Relational enforcement

Achim, Pe e ; Knoep le, Jan
A icle
Rela ional en o cemen
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Achim, Pe e ; Knoep le, Jan (2024) : Rela ional en o cemen , Theo e ical
Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 19, Iss. 2, pp. 823-863,
h ps://doi.o g/10.3982/TE5183
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Theo e ical Economics 19 (2024), 823–863 1555-7561/20240823
Rela ional en o cemen
Pe e Achim
Depa men o Economics, Uni e si y o Yo k
Jan Knoep le
School o Economics and Finance, Queen Ma y Uni e si y o London
A p incipal incen i izes an agen o main ain compliance and o u h ully an-
nounce any b eaches o compliance. Compliance is impe ec ly con olled by he
agen ’s p i a e e o choices, is pa ially pe sis en , and is e i iable by he p inci-
pal only h ough cos ly inspec ions. We show ha in p incipal-op imal equilib ia,
he p incipal en o ces maximum compliance using de e minis ic inspec ions. Pe-
iodic inspec ion cycles a e suspended du ing pe iods o sel - epo ed noncom-
pliance, du ing which he agen is ined. We show how commi men o andom
inspec ions would bene i he p incipal, and discuss possible ways o he p inci-
pal o o e come he commi men p oblem.
Keywo ds. Rela ional con ac s, dynamic en o cemen , pe sis ence, cos ly in-
spec ions.
JEL classi ica ion. C73, D83.
1. In oduc ion
In 2018 and 2019 wo plane c ashes killed 346 people and led o a wo ldwide g ound-
ing o he Boeing 737 MAX.1An in es iga ion by he U.S. Cong ess concluded ha he
acciden s we e o a la ge ex en due o “g ossly insu icien o e sigh by he FAA” (Fed-
e al A ia ion Adminis a ion).2S a ing om he ea ly 2000s, he FAA had inc easingly
us ed manu ac u e s o ce i y hei own planes o sa e cos s. By 2018, Boeing had
sel -ce i ied nea ly all o i s wo k (Ki oe , Gelles, and Nicas (2019)). Boeing ushed he
de elopmen o he 737 MAX a he expense o sa e y. This case illus a es he isk in
elying on sel - epo ed quali y assu ances wi hou su icien o e sigh .
Pe e Achim: [email p o ec ed]
Jan Knoep le: [email p o ec ed]
We a e hank ul o F ancesc Dilmé, Daniel Hause , Flo ian Ho mann, Ma in Poll ich, S en Rady, and Alex
Smolin o aluable commen s, and hank pa icipan s a he Canadian Economic Theo y Mee ing in Van-
cou e and he Econome ic Socie y Summe Mee ing in S Louis in 2017, as well as semina pa icipan s a
Bonn and HECER. Achim is g a e ul o Ch is Shannon and he Economics Depa men a UC Be keley o
hei hospi ali y du ing a p oduc i e isi in he all o 2016 and o he Hausdo Cen e o Ma hema ics in
Bonn o p o iding inancial suppo . Knoep le acknowledges inancial suppo by he Ge man Resea ch
Founda ion h ough CRC TR224-B04 and om he Academy o Finland (P ojec 325218).
1As a esul , Boeing su e ed an ope a ional loss o o e $20 billion. The es ima ed impac on he U.S.
economy as a whole was a 0.4 pe cen age poin s loss in g oss domes ic p oduc (GDP) g ow h (di Gio anni
e al. (2020)).
2See U.S. House o Rep esen a i es (2020).
©2024 The Au ho s. Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h ps://econ heo y.o g.h ps://doi.o g/10.3982/TE5183
824 Achim and Knoep le Theo e ical Economics 19 (2024)
In his pape , we s udy en o cemen ela ionships in which he agen p i a ely con-
ols and obse es he s a e o compliance and makes epo s o a p incipal wi hou
commi men powe . Compliance is pa ially pe sis en o e ime and can be obse ed
by he p incipal only h ough cos ly inspec ions. The p incipal schedules inspec ions
and imposes ines o incen i ize he agen o exe e o and o sel - epo ins ances
o noncompliance. We show ha he p incipal can induce he agen o exe ull e -
o and epo u h ully a all imes h ough ela ional incen i es. The p incipal ca ies
ou inspec ions despi e knowing he esul be o ehand. Ou analysis highligh s he im-
po ance o he pe sis en e ec o e o . Fu he , he p incipal canno gain om an-
domized inspec ions when she lacks commi men , bu andom inspec ions would be
op imal wi h commi men .
Public-sec o applica ions o ou model include banking supe ision o ensu e ha
banks main ain unc ioning in e nal isk assessmen s3and en i onmen al p o ec ion
whe e he co esponding go e nmen agency ensu es he en o cemen o egula ion by
i ms.4Simila ly, p i a e-sec o o ganiza ions mus ensu e in e nally ha employees
ollow egula ions.5
We conside p incipal-op imal equilib ia in which he agen u h ully discloses all
ins ances o noncompliance and exe s maximum e o h oughou . The p incipal-
op imal equilib ium we de i e in ou main esul (Theo em 1) en ails wo phases: a
moni o ing phase and a penal y phase. The agen is in he moni o ing phase when he
epo s compliance. Du ing he moni o ing phase, he agen is no ined, bu is subjec
o pe iodic inspec ions ha would esul in he maximal possible ine in he o -pa h
e en ha he inspec ion e ealed mis epo ing. The agen is in he penal y phase when
he epo s noncompliance. He pays a cons an low ine, bu is ne e inspec ed. He
also pays a lump sum ine each ime he s a e ansi ions om compliance o noncom-
pliance. C ucially, his ansi ion ine ea u es penal y educ ions o ea ly disclosu es
o noncompliance, an aspec ha is consis en wi h olun a y disclosu e schemes com-
monly used in p ac ice. The penal y educ ion p e en s he agen om delaying a epo
o an incidence o noncompliance in he hope ha he can egain compliance be o e he
nex inspec ion.6
No ably, inspec ion imes in his equilib ium a e en i ely p edic able o he agen ,
which implies ha he p incipal canno gain om andomized inspec ions. In ui i ely,
3See Sec ion 5 o a b ie discussion o banking supe ision p ac ices in Ge many.
4Fo he Uni ed S a es, Blundell, Gow isanka an, and Lange (2020) measu e he bene i s o dynamic
p ocedu es used by he EPA.
5Fo ins ance, he Eu opean Commission (2019) suppo s expo ing i ms in elabo a ing in e nal compli-
ance p og ams(ICP) o “mi iga e isks associa ed wi h dual-use ade con ols and o ensu e compliance”
in e nally. Dual-use goods ha e ci il and mili a y applica ions and all unde special egula ion o p o-
mo e in e na ional secu i y, e.g., by “coun e ing isks associa ed wi h he p oli e a ion o Weapons o Mass
Des uc ion” (Eu opean Commission (2019, p. 17)).
6Blundell, Gow isanka an, and Lange (2020) poin ou ha when de e mining he g a i y o ines, he
EPA akes in o accoun whe he a iola ion was sel - epo ed o no . See also Kapon (2022), who s udies
op imal design o ine educ ions (amnes ies) g an ed o sel - epo s o illegal ac i i y when de ec ions
a i e a an exogenous a e. Focusing on de e minis ic ine educ ion pa hs, Kapon (2022)also indsa
cyclical s uc u e o he op imal mechanism.
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Theo e ical Economics 19 (2024) Rela ional en o cemen 825
he p incipal”s mo i e o inspec is de i ed om he desi e o main ain a epu a ion o
igilance.7P edic able inspec ions p o ide he s onges incen i e o he p incipal o
inspec . As long as he p incipal inspec s as p esc ibed by he equilib ium s a egy, he
agen con inues o expec o be moni o ed and, hus, has an incen i e o exe e o
and epo u h ully. Howe e , when he p incipal delays inspec ions in a way ha is
de ec able by he agen , hen he agen will in e ha he p incipal has become non ig-
ilan . This in u n induces he agen o shi k, which ul ima ely leads o a b eakdown o
he ela ionship ha is cos ly o he p incipal. I he p incipal uses a andom s a egy
and mix u es a e unobse able o he agen , de ia ions by he p incipal a e ha de o
de ec o he agen . This des oys any po en ial bene i o he p incipal in equilib ium.
We exploi he op imali y o p edic able inspec ion schedules o he cons uc ion
o he p incipal-op imal equilib ium in Theo em 1: he equilib ium payo s coincide
wi h he alue o an auxilia y mechanism-design p oblem in which he p incipal is e-
s ic ed o non andom inspec ions. We hen ans o m his op imiza ion in o a dynamic
p og amming p oblem ha uses he agen ’s p omised u ili y as a s a e a iable.
Compa a i e s a ics e eal he impo ance o pe sis ence o ela ional en o cemen .
In equilib ium, he pe sis en e ec o e o on compliance allows he p incipal o de e
he agen om de ia ing h ough isola ed inspec ions. As he s a e’s pe sis ence an-
ishes, he inspec ion cos s necessa y o en o ce compliance g ow a bi a ily la ge.
We hen con as he ela ional en o cemen equilib ium wi h s ochas ic inspec ion
mechanisms. The abili y o commi o andom inspec ions dec eases he p incipal’s
inspec ion cos s ela i e o he de e minis ic inspec ions ha a e equi ed in he non-
commi men case. De e minis ic inspec ions a e mo e cos ly because o delay and noise
in he compliance p ocess, and due o he ansi ion penal ies ha a e needed o gen-
e a e incen i es o olun a y disclosu e. Compa a i e s a ics highligh he con as be-
ween ela ional en o cemen and he commi men case wi h andom inspec ions. As
he pe sis ence o he s a e o compliance anishes, he andom inspec ion cos s de-
c ease mono onically. We also discuss ways o o e come he p incipal’s commi men
p oblem, including ins i u ional sepa a ion o planning and execu ion o o e sigh and
inspec ion sampling combined wi h publicly accessible and e i iable eco ds.
The es o he pape is o ganized as ollows. A e discussing ela ed li e a u e, he
model se up is p esen ed in Sec ion 2.Sec ion3cha ac e izes he agen ’s incen i e con-
s ain s, shows ha he p incipal-op imal equilib ium can be de e mined by sol ing
an auxilia y mechanism-design p oblem, and ou lines how o sol e he auxilia y p ob-
lem. We p esen he p incipal-op imal equilib ium in Sec ion 4, ollowed by compa -
a i e s a ics. Sec ion 5discusses andom inspec ions. All p oo s a e con ained in he
Appendix.
