen o he Cu en Popula ion Su ey (CPS). This supplemen p o ides da a on UI
eceip s among he unemployed, as well as key cha ac e is ics o hei job o hose cu -
en ly wo king. Ou analysis encompasses da a om 2009 o 2022. We un linea p obabil-
i y eg essions o d aw compa isons be ween he labo ma ke ajec o ies o unemployed
indi iduals bene i ing om UI and hose wi hou such bene i s. Figu e 1p o ides he main
es ima es (see Appendix A o he ull eg essions).
Figu e 1(a) plo s he di e ence in he likelihood o being unemployed one yea ahead
o unemployed wo ke s who ecei e UI e sus hose who do no . UI ecipien s a e abou
3These equalizing di e ences, su eyed by Rosen [1987], ha e been shown o be quan i a i ely impo an
in ecen wo k by Mas and Pallais [2017], So kin [2018], Hall and Muelle [2018].
5
The planne canno o ce he agen o ind a job i
φ(cu)> φ(ce)−η(ne).
Hence, he p og am ha he planne sol es is
C(W) = max
p,ce,cu,ne,˜
W
p
1−βne−ce−κϱ(p)
p+ (1 −p)h−cu+βC(˜
W)i,
subjec o he p omise-keeping,
W=p
1−β[φ(ce)−η(ne)] + (1 −p)hφ(cu) + β˜
Wi,(5)
and he incen i e cons ain ,
φ(ce)−η(ne)
1−β≥φ(cu) + β˜
W. (6)
We can w i e he p og am abo e as he ollowing Kuhn-Tucke p oblem,6
C(W) = max
p,ce,cu,ne,˜
W
p
1−βne−ce−κϱ(p)
p+ (1 −p)h−cu+βC(˜
W)i+
µp
1−β[φ(ce)−η(ne)] + (1 −p)hφ(cu) + β˜
Wi−W+
λφ(ce)−η(ne)
1−β−φ(cu)−β˜
W
To p oceed, we i s assess whe he he mo al haza d and he p omise-keeping con-
s ain s bind a he op imum. Lemma 4.1 below s a es ha whene e agen s sea ch o a
job, hey a e indi e en be ween doing so and emaining unemployed o ano he pe iod.
Lemma 4.1 The p omise-keeping cons ain (5) binds in e e y pe iod, and µ > 0.In any
pe iod in which he e is posi i e sea ch, he mo al-haza d cons ain binds, φ(ce)−η(ne) =
[1 −β]hφ(cu) + β˜
Wi, and λ > 0.
Fo e e y pe iod in which he mo al haza d cons ain binds we ha e
µ +1 =µ −λ
1−p
,
6We can ely on Lemma B.3 o w i e he p oblem as such. This lemma e e s o he case in which con ac s
a e no obse ed, bu he a gumen is easily adap ed o he case wi h obse ed con ac s.
12

which implies ha unemploymen consump ion dec eases o e ime,
cu
−1= (φ′)−1µ−1
>(φ′)−1µ−1
+1=cu
.
Mo eo e , he consump ion p ocess is desc ibed by an in e se Eule equa ion,
1
φ′cu
−1=µ =p µ +λ p−1
+ (1 −p )µ +λ p−1
=p
φ′(ce
)+1−p
φ′(cu
).
Also, he i s -o de condi ions wi h espec o ceand nimply ha , in con as o ou
one-pe iod model wi h non-obse ed con ac s, he e o is no dis o ed a he op imum in
he dynamic model wi h obse able con ac s. We ga he hese indings in P oposi ion 4.1.
P oposi ion 4.1 The solu ion o he planne ’s p oblem when con ac s a e obse able has
he ollowing p ope ies:
1. I en ails a ze o ma ginal income ax a e.
2. The unemploymen insu ance is dec easing o e ime. Mo eo e , i he agen sea ches
in pe iod , hen he unemploymen insu ance is s ic ly lowe han he one om he
p e ious pe iod.
3. The consump ion p ocess is desc ibed by an in e se Eule equa ion.
To unde s and 1, no e ha he incen i e-compa ibili y cons ain (6) only depends on he
agen ’s u ili y when employed, no on how i is gene a ed. Since he go e nmen obse es
con ac s, i can choose hem o minimize he cos o p o iding his u ili y. Tha is, gi en
any u ili y le el, he e is no eason o he go e nmen o dis o e o , which implies
1. Second, unemploymen insu ance should dec ease o e ime o make i mo e cos ly
o u n down employmen oppo uni ies, which is he con en o 2. Finally, simila o
se e al dynamic mo al-haza d models—e.g., Roge son [1985] — he consump ion p ocess
is desc ibed by an in e se Eule equa ion.
4.2 Non-obse able Con ac s
Sec ion 4.1 adop ed he s ong assump ion ha he go e nmen obse es he con ac s cho-
sen by wo ke s and hence he disu ili y o e o om a pa icula job. We now conside
op imal policies unde non-obse able labo con ac s. In his se up, he op imal policy
13
mus be based only on whe he o no he agen is employed, on hei ea nings, and he
leng h o he unemploymen spell.
I he agen is p omised a su icien ly high u ili y, hen he e is no sea ch in equilib ium
a he solu ion o he planne ’s p og am; i is cheape o deli e he p omised u ili y i he
agen emains unemployed o e e ; we show his in Lemma B.1 in he Appendix. This is
an unin e es ing case, and we ins ead ocus on he case in which u ili y is no oo high.
To cha ac e ize he op imal unemploymen insu ance p og am in his case, we ely on
a i s -o de app oach. Lemma B.2 shows ha he solu ion o his elaxed p oblem is he
solu ion o he o iginal p og am. Hence, he planne ’s p oblem has a ecu si e s uc u e
and can be w i en as ollows,
C(W) = max
p,ce,cu,ye,˜
W
p
1−β(ye−ce) + (1 −p)h−cu+βC ˜
Wi,
subjec o a p omise-keeping cons ain
p
1−βφ(ce)−ηye+κϱ(p)
p+ (1 −p)hφ(cu) + β˜
Wi−W≥0,(7)
and an incen i e compa ibili y cons ain
1
1−βφ(ce)−ηye+κϱ(p)
p−φ(cu)−β˜
W
=pκ
1−βη′ye+κϱ(p)
pϱ(p)
p′
.(8)
Lemma B.3 shows ha he planne ’s p oblem is di e en iable, and hence he op imum
mus sa is y a cons ained op imiza ion in which we w i e µand λ o he mul iplie s el-
a i e o he cons ain s (42) and (43). Bo h mul iplie s a e s ic ly posi i e. I µwe e
no s ic ly posi i e, he planne would be able o sa e esou ces by lowe ing he u ili y
p omised o he agen in bo h s a es wi h no consequences o incen i es. λis s ic ly
posi i e because he wo ke does no in e nalize he iscal ex e nali y when unemployed.
