Contracting over persistent information

Zhao, Wei; Mezze i, Claudio; Renou, Ludo ic; Tomala, T is an
A icle
Con ac ing o e pe sis en in o ma ion
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Zhao, Wei; Mezze i, Claudio; Renou, Ludo ic; Tomala, T is an (2024) : Con ac ing
o e pe sis en in o ma ion, Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y, New
Ha en, CT, Vol. 19, Iss. 2, pp. 917-974,
h ps://doi.o g/10.3982/TE5056
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Theo e ical Economics 19 (2024), 917–974 1555-7561/20240917
Con ac ing o e pe sis en in o ma ion
Wei Zhao
School o Economics, Renmin Uni e si y o China
Claudio Mezze i
School o Economics, The Uni e si y o Queensland
Ludo ic Renou
School o Economics and Finance, Queen Ma y Uni e si y o London, CEPR, and Depa men o
Economics, Uni e si y o Adelaide
T is an Tomala
HEC Pa is and GREGHEC-CNRS
We conside a dynamic p incipal-agen p oblem, whe e he sole ins umen he
p incipal has o incen i ize he agen is he disclosu e o in o ma ion. The p inci-
pal aims a maximizing he (discoun ed) numbe o imes he agen chooses he
p incipal’s p e e ed ac ion. We show ha he e exis s an op imal policy, whe e
he p incipal ecommends i s mos p e e ed ac ion and discloses in o ma ion as
a ewa d in he nex pe iod, un il ei he his ac ion becomes s a ically op imal o
he agen o he agen pe ec ly lea ns he s a e.
Keywo ds. Dynamic, con ac , in o ma ion, e ela ion, disclosu e, sende , e-
cei e , pe suasion.
JEL classi ica ion. C73, D82.
1. In oduc ion
We conside a dynamic “p incipal-agen ” model, whe e he sole ins umen he p in-
cipal has is in o ma ion.1P incipal and agen a e engaged in a long- e m ela ionship.
The p incipal aims a inducing he agen o choose an ac ion— he p incipal’s mos p e-
e ed ac ion—as o en as possible, and can only do so by disclosing in o ma ion abou
Wei Zhao: [email p o ec ed]
Claudio Mezze i: [email p o ec ed]
Ludo ic Renou: [email p o ec ed]
T is an Tomala: [email p o ec ed]
Wei Zhao g a e ully acknowledges he suppo o he HEC Founda ion. Claudio Mezze i hank ully ac-
knowledges inancial suppo om he Aus alian Resea ch Council Disco e y g an DP190102904. Lu-
do ic Renou g a e ully acknowledges he suppo o he Agence Na ionale pou la Reche che unde g an
ANR CIGNE (ANR-15-CE38-0007-01) and h ough he ORA P ojec “Ambigui y in Dynamic En i onmen s”
(ANR-18-ORAR-0005). T is an Tomala g a e ully acknowledges he suppo o he HEC Founda ion and
ANR/In es issemen s d’A eni unde g an ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047.
1Tha is, he p incipal canno make ans e s, e mina e he ela ionship, choose alloca ions, o cons ain
he agen ’s choices.
©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/TE5056
918 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
an unknown s a e. To gi e examples, he p incipal is: (i) an ex e nal consul an wi h a
clea agenda abou wha a company ( he agen ) should do, (ii) a depa men in a co po-
a ion aiming o main ain a cen al ole while ad ising he CEO, (iii) a echnology lead-
ing, mul ina ional i m in a join en u e wi h a local i m in a less de eloped coun y;
(i ) a lobbyis a emp ing o in luence a poli ician.
We assume ha he p incipal commi s o a disclosu e policy, which we e e o as he
o e o a “con ac .” The dynamic con ac ing p oblem we s udy is, he e o e, a dynamic
pe suasion p oblem.
The s anda d app oach in he s udy o dynamic con ac ing models (e.g., Spea and
S i as a a (1987)) is o use he agen ’s con inua ion alue, o p omised u ili y, as a s a e
a iable. The p incipal’s Bellman equa ion is hen he ixed poin o an ope a o , which
sa is ies a p omise-keeping cons ain in addi ion o incen i e cons ain s. Howe e , in
dynamic pe suasion models, he e a e addi ional complica ions.
Fi s , since he belie o he agen changes o e ime due o in o ma ion disclosu e,
we mus ea i as an addi ional s a e a iable. This inc eases he dimensionali y o he
p incipal’s p oblem. Second, any in o ma ion disclosu e policy, o which he p incipal
commi s, gene a es a ma ingale o belie s. We mus he e o e impose he cons ain
ha he belie p ocess is a ma ingale. To he bes o ou knowledge, we a e he i s o
be able o p o ide a comple e cha ac e iza ion o an op imal con ac by sol ing o he
ixed poin o a Bellman equa ion wi h wo s a e a iables acking he e olu ion o he
agen ’s belie s and o his p omised u ili y.
We now illus a e he gene al p ope ies o ou op imal policy. Fi s , he p incipal
uses in o ma ion disclosu e as a “ca o ” o mo i a e he agen o ake he p incipal’s
mos p e e ed ac ion un il ei he he agen pe ec ly lea ns he s a e, o choosing he
p incipal’s mos p e e ed ac ion becomes s a ically op imal. Mo eo e , i he agen
lea ns he s a e, he lea ns i in ini e ime. A e he agen has lea ned he s a e, he will
ake his op imal ac ion in ha s a e. Al e na i ely, as long as he agen keeps ge ing
pieces o in o ma ion om he p incipal (and hus, has no lea ned he s a e ye ), he will
ake he p incipal’s p e e ed ac ion. By ickling down bi s o in o ma ion, he p inci-
pal is able o induce he agen o delay mo ing away om his a o i e cou se o ac ion.
In some ins ances, he p incipal will p omise e en ual ull disclosu e o he s a e wi h
p obabili y one. In o he ins ances, he p incipal will be able o s i he agen ’s belie s so
ha , wi h posi i e p obabili y, he agen will ake he p incipal’s a o i e ac ion o e e .
We p o ide a cha ac e iza ion o when his occu s.