7He e, “main aining a epu a ion” means ollowing equilib ium ac ions because de ia ing leads o a less
a o able con inua ion alue (see Chap e 22 in Ljungq is and Sa gen (2018)). This “his o y-dependence”
no ion o epu a ion is dis inc om he “ad e se-selec ion” app oach o epu a ion (Maila h and Samuel-
son (2006, p. 459)), in which incen i es s em om he desi e o con ince he opponen ha you a e o a
speci ic ype.
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826 Achim and Knoep le Theo e ical Economics 19 (2024)
Rela ed li e a u e Ou pape is closely ela ed o he li e a u e on cos ly s a e e i ica-
ion (CSV). Ea ly pape s, including Townsend (1979), Gale and Hellwig (1985), Mookhe -
jee and Png (1989), and Bo de and Sobel (1987), ocus on one-sho in e ac ions. One o
he main indings in his li e a u e is he op imali y o cu o e i ica ion p o ocols, an
insigh ha has been in luen ial in explaining he use o deb con ac s and he ole o
inancial in e media ies. A numbe o pape s conside dynamic ex ensions. In many o
hese, he p incipal’s obse a ion e eals he agen ”s cu en p i a e in o ma ion wi h
no in e empo al link o pas ac ions o s a es.8By con as , he s a e in ou model is
pa ially pe sis en , so inspec ions e eal in o ma ion abou pas beha io .
Inspec ions o a pe sis en s a e a e analyzed in Ra ikuma and Zhang (2012)and
Kim (2015). These pape s s udy pu e ad e se-selec ion p oblems wi h exogenous p i-
a e in o ma ion when he p incipal has commi men . In Ra ikuma and Zhang (2012),
he con ac ing ic ion is d i en by isk-sha ing conce ns. They ind ha andom in-
spec ions a e op imal, and, a e each inspec ion, he e is a g ace pe iod wi hou in-
spec ions. In Kim (2015), he con ac ing ic ion is d i en by he agen ’s limi ed liabili y.
They ind ha andom inspec ions a e op imal o incen i e p o ision when u h ul dis-
closu e is a ainable, bu pe iodic inspec ions a e op imal o guide en i onmen al p o-
ec ion ac i i ies when he ines a e insu icien o a ain u h- elling. Ou se ing ea-
u es an ad e se-selec ion and mo al-haza d p oblem, he p incipal lacks commi men
powe , and he agen is isk-neu al so ha he con ac ing ic ion s ems om limi ed
liabili y. We ind de e minis ic inspec ions a e op imal when he p incipals lacks com-
mi men . Ou esul o he commi men case is in line wi h hei indings ha andom
inspec ions p o ide incen i es mo e e ec i ely.
Mos closely ela ed is he pape by Va as, Ma ino ic, and Sk zypacz (2020), which
s udies a pu e mo al-haza d model wi h ull commi men and wi hou ines.9In hei
model, he agen is incen i ized by he desi e o main ain a good epu a ion and in-
spec ions make he agen ’s ype public. Addi ionally, inspec ions se e an in o ma ion-
acquisi ion pu pose o he p incipal. The au ho s ind andom inspec ions a e op imal
o incen i e p o ision, bu de e minis ic inspec ions a e op imal o in o ma ion acqui-
si ion. In con as , in ou model, he agen discloses he s a e o compliance, so ha in-
spec ions do no educe he unce ain y abou he s a e. Ball and Knoep le (2023)s udy
8Fo dynamic mo al-haza d p oblems in which moni o ing e eals he cu en ac ion, see An inol i and
Ca li (2015), Pisko ski and Wes e ield (2016), Dilmé and Ga e (2019), Chen, Sun, and Xiao (2020), Li and
Yang (2020), Dai, Wang, and Yang (2022), Wong (2022). Fo dynamic ad e se-selec ion p oblems in which
e i ica ion e eals he agen ’s cu en in o ma ion ha is independen and iden ically dis ibu ed (i.i.d.)
ac oss pe iods, see Chang (1990), Webb (1992), Monne and Quin in (2005), Wang (2005), Popo (2016),
Malenko (2019).
9In bo h pape s, s a e ansi ions a e based on he epu a ion o quali y model (Boa d and Meye - e -
Vehn (2013)). In Boa d and Meye - e -Vehn (2013) quali y becomes publicly obse able a andom imes. In
he p esen pape and in Va as, Ma ino ic, and Sk zypacz (2020), he p incipal chooses he imes a which
he s a e becomes publicly obse able a a cos . In Halac and P a (2016) and Dilmé and Ga e (2019), he
p incipal in es s in building he pe sis en moni o ing capabili ies, and moni o ing e eals in o ma ion
abou cu en ac ions o he agen s. In con as o he se up in he p esen pape , he p incipal’s ac ions
a e p i a e and she canno pe ec ly con ol he ime a which she signals igilance. In Halac and P a
(2016) his leads o a b eakdown o he ela ionship wi h posi i e p obabili y a e he agen ’s e o emains
un ecognized o oo long.
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Theo e ical Economics 19 (2024) Rela ional en o cemen 827
op imal inspec ions wi h commi men and show ha andom inspec ions a e op imal
o incen i es when he agen mus a oid a b eakdown and de e minis ic inspec ions
a e op imal when he agen mus achie e a b eak h ough. The d i e o non andom
inspec ions in ou pape is he p incipal’s lack o commi men .
The pe sis en e ec o e o is impo an o ela ional incen i es. This is also high-
ligh ed o a collabo a ion p oblem wi hou commi men in Ramos and Sadzik (2023).
Simila o ou compa a i e s a ics in Sec ion 4.2, he au ho s show ha ela ional incen-
i es anish wi hou pe sis ence. Wi hou pe sis ence, commi men is c ucial o en-
o cemen wi h cos ly inspec ions: when he agen is expec ed o comply, he p incipal
has no incen i e o pay he inspec ion cos o e eal in o ma ion she al eady knows. In-
deed, Reinganum and Wilde (1985) con i m o a non epea ed se ing ha compliance is
no achie able wi hou ull commi men . Wi h epea ed in e ac ions, con inua ion play
can p o ide punishmen o insu icien inspec ion. Ben-Po a h and Kahneman (2003)
p o e a olk heo em, showing ha ull compliance can be ob ained wi hou commi -
men in he undiscoun ed limi . In ou game, ull compliance is a ainable e en wi h
discoun ing. This di e ence s ems om he pe sis ence o he s a e and he obse abil-
i y o inspec ions by he agen in ou model.
2. Model
Playe s, ac ions, and s a e dynamics The e a e an agen and a p incipal. Time ∈[0, ∞)
is con inuous. The agen , a each ins an , p i a ely chooses e o η ∈[0, 1] o comply
wi h exogenously gi en egula ion as bes he can. The s a e o compliance a ime
is θ ∈{0, 1}, whe e we e e o s a e 0 as noncomplian and o s a e 1 as complian .
E o a ec s he ansi ions o he p ocess {θ } ≥0: he e a e pa ame e s λ>0andα∈
(0, 1)such ha he s a e changes om 0 o 1 a Poisson a e η λα and om 1 o 0 a
a e λ(1−η α). We may in e p e λand αas ollows. The e is a Poisson p ocess o
shocks a i ing a a e λ. Whene e he e is a shock a ime , he esul ing s a e is θ =1
wi h p obabili y η αand i is θ =0 wi h p obabili y 1 −η α; be ween shocks he s a e
emains unchanged. Thus, λmeasu es he a iabili y o compliance and αmeasu es he
esponsi eness o he agen ’s e o condi ional on a shock; α<1 implies ha he agen
canno always main ain compliance despi e his bes e o s. The agen obse es θ a all
imes and sends epo ˆ
θ ∈{0, 1} o he p incipal. The agen can exi he ela ionship
unila e ally a any ime.
The p incipal chooses inspec ions and ines o incen i ize he agen . We deno e
by NI
he cumula i e numbe o inspec ions and by F he cumula i e ines up o and
including ime .Tha is,dNI
≡N −lims NI
s∈{0, 1}is equal o 1 i and only i he e is
an inspec ion a ime and dF ≥0 is he ine paid by he agen a ime .
In o ma ion and iming The agen obse es he his o y o all pa hs
h =ηs,θs,ˆ
θs,NI
s,Fss∈[0, ].
The p incipal ne e obse es he agen ’s e o and is able o obse e he s a e θ only
by pe o ming an inspec ion a ime . To allow o andomized inspec ions, we equip
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828 Achim and Knoep le Theo e ical Economics 19 (2024)
he p incipal wi h a p i a e andom signal π, de ined on a su icien ly ich sample
space . The p incipal obse es his o ies o he o m hP
={π,ˆ
θs,NI
s,Fs,θs:NI
s=
1}s∈[0, ]. Heu is ically, we can desc ibe he iming o e en s wi hin each ins an [ , +d )
as ollows.10 Fi s , he agen chooses e o η . Subsequen ly, na u e de e mines whe he
a shock a i es and, condi ional on he a i al o a shock and he e o , d aws a new s a e
θ . The agen hen obse es he ealized θ and sends a epo ˆ
θ ∈{0, 1} o he p incipal.
The p incipal chooses whe he o inspec , dNI
∈{0, 1}, and se s a ine dF incu ed im-
media ely by he agen , whe e he ine can be con ingen on he ue s a e θ i and only
i he p incipal chose o inspec .
Payo s and equilib ium The p incipal and he agen a e isk-neu al and discoun u-
u e payo s a a common a e >0. The p incipal is asked wi h ensu ing ha he agen
complies wi h he egula ion. She incu s a lump-sum cos κ>0 om each inspec ion.