Combining he i s o de condi ions wi h espec o yeand ce, one ob ains
φ′(ce)−η′ye+κϱ(p)
p=λpκ
µp +λη′′ ye+κϱ(p)
pϱ(p)
p′
>0.(9)
The op imal alloca ion now displays a posi i e wedge a he in ensi e ma gin. The
dynamic model inhe i s he inding om ou one-pe iod model. I a i m o e s a be e job,
14
i.e., one equi ing less e o o he same ea nings, hen i will a ac mo e job candida es.
Wo ke s, in u n, will ind i ha de o land such a job, hus emaining unemployed o a
longe ho izon. Condi ional on ge ing one o hese jobs a wo ke would ha e a highe
willingness o exe e o compa ed o someone who go one o he jobs o e ed by i ms
along he equilib ium pa h. To make hese de ia ions less a ac i e, he planne dis o s
e o downwa ds by axing ea nings a he ma gin.
Since p e e ences a e sepa able in consump ion and e o , i is always easible o a y
he unemploymen consump ion u ili y in a pe iod and compensa e o i by a ying he
consump ion u ili y in all s a es o na u e in subsequen pe iods. Such a s a egy changes
nei he incen i es no expec ed u ili y. Thus, hese pe u ba ions canno sa e esou ces a
he op imum. Because he ma ginal cos o deli e ing u ili y is 1/φ′, he in e se Eule
equa ion ensues.
These indings a e summa ized in Theo em 4.1, which is p o ed in he Appendix.
Theo em 4.1 A he op imum, in e e y pe iod in which he wo ke sea ches,
1. he ma ginal income ax a e is always posi i e;
2. he mo al-haza d cons ain (43) binds, and he go e nmen bene i s om s ic ly
inc easing p, and;
3. condi ional on no inding a job a pe iod , he wo ke ’s ma ginal u ili y o consump-
ion sa is ies he in e se Eule equa ion,
1
φ′(cu
)=E1
φ′(c +1).
The planne can a oid dis o ing he e o ma gin. Taxes may be based on employmen ,
independen ly o ea nings. Mo eo e , he u ili y condi ional on inding a job depends on
φ(ce)−η(ne), ega dless o whe he ceand nea e e icien ly chosen. Wha is hen he
a ionale o dis o ing he in ensi e ma gin p esc ibed in Theo em 4.1? I is he same as in
he s a ic se ing. Conside a wo ke deciding whe he o apply o a job in a sub-ma ke ha
is sligh ly less igh han wha he planne has p esc ibed ˆp<p. The planne con ols yeand
ce, bu no he amoun o e o he agen mus make o ea n ye. Upon landing a job in a less
igh ma ke , he wo ke is equi ed o supply e o , ˆn=ye+κϱ(ˆp)/ˆp<ye+κϱ(p)/p =n
while ecei ing he same ce. This wo ke , he e o e, has a lowe ma ginal disu ili y o
e o han agen s who ollowed he op imal policy. To make his downwa d de ia ion
less aluable, which is he ele an de ia ion acco ding o 1, he planne dis o s e o
15
downwa d by in oducing a posi i e wedge. A li le less su p ising is he ac ha , as in
Roge son [1985], A keson and Lucas [1995], he In e se Eule equa ion cha ac e izes he
dynamics o consump ion o he unemployed.
When is sea ch op imal? Theo em 4.1 desc ibes he e icien alloca ion in pe iods in
which he e is sea ch. Bu when is i op imal o sea ch?
P oposi ion 4.2 The unemploymen bene i is dec easing o e ime wi h cu
> cu
+1 when-
e e he wo ke sea ches in pe iod + 1.
Mo eo e , whene e he wo ke sea ches in pe iod + 1, hei consump ion om em-
ploymen is s ic ly g ea e han he unemploymen bene i om any pe iod τ≥ .
When he p omised u ili y is e y high, he op imal con ac p o ides cons an bene i s
and asks he wo ke ne e o sea ch o a job. On he o he hand, job sea ch mus be
incen i ized when he go e nmen p omises a su icien ly low u ili y o he wo ke . These
wo possibili ies ende he go e nmen ’s cos o p o iding u ili y W o he wo ke no
con ex in W, in gene al. As a consequence, we canno ule ou he possibili y ha he
wo ke does no sea ch o a job in he i s pe iod o he op imal con ac .
To be e unde s and when i is op imal o sea ch in e e y pe iod, de ine z(W)by
z(W)≡a gminzzs. . maxye[φ(ye+z)−η(ye+ϕ)] ≥W, whe e, as we ecall,
ϕ= limp↓0ϱ(p)/p > 0. In ui i ely, z(W)is he minimum amoun o esou ces ha would
cos he go e nmen o mo i a e he wo ke o sea ch o employmen i hei unemploy-
men con inua ion u ili y we e W, assuming ha he labo ma ke was compe i i e. To see
his, we use he ac ha ϱ(p)/p is inc easing in p. Hence, o ind a job wi h p obabili y p
he wo ke would ha e o pay ϱ(p)/p > ϕ o he i m upon landing a job. Le also cu(W)
by φ(cu(W)) = W, he cos o p o iding u ili y W o a wo ke who ne e sea ches o a
job.
We show in Lemma B.1, in he Appendix, ha he e is a le el o u ili y, W∗, abo e
which z(W)> cu(W)and below which z(W)< cu(W). Lemma 4.2 below shows ha ,
i he ini ial unemploymen insu ance p o ides less u ili y han W∗, hen he wo ke mus
sea ch o a job in e e y pe iod.
Lemma 4.2 Assume ha φ(cu
0)< W∗. Then, p >0in e e y pe iod, .
When he ini ial u ili y, W0, is smalle han W∗, he ini ial con ac mus induce sea ch
in some pe iod. Mo eo e , whene e he wo ke sea ches in some pe iod he unemploymen
bene i s e en ually all so ha φ(cu
)< W∗ o some pe iod . Hence, he wo ke sea ches
in e e y pe iod, τ > , which is he con en o Lemma 4.3, below.
16
Lemma 4.3 The ollowing condi ions hold in any op imal con ac :
a) Assume ha W0< W∗. Then, he e is > 0such ha φ(cu
)< W∗. Hence, he
wo ke who is unemployed in any pe iod τ > sea ches o a job.
b) Assume ha he wo ke sea ches o a job in some pe iod . Then he e is T > such
ha he unemployed wo ke sea ches in any pe iod τ > T.
In his case, acco ding o P oposi ion 4.2,cu
> cu
+1 o all . The e o e, he unemploy-
men bene i con e ges o a non-nega i e numbe . P oposi ion 4.3 shows ha his numbe
is 0.
P oposi ion 4.3 Assume ha W0< W∗, hen unemploymen bene i s con e ge o ze o.
We ha e ocused hus a on he case o sepa able p e e ences be ween consump ion and
e o . This has been he mos equen ly s udied case in he li e a u e. In Appendix C, we
s udy non-sepa abili y o he case o GHH-CARA u ili y U(c, n) = −exp −αc−η(n).7
These p e e ences will be he ocus o ou analysis when we assume ha sa ings canno be
con olled by he planne . The esul s o his sec ion ca y o e o he GHH-CARA case. In
pa icula , he op imal policy o his case also p esc ibes a posi i e wedge be ween e o
and consump ion.