De ine he agen ’s oppo uni y cos a a s a e as he di e ence be ween he agen ’s
s age payo a his op imal ac ion and he s age payo when aking he p incipal’s p e-
e ed ac ion. Gene ically, he agen ’s oppo uni y cos , ela i e o he p incipal’s bene i
om his p e e ed ac ion, is di e en in di e en s a es, and ou op imal policy exploi s
hese di e ences. The second p ope y o ou op imal policy is ha , along he pa hs a
which he agen plays he p incipal’s mos p e e ed ac ion, his belie abou he likeli-
hood o he “high oppo uni y cos ” s a e is dec easing. In ui i ely, he op imal con ac
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 919
Figu e 1. E olu ion o ac ions and belie s o e ime.
exploi s he asymme y in oppo uni y cos s and lowe s he agen ’s expec ed oppo u-
ni y cos —hence making i easie o incen i ize he agen —by biasing in o ma ion dis-
closu e in he di ec ion o in o ming him when he oppo uni y cos is high.2
Figu e 1plo s ou ep esen a i e e olu ions o he agen ’s belie abou he high op-
po uni y cos s a e. In each panel, he g ey egion “OPT” indica es he egion a which
choosing he p incipal’s mos p e e ed ac ion is op imal o he agen . An a ow poin -
ing om one belie o ano he indica es how he agen e ises his belie wi hin he pe-
iod ollowing a signal’s ealiza ion. Mul iple a ows o igina ing om he same poin
hus ep esen he in o ma ion disclosed by he policy. Wi hin a pe iod, he agen akes
a decision a e ha ing e ised his belie s. A ows ha e di e en colo s/pa e ns. A all
belie s a he end o con inuous black a ows, he agen chooses he p incipal’s mos
p e e ed ac ion. A all belie s a he end o do ed magen a a ows, he chooses wha is
bes gi en his cu en belie .
Thi d, in panels (a), (b), and (c), he policy does no disclose in o ma ion o he agen
a he i s pe iod. S a ing om he second pe iod, he policy discloses jus enough in-
o ma ion o compensa e he agen o he oppo uni y cos o choosing he p incipal’s
p e e ed ac ion; no en is le o he agen . Howe e , as panel (d) illus a es, in some
cases he policy discloses in o ma ion in he i s pe iod, which may lea e a s ic ly posi-
i e en o he agen . Fo ins ance, i does so i he p omise o ull in o ma ion disclosu e
a he nex pe iod would no incen i ize he agen o choose he p incipal’s p e e ed
2To be p ecise, unde ou policy, upon ecei ing he signal “ he oppo uni y cos is high,” he agen lea ns
ha his is indeed ue. Howe e , he signal is no sen wi h p obabili y one. This co esponds o he (ma-
gen a/do ed) a ows poin ing a 1 in Figu e 1.
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920 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
ac ion. Disclosing in o ma ion a he i s pe iod may also be necessa y o educe he
agen ’s expec ed oppo uni y cos o ollowing he p incipal’s ecommenda ion.
Finally, wi h he excep ion o panel (b), he policy does no induce he agen o be-
lie e ha playing he p incipal’s mos p e e ed ac ion is op imal. This is ma kedly di -
e en om wha we would expec om he s a ic analysis o Kamenica and Gen zkow
(2011). In ui i ely, he “s a ic” pe suasion policy is subop imal because i does no ex-
ac all he in o ma ion su plus i c ea es. E en in panel (b), he belie s do no jump
immedia ely o he “OPT” egion. In ac , he belie p ocess may app oach he “OPT”
egion only asymp o ically.
These p ope ies highligh ha in ou dynamic en i onmen , in o ma ion is used as
a compensa ion ool o c ea ing in e empo al incen i es, mo e han as a pe suasion
ool o a ec he agen ’s myopic incen i es.
Rela ed li e a u e The pape is pa o he li e a u e on Bayesian pe suasion, pionee ed
by Kamenica and Gen zkow (2011), and ecen ly su eyed by Kamenica (2019). The
h ee mos closely ela ed pape s a e Ball (2023), Ely and Szydlowski (2020), and O lo ,
Sk zypacz, and Z yumo (2020). In common wi h ou pape , hese pape s s udy he
op imal disclosu e o in o ma ion in dynamic games and show how he disclosu e o
in o ma ion can be used as an incen i e ool. The obse a ion ha in o ma ion can be
used o incen i ize agen s is no new and da es back o he li e a u e on epea ed games
wi h incomple e in o ma ion, o example, Aumann, Maschle , and S ea ns (1995). See
Ga icano and Rayo (2017)andFudenbe g and Rayo (2019) o some mo e ecen pape s
explo ing he ole o in o ma ion p o ision as an incen i e ool.
The classes o dynamic games s udied di e conside ably om one pape o an-
o he , and his makes compa isons di icul . In Ely and Szydlowski (2020), he agen
has o epea edly decide whe he o con inue wo king on a p ojec o o qui (i.e., unlike
ou pape , he e a e only wo ac ions); qui ing ends he game. The p incipal aims a
maximizing he numbe o pe iods he agen wo ks on he p ojec and can only do so
by disclosing in o ma ion abou i s complexi y, modeled as he numbe o pe iods e-
qui ed o comple e he p ojec . Thus, hei dynamic game is a qui ing game, while ou s
is a epea ed game. When he p ojec is ei he easy o di icul (i.e., when he e a e wo
s a es), he op imal disclosu e policy ini ially pe suades he agen ha he ask is easy, so
ha he s a s wo king. (Na u ally, i he agen is su icien ly con inced ha he p ojec
is easy, he e is no need o pe suade him ini ially.) I he p ojec is in ac di icul , he
policy hen discloses i a a la e da e, when comple ing he p ojec is now wi hin each.
A main di e ence wi h ou op imal disclosu e policy is ha in o ma ion comes in lumps
in Ely and Szydlowski (2020), ha is, in o ma ion is disclosed only a he ini ial pe iod
and a a la e pe iod, while in o ma ion is epea edly disclosed in ou model.3Ano he
main di e ence is as ollows. In Ely and Szydlowski, only when he p omise o ull in o -
ma ion disclosu e a a la e da e is no enough o incen i ize he agen o s a wo king
3When he e a e mo e han wo s a es, he op imal policy discloses in o ma ion mo e equen ly in Ely
and Szydlowski (2020). The equency o disclosu e is hus a consequence o he dimensionali y o he s a e
space in hei model, while i is no so in ou model.
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 921
does he p incipal pe suade he agen ini ially. This is no so wi h ou policy: he p inci-
pal pe suades he agen in a la ge se o ci cums ances. This ini ial pe suasion educes
he cos o incen i izing he agen in u u e pe iods.
O lo , Sk zypacz, and Z yumo (2020) also conside a qui ing game, whe e he p in-
cipal aims a delaying he qui ing ime as a as possible.4The qui ing ime is when
he agen decides o exe cise an op ion, which has di e en alues o he p incipal and
he agen . The p incipal chooses a disclosu e policy in o ming he agen abou he op-
ion’s alue. When he p incipal is able o commi o a long- un policy, i is op imal o
ully e eal he s a e wi h some delay. This policy is no op imal in Ely and Szydlowski
(2020), o in ou pape . See Au (2015), Bizzo o, Rüdige , and Vigie (2021), Che, Kim,
and Mie endo (2023), Hen y and O a iani (2019), and Smolin (2021) o o he pape s
on in o ma ion disclosu e in qui ing games, whe e he agen ei he wai s and ob ains
addi ional in o ma ion, o akes an i e e sible ac ion and s ops he game.