Fo a ealized his o y h={η ,θ ,ˆ
θ ,NI
,F } ∈[0,∞), he discoun ed ne p esen cos o he
p incipal a ime is
k =∞
e− (s− )κdNI
s.(1)
The p incipal does no bene i di ec ly om compliance o om ining he agen . Fines
a e in e p e ed as emedial ac ions ha nega i ely impac he agen .11 To ensu e ha
he p incipal is willing o bea he inspec ion cos s, assume ha when he ela ionship
b eaks down, i.e., he agen exi s o ceases o exe e o , he p incipal su e s cos ¯
K.We
assume h oughou ha he bound ¯
Kis la ge enough such ha i exceeds he expec ed
inspec ion cos s necessa y o incen i ize he agen .12
The agen incu s e o cos o cη d wi h c>0 and disu ili y dF om ines. His
discoun ed ne p esen payo a ime is gi en by
u =∞
e− (s− )(−cηsds−dFs).(2)
The agen is p o ec ed by limi ed liabili y. I he chooses o exi , he ela ionship ends
pe manen ly, which esul s in a con inua ion payo o −B. This implies a cons ain on
he se e i y o ines he p incipal can impose. We assume ha he exogenously gi en
10We ou line he sequen iali y a a gi en ins an o gi e an in ui ion abou he o de o mo es. Fo mally,
he o de is cap u ed by con inui y p ope ies o he espec i e ac ion and s a e pa hs. I is well known
ha in con inuous- ime games wi h obse able ac ions, s a egies may no p oduce well de ined ac ion
pa hs. To ocus he exposi ion in he main ex on he main economic o ces, we de e a mo e o mal ea -
men o Supplemen al Appendix A (a ailable a h p://econ heo y.o g/supp/5183/supplemen .pd ), whe e
we adop an app oach by Kamada and Rao (2023) o impose es ic ions on s a egies ha gua an ee well
de ined ac ion pa hs.
11Ou esul s do no ely on his assump ion. When he p incipal bene i s om ines, he p e e ed equi-
lib ium di e s om he one we p esen only in an ini ial ine paid by he agen (see Sec ion 6).
12This se es as a concise way o deli e incen i es o inspec ion o he p incipal when analyzing he
equilib ium p oblem wi hou commi men . Al e na i ely, we could explici ly inco po a e an (unobse ed)
low ewa d θ Ro η Rin ou model so ha , o R>0 la ge enough, he p incipal’s expec ed payo om
inducing e o by he agen exceeds he necessa y inspec ion cos s. Ou esul s would be una ec ed.
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Theo e ical Economics 19 (2024) Rela ional en o cemen 829
bound Bis la ge enough: B> ¯
B≡c( +λ)/(λα ). O he wise, he maximal punishmen
is insu icien o incen i ize e o e en i θ we e public a all imes.
Gi en a s a egy p o ile, he p incipal and he agen o m expec a ions abou his o y
hbased on hei pas obse a ions. Fo s a egies ha induce measu able ac ion p o-
cesses on pa h, we deno e he expec ed cos o he p incipal and payo o he agen a
ime by
K =EP
−[k ]and U =EA
−[u ].
The expec a ion is wi h espec o he p ocess {θs}s∈[0,∞)and he andomiza ion de ice
π, and i is condi ional on he in o ma ion ha is a ailable o he p incipal and he agen ,
espec i ely.13 In con inuous- ime games wi h obse able ac ions and s ochas ic en i-
onmen s, playe s’ beha io may be nonmeasu able. We do no impose es ic ions on
s a egies ha ule ou nonmeasu able beha io . Ins ead, ou equilib ium de ini ion
below equi es ha s a egies lead o measu able ac ions on pa h. His o ies away om
he equilib ium pa h may lead o nonmeasu abili y. Payo s a such his o ies can be
assigned eely wi hin he easible bounds. In ou game, he lowe bounds on payo s
can be eached by ei he playe unila e ally h ough exi o by imposing he maximal
ine. The e o e, po en ial nonmeasu abili ies o pa h and he assigned payo s canno
be used as a h ea o enla ge he equilib ium se (see also he discussion o his ap-
p oach in Kamada and Rao (2023)).
We de ine a s a egy p o ile, oge he wi h p ocesses {K ,U } ≥0, obeape ec
Bayesian equilib ium i he ollowing s a emen s hold.
•The s a egies o he p incipal and he agen a e sequen ially a ional.
•Along he equilib ium pa h, K and U a e equal o he condi ional expec a ions
gi en abo e. Away om he equilib ium pa h, K and U a e equal o he condi ional
expec a ions whene e hese a e well de ined.
•A all his o ies and all imes, K ∈[0, ¯
K]and U ∈[−B,0
].
We say ha he agen ’s s a egy is u h ul i ˆ
θ =θ a all his o ies along he equi-
lib ium pa h. Fu he , we call he agen ’s s a egy maximally complian i η =1a all
his o ies along he equilib ium pa h. No e ha wi h ull e o by he agen , he p oba-
bili y o compliance a any gi en ime is maximized. We e e o an equilib ium as u h-
ul o maximally complian i he agen ’s s a egy in his equilib ium has he espec i e
p ope y. Hence o h, we es ic a en ion o such equilib ia (see he discussion in Sec-
ion 6).
13Fo he p incipal, he expec a ion is wi h espec o he na u al il a ion gene a ed by he p ocess
{NI
s,Fs,θs:dNs=1}s∈[0, )∪{ˆ
θs}s∈[0, ]when aking he inspec ion decision, and wi h espec o he na u al
il a ion gene a ed by {ˆ
θs,NI
s,Fs−,θs:dNs=1}s∈[0, ] o he ine. Fo he agen , he expec a ion is wi h e-
spec o he na u al il a ion gene a ed by he p ocess {ηs,θs,ˆ
θs,NI
s,Fs}s∈[0, ) o his e o choice, and wi h
espec o he na u al il a ion gene a ed by {ˆ
θs,NI
s,Fs}s∈[0, )∪{ηs,θs}s∈[0, ] o his epo . As men ioned
abo e, see Supplemen al Appendix A o a o mal ea men o pe missible s a egies.
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830 Achim and Knoep le Theo e ical Economics 19 (2024)
We say ha inspec ions a e p edic able o he agen i he knows o ce ain whe he
o no his cu en epo will lead o an inspec ion a any his o y.14 Hence o h, we e e
o inspec ions as andom whene e hey a e nonp edic able o he agen .
3. Agen ’s and p incipal’sp oblem
3.1 Agen : Incen i e compa ibili y
Fix an a bi a y p incipal s a egy o ines and inspec ions, and le U be he agen ’s asso-
cia ed expec ed discoun ed con inua ion payo a ime unde he assump ion ha he
exe s ull e o and epo s u h ully. We cha ac e ize ecu si ely he condi ions unde
which u h ul epo ing and maximal compliance a e a bes esponse o he agen in
e ms o he e olu ion o his p omised u ili y a all imes. Due o he pe sis ence in he
agen ’s p i a e in o ma ion, he ecu si e cha ac e iza ion o incen i e compa ibili y e-
qui es acking wo s a e a iables:15 he agen ’s expec ed con inua ion u ili y gi en ha
θ =0andgi en ha θ =1. Fo mally, ix a p incipal s a egy and de ine o any s ic
his o y a ime ,
U0
=EA
−[U |θ =0]and U1
=EA
−[U |θ =1].(3)
These a e he agen ’s expec ed con inua ion u ili ies when his o y h −is ollowed by he
ealiza ion o θ =0o θ =1. He e, EA
− ep esen s he expec a ion condi ional on all
a ailable in o ma ion be o e ime . Following Zhang (2009), we call U1
he pe sis en
payo i θ −=1and he ansi ional payo in case θ −=0, and ice e sa o U0
.
Ou i s esul p o ides a comple e cha ac e iza ion o he agen ’s incen i e-
compa ibili y cons ain s in e ms o he e olu ions o U0
and U1
. The cons uc ion
is based on he ma ingale ep esen a ion o ma ked poin p ocesses (Las and B and
(1995)), which is p esen ed in de ail in Appendix A. We exploi he ac ha he agen ’s
ime- expec a ion o his o al discoun ed li e ime u ili y is a ma ingale. Fo he inspec-
ion coun ing p ocess NI, hecompensa o is a p edic able p ocess νI={νI
} ≥0such
ha he compensa ed p ocess NI
−νI
is a ma ingale. The compensa o exis s unde
e y gene al condi ions and can be in e p e ed as he p edic able d i o he unde ly-
ing (nonp edic able) s ochas ic p ocess. We can hink o he compensa o as a gene -
aliza ion o he cumula i e haza d unc ion, and, consequen ly, hink o dνI/d as he
haza d a e o inspec ions (whene e i exis s). Fu he mo e, le he p edic able p ocess
I={I
} ≥0measu e he jump in he pe sis en payo i an inspec ion is pe o med a
ime .16
Lemma 1. A p incipal’s s a egy induces maximal compliance and u h ul epo ing i
and only i i gene a es he p ocesses {U1
,U0
} ≥0o p omised u ili ies sa is ying o i=θ −
and j=1−θ −, and a all wi h dNI
=0and θ −=θ ,
14Fo mally, p edic abili y means ha he p ocess NIis measu able wi h espec o he in o ma ion a ail-
able o he agen (see Da is (1993, p. 67, o a de ini ion in he con ex o jump p ocesses)).
15This is based on Fe nandes and Phelan (2000), who in oduce a ecu si e app oach wi h se ially co e-
la ed s a es in disc e e ime. See Zhang (2009) o a ea men in con inuous ime.
16Tha is, gi en ha an inspec ion occu s a ime (and θ −=1), hen I
=U1
−U1
−, supposing ha he
inspec ion con i ms ha he agen epo ed u h ully, which is he case along he equilib ium pa h.
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Theo e ical Economics 19 (2024) Rela ional en o cemen 837
Figu e 3. The e olu ion o an example pa h ealiza ion s a ing in he complian s a e. Solid
cu es depic he agen ’s pe sis en payo in he cu en s a e; dashed cu es depic he ansi-
ional payo , o which he agen ’s payo jumps when he s a e changes.
agen o he p incipal’s con inued o e sigh . Demons a ed igilance shapes he agen ’s
pe cep ion ha he will e en ually be de ec ed i he we e o de ia e. While andom in-
spec ions may be suppo ed in a ela ional con ac , such a angemen s equi e s ong
de e en s o he p incipal o ensu e he adhe ence o he equilib ium s a egy. This e-
qui emen ul ima ely ende s andomiza ion nonbene icial o he p incipal (Lemma 2).