4.3 Quan i a i e Analysis
In his subsec ion, we analyze quan i a i ely he impac o implemen ing he op imal un-
employmen insu ance (UI) con ac de i ed abo e. We use he Uni ed S a es as ou bench-
ma k. Ou ini ial s ep in ol es he calib a ion o model pa ame e s based on he p e ailing
policy amewo k. To do his, we i s w i e he p oblem o he agen unde such a policy.
An unemployed wo ke is en i led o UI o a ixed du a ion o Tpe iods, wi h a cons an
bene i payou o b. A e T, i he indi idual emains unemployed, hey ecei e a gua an-
eed minimum consump ion loo o . The wo ke chooses in which ma ke o sea ch; ha
is, hey choose he job- inding a e p. Plus, hey choose hei p e e ed consump ion and
sa ings bundle, (c, a′).
7The cons an absolu e isk a e sion (CARA) case is he only one o which Shime and We ning [2007]
ha e heo e ical esul s o he non-obse able sa ings scena io. They o e nume ical explo a ions o he
cons an ela i e isk a e sion (CRRA) case. Because we a e also in e es ed in unde s anding choices a he
in ensi e ma gin, we supp ess income e ec s a his ma gin h ough he assump ion o quasi-linea i y, as in
G eenwood e al. [1988].
17

Deno e by Vu( , a) he alue unc ion o an unemployed wo ke ha s ill has pe iods
o UI and owns asse s a. I he wo ke is s ill eligible o UI (i.e., ≥0), hei alue
unc ion eads:
Vu( , a) = max
p,c,a′pVe(a, p) + (1 −p) [φ(c) + βVu( −1, a′)]
s. . c+a′= (1 + )a+b,
whe e Ve(a, p)deno es he alue o being employed in a ype-pjob wi h asse le el a. The
con inua ion alue Vu( −1, a′) e lec s he ac ha he wo ke will ha e one ewe pe iod
o UI nex pe iod i hey do no ind a job in he cu en pe iod.
The alue unc ion o an unemployed wo ke wi hou UI (i.e., a e Tpe iods o un-
employmen ) eads:
Vu(0, a) = max
p,c,a′pVe(a, p) + (1 −p) [φ(c) + βVu(0, a)]
s. . c+a′= (1 + )a+ .
The alue unc ion o an employed wo ke is gi en by:
Ve(a, p) = max
c,a′φ(c)−ηy(p) + κϱ(p)
p+βVe(a′, p)
s. . c+a′= (1 + )a+y(p),
whe e he income y(p)is de e mined by:
η′y(p) + κϱ(p)
p=φ′(c)
We mus now se unc ional o ms and pa ame e alues o pe o m coun e ac uals. We
assume he u ili y unc ion o consump ion is loga i hmic: φ(c) = log c. The disu ili y o
e o is gi en by: η(n) = η1nη2. Mo eo e , he labo ma ke igh ness is de e mined by he
unc ion: ϱ(p) = 1/(1/p −1). Each model pe iod co esponds o one week. Acco dingly,
we se he discoun ac o β= 0.961/52, a s anda d alue. We se he UI in he benchma k,
b, o 40% o he a e age income, he same a io as in Shime [2005]. We assume his bene i
las s o T= 26 weeks, as i does in he Uni ed S a es. Addi ionally, he consump ion loo
is ixed a 10% o he a e age income.
Th ee pa ame e s a e chosen in e nally so ha he benchma k model ma ches ce ain
18
da a a ge s: he pa ame e s ha con ol he disu ili y o e o , η1and η2, and he acancy
pos ing cos κ. These pa ame e s a e join ly chosen o ma ch h ee da a a ge s. The i s is
he mass o unemployed wo ke s who ind a job be o e he UI expi es: 86.8% acco ding o
Shime [2008]. The second da a a ge is he wage ma kdown; ha is, how much lowe is
he wage ela i e o he wo ke ’s p oduc i i y. Be ge e al. [2022] epo an a e age wage
ma kdown be ween 11% and 22%. We a ge he in e media e alue o 16.5%. Finally,
we ma ch he ela i e sea ch e o spen by an unemployed wo ke a week 26 o unem-
ploymen ( igh be o e losing he UI bene i ) e sus week 1. We a ge 50%, he numbe
epo ed by Ma inescu and Skandalis [2020]. Table 1 epo s he pa ame e alues and he
model i .
Table 1: Calib a ed Pa ame e s and Model Fi
Pa ame e s
κ η1η2
0.1352 0.200 5
Momen s
Ma kdown % Reemployed Rel. sea ch e o
Model 0.165 0.867 1.549
Da a 0.165 0.868 1.500
The i o he model is qui e good, as epo ed in Table 1. F om his benchma k, we ake
he alue unc ion o a wo ke ha s ill has all o hei UI paymen s o ecei e and owns he
a e age le el o asse s as in he da a: Vu(26,¯a).8We se his alue as he baseline u ili y
ha he planne will p omise he wo ke in he op imal unemploymen insu ance con ac s:
W0. We sol e o hese op imal con ac s unde wo scena ios: one in which he con ac s
a e obse able ( he con ac cha ac e ized in Sec ion 4.1) and ano he in which hey a e no
(Sec ion 4.2).
Unde obse able con ac s, he planne can p o ide he co esponding con ac mo e
cheaply, hough he p omised u ili y is he same. Quan i a i ely, wi h non-obse able con-
ac s, he cos o he p og am is 10.5% highe , a subs an ial inc ease.
Figu e 2 epo s he compa ison o di e en ou comes unde obse able e sus non-
obse able con ac s. UI declines wi h he du a ion o unemploymen in bo h cases. How-
e e , he decline is s eepe unde non-obse abili y (Panel a). This e lec s a dec easing
p omised u ili y o an unemployed wo ke h oughou he unemploymen spell (Panel b).
8The le el o ¯ais calib a ed by a ge ing he le el o liquid asse s o he median indi idual in he Uni ed
S a es as epo ed in Kaplan and Violan e [2014]. We ake hei numbe and di ide i by eal GDP pe wo ke
(USARGDPE om he S . Louis FRED da abase). The co esponding a io is 1.59.
19
Figu e 2: Ou comes unde he Op imal UI Con ac , Obse able e sus Non-obse able
(a) Unemploymen Insu ance
20 40 60 80 100
Week ( )
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1Obs.
Non obs.
(b) Wel a e o Wo ke s
20 40 60 80 100
Week ( )
-580
-560
-540
-520
-500
-480
-460
-440
Obse able
Unemployed - Non obs.
Employed - Non obs.
(c) P obabili y o Finding a Job
20 40 60 80 100
Week ( )
5.85
5.86
5.87
5.88
5.89
5.9
5.91
5.92
Obs.