Ball (2023) s udies a con inuous ime model o in o ma ion p o ision, whe e he
s a e changes o e ime and payo s a e he ones o he quad a ic example o C aw o d
and Sobel (1982). Ball shows ha he op imal disclosu e policy equi es he sende o
disclose he cu en s a e a a la e da e, wi h he delay sh inking o e ime. The main
di e ence be ween his wo k and ou s is he pe sis ence o he s a e (also, we conside
wo di e en classes o games). When he s a e is ully pe sis en , as in Ely and Szyd-
lowski (2020) and ou model, ull in o ma ion disclosu e wi h delay is no op imal in
gene al. (See he discussion o Example 1in Sec ion 3.)
Finally, he e a e a ew pape s on dynamic pe suasion, whe e he agen akes an ac-
ion epea edly. Howe e , ei he he agen is myopic, o example, Ely (2017)andRe-
naul , Solan, and Vieille (2017), o he p incipal canno commi , o example, Escude
and Sinande (2023).
2. The p oblem
2.1 The model
A p incipal and an agen in e ac o e an in ini e numbe o pe iods, indexed by ∈
{1, 2, }. A he i s pe iod, he p incipal lea ns a payo - ele an s a e ω∈=
{ω0,ω1}, while he agen emains unin o med. The p io p obabili y o ωis p0(ω)>0.
A each pe iod , he p incipal sends a signal s∈Sand, upon obse ing s, he agen akes
decision a∈A.These sAand Sa e ini e. The ca dinali y o Sis as la ge as necessa y
o he p incipal o be uncons ained in his in o ma ion disclosu e policy.5Th oughou ,
we in e changeably use he wo ds “pe iod” and “s age.”
We assume ha he e exis s a∗∈Asuch ha he p incipal’s s age payo is s ic ly
posi i e whene e a∗is chosen, and ze o o he wise. The p incipal’s s age payo unc ion
is hus :A×→R,wi h (a∗,ω0)>0, (a∗,ω1)>0, and (a,ω0)= (a,ω1)=0 o all
a∈A {a∗}. The agen ’s s age payo unc ion is u:A×→R. The (common) discoun
ac o is δ∈(0, 1).
4We e e o wha O lo , Sk zypacz, and Z yumo (2020) call he agen as he p incipal, and ice e sa.
5F om Mak is and Renou (2023), i is enough o ha e he ca dinali y o Sas la ge as he ca dinali y o A.
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922 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
We w i e A −1 o A×···×A
 
−1 imes
and S −1 o S×···×S
 
−1 imes
, wi h gene ic elemen s a and
s , espec i ely. A beha io al s a egy o he agen is a collec ion o maps σ=(σ )∞
=1
wi h σ :A −1×S →(A).
Be o e lea ning he s a e, he p incipal commi s o a s a egy, o con ac , speci ying,
as a unc ion o he s a e, he in o ma ion o be disclosed (i.e., he s a is ical expe imen
o be conduc ed) a each his o y o ealized signals and ac ions. Fo mally, he p incipal
commi s o a collec ion o maps (a con ac ) τ=(τ )∞
=1,wi hτ :A −1×S −1×→(S).
The con ac enables he p incipal o use in o ma ion disclosu es o ewa d o punish
he agen o choosing he “ igh ” o he “w ong” ac ion.
We deno e by V(τ,σ)and U(τ,σ) he p incipal’s and he agen ’s o e all expec ed
payo unde he p o ile (τ,σ).Le Pτ,σ(·|ω)be he dis ibu ion o e sequences o sig-
nals and ac ions induced by (τ,σ)condi ional on ω. The p incipal’s expec ed payo
V(τ,σ)is

ω
p0(ω)

s ,a −1
(1−δ)δ −1Pσ,τs −1,a −1|ωτ s |s −1,a −1,ωσ a∗|s ,a −1
× a∗,ω.(1)
The agen ’s expec ed payo is de ined simila ly. The objec i e is o cha ac e ize he max-
imal expec ed payo Vmax he p incipal can achie e by commi ing o a con ac τbe o e
lea ning he s a e, ha is,
Vmax =⎧
⎨
⎩
sup
(τ,σ)
V(τ,σ)
subjec o U(τ,σ)≥Uτ,σ o all σ.
Se e al commen s a e wo h making. Fi s , an al e na i e in e p e a ion o ou
model is ha nei he he p incipal no he agen know he s a e, bu he p incipal has
he abili y o conduc s a is ical expe imen s con ingen on he s a e and pas signals
and ac ions. Second, he only addi ional in o ma ion he agen ob ains each pe iod is
he ou come o he s a is ical expe imen . Thi d, he s a e is ully pe sis en and he p in-
cipal pe ec ly moni o s he ac ion o he agen . Finally, he only ins umen a ailable o
he p incipal is in o ma ion. The p incipal can nei he emune a e he agen no e mi-
na e he ela ionship no alloca e di e en asks o he agen . We pu pose ully make all
hese assump ions o add ess ou main ques ion o in e es : Wha is he op imal way o
incen i ize he agen wi h in o ma ion only?
2.2 An example
Th oughou he pape , we illus a e ou esul s wi h he help o he ollowing example.
Example 1. The agen has h ee possible ac ions a0,a,1and a∗,wi ha0( esp., a1) he
agen ’s op imal ac ion when he s a e is ω0( esp., ω1). The p io p obabili y o ω1is
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 923
Table 1. Payo able.
a0a1a∗
ω00, 1 0, 0 1, 1/2
ω10, 0 0, 2 1, 1/2
1/3 and he discoun ac o is 1/2. The payo s a e in Table 1, wi h he i s coo dina e
co esponding o he p incipal payo .
We s a wi h ew p elimina y obse a ions. Fi s , ega dless o he agen ’s belie ,
ac ion a∗is ne e op imal. Second, he oppo uni y cos o playing a∗is highe when he
s a e is ω1 han ω0, ha is,u(a1,ω1)−u(a∗,ω1)>u
(a0,ω0)−u(a∗,ω0). I is, he e o e,
ha de o incen i ize he agen o play a∗when he is mo e con iden ha he s a e is ω1.
As we shall see, he op imal policy exploi s his asymme y.
We now conside some simple s a egies he p incipal may commi o. To s a wi h,
assume ha he p incipal commi s o disclose in o ma ion a he ini ial s age only. We
call i he KG policy, in e e ence o Kamenica and Gen zkow (2011). Clea ly, since a∗is
ne e op imal, he p incipal’s payo is 0. To ob ain a posi i e payo , he p incipal mus
condi ion his in o ma ion disclosu e on he agen ’s ac ions.