The equilib ium in Theo em 1na u ally ea u es penal y educ ions o ea ly dis-
closu es o noncompliance. This is consis en wi h olun a y disclosu e schemes ha
a e commonly used in p ac ice. The U.S. en i onmen al p o ec ion agency (EPA) em-
ploys a sel - epo ing p og am called “Incen i es o Sel -Policing” ha equi es i ms
olun a ily disclose any iola ions ha a e de ec ed in e nally. Simila o he way he
agen is incen i ized in he abo e equilib ium, i ms ha disclose iola ions ea ly a e e-
wa ded by a educ ion in penal ies and a suspension o inspec ions un il compliance is
es o ed. Theo em 1p o ides insigh s in o how en o cemen agencies can bene i om
o e ing egula ed i ms incen i es o olun a y disclosu e. Volun a y disclosu e allows
he p incipal o limi inspec ion o pe iods o compliance and, hus, lowe s he o e -
all inspec ion cos s. The EPA poin s ou ha he ad an age o hese incen i es lies in
“making o mal EPA in es iga ions and en o cemen ac ions unnecessa y.”22 In he he-
o e ical li e a u e, he obse a ion ha olun a y disclosu e educes moni o ing cos s
da es back o Kaplow and Sha ell (1994), who in oduced sel - epo ing in o he en o ce-
men model by Becke (1968). Wi hou he agen ’s disclosu e, he p incipal in ou model
would no be able o consis en ly a oid inspec ions du ing phases o noncompliance.23
22h ps://www.epa.go /compliance/how-we-moni o -compliance.
23See Va as, Ma ino ic, and Sk zypacz (2020) o a model wi hou epo s. Ou esul s con i m he con-
jec u e in ha pape ha olun a y disclosu e can a oid unnecessa y inspec ions (see Va as, Ma ino ic,
and Sk zypacz (2020, p. 2921)).
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838 Achim and Knoep le Theo e ical Economics 19 (2024)
4.2 Compa a i e s a ics
How do a ia ions in he pa ame e s a ec he leng h o inspec ion cycles and he in-
spec ion cos s? As one would expec , i he penal y bound Binc eases o he e o cos
cdec eases, he agency p oblem becomes less se e e: he inspec ion cycle T∗becomes
la ge and he expec ed cos s dec ease.24 The e ec o a change in he a i al a e o
shocks λis mo e in ica e. An inc ease in λdec eases he s a e’s pe sis ence and has
a non-mono one e ec on he leng h o inspec ion cycles and he o e all cos s. The
ollowing esul makes hese s a emen s p ecise. To ensu e ha he equilib ium in The-
o em 1always exis s, we equi e ha λ>λ≡c /(B α −c)>0, ixing all o he pa ame-
e s.25
Lemma 3. As hea i al a eo shocksλinc eases, he ollowing s a emen s hold.
•The inspec ion cycle leng h inc eases o low λand dec eases o high λ,wi h
limλ↓λT∗(λ)=limλ↑∞ T∗(λ)=0.
•The discoun ed inspec ion cos s dec ease o low λand inc ease o high λ,wi h
limλ↓λK∗
0(λ)=limλ↑∞ K∗
0(λ)=∞.
Fo he cycle leng h T∗, he e a e wo opposing e ec s i λinc eases. Fi s , a any
gi en ins ance, he s a e is mo e likely o change in esponse o cu en e o . The
ma ginal bene i om e o is highe and i is easie o incen i ize he agen , allowing
o an inc ease in T∗. Second, he s a e becomes mo e agile: he link be ween cu en
e o and u u e compliance weakens. Delayed inspec ions ha e less incen i e powe ,
o cing he p incipal o sho en inspec ion cycles. Lemma 3shows ha he i s e ec
domina es o low λand he second e ec domina es o high λ.
Fo any ixed T∗, he o al inspec ion cos s dec ease in λas any cycle o ixed leng h
is mo e likely o be in e up ed by a b each o compliance, so he inspec ion is less likely
o be ca ied ou . Thus, o low λ, his e ec and he inc ease in T∗wo k in he same
di ec ion. Inspec ion cos s dec ease in λ.Fo highλ, he wo e ec s wo k in opposi e
di ec ions. Lemma 3shows ha he dec ease in T∗is as enough o ou do he second
e ec ; he inspec ion cos s inc ease in λ. Bo h inspec ion in ensi y and inspec ion cos s
g ow a bi a ily la ge a bo h ex emes.
As λgoes o in ini y and s a e pe sis ence anishes, inspec ions mus be immedia e
o de e de ia ions. This highligh s a key disad an age o non andom inspec ions and
he absence o commi men . In ui i ely, a shi king agen aces an “e ec i e” discoun
a e o +λwhen conside ing he impac o he nex inspec ion. This is because he
s a e oday de e mines he s a e a he nex inspec ion only wi h p obabili y e−λT .To
ensu e inspec ion e ec i eness, T∗(λ)mus app oach ze o as enough so as o keep
limλ→∞ λe−( +λ)T∗(λ)s ic ly posi i e. Fo he p inciple, in con as , he e ec i e dis-
coun a e is +λ(1−α), which is smalle han ha o he agen . The limi o he
24A o mal p oo o he changes in Band c, as well as compa a i e s a ics wi h espec o α,canbe ound
in a wo king pape e sion, which is a ailable upon eques .
25Obse e ha he lowe bound on B equi ed o easibili y o e o , ¯
B=c( +λ)
λα , g ows a bi a ily la ge
as λgoes o 0.
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Theo e ical Economics 19 (2024) Rela ional en o cemen 839
p incipal’s cos is p opo ional o limλ→∞ λe−( +(1−α)λ)T∗(λ). I is hen easy o see ha
his cos mus be in ini e o α<1whenlimλ→∞ λe−( +λ)T∗(λ)>0.
The high compliance cos o la ge λa ises om he agen ’s oppo uni y o egain
compliance wi h high p obabili y unless he nex inspec ion is imminen . Imminen
inspec ions (T∗nea 0) in la e cos s. Random inspec ions may be aluable, allowing
he p incipal o h ea en immedia e inspec ions wi hou pe o ming hem cons an ly.
We now con i m ha andom inspec ion schedules ou pe o m p edic able ones when
easible. Wi h andomiza ion, he p incipal s ic ly p e e s highe a i al a es λ.
5. Commi men
Ou esul s show ha wi hou commi men , he p incipal canno bene i om andom-
iza ion. In his sec ion, we con i m ha i he p incipal could commi o ollow h ough
wi h andom inspec ion schedules, his would dec ease inspec ion cos s.
One way o enhance commi men o a p o i able andom p ocedu e in a m’s-leng h
en o cemen is o sepa a e planning and execu ion o inspec ions, as seen in Ge man
banking supe ision. The Eu opean Cen al Bank o he supe iso y agency a he Fi-
nance Minis y (BaFin) schedules audi s, while he Ge man Bundesbank execu es hem
(BaFin (2016)). The inspec ion cos is no incu ed by he pa y making he inspec ion
decision, elimina ing he emp a ion o delay o skip inspec ions. This sepa a ion di e s
signi ican ly om wo seemingly simila al e na i es: ou sou cing all o e sigh o com-
pensa ing he p incipal o inspec ion cos s. Ou sou cing only shi s he p oblem one
laye u he ; compensa ion equi es p ecise knowledge o he cos o a oid ine icien
inspec ions.26
Al e na i ely, he lack o de ec abili y, which hinde s p o i able andomiza ion, can
be o e come i he p incipal is esponsible o o e seeing a la ge pool o independen
agen s and he e a e public eco ds. The p incipal can hen egula ly inspec a ixed
p opo ion o agen s and make he esul s publicly a ailable o c ea e a e i iable signal
o con inued igilance. Fo example, he EPA’s da abase “En o cemen and Compliance
His o y Online” collec s o e 44,000 inspec ed acili ies wi hin he 12 mon hs up o Ap il
2021;27 he Public Company Accoun ing O e sigh Boa d (PCAOB) publicizes app oxi-
ma ely 100–300 inspec ion epo s pe yea .28
To con i m he bene i o andomiza ion, conside he ollowing mechanism, which
is op imal in he class o s a iona y andom mechanisms.29
•Inspec ions a e pe o med only du ing compliance wi h cons an Poisson a i al
a e
m∗
R=
¯
B
B−¯
B.
26I he compensa ion alls sho o he ac ual cos and e o equi ed o an inspec ion, he incen i e o
skip i pe sis s. I he compensa ion exceeds he cos , his c ea es an incen i e o inspec ine icien ly o en.
27h ps://echo.epa.go .
28h ps://pcaobus.o g/o e sigh /inspec ions/ i m-inspec ion- epo s.
29Fo a p oo o his claim, see Appendix C. No e ha we do no claim ha he mechanism p esen ed
he e is he op imal commi men mechanism.
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840 Achim and Knoep le Theo e ical Economics 19 (2024)
•Fines a e le ied only du ing noncompliance wi h a cons an low ine
∗
R= ¯
B.
•I an inspec ion e eals noncompliance, hen he agen pays he maximal ine.
Simila o he equilib ium wi h p edic able inspec ions, he e a e wo phases. In-
spec ions bu no ines du ing compliance, and ines bu no inspec ions du ing noncom-
pliance. The di e ences a e ha inspec ions a i e a andom and he e is no lump-sum
ine a ansi ions o noncompliance.
Inse ing in o Lemma 1 he alues dF =0anddνI
=m∗
Rd in case i=1, and he
alues dF = ∗
Rd and dνI
=0incasei=0, i is s aigh o wa d o e i y ha he payo s
o he agen a e cons an a
U1
R=− c
α,U0
R=− c
α −c
λα.(8)
He e, U1
Ris he pe sis en payo and U0
Ris he ansi ional payo when he agen epo s
compliance, and ice e sa when he agen epo s noncompliance. I is s aigh o wa d
ha all cons ain s a e sa is ied a all imes, wi h (H) binding in i=1and(O) binding in
bo h s a es. The nex esul shows ha he p incipal’s inspec ion cos s wi h p edic able
inspec ions a e gene ally highe han wi h andom inspec ions. In con as o he p e-
dic able inspec ion schedule, a high a i al a e λbene i s he p incipal in his andom
mechanism.
Theo em 2. The inspec ion cos s in he s a iona y andom mechanism de ined abo e a e
s ic ly lowe han in he p incipal-op imal equilib ium in Theo em 1.Fu he mo e, he
cos s in his andom mechanism a e dec easing in λ o all λ,wi h
lim
λ↓λKR(λ)=∞ and lim
λ↑∞ KR(λ)=cα
B α −cκ.
Random inspec ions domina e p edic able inspec ion p ocedu es o wo easons.