Non obs.
(d) P oduc ion o Employed Wo ke
20 40 60 80 100
Week ( )
1.03
1.035
1.04
1.045
1.05 Obs.
Non obs.
(e) Income o Employed Wo ke
20 40 60 80 100
Week ( )
1
1.005
1.01
1.015
1.02
Obs.
Non obs.
( ) Consump ion o Employed Wo ke
20 40 60 80 100
Week ( )
0.94
0.95
0.96
0.97
0.98
0.99
1
1.01 Obs.
Non obs.
No es: The ou comes o an unemployed wo ke a e epo ed o each week du ing he unemploymen
spell. The ou comes o an employed wo ke a e hose o an indi idual who ound a job a exac ly week .
20
Wel a e upon employmen also goes down wi h he du a ion o unemploymen , as he plan-
ne wan s o incen i ize he agen o sea ch ha de ea lie on. The planne achie es his by
inc easing axes wi h he du a ion o he unemploymen spell, as we will see momen a -
ily. In he obse able con ac case, due o a binding incen i e compa ibili y cons ain , he
wel a e o he unemployed and he employed coincide (so ha only one line is displayed in
he igu e o he obse able case).
Lowe unemploymen insu ance and wel a e o e ime incen i ize he wo ke o sea ch
ha de o a job (Panel c). The e o e, wi h non-obse able con ac s, he sea ch e o he
wo ke engages in inc eases as e o e ime. This highe sea ch e o (highe p obabili y
o inding a job) ma e ializes because he indi idual is sea ching o jobs in which hey
ha e o wo k ha de and p oduce mo e (Panel d). Consequen ly, he wo ke is compensa ed
o his highe e o , and hei income as an employed wo ke is highe when hey ind a
job la e (Panel e). This happens because sea ch e o inc eases wi h ime and he wo ke
is compensa ed o his. Howe e , he consump ion o he employed wo ke is lowe o
hose who ind a job la e (Panel ), implying he ax inc eases wi h he du a ion o unem-
ploymen . As seen in he p e ious sec ions, he planne inds i op imal o inc ease axes
o e he du a ion o unemploymen o incen i ize sea ch. Again, unde non-obse able
con ac s, he a ia ion h oughou unemploymen is s eepe .
Wi h non-obse able con ac s, he ax collec ed by he planne inc eases wi h he un-
employmen spell. This anspi es by pu ing oge he Panels (e) and ( ) in Figu e 2. In
he p e ious sec ion, we p o ed ha , in his non-obse able case, he planne imposes a
dis o iona y ax a e o incen i ize he e o supplied by he wo ke . We can compu e
such a ax a e in ou nume ical exe cise: 0.03%. So, hough he planne does dis o he
in ensi e ma gin o he wo ke , i does so wi h a somewha low ax a e. This low le el o
he dis o iona y ax ma e ializes because, in ou model, once employed, he wo ke ne e
loses hei job. Hence, he planne mus ake in o accoun ha , by imposing his dis o ion,
he wo ke will ace i o e e . We e he wo ke a isk o losing hei job and sea ching
again, his ax a e would ha e been highe . This dis o iona y ax a e also inc eases wi h
he du a ion o he unemploymen spell.
In sum, he non-obse abili y o he employmen con ac has an impo an quan i a i e
e ec on he op imal UI con ac o e ed by he planne . Wi h non-obse able con ac s,
he planne dec eases UI as e and inc eases axa ion upon employmen as e wi h he
du a ion o unemploymen (including adding a dis o iona y componen o he ax). These
changes all add up o a conside ably mo e expensi e UI p og am.
21
he ma gin unde he non-obse abili y o employmen con ac s. This dis o ion occu s
because unobse ed high-quali y jobs a e e ec i ely subsidized by he unemploymen in-
su ance p og am. In oducing a ma ginal ax on ea nings discou ages such jobs, ende -
ing hem less a ac i e despi e p o iding highe non-wage quali y. This dis o iona y ax
eme ges whe he o no he agen can hide hei sa ings om he planne .
We calib a e ou model o he U.S. economy. When employmen con ac s a e un-
obse able o policymake s, se e al consequences a ise: unemploymen bene i s decline
mo e apidly, axes upon e-employmen inc ease mo e s eeply wi h unemploymen du a-
ion, and dis o iona y ax policies become necessa y. Collec i ely, hese adjus men s esul
in an unemploymen insu ance p og am ha , when compa ed o a scena io whe e con ac s
a e ully obse able, is 10.5% cos lie while p o iding he same le el o wel a e.
Ou indings highligh he need o e ec i e moni o ing o he non-wage dimensions
o jobs o alida e bene i eligibili y and de e job seeke s om solely pu suing highly de-
si able ye imp obable posi ions. Implemen ing obus o e sigh policies can help mi iga e
mo al haza d and acili a e a mo e e icien unemploymen insu ance sys em aligned wi h
he eali ies o oday’s complex labo ma ke .
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A Da a Appendix
This appendix p o ides he ull esul s o he eg essions ha yield he coe icien s om
Figu e 1in Sec ion 2. See Tables 2and 3. These eg essions use U.S. da a om he
Ma ch Supplemen o he Cu en Popula ions Su eys (CPS) be ween 2009 and 2022.
The con ols used in some o he eg essions a e age, gende and educa ion.
Table 2: Linea P obabili y Model, P obabili y o Being Unemployed One Yea La e
(1) (2)
Unemployed Unemployed
insu ance 0.0820∗∗∗ 0.0691∗∗∗
(0.00995) (0.0101)
Con ols No Yes
N 11804 11804
S anda d e o s in pa en heses
∗p < 0.05,∗∗ p < 0.01,∗∗∗ p < 0.001
30
Table 3: Linea P obabili y Model, P obabili y o Ha ing a Job wi h some Cha ac e is ics
One Yea La e
(1) (2) (3) (4)
Unionized Unionized Heal h Heal h
insu ance 0.0167∗∗∗ 0.0148∗∗∗ 0.0310 0.0149
(0.00431) (0.00439) (0.0366) (0.0373)
Con ols No Yes No Yes
N 7422 7422 670 670
S anda d e o s in pa en heses
∗p < 0.05,∗∗ p < 0.01,∗∗∗ p < 0.001
B Theo e ical Appendix
B.1 P oo s o Sec ion 4.1
P oo o Lemma 4.1.Fi s , we show ha he cons ain (5) binds. The i s o de condi ion
wi h espec o cu eads
φ′(cu) = 1−p
µ(1 −p)−λ>0.
I µ≤0 hen λ < 0and hus p > 0and hen using he i s o de condi ion w. . . cewe
ob ain
φ′(ce) = p
pµ +λ<0,
a con adic ion.