The simples such policy is o “ ewa d” he agen wi h ull disclosu e o he s a e o
playing a∗a he beginning o he ela ionship, say up o pe iod T∗. I he agen de ia es,
he ha shes punishmen he p incipal can impose is o e eal no in o ma ion in sub-
sequen pe iods, inducing a no malized expec ed payo o 2/3. We a e hus looking o
he la ges T∗such ha
(1−δ)1
2δ0+δ1+···+δT∗−1+1
3·2+2
3·1δT∗+···≥2
3,
which is T∗=ln(5)/ln(2)=2, yielding he p incipal a payo o (1−1
2)·(1+1
2)=3
4.
Ano he simple s a egy he p incipal can commi o is a “ andom ull-disclosu e
policy,” whe e he ully discloses he s a e wi h p obabili y αa pe iod (and wi hholds
all in o ma ion wi h he complemen a y p obabili y) i he agen plays a∗a pe iod −1.6
(Again i he agen de ia es, he ha shes punishmen is o wi hhold all in o ma ion in all
subsequen pe iods.) Thus, i we w i e V( esp., U) o he p incipal ( esp., agen ) payo ,
he bes ecu si e policy is o choose αso as o maximize
V=1
21+1
2(1−α)V,subjec o:
U=1
21
2+1
2(1−α)U+α4
3≥2
3.
The p incipal’s bes payo is V=4/5wi hα=1/4. The andom ull-disclosu e pol-
icy does be e han he policy o ully disclosing he s a e wi h delay since i ci cum en s
6Full in o ma ion wi h delay plays an impo an ole in he wo k o Ball (2023) and O lo , Sk zypacz, and
Z yumo (2020).
15557561, 2024, 2, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5056 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
924 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
he in ege cons ain on T. In ui i ely, i makes i possible o incen i ize he agen o
play a∗a discoun ed numbe o pe iods sligh ly la ge han 2, namely ln(5)/ln(2).
As we will see in Sec ion 3.5, he andom ull-disclosu e policy is s ill subop imal
since i does no exploi he asymme y in he agen ’s oppo uni y cos o choosing a∗in
he wo s a es. The op imal policy exploi s such asymme y by disclosing no in o ma-
ion in he i s pe iod and hen ei he e ealing ha he s a e is ω1, he high oppo u-
ni y cos s a e, o lowe ing he agen ’s belie ha he s a e is ω1. By doing so, he policy
incen i izes he agen o ake ac ion a∗ o a longe expec ed ime. ♦
3. Op imal con ac s
This sec ion cha ac e izes op imal con ac s and discusses hei mos salien p ope ies.
3.1 A ecu si e o mula ion
The i s s ep owa d cha ac e izing op imal con ac s is o e o mula e he p incipal’s
p oblem as a ecu si e p oblem. To do so, we in oduce wo s a e a iables. The i s
s a e a iable is p omised payo . I is well known ha classical dynamic con ac ing
p oblems admi ecu si e o mula ions i we in oduce p omised payo as a s a e a i-
able and impose p omise-keeping cons ain s, o example, Spea and S i as a a (1987).
The second s a e a iable we in oduce is belie s. We now u n o he o mal e o mula-
ion o he p oblem.
We i s need some addi ional no a ion. We deno e by p∈[0, 1]a gene ic belie ,
wi h p he p obabili y o ω1.Wele u(a,p):=p[u(a,ω1)−u(a,ω0)] +u(a,ω0)be he
agen ’s expec ed s age payo o choosing awhen his belie is p. We de ine m(p):=
maxa∈Au(a,p)as he agen ’s op imal s age payo when his belie is p,andM(p):=
p[m(1)−m(0)] +m(0)as he agen ’s expec ed s age payo i he lea ns he s a e p io o
choosing an ac ion. No e ha mis a piecewise linea con ex unc ion ha Mis linea
and ha m(p)≤M(p) o all p. Simila ly, we le (a,p)be he p incipal’s expec ed s age
payo when he agen chooses aand he p incipal’s belie is p. Finally, le P:={p∈
[0, 1]:m(p)=u(a∗,p)}be he se o belie s a which a∗is op imal. I nonemp y, he se
Pis he closed in e al [p,¯
p].
Le W⊆[0, 1]×Rbe such ha (p,w)∈Wi and only i w∈[m(p),M(p)]. Th ough-
ou , we conside he comple e me ic space o bounded, con inuous unc ions V:W→
R, wi h he in e p e a ion ha V(p,w)is he p incipal’s payo i he p omises a payo o
w o he agen when he agen ’s cu en belie is p. Conside he ollowing maximiza ion
p og am:
T(V)(p,w):=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
max
((λs,(ps,ws),as)∈[0,1]×W×A)s∈S
s∈S
λs(1−δ) (as,ps)+δV (ps,ws),
subjec o:
(1−δ)u(as,ps)+δws≥m(ps) o all ssuch ha λs>0,

s∈S
λs(1−δ)u(as,ps)+δws≥w,

s∈S
λsps=p,
s∈S
λs=1.
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 931
(ii) When (a∗,0)
(a∗,1)=m(0)−u(a∗,0)
m(1)−u(a∗,1), he andom ull-disclosu e policy is op imal.
We has en o s ess ha bo h he KG and andom ull-disclosu e policies a e no op-
imal in gene al, as Example 1demons a es. See Sec ion 3.5.
3.4 Op imal policy: A o mal desc ip ion
We de ine a amily o policies (τq)q∈[q1,q1]indexed by a belie q, and p o e la e he ex-
is ence o q∗∈[q1,q1]such ha he policy τq∗is op imal. A each (p,w)∈W,apolicy
p esc ibes a easible uple (λs,(ps,ws),as)s∈S, ha is, a spli ing (λs,ps)s∈S, a p o ile o
ecommenda ions (as)s∈Sand a p o ile o con inua ion payo s (ws)s∈S.The ea e ou
di e en ypes o p esc ip ion, depending on which o ou egions he s a e a iables
(p,w)belong o; he belie qpa ame e izes hese egions. The ou egions a e
W1
q:=(p,w):p∈0, q1),w≤q1−p
q1m(0)+p
q1mq1,
W2
q:=(p,w):p∈(q,1
,1−p
1−qm(q)+p−q
1−qm(1)<w≤1−p
1−q1mq1+p−q1
1−q1m(1)
∪(p,w):p∈q1,q,w≤1−p
1−q1mq1+p−q1
1−q1m(1),
W3
q:=(p,w):p∈(q,1
],w≤1−p
1−qm(q)+p−q
1−qm(1),
W4
q:=W W1
q∪W2
q∪W3
q.
Figu e 3illus a es he ou egions, wi h W1
q he black egion, W2
q he egion wi h e ical
lines, W3
q he g ay egion, and W4
q he egion wi h slan ed lines. Obse e ha egions W1
q
and W4
qdo no depend on he pa ame e q, while he o he wo do.