Fi s , by he a gumen a he end o Sec ion 4.2, noise and delay make pe iodic inspec-
ions less e ec i e. The h ea o an imminen inspec ion a all imes is mo e e ec i e in
ou se ing, e en when holding he payo impac o each inspec ion ixed.30 Tha is, i
he agen ’s ini ial u ili y is ixed a some le el u, he cos s om he andom mechanism
implemen ing his u ili y le el a e s ic ly below he cos s in he p edic able equilib ium
implemen ing he same le el. Second, in ou se ing wi h ines and sel - epo ed com-
pliance, andom inspec ions allow o a g ea e payo impac o an inspec ion on he
de ia ing agen : wi h p edic able inspec ions, sel - epo ing equi es a ansi ion ine
whene e a b each o compliance occu s. The isk o he ansi ion ine educes he
agen ’s o e all equilib ium payo . Since he lowe bound on payo s is ixed a B, his e-
duc ion dec eases he maximum loss ha he p incipal can impose when an inspec ion
30See also Va as, Ma ino ic, and Sk zypacz (2020), who show ha a cons an inspec ion a e p o ides
incen i es mos e ec i ely unde commi men when he payo consequence o each inspec ion is ixed.
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Theo e ical Economics 19 (2024) Rela ional en o cemen 841
e eals a mis epo . Thus, p edic able inspec ions ha e a smalle payo impac , making
hem o e all less powe ul.
Ou inding ha andom inspec ions p o ide incen i es mo e e ec i ely is consis-
en wi h Va as, Ma ino ic, and Sk zypacz (2020), who s udy a se ing wi hou olun a y
disclosu e. They show ha (pa ially) p edic able inspec ions can be op imal when he
p incipal de i es di ec alue om in o ma ion, i.e., when he low payo is con ex in
he pos e io belie . In ou model, he p incipal induces hones sel -disclosu e. Along
he equilib ium pa h, he p incipal always knows he ue s a e. The e o e, in oducing
con exi y in he p incipal’s alue as a unc ion o he belie would no a ec ou esul s;
he belie is always 0 o 1. Va as, Ma ino ic, and Sk zypacz (2020) iden i y a ade-o
acco ding o which incen i e p o ision ecommends andomiza ion while in o ma ion
acquisi ion makes p edic able inspec ions mo e p o i able. Ou analysis sugges s ha
sel - epo ing can esol e his ade-o in a o o andomiza ion when he cu en s a e
is known o he agen and mone a y incen i es a e easible.
6. Conclusion
We s udy en o cemen h ough inspec ions and ines. In ela ional en o cemen , maxi-
mum compliance and u h ul disclosu e a e a ained h ough non andom inspec ions.
A ully commi ed p incipal would bene i om andom inspec ions.
The pe sis en e ec o he agen ’s e o on compliance makes i possible o c ea e
incen i es h ough isola ed inspec ions. An in e media e le el o pe sis ence is op imal
in he case o ela ional en o cemen . I he p incipal can commi o andom inspec-
ions, inspec ion cos s a e inc easing in he le el o pe sis ence as compliance becomes
less esponsi e o e o . This highligh s he impo ance o pe sis ence in ela ional en-
o cemen .
Th oughou he analysis, we assume ha he p incipal does no bene i om he
ines imposed on he agen . This assump ion is innocuous. Gi en ha he agen exe s
ull e o a all imes, when his ini ial p omised u ili y is u, he expec ed discoun ed sum
o ines paid by he agen is equal o −u−c/ . I he p incipal we e o bene i om ines
a a e β∈(0, 1], he objec i e would be o maximize −K(u)+β(−u−c/ )ins ead o
maximizing −K(u)in he baseline model. Deno ing he maximize o −K(u)by u∗,i
is easy o see ha he op imal equilib ium consis s o an ini ial ine B+u∗paid o he
p incipal, ollowed by he equilib ium o Theo em 1.31
A possible a ia ion o ou model is o allow he p incipal o pay subsidies o he
agen when success ully passing inspec ions. I he p incipal could ewa d he agen
o passed inspec ions, he uppe bound on he agen ’s con inua ion u ili y would in-
c ease. The p incipal could hen dec ease he inspec ion equency as he maximal
punishmen inc eases. Wi h commi men o andom inspec ions, he p incipal could
essen ially a oid all inspec ion cos s i ewa ds we e unbounded. She could o e an
a bi a ily la ge ewa d a e inspec ing wi h anishing p obabili y.
31The agen ’s ini ial u ili y (be o e paying he ine) is a his ou side op ion −Band hen jumps o u∗.The
p incipal’s payo −K(u∗)+β(B−c/ )is clea ly an uppe bound o −K(u)+β(−u−c/ )among u≥−B.
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842 Achim and Knoep le Theo e ical Economics 19 (2024)
The assump ions ha he p incipal implemen s ull e o is na u al in many si ua-
ions, o example, when he p incipal, asked wi h moni o ing compliance, is no he
same ins i u ion as he one designing he egula ion. The assump ion is also impo an
o ac abili y. To le he p incipal choose e o , he model would need o accoun ex-
plici ly o he p incipal’s bene i om compliance.32 Mo e impo an ly, he op imiza-
ion p oblem would become subs an ially mo e complex. Maximizing o e he e o
le el would add a con inual con ol a all imes.33
Simila ly, we ocus on equilib ia wi h u h ul sel - epo s. This ocus is na u al in
many applica ions in which i is essen ial o egula o s o accu a ely iden i y compli-
ance iola ions. In he auxilia y mechanism design p oblem wi h p incipal commi -
men , ha ing only one agen ensu es ha he e ela ion p inciple applies. Wi h com-
mi men , he p incipal can eplica e he ou come o any combina ion o mechanism
and epo ing s a egy wi h he co esponding di ec mechanism and a u h ul epo -
ing s a egy. Howe e , in he equilib ium p oblem wi hou commi men , we do no ule
ou po en ial bene i s om non u h ul beha io . Since we ind he op imal p edic able
equilib ium ia he auxilia y mechanism design p oblem, he only emaining conce n
is whe he he p incipal could exploi non u h ul epo ing o bene i om andom in-
spec ions in equilib ium. We suspec his is no he case, bu e i ying he conjec u e is
beyond his a icle’s scope.
Appendix A: P oo s o Sec ion 3
P oo o Lemma 1. The p oo o Lemma 1consis s o wo in e media e esul s. Lem-
ma Ap o ides a ma ingale ep esen a ion o he agen ’s li e ime expec ed u ili y, and
Lemma Bp o ides necessa y and su icien condi ions o he pa h o expec ed payo s
such ha ull e o and u h ul epo ing a e a bes esponse o he agen .
De ine W o be he agen ’s li e ime expec ed u ili y, wi h expec a ions aken wi h
espec o he in o ma ion ha is a ailable a ime :
W =
0
e− s(−dFs−cηsds)+e− U .
By cons uc ion, he p ocess {W } ≥0is a ma ingale (Da is (1993, p. 20)). The e a e h ee
ypes o e en s: changes in he s a e, changes in epo s, and inspec ions. Inspec ions
a e go e ned by he p ocess NIgi en by he p incipal’s s a egy. Fo consis ency, we in-
oduce he coun ing p ocesses Nθ={Nθ
} ≥0and Nˆ
θ={Nˆ
θ
} ≥0 ha coun he numbe
o changes in he s a e o compliance and in he epo s, espec i ely. Fo each p ocess
32In some cases, when he p incipal’s bene i is la ge enough, implemen ing ull e o is op imal and he
analysis would be una ec ed.
33Indeed, o p edic able inspec ions, we sides ep he p oblem o con inual con ols by showing ha i
is wi hou loss o le y no ines be ween inspec ions. This di icul y is also he eason why we do no claim
ha he andom mechanism in Sec ion 5is op imal among all inspec ion mechanisms. While i is op imal
among any Ma ko ian p ocedu e, con i ming ha i is op imal gene ally would equi e a e i ica ion a -
gumen ha deals wi h con inual con ols ha can change bo h con inuously o impulsi ely. We a e no
awa e o exis ing dynamic p og amming esul s o e i y ha he ecu si e app oach we employ emains
alid in his class o op imiza ion p oblems.
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Theo e ical Economics 19 (2024) Rela ional en o cemen 843
Nawi h a∈{θ,ˆ
θ,I}, de ine he compensa o o be a p edic able p ocess νa={νa
} ≥0
such ha he compensa ed p ocess Na
−νa
is a ma ingale. The compensa o exis s
unde e y gene al condi ions and can be in e p e ed as he p edic able d i o he un-
de lying (nonp edic able) s ochas ic p ocess. Fo he haza d a e o ansi ions in com-
pliance, we shall w i e q (η ):=dνθ
/d o , mo e explici ly,
q (η )=θ −λ(1−αη )+(1−θ −)λαη .(9)
The ma ingale ep esen a ion heo em o ma ked poin p ocesses (Las and B and
(1995)) implies he ollowing esul .34
Lemma A. The e exis p edic able p ocesses θ,ˆ
θ,andIsuch ha he e olu ion o he
agen ’s expec ed u ili y is gi en by
dU = U d +dF +cη d +
a∈{θ,ˆ
θ,I}
a
dNa
−dνa
. (10)
The p ocesses θ,ˆ
θ,andIha e an in ui i e in e p e a ion: They ep esen he
jump in u ili y a ime ha esul s om a change in compliance, a change in epo ed
compliance, o an inspec ion.
The ollowing lemma will comple e he p oo o Lemma 1.
Lemma B. A mechanism ha induces he payo s {U } ≥0is incen i e compa ible wi h ull
e o and u h ul epo ing i and only i o all ≥0,
(i) ( +q (1))ˆ
θ
−dνI
(I
−ˆ
θ
)≥dˆ
θ
when θ = ˆ
θ
(ii) (1−2θ −)λα(θ
+ˆ
θ
)≥cwhen θ =ˆ
θ
(iii) U ∈[−B,0
].
P oo . De ine
W =
0
e− s(−dFs−cηsds)+e− ˜
U
o be he agen ’s expec ed payo om choosing e o {˜ηs}and epo {ˆ
θs}up o ime
wi h maximum e o and u h ul epo ing he ea e . He e ˜
U is he expec ed con inu-
a ion payo . We may ha e ˜
U = U i he agen has epo ed non u h ully, i.e., ˆ
θ −= θ −.
Conside i s he case in which he agen ’s epo ega ding his ype a ime is u h ul,
so ha ˜
U =U . Di e en ia ing wi h espec o yields
dW =e− (−dF −cη d )− e− U d +e− dU .
34A o mal p oo o ou se ing, which is a s aigh o wa d adap a ion o he p oo o Theo em 1.13.15 in
Las and B and (1995) p. 25, is p o ided in Supplemen al Appendix B.
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844 Achim and Knoep le Theo e ical Economics 19 (2024)
Using Lemma A o eplace dU yields
dW =e− (1−η )cd +
a∈{θ,ˆ
θ}
a
dNa
−q (1)d +I
dNI
−dνI
.