Nex , owa ds a con adic ion, assume ha , wi hou loss o gene ali y, he cons ain (6)
does no bind a = 0,φ′(cu
0) = φ′(ce
0) = µ−1
0=η′(n0).In his case,
φ(ce
0)−η(n0)< φ(cu
0).(13)
The mo al haza d cons ain mus bind o some > 0, o he wise, φ(cu
) = µ−1
0, o
e e y . This means ha ge ing a job in pe iod ze o is wo se han being unemployed
o e e .
Assume ha he i s pe iod in which he cons ain binds is = 1 ( he o he case is
analogous). We ha e µ1=µ0,λ1>0and, hence,
φ′(ce
1) = p1
p1µ0+λ1
=η′(n1).
31
The e o e,
φ(ce
0)−η(n0)< φ(ce
1)−η(n1)(14)
Hence, using (13) and (14) we ob ain
φ(ce
0)−η(n0)
1−β< φ(cu
0) + βφ(ce
1)−η(n1)
1−β,
which, using he ac ha he mo al haza d cons ain was binding in he second pe iod,
implies ha he wo ke s ic ly p e e s being unemployed o ge ing a job a ze o, a con a-
dic ion.
B.2 P oo s o Sec ion 4.2
Lemma B.1 Bo h mappings, z(·)and cu(·), a e s ic ly inc easing, wice di e en iable,
and s ic ly con ex. Mo eo e , he e exis s W∗such ha z(W∗) = cu(W∗),z(W)>
cu(W), o all W > W∗, and z(W)< cu(W), o all W < W∗.
P oo o Lemma B.1.Le ye(W)be gi en by
a gmax
ye
[φ(ye+z(W)) −η(ye+ϕ)] ,
and no e ha i φ′(z(W)) −η′(ϕ)≤0, hen ye(W) = 0. O he wise, ye(W)is gi en by
φ′(ye+z(W)) −η′(ye+ϕ) = 0.
Hence, because z(W) + ye(W)> cu(W), we ha e
z′(W) = 1
φ′(z(W) + ye(W)) >1
φ′(cu(W)) =cu′(W).
This implies ha i z(·)and cu(·)c oss a mos once, and z(W)> cu(W)( esp. z(W)<
cu(W)) o e e y u ili y g ea e ( esp. lowe ) han his u ili y le el.
Since z(W)→ ∞ as W→ ∞, we ha e ye(W)=0 o Wla ge enough, which
implies z(W)> cu(W). The exis ence o a small Wsuch ha z(W)< cu(W)holds by
assump ion. The e o e, W∗exis s by con inui y.
I emains o show ha bo h mappings a e s ic ly con ex. Since ce(W) := z(W) +
32
ye(W)is s ic ly inc easing wi h posi i e de i a i e, we ha e
z′′ (W) = −φ′′ (ce(W))
φ′(ce(W))2ce′(W)>0,
and
cu′′ (W) = −φ′′ (cu(W))
φ′(cu(W))2cu′(W)>0.
Lemma B.2 Suppose ha , i a wo ke ge s a job, hen hey mus ea n ce+T, paying T
o he go e nmen , o consume ce, whe eas i he wo ke ails o ge a job hen hey ob ain
he con inua ion u ili y W. Then his p oblem admi s a unique solu ion. I he solu ion is
in e io , i is gi en by he associa ed i s -o de condi ions.
P oo .
Conside he p oblem
max pφ(ce)−ηce+T+κϱ(p)
p−W
This p oblem admi s an in e io solu ion i and only i φ(ce)−η(ce+T)> W.
Assume ha his is he case and conside p ha makes i s de i a i e equal o ze o:
φ(ce)−ηce+T+κϱ(p)
p−W−pη′ce+T+κϱ(p)
pκd
dp ϱ(p)
p= 0
Di e en ia e he le -hand side again o ob ain
−2η′ce+T+κϱ(p)
pκd
dp ϱ(p)
p−pη′ce+T+κϱ(p)
pκd2
dp2ϱ(p)
p
−pη′′ ce+T+κϱ(p)
pκd
dp ϱ(p)
p2
.
To show ha he exp ession abo e is nega i e, i su ices o show ha
−2d
dp ϱ(p)
p−pd2
dp2ϱ(p)
p<0⇔2ϱ′(p)p−ϱ(p)
p2+pd
dp ϱ′(p)p−ϱ(p)
p2>0
⇔p2d
dp [ϱ′(p)p−ϱ(p)] >0⇔ϱ′′(p)
p>0.
33

Lemma B.3 Fo e e y W, le C(W)be he planne ’s cos o p o iding u ili y W. The
mapping C(·)is di e en iable a W o e e y > 0.
P oo . We p o e ha Cis di e en iable a W . Fo ha , we assume ha p >0as he o he
case is analogous. Conside any small ϵ∈Rand no e ha he ollowing pe u ba ion is
easible:
˜u −1,˜u ,˜ce
=u −1+ϵ, u −ϵβ−1, φ−1(φ(ce
) + ϵ).
One can hus apply he a gumen in Clausen and S ub o conclude ha
C′(W ) = −c′(u ) = 1
φ′(u ).
Lemma B.4 The mul iplie s, µand λ, a e s ic ly posi i e i he e is a sea ch.
P oo . Fi s , no ice ha
[µ(1 −p)−λ]φ′(cu) = 1 −p
and pµ +λ
1−βφ′(ce) = p
1−β
Hence, µ= 0 implies φ′(cu)φ′(ce)≤0, which is absu d.
Hence assume owa ds a con adic ion ha λ0≤0.Clea ly, he e is a las pe iod a
which λ ≤0and λ +1 >0. O he wise, as we will e i y below, cu
≥ce
o e e y , and
hence he e is no sea ch. Assume ha λ1>0(case in which λs≤0 o all s < and
λ >0 o some > 1can be handled analogously).
F om he i s -o de condi ion wi h espec o p, we ge
φ′(cu) = 1
µ−λ(1 −p)−1≤1
µ+λp−1=φ′(ce).
Hence, cu≥ce.
Mo eo e , no ice ha om he i s o de condi ion we ha e
C′(W1) = −µ0+λ0
(1 −p)=−µ1,
34
which implies
µ1=µ0−λ0
(1 −p)≥µ0.
This and λ0≤0< λ1imply
φ′(ce
1) = 1
µ1+p−1
1λ1
<1
µ0+p−1
0λ0
=φ′(ce
0).
Hence,
ce
1> ce
0.(15)
We can ea ange he i s o de condi ion wi h espec o ye o ge
µη′ye+κϱ(p)
p= 1 −λη′′ ye+κϱ(p)
pκd
dp ϱ(p)
p−λ
pη′ye+κϱ(p)
p.
The e o e, λ0≤0< λ1imply
η′ye
1+κϱ(p1)
p1< µ−1
1.
Simila ly,
η′ye
0+κϱ(p0)
p0≥µ−1
0.
Since µ1≥µ0, his implies
ye
1+κϱ(p1)
p1
< ye
0+κϱ(p0)
p0
,
and
ηye
1+κϱ(p1)
p1< η ye
0+κϱ(p0)
p0,
because ηis s ic ly con ex.