We begin wi h an in o mal o e iew o ou op imal policy. In egion W2
q, he p inci-
pal ecommends a∗and discloses in o ma ion in he nex pe iod as a ewa d. The belie
pdec eases o e ime, un il ei he i eaches a poin a which he agen will choose a∗
o e e , o i en e s egion W4
q.
Figu e 3. Regions W1
q(black), W2
q( e ical lines), W3
q(g ay), and W4
q(slan ed lines).
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932 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
In egion W4
q, he p incipal discloses he s a e wi h su icien ly high p obabili y so
ha , when disclosu e does no occu , he agen ’s belie is q1. A he belie q1, he agen
plays a∗one inal ime. (Recall ha a q1, he agen plays a∗i ewa ded wi h ull in o -
ma ion disclosu e a he nex pe iod.)
In egion W1
q, he belie pis so low ha e en he p omise o ull in o ma ion disclo-
su e a he nex pe iod does no incen i ize he agen o play a∗, no e en once. In his
egion, he p incipal sends ei he he signal s∗o he signal s0.Thesignals0pe ec ly
in o ms he agen ha he s a e is ω0, while he signal s∗induces he belie q1,a which
he agen plays a∗one inal ime.
In egion W3
q, he belie pis highe han qand, e en possibly, highe han q1.In his
egion, he p incipal sends ei he he signal s∗o he signal s1.Thesignals1pe ec ly
in o ms he agen ha he s a e is ω1, while he signal s∗induces he belie q≤q1,a
which he agen plays a∗. I is almos he mi o image o wha he policy does in egion
o W1
q; he only concep ual di e ence is ha he policy induces he belie q a he han
q1, he na u al coun e pa o q1. This asymme y is a consequence o ying o induce
a∗a he lowes possible a e age belie .
We now o mally de ine he policy τq, s a ing wi h egion W2
q. De ine he unc ions
λ:W→[0, 1]and ϕ:W→[0, 1]so ha (λ(p,w),ϕ(p,w)) is he unique solu ion o
p
w=λ(p,w)ϕ(p,w)
mϕ(p,w)+1−λ(p,w)1
m(1)(5)
o all w>m
(p)and (λ(p,m(p)),ϕ(p,m(p))) =(1, p).When
(p,w)is in egion
W2
q, he policy spli s pin o wo belie s ϕ(p,w)and 1, wi h p obabili y λ(p,w)and
1−λ(p,w), espec i ely. When he pos e io belie is ϕ(p,w), he policy ecommends
a∗and p omises he con inua ion payo w(ϕ(p,w)) i he ecommenda ion is ollowed.
The e o e, i he agen ollows he ecommenda ion, his discoun ed expec ed payo is
m(ϕ(p,w)) =(1−δ)u(a∗,ϕ(p,w)) +δw(ϕ(p,w)). When he pos e io belie is 1, he
policy ecommends a1and p omises he con inua ion payo m(1),wi ha1an op i-
mal ac ion a s a e ω1. The e o e, i he agen ollows he ecommenda ion, he achie es
he discoun ed expec ed payo m(1). No e ha when w=m(p), he p incipal ecom-
mends a∗wi h p obabili y one, and p omises he con inua ion payo w(p)in he u-
u e. Upon ollowing he ecommenda ion, he agen achie es he discoun ed expec ed
payo m(p).
The key ea u e o he policy in egion W2
qis o disclose, wi h some p obabili y, ha
he s a e is ω1. As we al eady sugges ed, he a ionale o disclosing when he s a e is
ω1is wo- old. Fi s , he lowe he agen ’s belie , he lowe he cos o incen i izing he
agen o play a∗ ela i e o he p incipal’s bene i . Second, o sa is y he p omise-keeping
cons ain , he policy needs o compensa e he agen o playing a∗. Since he p incipal’s
payo is ze o when he agen akes any ac ion di e en om a∗, he bes is o choose a
compensa ion, which gua an ees he highes p obabili y o playing a∗. Pu ing hese
wo obse a ions oge he , a (p,w),policyτq(p,w) inds wo belie s (p,p )such ha
(i) he agen is asked o play a∗a p, (ii) p<psince he agen should play a∗a he
lowes belie , and (iii) he p obabili y o pis as high as possible. The bes spli ing is o
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 933
Figu e 4. Cons uc ion o λand ϕ:p=λϕ +(1−λ)1; w=λm(ϕ)+(1−λ)m(1).
ha e pas close as possible o pand p as a as possible, ha is, equal o 1. Obse e
ha since (1−λ(p,w))m(1)+λ(p,w)m(ϕ(p,w)) =w, he p omise-keeping cons ain
binds in egion W2
q. See Figu e 4 o an illus a ion.
No e ha s a ing wi h (p,w)∈W2
q, he dec easing sequence o belie s (ϕ(p,w),
ϕ2(p,w),)(and co esponding payo s) eaches ei he egion W4
q—as in panels (A)
and (C)o Figu e 1—o a belie in Pa which i is s a ically op imal o he agen o play
a∗—as in panel (B)o Figu e 1.14 In he la e case, he policy ecommends a∗and s ops
disclosing in o ma ion (i.e., he belie s ays cons an ).
When (p,w)is in egion W4
q, he agen canno be incen i ized o play a∗a (p,w).15
In ha case, he policy spli s pin o pos e io s 0, q1, and 1 wi h espec i e p obabili ies
λ0,λq1,andλ1. Condi ional on 0 ( esp., 1), he policy ecommends an ac ion op imal
a 0, ( esp., an ac ion op imal a 1), and p omises a con inua ion payo o m(0)( esp.,
m(1)). Condi ional on q1, he policy ecommends ac ion a∗and p omises a con inua ion
payo o w(q1). Doing so, he p incipal ensu es ha he agen plays a∗one mo e ime.
The p obabili ies (λ0,λq1,λ1)∈R+×R+×R+a e he unique solu ion o
λ0⎛
⎜
⎝
0
m(0)
1⎞
⎟
⎠+λq1⎛
⎜
⎝
q1
mq1
1
⎞
⎟
⎠+λ1⎛
⎜
⎝
1
m(1)
1⎞
⎟
⎠=⎛
⎜
⎝
p
w
1⎞
⎟
⎠.
A solu ion exis s since W4
qis he con ex hull o (0, m(0)),(q1,m(q1)),and(1, m(1)).In
his egion, he p omise-keeping cons ain is also binding.
When (p,w)is in egion W1
q, he policy spli s pin o 0 (i.e., discloses ha he s a e is
ω0)andq1wi h espec i e p obabili ies q1−p
q1and p
q1. I he ealized belie is 0, he pol-
icy ecommends an ac ion op imal a 0 and p omises a con inua ion payo o m(0).I
he ealized belie is q1, he policy ecommends a∗and p omises a con inua ion payo
o w(q1). The agen is hus made indi e en be ween playing a∗and ecei ing w(q1)in
he u u e, and playing a bes eply o he belie q1 o e e . In ui i ely, in egion W1
q, he
p incipal canno incen i ize he agen o ake ac ion a∗by p omising u u e in o ma ion
14We w i e ϕ2(p,w) o ϕ(ϕ(p,w),m(ϕ(p,w))).