I he agen de ia es o an addi ional ins an (bu s ill epo s u h ully), hen
dNθ
=dNˆ
θ
=1 wi h p obabili y q (˜η )d
0 wi h p obabili y 1 −q (˜η )d .
Taking expec a ions he e o e yields
EA
[dW ]=e− EA(1−η )cd +θ
+ˆ
θ
q (˜η )−q (1)d .
I ollows om condi ion (ii) ha
θ
+ˆ
θ
q(˜η )−cη ≤θ
+ˆ
θ
q (1)−c.
Thus EA
[dW ]≤0. We hus ob ain he chain o inequali ies
EA
0[W ]=EA
0
0
dWs+W0=
0
EA
0[dWs]+EA
0[W0]
=
0
EA
0EA
s[dWs]+W0≤W0. (11)
Now conside he case in which he agen ’s mos ecen epo a ime is alse, ha is,
θ −= ˆ
θ −, and he con inues he non u h ul s a egy o an addi ional momen a ime .
I no change in he s a e occu s a he addi ional momen , hen he agen mus co ec
his epo immedia ely he ea e . I a change occu s, hen he p e iously alse s a emen
becomes u h ul, and hus his epo does no change. The e o e, we ha e
d˜
U =˜
U −˜
U −d
=dNθ
U −U −d −ˆ
θ
−d +dNI
U +I
−U −d −ˆ
θ
−d 
+1−dNθ
−dNI
U +ˆ
θ
−U −d −ˆ
θ
−d 
=dNθ
dU +dˆ
θ
−ˆ
θ
+dNI
dU +dˆ
θ
−ˆ
θ
+I

+1−dNθ
−dNI
dU +dˆ
θ

=dU +dˆ
θ
−dNθ
ˆ
θ
+dNI
I
−ˆ
θ
.
Again using Lemma A o eplace dU ,weob ain
dW =e− (−dF −cη d )− e− U +ˆ
θ

+e−  U d +dF +cd +θ
dNθ
−q∗
+dˆ
θ
−dNθ
ˆ
θ
+dNI
I
−ˆ
θ
.
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Theo e ical Economics 19 (2024) Rela ional en o cemen 845
I ollows om he hones y cons ain (i) ha , in expec a ion, dˆ
θ
≤( +q (1))ˆ
θ
−
dν (I
−ˆ
θ
). Subs i u ing i in o dW and simpli ying, again using ˜
U =U +ˆ
θ
,gi es
EA
[dW ]=e− (1−η )cd +θ
−ˆ
θ
q(˜η )−q (1)θ
−ˆ
θ
.
Now θ
−ˆ
θ
=(θ
+U )−(ˆ
θ
+U )is he payo di e ence om a change in he s a e
wi hou a change in epo and a change in epo wi hou a change in he s a e. Since
θ −= ˆ
θ −by hypo hesis, his is iden ical o ˜
θ
+˜
ˆ
θ
a e he his o y in which he ue
s a e was iden ical o his epo . Thus (ii) implies ha η =1 maximizes he igh -hand
side, so ha EA
[dW ]≤0.By hesamea gumen asin(11), we ha e EA
0[W ]≤W0=U0,
so ha he agen canno p o i om de ia ing. Taking he limi , we ind ha
lim
→∞
EA
0[W ]≤U0,
which implies ha he agen canno gain om de ia ing om maximum e o and
u h ul epo ing. Con e sely, i he incen i e cons ain (i) is iola ed, hen he abo e
inequali ies a e in e ed, so ha he agen has a s ic incen i e o be dishones . Like-
wise, i (ii) is iola ed, he agen has a s ic incen i e o exe no e o , and a iola ion
o (iii) leads o exi by he agen .
To comple e he p oo o Lemma 1, we show ha condi ion (Pk) ollows om
Lemma A,and(H),(O),and(P) a e equi alen o condi ions (i), (ii), and (iii) in Lemma B.
Conside a mechanism and a s a egy o he agen ha join ly gene a e he payo
p ocess {U } ≥0 o he agen , and deno e by {U1
,U0
} ≥0 he associa ed pai o p omised
u ili ies de ined in (3).
S ep 1. By he de ini ion o U0
,U1
,weha e
θ
+ˆ
θ
=U1
−U0
i θ −=ˆ
θ −=0
U0
−U1
i θ −=ˆ
θ −=1,
q (1)=q (1)=λα i θ −=0
(1−α)λi θ −=1.
(12)
Combining hese wo exp essions, we can w i e mo e succinc ly
q (1)θ
+ˆ
θ
=λ(θ −−α)U1
−U0
.
Lemma A hen implies ha , condi ioning on he e en ha dNθ
=dNˆ
θ
=dNI
=0, we
ge exac ly condi ion (Pk) in Lemma 1.
S ep 2. Nex , suppose ha he agen is no u h ul a e some his o y a ime .Le
i=θ be he ue s a e and suppose he agen epo s j=1−θ .ThenUi
=U +ˆ
θ
and
dUi
=U +d +ˆ
θ
+d −U +ˆ
θ
= U d +dF +cd −q (1)θ
+dˆ
θ
≤ U d +dF +cd −q (1)θ
+ +q (1)ˆ
θ
−dν I
−ˆ
θ

= Ui
+λ(i−α)U1
−U0
−dν I
−ˆ
θ
+dF +cd . (13)
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846 Achim and Knoep le Theo e ical Economics 19 (2024)
The second line ollows om Lemma A, and he inequali y in he hi d line ollows om
condi ion (i) in Lemma B, whe e we ake expec a ions condi ional on he e en ha
dNθ
=dNˆ
θ
=0. The las equali y in (13) holds since
q (1)θ
−ˆ
θ
=q (1)U +θ
−U +ˆ
θ
=q (1)Uj
−Ui
=λ(i−α)U1
−U0
.
Punishmen is wi hou cos o he p incipal and, he e o e, i is op imal o impose he
mos se e e punishmen a e an inspec ion e eals a dishones epo . The se e i y o
punishmen s is es ic ed by he limi s o en o cemen ha equi e he agen ’s con inu-
a ion alue no o all below he lowe bound −B<0. The e o e, we ha e
I
−ˆ
θ
=U +I
 
=−B
−U +ˆ
θ
 
=Ui
=−
B+Ui
.
Subs i u ing his las equa ion in o (13) yields
dUi
= Ui
+λ(i−α)U1
−U0
d +dν B+Ui
+dF +cd ,
which is equal o condi ion (H) in Lemma 1. Con e sely, i (i) does no hold a some ,
hen using he same s eps as abo e, he inequali y is e e sed, so ha (H) is iola ed.
S ep 3. Subs i u ing (12) in o he obedience cons ain (ii), we ob ain, o each θ −,
θ
+ˆ
θ
(1−2θ −)λα =λαU1
−U0
≥c.
The las inequali y is iden ical o (O) in Lemma 1. Con e sely, i (ii) is iola ed a some ,
hen he inequali y is e e sed, so ha (O) is iola ed.
P oo o Lemma 2. The p oo o Lemma 2consis s o wo esul s, s a ed and p o en
o mally below.
Lemma C. Fo any u h ul and maximally complian equilib ium, he e exis s a p inci-
pal s a egy such ha u h ul epo ing and maximal compliance a e a bes esponse o
he agen and
(i) inspec ions a e p edic able o he agen whene e he epo s compliance
(ii) i gene a es weakly lowe inspec ion cos s o he p incipal.
P oo . Take any u h ul maximally complian equilib ium. Le U0
and U1
be he con-
inua ion payo s o he agen in his equilib ium. The ollowing s eps p esen a modi ied
inspec ion schedule sa is ying he p ope ies s a ed in Lemma C. As he o iginal equi-
lib ium is u h ul and maximally complian , U0
and U1
sa is y he cons ain s om
Lemma 1. Fi s , we a gue ha o any inspec ion ollowing a high epo , i is wi hou
loss o assume ha u h ul epo s a e ne e punished mo e han mis epo s o he low
s a e a he ime o an inspec ion. Tha is, i he pe sis en payo U1
jumps downwa d on
he pa h a e an inspec ion, i will do so by less han he dis ance om he ansi ional
u ili y o he lowe bound −B. Fo mally, le ¯
U1
be he agen ’s pe sis en payo igh a e
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Theo e ical Economics 19 (2024) Rela ional en o cemen 853
P oo . Using Claim 5, he e is no loss in gene ali y in assuming ha ˆ
θ0=1. Conside
he op imal ini ial u ili ies (u0,u1), whe e we assume o he con a y ha u1−u0>
c/(λα).Deno eby ∗ he minimize o U1
.Le Tbe he i s inspec ion ime condi-
ional on no ansi ion, and le he p omised u ili ies a ha ime be ˆ
u1and ˆ
u0.Now ix
>0 su icien ly small, and conside an al e na i e mechanism iden ical o he o iginal
mechanism, excep ha he i s ime o inspec ion is (T+), and wi h ini ial u ili ies
(˜
u1,˜
u0).I T<
∗, henle ˜
u0=˜
u1−u1+u0and le ˜
u1sol e
ˆ
u1=e (T+)˜
u1+(1−α)eλ(T+)−1u1−u0−c1−e (T+)/ .
Thus, by shi ing he ini ial p omised u ili ies up, he i s inspec ion da e is pos poned,
while main aining incen i e compa ibili y and keeping he e minal alues cons an .
Consequen ly, he ini ial u ili ies could no ha e been op imal. I T≥ ∗, henle ˜
u1=u1
and le ˜
u0sol e
ˆ
u1=e (T+)˜
u1+(1−α)eλ(T+)−1˜
u1−˜
u0−c1−e (T+)/ .
Thus, by shi ing u0up while keeping u1cons an , he i s inspec ion da e can be pos -
poned while main aining incen i e compa ibili y and keeping he e minal alues con-
s an . In ei he case, a pai o ini ial u ili ies wi h u1−u0>c/
(λα)canno be op imal.
Wi hou loss, we can now es ic a en ion o ini ial pai s o u ili y (u0,u1)such ha
u1−u0=c/(λα).Le u=u1deno e he ini ial u ili y o he agen in he high s a e. The
pa hs o p omised u ili ies a e hen desc ibed by φ0( ,u)and φ1( ,u). De ine
K1
n(u)=min
0≤ ≤T(u)
u≥φ1( ,u)
0
e−( +λ(1−α))sλ(1−α)K0
nds+e−( +λ(1−α)) K1
n−1u+κ(19)
o be he maximum payo o he p incipal a ini ial u ili y u o he agen , whe e he
p incipal maximizes o e s opping imes and he pos inspec ion u ili y u esul ing om
he e minal p omised u ili y φ1( ,u)and a po en ial ine a he ime o an inspec ion.