35
Since p0>0, by he assump ion o he lemma, we ha e
0<1
1−βφ(ce
0)−ηye
0+κϱ(p0)
p0−[φ(cu
0) + βW1]
=1
1−βφ(ce
0)−ηye
0+κϱ(p0)
p0−φ(cu
0)
−βp1
1
1−βφ(ce
1)−ηye
1+κϱ(p1)
p1+ (1 −p1) [φ(cu
1) + βW2]
=φ(ce
0)−ηye
0+κϱ(p0)
p0−φ(cu
0) + β"1
1−βφ(ce
0)−ηye
0+κϱ(p0)
p0
−p1
1
1−βφ(ce
1)−ηye
1+κϱ(p1)
p1−(1 −p1) [φ(cu
1) + βW2]#
=1
1−βφ(ce
0)−ηye
0+κϱ(p0)
p0−φ(cu
0)
−βp1
1
1−βφ(ce
1)−ηye
1+κϱ(p1)
p1+ (1 −p1) [φ(cu
1) + βW2](16)
Since p1>0, due o λ1>0, we ha e
1
1−βφ(ce
1)−ηye
1+κϱ(p1)
p1> φ(cu
1) + βW2
Hence,
φ(ce
0)−ηye
0+κϱ(p0)
p0−φ(cu
0) + β(1
1−βφ(ce
0)−ηye
0+κϱ(p0)
p0
−p1
1
1−βφ(ce
1)−ηye
1+κϱ(p1)
p1−(1 −p1) [φ(cu
1) + βW2])
< φ(ce
0)−ηye
0+κϱ(p0)
p0−φ(cu
0)+β1
1−βφ(ce
0)−ηye
0+κϱ(p0)
p0−[φ(cu
1) + βW2]
Since he i s line om he las e m is nega i e, he en i e e m is less han
β1
1−βφ(ce
0)−ηye
0+κϱ(p0)
p0−[φ(cu
1) + βW2],
36
which is less han
1
1−βφ(ce
0)−ηye
0+κϱ(p0)
p0−[φ(cu
1) + βW2],
since he e m is posi i e.
Since φ(ce
0)< φ(ce
1), and
ηye
0+κϱ(p0)
p0> η ye
1+κϱ(p1)
p1,
his is less han
1
1−βφ(ce
1)−ηye
1+κϱ(p1)
p1−[φ(cu
1) + βW2].
Hence, using he i s -o de condi ions wi h espec o p, he algeb a jus pe o med
means ha
p1
1−βη′ye
1+κϱ(p1)
p1κϱ(p1)
p1>p0
1−βη′ye
0+κϱ(p0)
p0κϱ(p0)
p0.(17)
Since
ye
1+κϱ(p1)
p1
< ye
0+κϱ(p0)
p0
,
I ye
1≥ye
0,we will ha e p1< p0which oge he con adic (17). We conclude ha ye
1< ye
0.
Finally, no ice ha λ1>0and he i s o de condi ion wi h espec o pand he ac
ha pis a local maximum imply
ye
0−ce
0
1−β≤ −cu
0+βC (W1).(18)
Analogously, in pe iod 1, using λ0≤0, he i s o de condi ion wi h espec o p
implies ye
1−ce
1
1−β≥ −cu
1+βC (W2).
Bu no ice ha
C(W1) = p1
ye
1−ce
1
1−β+ (1 −p1) [−cu+βC (W2)] ≤ye
1−ce
1
1−β(19)
37
P oo o Theo em 5.2.Le {(p∗
, ye∗
, cu∗
, ce∗
)}∞
=0 be he op imal alloca ion.
No ice ha W∗
0=p∗
0We∗
0+ (1 −p∗
0)Wu∗
0. I We∗
0≤Wu∗
0, hen p∗
0= 0. In his case, he
op imal alloca ion can be implemen ed by asse s
a0=−α−1log (−(1 −β)W0)
1−β
and a some pai (ye, Te)wi h ye=Te. The wo ke bes esponds by ne e sea ching
o a job and consuming −(1 −β)α−1log (−(1 −β)W0)in e e y pe iod. Acco ding o
Lemma B.8 his is op imal.
Nex , assume ha We∗
0> Wu∗
0. Conside he i s o de condi ion:
−1
1−βexp −αce∗
0−ηye∗
0+κϱ(p∗
0)
p∗
0−Wu∗
0−
αp∗
0
1−βexp −αce∗
0−ηye∗
0+κϱ(p∗
0)
p∗
0η′ye
0+κϱ(p∗
0)
p∗
0κϱ(p∗
0)
p∗
0′
= 0,
and he ollowing p omise-keeping condi ion, W∗
0=p∗
0We∗
0+ (1 −p∗
0)Wu∗
0.
By sol ing hese wo equa ions we ob ain:
We∗
0
W∗
0
=1 + αp∗
0(1 −p∗
0)η′ye∗
0+κϱ(p∗
0)
p∗
0κϱ(p∗
0)
p∗
0′−1
Wu∗
0
W∗
0
= 1 +
αp∗2
0η′(ne∗
0)κϱ(p∗
0)
p∗
0
1 + αp∗
0(1 −p∗
0)αη′(ne∗
0)κϱ(p∗
0)
p∗
0.
Nex , no ice ha We∗
0deli e s ce∗
0by
−1
1−βexp −αce∗
0−ηye∗
0+κϱ(p∗
0)
p∗
0=We∗
0,
which implies
ce∗
0=−α−1log (−(1 −β)We∗
0) + ηye∗
0+κϱ(p∗
0)
p∗
0.
44

We claim ha he e exis s (a∗
0, Te∗) ha sol es he sys em:
ce∗
0= (1 −β)a0+ye∗
0+Te(22)
Wu∗
0= max
c−exp {−αc}+βU β−1(a0−c), ye∗
0, Te,(23)
whe e U(a, ye∗
0, Te)is he u ili y o an agen who s a s a pe iod unemployed and aces a
simple policy, (a, ye∗
0, Te).
I Te=yeand a0=ce∗
0
1−β, hen
W∗
1<max
c−exp {−αc}+βU β−1(a0−c), ye∗
0, Te,(24)
as he agen can keep consump ion cons an a ce∗e en wi hou aking a job.
The indi idual bes esponds o ha con ac by choosing p= 0 in e e y pe iod. F om
his poin , i we dec ease a0by −ε
1−βand dec ease Teby ε, he planne ’s payo is inc eased
by
ε
1−β1−p(a, ye∗
0, Te)
1−(1 −p(a, ye∗
0, Te)) β>0.(25)
Nex , no ice ha , by cons uc ion,
−1
1−βexp −α(1 −β)a0+ye∗
0+Te−ηye∗
0+κϱ(p∗
0)
p∗
0=We∗
0.