15Recall ha q1is he lowes belie a which he agen can be incen i ized o play a∗.
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934 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
disclosu e (since p<q
1). Hence, he p incipal mus i s pe suade he agen by disclos-
ing some in o ma ion. No e ha he p omise-keeping cons ain is slack in his egion
whene e (p,w)sa is ies q1−p
q1m(0)+p
q1m(q1)>w.
When (p,w)is in egion W3
q, he policy spli s pin o qand 1 wi h espec i e p oba-
bili ies 1−p
1−qand p−q
1−q. Condi ional on 1, he policy ecommends an ac ion op imal a 1
and p omises a con inua ion payo o m(1). Condi ional on q, he policy ecommends
a∗and p omises a con inua ion payo o w(q). The agen is hus made indi e en be-
ween playing a∗and ecei ing w(q)in he u u e, and playing a bes - eply o he belie q
o e e . The policy in his egion is analogous o he one in egion W1
q— he policy s a s
by disclosing some in o ma ion. When q=q1, he eason o he analogy is immedia e,
as q1is he highes belie a which he agen is willing o ake ac ion a∗a he cu en
pe iod in exchange o ull in o ma ion a he nex pe iod. As we shall see la e , he
op imal policy τq∗may equi e q∗< q1, in o de o gua an ee ha he p incipal’s alue
unc ion is conca e, a necessa y equi emen o minimize he cos o incen i izing he
agen ela i e o he bene i o he p incipal. As in egion W1
q, he p omise-keeping con-
s ain is also slack in his egion whene e (p,w)sa is ies w<1−p
1−qm(q)+p−q
1−qm(1).This
comple es he desc ip ion o he policy τq.
Be o e mo ing on, we i s e i y ha ou policy τq∗is op imal unde he wo bench-
ma k scena ios discussed in Sec ion 3.3. Gi en he alue unc ion, we jus need o check
whe he τq∗sol es he Bellman equa ion.
Co olla y 2. The policy τq∗is op imal bo h when |A|=2and when (a∗,0)
(a∗,1)=
m(0)−u(a∗,0)
m(1)−u(a∗,1).
We now illus a e ou cons uc ion by e isi ing Example 1.
3.5 Example 1 e isi ed
We ha e ha M(p)=1+p,m(p)=max(1−p,2p)and w(p)=2max(2p,1−p)−(1/2).
The e o e, Q1=[1/6, 1/2]. Assume ha q=1/3 (we will show ha his choice is he
op imal’s one). Remembe ha he p io p obabili y o ω1is 1/3 and he discoun ac o
is 1/2. Le us s a wi h he pai (p,m(p)) =(1/3, 2/3), which is in egion W2
1/3.The
policy ecommends a∗ o he agen and p omises a con inua ion payo o w(1/3)=5/6.
The nex alue o he s a e a iables is he e o e (1/3, 5/6), which is again in W2
1/3.I he
agen had been obedien , he policy hen spli s he p io p obabili y 1/3 in o 3/11 and 1
wi h p obabili y 22/24 and 2/24, espec i ely. Indeed, we ha e
⎛
⎜
⎝
1
3
5
6
⎞
⎟
⎠=22
24 ⎛
⎜
⎝
3
11
m3
11⎞
⎟
⎠+2
24 1
m(1).
Condi ional on he pos e io 3/11, he policy ecommends a∗ o he agen and
p omises a con inua ion payo o w(3/11)=21/22. Condi ional on he pos e io 1, he
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 935
Figu e 5. E olu ion o he belie s.
policy ecommends a1and p omises a con inua ion payo o m(1)=2. The e o e, he
nex alue o he s a e a iables is ei he (3/11, 21/22)o (1, 2), wi h he o me again in
W2
1/3.
I he alue o he s a e a iables is (1, 2), he policy ye again ecommends a1and
a con inua ion payo o 2. I he alue o he s a e a iables is (3/11, 21/22), he policy
spli s 3/11 in o 7/39 and 1, wi h p obabili y 39/44 and 5/44, espec i ely. Condi ional on
he pos e io 7/39, he policy ecommends a∗ o he agen and p omises a con inua ion
payo o w(7/39)=89/78. Condi ional on he pos e io 1, he policy ecommends a1
and p omises a con inua ion payo o m(1)=2.
Finally, a he s a e a iables alue o (7/39, 89/78), which is in egion W4
1/3, he pol-
icy does a penul ima e spli o 7/39 in o 0, 1/6 and 1 wi h p obabili y 113/156, 18/156
and 25/156, espec i ely. Condi ional on he pos e io 1/6, he policy ecommends a∗
and p omises a con inua ion payo o 7/6, ha is, ull in o ma ion disclosu e a he nex
pe iod. The policy ully discloses he s a e in ini e ime o he agen . See Figu e 5 o he
e olu ion o he belie s a he beginning o each pe iod. A all belie s o he han 0 and 1,
he agen is ecommended o play a∗. The p incipal’s expec ed payo is 1285/1536, ha
is, abou 0.83.
Ou op imal policy pe o ms s ic ly be e han he andom ull-disclosu e policy
because i exploi s he asymme y in he agen ’s oppo uni y cos o choosing a∗in he
wo s a es. A each pe iod in which in o ma ion is disclosed and a∗is played, ou policy
dec eases he belie a which a∗is played; he a e age discoun ed belie s is p∗≈0.197 <
1/3.
On he con a y, he andom ull-disclosu e policy does no al e he belie ha he
s a e is ω1when a∗is played; he belie s ays ixed a he p io p0=1/3.
I emains o explain how o choose he pa ame e q∗ o gua an ee he op imali y o
τq∗.
3.6 Cons uc ion o q∗and op imali y
Fo all q∈[q1,q1],le Vq:W→Rbe he alue unc ion induced by he policy τq.Fo
all q,no e ha Vq(1, m(1)) =0 since a∗is no op imal a p=1, and Vq(0, m(0)) =0i
a∗is no op imal a p=0( esp.,Vq(0, m(0)) = (a∗,0
)i a∗is op imal a p=0). Also,
Vq(q1,m(q1)) =(1−δ) (a∗,q1)i q1>0( esp.,Vq(0, m(0)) = (a∗,0
)i q1=0, since a∗
is hen op imal a p=0). The e o e, any wo policies τqand τqinduce he same alues
a all (p,w)∈W1
q∪W4
q=W1
q∪W4
q. (Remembe ha he egions W1
qand W4
qdo no a y
wi h q; see Figu e 3.)
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936 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
Simila ly, any wo policies τqand τqinduce he same alues a all (p,w)∈W2
min(q,q).