Le u∗
nbe a minimize o K1
nand deno e by ∗
n he associa ed i s inspec ion da e.
Claim 7. Le u∗
n−1be a minimize o K1
n−1(u)and suppose K1
n−1
(u)>0 o all u>u
∗
n−1.
Then ∗
n=T0(u∗
n)and φ1( ∗
n,u∗
n)>u
∗
n−1.
P oo . Fi s we show ha ∗
n=T0(u∗
n). Suppose, o he con a y, ha ∗
n<T(u∗
n).I
φ1( ∗
n,u∗
n)>u
∗
n−1, hen because φ1is s ic ly inc easing in i s second a gumen , we can
ind a lowe ini ial u ili y u<u
∗
nsuch ha φ1( ∗
n,u)<φ
1( ∗
n,u∗
n). Since K1
n−1
(˜
u)>0 o
˜
u>u
∗
n−1,weha eK1
n(u)<K
1
n(u∗
n), con adic ing op imali y o u∗
n.I φ1( ∗
n,u∗
n)≤u∗
n−1,
hen he op imal ini ial u ili y in s ep n−1isu=−u∗
n−1. We can hus ind >
∗
nsuch
ha φ1( ,u∗
n)<u
∗
n−1. Thus, he i s inspec ion was delayed, while he con inua ion
u ili y o he agen emains cons an , con adic ing op imali y o u∗
n. Thus, we ha e
∗
n=T0(u∗
n). Now suppose φ1(T0(u∗
n),u∗
n)<u
∗
n−1. Then we can ind a new ini ial u ili y
u>u
∗
nsuch ha φ1(T0(u),u)=u∗
n−1. Since T0(·)is inc easing we ha e T0(u)>T0(u∗
n),
con adic ing he op imali y o u∗
n.
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854 Achim and Knoep le Theo e ical Economics 19 (2024)
In ligh o he esul o Claim 7, he e will be no loss in limi ing ou a en ion o he
case =T(u)and u=φ1( ,u). The p incipal’s expec ed cos s o gi en u ili y ua e,
he e o e,
K1
n(u)=T(u)
0
e−( +λ(1−α))sλ(1−α)K0
nds+e−( +λ(1−α))T(u)K1
n−1φ1( ,u)+κ.
De ine β0=λα/( +λα)and β1=λ(1−α)/( +λ(1−α)). Sol ing he in eg als and e-
a anging, he p incipal’s payo can be exp essed mo e succinc ly as
K1
n(u)=a(u)+b(u)K1
n−1φ1T(u),u,
whe e
a(u)=e−( +λ(1−α))T(u)
1−β0β1+β0β1e−( +λ(1−α))T(u)κ
b(u)=e−( +λ(1−α))T(u)
1−β0β1+β0β1e−( +λ(1−α))T(u).
Simple calculus e eals
a(u)=−e( +λ−αλ)T(u)( +λ−αλ)2( +αλ) ( +λ)
(1−α)αλe( +λ−αλ)T(u) ( +λ)2κT(u)and
b(u)=−e( +λ−αλ)T(u) ( +λ)( +λ−αλ)2( +αλ)
(1−α)αλ2+e( +λ−αλ) ( +λ)2T(u),
so ha sign a(u)=sign b(u)=sign T(u).F om(17), i ollows ha
a(u)<0i u<u
∗
1
>0i u>u
∗
1
,b(u)<0i u<u
∗
1
>0i u>u
∗
1.
S ep 0 Conside he case n=0, so he p incipal canno pe o m any inspec ions. The
p incipal has no way o incen i ize he agen so ha he alue unc ion is equal o he
lowe bound
Kθ
0=¯
K.
S ep 1 Suppose he p incipal can inspec a mos once, so ha n=1.Le be he i s
inspec ion i no ansi ion occu s; le ube he ini ial u ili y o he agen . The expec ed
cos s o he p incipal when inspec ing a ime a e
K1
1(u)=a(u)+b(u)¯
K.
The ma ginal cos inc ease in u ili y uis
K1
1
(u)=a(u)+b(u)¯
K.
We ha e K1
1
(u)<0 o u<u
1∗and K1
1
(u)>0 o u>u
∗
1,wi hu∗
1>¯
u.Thusu1∗mini-
mizes K1
1.
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Theo e ical Economics 19 (2024) Rela ional en o cemen 855
S ep 2 Suppose he e a e wo inspec ions le o be pe o med. The p incipal’s payo can
be w i en as
K1
2(u)=a(u)+b(u)K1
1φ1T(u),u.
When u>u
∗
1, hen(K1
2)(u)>0 and, he e o e, he op imize does no exceed u∗
1.Be-
cause K1
2(u)is maximal when ulies a he pa icipa ion bounda y and is con inuous in
be ween, he e mus be a minimize u∗
2. The ma ginal cos inc ease is
K1
2
(u)=a(u)+b(u)K1φ1T(u),u+b(u)Duφ1T(u),u)K1
1
φ1T(u),u.
He e Duφ1(T(u),u)is he o al de i a i e o φ1(T(u),u)wi h espec o u, which can be
shown o be
Duφ1T(u),u=e T(u)1+T(u)ceλT (u)−11−α
α
λ+ u+ceλT(u)1−α
α+1>0.
Thus, o u>u
∗
1(>¯
u),
K1
2
(u)=a(u)+b(u)K1φ1T(u),u+b(u)Duφ1T(u),u)K
1φ1T(u),u
>a
(u)+b(u)K1φ1T(u),u>a
(u)+b(u)¯
K=K1
1
(u).
In pa icula , his means u∗
2<u
∗
1.
S ep nK
1
n(u)=a(u)+b(u)K1
n(φ1(T(u),u)) has a minimum a u∗
n. The ma ginal cos
a u>u
∗
n−1(>¯
u)is
K1
n
(u)=a(u)+b(u)Kn−1φ1T(u),u+b(u)Duφ1T(u),u)K
n−1φ1T(u),u
<a
(u)+b(u)K1
n−1(u)+b(u)Duφ1T(u),u)K
n−2φ1T(u),u
<a
(u)+b(u)K1
n−2(u)+b(u)Duφ1T(u),u)K
n−2φ1T(u),u,
whe e he i s line ollows om ou induc ion hypo hesis. The e o e, u≥u∗
n−1implies
K1
n
(u)<K
1
n−1
(u)<0.
The induc ion shows ha u∗
n<u
∗
n−1 o all n≥0. I ollows immedia ely om he
de ini ion o ¯
u ha u∗
n>¯
u o all n.Hence,{u∗
n}is a dec easing and bounded sequence,
so ha by he mono one con e gence heo em, he sequence con e ges o a limi ˆ
u≥¯
u.
Since {u∗
n}is con e gen , i is a Cauchy sequence, so ha by Claim 3,
lim
n→∞u∗
n−u∗
n−1=lim
n→∞u∗
k−φ1Tu∗
n,u∗
n=0⇒ˆ
u=¯
u.
This es ablishes he mechanism cha ac e ized in Theo em 1as he op imum in he
auxilia y p oblem. We now e i y ha i is also op imal in he o iginal p oblem.
B.2 Ve i ica ion o he p oo o Theo em 1
B.2.1 No ines be ween inspec ions We now show ha he mechanism desc ibed in he
p e ious sec ion emains op imal when we emo e assump ion (A). To his end, we show
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856 Achim and Knoep le Theo e ical Economics 19 (2024)
ha when pe o ming he i e a ion o e he numbe o a ailable inspec ions n, he p in-
cipal canno gain om imposing ines be ween inspec ions when ninspec ions a e le .
Conside again S ep no he i e a ion in he p e ious sec ion. By he same a gumen as
be o e, we ha e u1−u0=c/(λα)and he i s ime o inspec ion is a he i s ime a
which U0
=−B. The e olu ion o he pa hs o p omised u ili ies a e gi en by
dU1
= U1
d −λ(1−α)U1
−U0
d +cd +dF
dU0
= U0
d −λαU1
−U0
d +cd +dF −dμ ,
whe e we le dμ ≥0 deno e he slacking in he hones y cons ain . The e olu ion o he
di e ence in u ili ies is
dU1
−U0
=( +λ)u1−u0+dμ ,
which implies ha he u ili y pa hs di e ge a leas exponen ially, and a e independen
o any ines and inc easing in h ea s. I he i s inspec ion akes place a , condi ional
on no ansi ion be o e , his means ha U0
=−Band
U1
=−B+e( +λ) c
λα +
0
e( +λ)sdμs.
The las e m has o be ze o because o he wise we could ind a pai o ini ial p omised
u ili ies wi h ˆ
u1<u
1and se dμs=0 o alls∈(0, ), and a ime > such ha he
p omised u ili ies a ime unde he new ini ial condi ions a e as wi h he o iginal pai
a ime , hus inc easing he p incipal’s payo . The e o e, a he i s ime o inspec ion,
U1
=−B+e( +λ) c
λα.
Gi en ha U
1is independen o any ines in s ep n, and he e no ines in s ep n−1
onwa d, we mus ha e U1
=φ1(T(u∗
n),u∗
n). This means ha he policy o he p e ious
sec ion wi h ini ial p omised u ili ies (u∗
n,u∗
n−c/(λα)) emains op imal e en when ines
be ween inspec ions a e a ailable.