Recall ha he inequali y (24) implies ha p(a, ye∗
0, Te)< p∗. We claim ha , i we
keep dec easing a0by −ε
1−βand Teby ε, we can gene a e (a∗
0, Te∗)sa is ying (22) and
(23). O he wise, as we ake a0 o −∞, he planne ’s e enue goes o in ini y while he
wo ke ’s u ili y a he beginning emains abo e W∗
0, a con adic ion. F om he i s o de
condi ion, we know ha p emains bounded below p∗(and by lemma B.5, his holds in
e e y u u e pe iod) and he p incipal ob ains in ini e p o i s because o (25). A he same
ime, he wo ke ’s u ili y emains g ea e han pWe∗
0+ (1 −p)Wu∗
0, a con adic ion.
The easoning abo e shows ha o e ing (a∗
0, ye∗, Te∗)in he i s pe iod is op imal
o gene a e u ili y W∗
0. In his case, Lemma B.8 implies ha (a∗
1, ye∗, Te∗)is op imal o
gene a e u ili y W∗
1,whe e a∗
1is he asse holdings chosen by he agen . Induc i ely, we
conclude ha (a∗
, ye∗, Te∗)is op imal o gene a e u ili y W∗
o e e y and hence he
simple policy (a∗
0, ye∗, Te∗)is op imal.
Lemma B.9 We ha e ∂p
∂ce>0and ∂p
∂ye<0.
P oo . We mus calcula e ∂p
∂ceand ∂p
∂ye. Le ce:= ye−Te, assume wi hou a loss ha he
45
agen s a s wi h ze o asse s (Lemma B.5) and w i e W1 o he payo o an agen who
s a s a pe iod o unemploymen wi h ze o asse s. S a wi h he i s o de condi ion wi h
espec o p:
−1
1−βexp {−α[ce−η(ne)]} − max
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]
−αp
1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p)
p′
= 0.(26)
Nex , we ema k ha he p oblem is s ic ly conca e in p, and hence he de i a i e o
(26) w. . . pis s ic ly nega i e. Di e en ia ing his condi ion w. . . cewe ob ain
α
1−βexp {−α[ce−η(ne)]} − d
dcehmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i
+α2p
1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p)
p′
.
Now, no ice ha
d
dcehmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i<
−αmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}],(27)
whe e he las numbe is ob ained by he de i a i e o an inc ease in cin e e y s a e o
na u e.
46
The e o e, we ha e
α
1−βexp {−α[ce−η(ne)]} − d
dcehmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i
+α2p
1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p)
p′
=
α
1−βexp {−α[ce−η(ne)]}+αhmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i
+α2p
1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p)
p′
−αhmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i−
d
dcehmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i=
−αhmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i−
d
dcehhmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]ii>0,
whe e we ha e used (26) and (27). The e o e, ∂p/∂ce>0.
Nex , di e en ia ing he i s o de condi ion wi h espec o ye, we ge
−αη′(ne)
1−βexp {−α[ce−η(ne)]}− d
dyehmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i
−α2p
1−βexp {−α[ce−η(ne)]}η′(ne)2κϱ(p)
p′
−αpη′′(ne)
1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p)
p′
.
No ice ha
d
dyehmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i>
αη′(ne)hmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i.(28)
47
Hence,
−αη′(ne)
1−βexp −αce−ηye+κϱ(p)
p−
d
dyehmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i
−α2pη′(ne)
1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p)
p′
−αpη′′(ne)
1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p)
p′
=−αη′(ne)
1−βexp {−α[ce−η(ne)]} −
αη′(ne)hmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i
−α2pη′(ne)
1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p)
p′
αη′(ne)hmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i−
d
dyehmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i
−αpη′′ (ne)1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p)
p′
=
αη′(ne)hmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i−
d
dyehmax
a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i
−αp
1−βexp {−α[ce−η(ne)]}η′′ (ne)η′(ne)κϱ(p)
p′
<0,
whe e we ha e used (26) and (28).
Lemma B.10 We ha e
∞
X
=0
pβ (1 −p) −11 + exp {−α(1 −β)a }
W0(1 −β)exp {−α[ce−η(ne)]}>0.
48
P oo . We ha e
∞
X
=0
pβ (1 −p) −1−1
1−β−exp {−α(1 −β)a }
(1 −β)2W0
exp {−α[ce−η(ne)]}<0
⇔
∞
X
=0
pβ (1 −p) −1"−exp {−α(1 −β)a }
(1 −β)P∞
=0 pβ (1 −p) −1exp {−α[ce−η(ne)]}#> W0,
since z→ − exp{−αz}is s ic ly inc easing. No ice ha U0is he mix u e o he dis i-
bu ion Feo e employed payo s de ined abo e and he dis ibu ion o e −exp{−αcu
},
which we call Fu. I ollows ha i Fe i s o de s ochas ic domina es Fu. Hence o any
λ∈(0,1),
Zxd [λFe(x) + (1 −λ)Fe(x)] <ZxdFe(x).
I , he e o e, su ices o show ha W0<RxdFe(x).
We ha e
W0(1 −β) = p(1 −β)W0
e+ (1 −p) (1 −β)h−exp{−αcu
0}+
β[pWe
1+ (1 −p) [−exp{−αcu
1}+βWu
2]] i.
Using W0
e>−exp{−αcu
0}
1−βand −exp{−αcu
0}
1−β=pWe
1+ (1 −p) [−exp{−αcu
1}+βWu
2],
we ha e
W0<pW0
e+β(1 −p) [pWe
1+ (1 −p) [exp{−αcu
1}+βWu
2]]
1−(1 −p) (1 −β).
P oceeding analogously, i ollows ha he las exp ession is less han
pW0
e+β(1 −p) [pWe
1+ (1 −p)βWu
2]
1−(1 −p) (1 −β)−(1 −p)2β2.
P oceeding analogously and aking he limi , we ob ain he desi ed inequali y.
P oo o Theo em 5.3. Pa (i). Recall om (11)
∂
∂p p
1−(1 −p)βye−ce
1−β∂p
∂ce=p(ye, ce)
1−(1 −p(ye, ce)) β+Uce(ye, ce)
eα(1−β)a0α(1 −β)W0
.
Since ∂
∂p p
1−(1 −p)β>0and ∂p
∂ce>0,
49

ye−cehas he same sign as
−
∞
X
=0
pβ (1 −p) −1"−1
1−β−exp −αce−ηye+κϱ(p)/p
(1 −β)2W0#,
by Lemma B.9, which is s ic ly posi i e by Lemma B.10.
Pa (ii). Conside he p oblem
C(W0) = max
W1,ce,ye
pye−ce
1−β+ (1 −p)βC(e−αa(1−β)W1),
subjec o
−p
1−βexp −αce−ηye+κϱ(p)
p+
(1 −p) max
a′−exp{αa′β}+ exp{−αa′(1 −β)}βW1−W0= 0
and
−1
1−βexp −αce−ηye+κϱ(p)
p
−max
a′−exp{αa′β}+ exp{−αa′(1 −β)}βW1
−αp 1
1−βexp −αce−ηye+κϱ(p)
pη′ye+κϱ(p)
pκϱ(p)
p′
= 0.