Thus, in pa icula , τqand τq1induce he same alues a all (p,w)∈W W3
q. Finally, a all
(p,w)∈W3
q,Vq(p,w)=1−p
1−qVq(q,m(q)) =1−p
1−qVq1(q,m(q)). Hence, cha ac e izing Vq1is
enough o cha ac e ize Vq. (See Appendix B o mo e de ails.)
Recall ha V∗is he unique solu ion o he ixed-poin p oblem— o be op imal, a
policy mus he e o e induce he alue unc ion V∗.Le
q∗=supp∈q1,q1:Vq1p,m(p)≥Vq1(p,w) o all w.
We a e now eady o s a e ou main esul .
Theo em 1. The policy τq∗is op imal: Vq∗=V∗.
To unde s and he ole o q∗, ecall ha o all p∈[q∗,1
], he policy lea es en s o he
agen .16 To minimize hese en s, he p incipal he e o e would like o ha e q∗as high
as possible, ha is, equal o q1, he highes belie a which he agen is willing o play
a∗in exchange o ull in o ma ion disclosu e a he nex pe iod. Howe e , Vq1(·,m(·))
is no gua an eed o be conca e in p, a necessa y condi ion o op imali y. To see ha
V∗(·,m(·)) mus be conca e in p, conside any pai (p,p)∈[0, 1]×[0, 1]and α∈[0, 1].
We ha e
αV ∗p,m(p)+(1−α)V∗p,mp≤V∗αp +(1−α)p,αm(p)+(1−α)mp
≤V∗αp +(1−α)p,mαp +(1−α)p,
whe e he i s inequali y ollows om he conca i y o V∗in bo h a gumen s and he
second om V∗dec easing in wand he con exi y o m. The op imal choice o q∗is hus
he la ges q, which gua an ees Vq(·,m(·)) o be conca e.
Mo e p ecisely, as we show in Appendix A.5, he de ini ion o q∗gua an ees ha Vq∗
is conca e in bo h a gumen s and dec easing in w,so ha Vq∗(·,m(·)) is a conca e unc-
ion o p.Wealsop o e ha Vq∗(p,m(p)) ≥Vq1(p,m(p)) o all p. Since i is clea ly
he smalles such unc ion, Vq∗is he conca i ica ion o Vq1.Inpa icula ,q∗=q1i
Vq1(·,m(·)) is al eady conca e. Figu e 6illus a es he conca i ica ion o Example 1.In
dashed ed is he alue unc ion o policy τq1; in solid blue i s conca i ica ion— he alue
unc ion o policy τq∗,wi hq∗=1
3.
The policy τq∗lea es en s o he agen , ha is, he (ex an e) pa icipa ion cons ain
does no bind, o all p io s in [0, q1)∪(q∗,1
]. Thisisqui ena u al o allp io sin
[0, 1] Q1since he agen canno be incen i ized o play a∗e en once. In he language
o Ely and Szydlowski (2020), “ he goalpos s need o mo e,” ha is, one needs o dis-
close in o ma ion a he ex an e s age o pe suade he agen o play a∗. Howe e , ou
policy also lea es en s o all p io s in (q∗,q1]. The in ui i e eason is ha he ini ial in-
o ma ion disclosu e educes he cos o incen i izing he agen in subsequen pe iods
su icien ly enough o compensa e o he ini ial loss. (When he ealized pos e io is 1,
he agen ne e plays a∗, hus c ea ing he loss.)
16Tha is, he agen is p omised a payo o 1−p
1−q∗m(q∗)+p−q∗
1−q∗m(1)>m
(p).
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 937
Figu e 6. The conca i ica ion o Vq1(·,m(·)) in Example 1.
4. E olu ion o belie s in he op imal policy
The op imal policy discloses in o ma ion g adually o e ime, wi h belie s e ol ing un il
ei he he agen lea ns he s a e o belie es ha a∗is s a ically op imal. We can be mo e
speci ic. Fi s , we conside he ins ances when he policy con e ges wi h posi i e p oba-
bili y o a belie p∈P=[p,p], he se o belie s a which a∗is op imal. Le Q∞=[p,q∞],
wi h q∞ he solu ion o
mq∞=(1−δ)ua∗,q∞+δ1−q∞
1−pm(p)+q∞−p
1−pm(1),
i Pis nonemp y, and Q∞=∅,o he wise.No e ha P⊆Q∞. See Figu e 7 o a g aphical
illus a ion.
In ui i ely, he se Q∞has he “ ixed-poin p ope y,” ha is, i one s a s wi h a be-
lie p∈Q∞and p omised u ili y w(p), hen he belie ϕ(p,w(p)) ∈Q∞.Tosee his,no e
ha he pai (p,w(p)) is in egion W2
q. Since ϕ(p,w(p)) ≤p(wi h a s ic inequali y
i p/∈P), we hen ha e a dec easing sequence o belie s con e ging o an elemen in
P. This is because, a all belie s p∈Q∞, he policy spli s pin o p=ϕ(p,w(p)) and 1,
Figu e 7. Cons uc ion o q∞.
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938 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
hen spli s pin o p =ϕ(p,w(p)) and 1, e c. The dec easing sequence (p,p,p,)
con e ges, ei he in ini e ime o asymp o ically, o a belie in P, a which no u he
spli ing occu s and he agen plays a∗ o e e . See panel (b) o Figu e 1 o an an illus-
a ion.
Recall ha i he p io p0is la ge han q∗, he policy i s spli s p0in o q∗and 1.
Hence, i q∗≤q∞, he agen ’s belie en e s he se Q∞wi h s ic ly posi i e p obabil-
i y.17 The e o e, i he agen ’s p io belie is in he se Q∞
q∗, hen he e is a s ic ly posi i e
p obabili y ha he agen chooses ac ion a∗ o e e , whe e
Q∞
q∗:=!Q∞i q∗> q∞,
[p,1
)o he wise.
Second, a all p io s in [0, 1] Q∞
q∗, he eexis sTδ<∞such ha he belie p ocess
is abso bed in he degene a e belie s 0 o 1 a e a mos Tδpe iods. In o he wo ds, he
agen lea ns he s a e o su e in ini e ime. The numbe o pe iods Tδco esponds o
he maximal numbe o pe iods he agen can be incen i ized o play a∗.Wep o idean
explici compu a ion in Appendix B.InExample1,Tδ=3. Mo eo e , he numbe Tδis
inc easing in δand con e ges o +∞ as δcon e ges o 1. (No e ha he con e gence
is uni o m in ha i does no depend on p0∈[0, 1] Q∞
q∗.) Thus, we ha e he ollowing
co olla y.