B.2.2 Gene al mechanisms in he elaxed p oblem Pa s B.1 and B.2.1 demons a e
ha he mechanism desc ibed in he heo em is an op imal Ma ko ian mechanism un-
de he elaxing o assump ion (B). I emains o e i y ha no (non-Ma ko ian) mecha-
nism can do be e . Le Kθ (U)deno e he expec ed cos s o he p incipal in ou mech-
anism ha deli e s he agen wi h p omised payo s o U=(U0,U1). We show ha he
expec ed alue in s a e θ om any incen i e-compa ible mechanism ha deli e s he
ini ial p omised payo U0=(U0
0,U1
0) o he agen canno exceed Kθ (U0). Since bo h
he inspec ion cos and he se o easible con inua ion u ili ies do no depend on hei
alues p io o inspec ion, we can apply P oposi ion 54.18 and Theo em 54.28 in Da is
(1993, pp. 235 & 242) o conclude ha Kn, he alue unc ion wi h no mo e han nin-
spec ions, con e ges o alue unc ion Ko he p oblem wi hou bound on he numbe
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Theo e ical Economics 19 (2024) Rela ional en o cemen 857
o inspec ions, and ha Kis he unique bounded and con inuous unc ion ha sol es
he quasi- a ia ional inequali y
UKθ(u)− Kθ(u)≥0
WK
θ(u)−Kθ(u)≥0
UKθ(u)− Kθ(u)WK
θ(u)−Kθ(u)=0
on he s a e space {(θ,u0,u1)|θ∈{0, 1},(u0,u1)∈[−B,0
]2,u1−u0≥c/(λα)}.He e,
U
deno es he ex ended gene a o o he piecewise de e minis ic Ma ko p ocess ha is
de ined by he ela ionship35
EP
0Kθ (U )=Kθ0(u)+EP
0
0
UKθs(us)ds
in case no inspec ion occu s be o e ,andWis he expec ed cos a an inspec ion ime:
WK
θ=min
u0,u1Kθ(u0,u1)+κ.
Conside an a bi a y incen i e-compa ible mechanism wi h inspec ion p ocess {dNI
}
and de ine he expec ed alue a ime by
G =
0
e− sκdNI
s+e− Kθ (U ).
Fo =0, we ha e G0=Kθ0(U0).Fo >0, we can ep esen G by he di e en ial o -
mula (see Theo em 31.3 in Da is (1993, p. 83)) as
Es[G ]−Gs=
s
e− (z−s)UKθz(Uz)− Kθz(Uz)dz
+Es
s
e− (z−s)WK
θz(Uz)−Kθz(Uz)dNI
z.
By he a ia ion inequali y abo e, bo h in eg als a e posi i e so ha he p ocess (G ) ≥
0 is a subma ingale bounded by 0. This implies ha E0[G ]≥G0 o any ≥0.
In pa icula , aking he limi as app oaches in ini y, we ge E0[∞
0e− s(κdNI
s)] =
E0[lim →∞ G ]≥G0=Kθ0(U0). Hence, any incen i e-compa ible maximal-compliance
mechanism leads o weakly highe inspec ion cos s.
B.2.3 Op imali y in he o iginal p oblem We now conside he o iginal model, in
which we emo e assump ion (A) so ha he hones y cons ain mus hold in bo h s a es.
We show ha du ing noncompliance, he hones y cons ain does no bind and, he e-
o e, he solu ion o he elaxed p oblem is also a solu ion o ou o iginal p oblem. The
p oo is cons uc i e. In he op imal mechanism o he elaxed p oblem, he pai o
p omised u ili ies a he ou se and du ing noncompliance is (u0,u1):=(¯
u,¯
u−c/(λα)).
35See Da is (1993, pp. 27–33).
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858 Achim and Knoep le Theo e ical Economics 19 (2024)
Since dU0
≤0, we ha e U0
≤u0.Se dF =u0−U0
and dμ =u1−U1
+u0−U0
.Nex ,
while θ=0, se dF =− u0+λα(u1−u0)and dμ =c+( +λ)(u1−u0). Subs i u -
ing in o he p omise-keeping and u h- elling cons ain s, i ollows ha dUi
=0 o
each i=0, 1 and dNI
=0 while θ =0, which is iden ical o he solu ion in he elaxed
p oblem.
P oo o Lemma 3
De ine
(T)≡(B−c/ )1−e− T −c/(λα)eλT e T −α+c/(λα)(1−α). (20)
By Theo em 1,weha eT∗=in {T>0:(T)=0}. This exis s and is unique whene e
B> ¯
B(is inc easing om 0 a T=0 and c osses 0 om abo e exac ly once). The
unc ion is con inuously di e en iable in λand Ton a neighbo hood o T∗.By he
implici unc ion heo em, we ha e
∂T∗
∂λ =−λ
TT=T∗
,
whe e xdeno es he pa ial de i a i e o wi h espec o x. As men ioned abo e,
(T)c osses 0 om abo e a T=T∗so ha T|T=T∗<0. Hence, o all pa ame e s, we
ha e
sign∂T∗
∂λ =sign(λ|T=T∗).
Conside in (20)asλλ=c /(B α −c),whichis helowe boundonλsuch ha
he easibili y assump ion B> ¯
B=c( +λ)
λα is ul illed. Then is equal o
B−c
1−e− T−B−c
αeλT e T −α−(1−α).
This can be equal o 0 only i T=0 because i is conca e in Tand he Tde i a i e is
0a T=0. Hence, limλ↓λT∗(λ)=0andT∗is ini ially inc easing in λ.
Finally, o show ha T∗(λ)λ→∞
−→ 0, conside (20)andobse e ha (T∗)=0implies
lim
λ→∞
e( +λ)T∗(λ)
λ=0.
This implies ha λT∗(λ)is ei he ini e o g ows a lowe han loga i hmic a e as λbe-
comes a bi a ily la ge. In pa icula , T∗(λ)mus go o 0.
Conside ing he cos s, le Le K0
EQ and K1
EQ deno e he expec ed discoun ed inspec-
ion cos when s a ing in s a e 0 o 1, espec i ely. Fo ixed inspec ion cycle leng h T,
hey ollow he nes ed equa ions
K0
EQ =∞
0
e−( +λα) λαK1
EQ d =λα
+λαK1
EQ
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Theo e ical Economics 19 (2024) Rela ional en o cemen 859
and
K1
EQ =T
0
e−( +λ(1−α)) λ(1−α)K0
EQ d +e−( +λ(1−α))Tκ+K1
EQ
=1−e( +λ(1−α))Tλ(1−α)
+λ(1−α)K0
EQ +e−( +λ(1−α))Tκ+K1
EQ.
Inse ing K0
EQ and sol ing o K1
EQ gi es
K1
EQ = +λα
( +λ)· +λ(1−α)e−( +λ(1−α))T
1−e−( +λ(1−α))T·κ.
No e ha K1
EQ is dec easing in Tand dec easing in λ o ixed T. Thus, gi en ha
T∗is inc easing in λ o low λ, i ollows immedia ely ha he cos s dec ease o low λ.
Fu he , K1
EQ app oaches ∞as Tgoes o ze o o any posi i e and ini e λ. Since
lim
λλT∗(λ)=0,
i ollows ha
lim
λλKEQ(λ)=0.
Fo he limi as λg ows a bi a ily la ge, no e ha he o al cos in he limi is gi en
by
lim
λ→∞ K1
EQ =(1−α)α
lim
λ→∞
λ
e(1−α)λT∗(λ).
Recall om abo e ha λT ∗(λ)g ows o ∞a lowe han loga i hmic a e so ha he e m
abo e mus be ∞.
Appendix C: P oo s o Sec ion 5
P oo o Theo em 2. We i s a gue ha as he mechanism desc ibed in he main
ex is op imal among he class o s a iona y mechanisms, conside an al e na i e
s a iona y s ochas ic mechanism ha deli e s some gi en p omised u ili y u.F om
he p omise-keeping cons ain (Pk) and he hones y cons ain (H) o i=1, i is
s aigh o wa d o ob ain ha he cons an a e mR(u)o inspec ion ha keeps he
p omised u ili y in s a e 1 s a iona y a he le el u∈[−B+c/(λα),−c/( α)] is mR(u)=
(c−αλu)/(αλ(B+u)−c).
The p incipal’s expec ed moni o ing cos s in he s a iona y andom mechanism ha
p o ides p omised u ili y U1
=u h oughou can be de e mined ecu si ely. Deno ing
by K1
R(u) he expec ed cos s while in compliance, we ha e
K1
R(u)= +λα
mR(u)
+λ= +λα
(c−αλu)
( +λ)αλ(B+u)−c. (21)
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860 Achim and Knoep le Theo e ical Economics 19 (2024)
I is easy o see ha KR(u)is dec easing in u. Gi en ha −c
α is an uppe bound on he
p omised u ili y o he agen du ing compliance (i is he maximum payo o he agen
subjec o sa is ying he obedience cons ain ) and i is he p omised u ili y deli e ed
by he mechanism cha ac e ized abo e, i ollows ha his mechanism is indeed he
op imal s a iona y mechanism.
To show ha he cos s o he andom mechanism a e s ic ly below he equilib ium
cos s wi h p edic able inspec ions, exp ess he la e as
K0
EQ =∞
0
e−( +λ) λαK1
EQ +(1−α)K0
EQd
K1
EQ =∞
0
e−( +λ) λα˜
K1(τ )+(1−α)K0
EQd +
∞

n=1
e−( +λ)kT∗κ,
whe e K0
EQ deno es he expec ed cos s while in noncompliance and
˜
K1(τ)=e−( +λ(1−α))(T∗−τ)κ+K1
EQ+T∗
τ
e−( +λ(1−α))(s−τ)λ(1−α)K0
EQ ds
deno es he expec ed cos s while in compliance and ime τ∈[0, T∗]has passed since
he las inspec ion o ansi ion. No e ha ˜
K1(τ)is inc easing in τwi h ˜
K1(0)=K1
EQ
and ˜
K1(T∗)=κ+K1
EQ. Thus, eplacing ˜
K1(τ )by K1
EQ in he ecu si e exp ession abo e,
and sol ing he sys em gi es an uppe bound on he equilib ium cos s K1
EG:
K1
EQ ≤˜
K= +λα
e−( +λ)T∗
1−e−( +λ)T∗κ. (22)
To see ha K1
Rin (21)islowe ,use(7) ow i eT∗in (22) as a unc ion o u1∗:
˜
Ku1∗= +λα
e−( +λ)T∗
1−e−( +λ)T∗κ= +λα
c
αλB+u1∗−cκ.
Now i is immedia e o check ha ˜
K(−c
α )=KR(−c
α )and ˜
K(u)<K

R(u) o u<−c
α
and B> ¯
B. Since −c/( α)is an uppe bound on u, i ollows ha KR(u1∗)<˜
K(u1∗).I
ollows om (22) ha K1
EQ >K
1
R.
Fo he compa a i e s a ics in λ,conside (21)andobse e ha K1
Ris dec easing in
λ o ixed mRand dec easing in mR(u). The op imal p omised u ili y −c/( α)does no
change wi h λand mR(u)is dec easing in λ o any u.
Fo he limi , obse e ha
lim
λ→∞ mR−c
α=c
B α −c
and, hus,
lim
λ→∞ K∗
R=α
c
B α −cκ=cα
B α −cκ.
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Theo e ical Economics 19 (2024) Rela ional en o cemen 861
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