Plugging he las cons ain in o he p oblem, one ob ains he ollowing Lag angian
C(W0) = max
W1,ce,ye
pye−ce
1−β+ (1 −p)βC(e−αa(1−β)W1)+
µ"−1
1−βexp −αce−ηye+κϱ(p)
p−
α(1 −p)p1
1−βexp −αce−ηye+κϱ(p)
pη′ye+κϱ(p)
pκϱ(p)
p′
−W0#.
50
The e o e, we ha e he i s o de condi ions wi h espec o ce,
p=µα exp −αce−ηye+κϱ(p)
p1 + α(1 −p)pη′ye+κϱ(p)
pκϱ(p)
p′,
and wi h espec o ye,
p=µα exp −αce−ηye+κϱ(p)
p"η′ye+κϱ(p)
p+
α(1 −p)pη′ye+κϱ(p)
p2
κϱ(p)
p′#
+µ(1 −p)pexp −αce−ηye+κϱ(p)
pη′′ ye+κϱ(p)
pκϱ(p)
p′
.
The e o e, we ha e
η′ye+κϱ(p)
p= 1 −
(1 −p)pη′′ye+κϱ(p)/pκϱ(p)
p′
α1 + α(1 −p)pη′ye+κϱ(p)/pκϱ(p)
p′.
Pa (iii). No ice ha W∗
< We∗
and hence i su ices o show ha lim We∗
=−∞. We
ha e
lim(1 −β)We∗
=
−lim exp −αc∗
0+ (1 −β) ¯a0−ηye∗+κϱ(p∗)
p∗α( −1) ∆c=−∞.
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C Ex ension: GHH-CARA ype and Obse able Sa ings
In his sec ion, we conside he case o obse able sa ings wi h pe iod u ili y o he o m
U(c, n) = −exp {−α[c−η(n)]}.
We can w i e he Lag angean as
C(W0) = max p
1−β(ye−ce) + (1 −p) [−cu+βC (W1)] ,
subjec o
p
1−β[−exp {−α[ce−η(ne)]}] + (1 −p) [−exp {−α[cu]}+βW1]−W0≥0,
and
1
1−β−exp −αc−ηye+κϱ(p)
p+ exp {−α[cu]} − βW1
=1
1−βαη′ye+κϱ(p)
pκϱ(p)
p′
exp −αc−ηye+κϱ(p)
p.
Le
Ue:= exp −αc−ηye+κϱ(p)
p
Uu:= exp {−αcu}.
The i s o de condi ion o ce
0is
−p+µpαUe+λαUe= 0.
The i s o de condi ion o cu
0is
−(1 −p) + (1 −p)µUu−λαUu= 0.
52
The i s o de condi ion o yeis
p−pµαUeη′(ne)−λαUeη′(ne)
−λUeαη′′(ne)κϱ(p)
p′
+α2Ue[η′(ne)]2κϱ(p)
p′= 0
F om hese, we ha e
Ue=p
µpα +λα
Uu=1−p
µ(1 −p)α−λα
p−η′(ne)Ue[pµα +λα] = λUeαη′′(ne)κϱ(p)
p′
+α2Ue[η′(ne)]2κϱ(p)
p′
1−η′(ne) = λUe
pαη′′(ne)κϱ(p)
p′
+α2[η′(ne)]2κϱ(p)
p′(29)
C′(W1) = −µ0+λ0
(1 −p0)=−µ1,
which implies
µ1=µ0−λ0
(1 −p).
Mo eo e , he de i a i e wi h espec o pimplies
p
1−β(ye−ce) = λαUe
1−β"η′(ne)κϱ(p)
p′
+η′′(ne)κϱ(p)
p′
+η′(ne)κϱ(p)
p′′
+η′(ne)κϱ(p)
p′2#.(30)
Lemma C.1 The mul iplie s µand λa e s ic ly posi i e i he e is sea ch.
P oo . Fi s no ice ha
Uu
0=1−p0
µ0(1 −p0)α−λ0α
Ue
0=p0
µ0p0α+λ0α
hence µ0= 0 implies Uu
0Ue
0≤0, which is an absu d.
Now, assume owa ds a con adic ion ha λ0≤0.Clea ly, he e is a las pe iod a which
53
which is no possible, a con adic ion.
Bu hen, by a con inui y a gumen , o e e y ε > 0, he e exis s a pe iod ∗such ha
≥ ∗implies ha he planne ’s u ili y is εaway om −cu
∞/(1 −β), while he wo ke ’s
u ili y is εaway om φ(cu
∞)/(1 −β).
I ollows by Assump ion DS ha he e is ε > 0such ha , i he planne demands
p oduc ion y∗in exchange o consump ion η(y∗+κϕ) + χ(φ(cu
∞)/(1 −β)) + ε, hen he
wo ke sea ches wi h p obabili y bounded away om some p > 0 o e e y la ge enough.
Mo eo e , his εcan be chosen o make bo h playe s be e o , a con adic ion.
D Va iable E o and Ameni ies
Thus a we ha e alked abou ameni ies sugges ing ha hei supply plays an analogous
ole o e o equi emen s. In eali y bo h dimensions will simul aneously help de ine wha
a desi able job is. In his ex ension, we add ameni ies o he one-pe iod model explici ly
connec ing i o he e o model we ha e p esen ed.
Assume ha ameni ies cos a o he i m and lead o a bene i ϕ(a)by making he
wo king en i onmen mo e pleasan . We can w i e he p oblem as
C(W0) = max p
1−β(ye−ce)−(1 −p)cu
subjec o12
pφ(ce)−ηye−ϕ(a)+(κ+a)ϱ(p)
p+ (1 −p)c(u)≥0,(42)
and
φ(ce)−ηye−ϕ(a)+(κ+a)ϱ(p)
p−c(u) =
p(κ+a)η′ye−ϕ(a)+(κ+a)ϱ(p)
pϱ(p)
p′
.(43)
The e icien le el o ameni ies is he solu ion o
a b := a gmax
aϕ(a) + aϱ(p)
p.
12We a e assuming ha he cos o ameni ies mus be paid ega dless o whe he he acancy is illed.
60

This implies
a b (p) := (ϕ′)−1ϱ(p)
p,
which is dec easing in p.
Suppose he go e nmen can choose a. The i s o de condi ion implies
"(µ+pλ)η′ye−ϕ(a)+(κ+a)ϱ(p)
p
+λp (κ+a)η′′ ye−ϕ(a)+(κ+a)ϱ(p)
pϱ(p)
p′#×"ϕ′(a)−ϱ(p)
p#
=pλη′ye−ϕ(a)+(κ+a)ϱ(p)
pϱ(p)
p′
The e o e, he op imal policy implies ϕ′(a)> ϱ(p)/p, a posi i e wedge on he op imal
le el o ameni ies. The posi i e wedge on ameni ies a ises whe he he e o is ano he
in ensi e adjus men ma gin o no .
61