Co olla y 3. Unde he op imal disclosu e policy τq∗, he e is a s ic ly posi i e p ob-
abili y ha he agen chooses ac ion a∗ o e e i , and only i , p0∈Q∞
q∗. Al e na i ely, i
p0/∈Q∞
q∗, hen he e exis s Tδsuch ha he agen pe ec ly lea ns he s a e (i.e., p eaches
ei he 0 o 1) wi h p obabili y 1a e a mos Tδpe iods.
The in e al Q∞
q∗includes he subin e al [p,¯
p], whe e he agen akes ac ion a∗wi h
p obabili y one. In he complemen a y se Q∞
q∗ [p,¯
p], he p obabili y ha he agen
akes ac ion a∗ o e e is s ic ly less han 1. Tha is, he p incipal discloses he s a e
wi h posi i e p obabili y, and wi h he complemen a y p obabili y he lowe s he agen ’s
belie so ha i con e ges o he egion whe e aking ac ion a∗is s a ically op imal. Con-
e gence may be asymp o ic o may happen in ini e ime.
As al eady men ioned, he p omise-keeping cons ain binds in egions W2
q∗and
W4
q∗, bu may no bind in he o he wo egions. We now a gue ha unde ou policy
τq∗, he p omise-keeping cons ain can only be slack in he i s pe iod. In o he wo ds,
he p omised-keeping cons ain binds om pe iod wo onwa ds. To see his, suppose
ha (p0,m(p0)) is in egion W3
q∗, hence he p io belie p0∈(q∗,1
). Wha he policy τq∗
does is o spli p0in o q∗and 1, so ha he s a e a iable ansi o ei he (q∗,m(q∗)) o
(1, m(1)). In he la e case, he p omise-keeping cons ain clea ly binds and will con-
inue o bind in all subsequen pe iods, since he agen has lea ned ha he s a e is ω1.
In he o me case, since (q∗,m(q∗)) ∈W2
q∗, he p omise-keeping cons ain binds and
will con inue o bind in all subsequen pe iods since he subsequen s a e a iables will
17F om he de ini ion o q∗,weha e ha q∗≥psince Vq1(p,m(p)) =u(a∗,p) o all p∈P.
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 939
ei he be in egions W2
q∗o W4
q∗o equal o (1, m(1)). A symme ic a gumen holds when
(p0,m(p0)) is in egion W1
q∗.
Co olla y 4. Unde he op imal policy τ∗
q, he p omise-keeping cons ain can only be
slack in he i s pe iod.
All in all, in o ma ion disclosu e plays wo oles in ou op imal policy. Fi s , he
p omise o u u e in o ma ion disclosu e mo i a es he agen o ake ac ion a∗in ea ly
pe iods. The in e empo al incen i es make i possible o mo i a e o play a∗a belie s
ou side P. Second, in o ma ion disclosu e dec eases he discoun ed a e age belie ha
he s a e is he high oppo uni y cos s a e ω1and, he e o e, makes i easie o incen-
i ize he agen o ake ac ion a∗ o a longe expec ed ime.
Appendix A: P oo s
A.1 Ma hema ical p elimina ies
We collec wi hou p oo s some use ul esul s abou conca e unc ions. Le :[a,b]→R
be a conca e unc ion and a≤x<y<z≤b. The ollowing p ope ies hold:
(a) (y)− (x)
y−x≥ (z)− (y)
z−y.
(b) (y)− (a)
y−a≥ (z)− (a)
z−a.
(c) (b)− (x)
b−x≥ (b)− (y)
b−y.
(d) (y)− (x)
y−x≥ (y+)− (x+)
y−x o all ≥0such ha y+≤b.
No e ha p ope y (a) implies (d) and is ue i espec i e o whe he x+y.Wewill
epea edly use hese p ope ies in mos o he ollowing p oo s.
To p o e Lemma 3, we will use he ollowing p ope y: i :[a,b]→Rsa is ies
(x)− (a)
x−a≥ (y)− (a)
y−a o all a<x≤y≤b, hen is conca e.
A.2 P oposi ion 2
P oo o P oposi ion 2(i). By con adic ion, assume ha he e exis s s∈Ssuch ha
λs>0and
(1−δ) (as,ps)+δV ∗(ps,ws)<V∗ps,(1−δ)u(as,ps)+δws.
Le (λ∗
s,p∗
s,w∗
s,a∗
s)s∈Sbe he policy, which achie es V∗(ps,(1−δ)u(as,ps)+δws),and
conside he new policy
(λs,ps,ws,as)s∈S {s},λsλ∗
s,p∗
s,w∗
s,a∗
ss∈S.
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940 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
By cons uc ion, he new policy is easible. Mo eo e , we ha e ha

s∈S {s}
λs(1−δ) (as,ps)+δV ∗(ps,ws)+λs
s∈S
λ∗
s(1−δ) a∗
s,p∗
s+δV ∗p∗
s,w∗
s
=
s∈S {s}
λs(1−δ) (as,ps)+δV ∗(ps,ws)+λsV∗ps,(1−δ)u(as,ps)+δws
>
s∈S
λs(1−δ) (as,ps)+δV ∗(ps,ws),
a con adic ion wi h he op imali y o (λs,ps,ws,as)s∈S. Thus, we mus ha e (1−δ) (as,
ps)+δV ∗(ps,ws)≥V∗(ps,(1−δ)u(as,ps)+δws) o all ssuch ha λs>0.
Since he ixed poin sa is ies V∗(ps,(1−δ)u(as,ps)+δws)≥(1−δ) (as,ps)+
δV ∗(ps,ws),weha e hedesi ed esul .
P oo o P oposi ion 2(ii). Le s∈Ssuch ha λs>0andas= a∗.Weha e
(1−δ) (as,ps)+δV ∗(ps,ws)=δV ∗(ps,ws)≥V∗ps,(1−δ)u(as,ps)+δws
≥V∗(ps,ws),
whe e he i s inequali y ollows om P oposi ion 2(i) and he second ollows om V∗
dec easing in wand ws≥u(as,ps) o
(1−δ)u(as,ps)+δws≥m(ps),
o hold. I ollows ha V∗(ps,ws)=0.
P oo o P oposi ion 2(iii). The p oo is by con adic ion. Suppose o he con a y
ha V∗(ps,ws)=V∗(ps,w
s) o some w
s∈(ws,M(ps)] and as=a∗. By P oposi ion 2(i),
we ha e
V∗ps,(1−δ)ua∗,ps+δws=(1−δ) a∗,ps+δV ∗(ps,ws)
=(1−δ) a∗,ps+δV ∗ps,w
s
≤V∗ps,(1−δ)ua∗,ps+δw
s.
Since V∗is dec easing in w, he inequali y canno be s ic , hence
V∗ps,(1−δ)ua∗,ps+δws=V∗ps,(1−δ)ua∗,ps+δw
s.(6)
Wenowshow ha
V∗ps,(1−δ)ua∗,ps+δws=V∗(ps,ws),(7)
hence
V∗ps,(1−δ)ua∗,ps+δw
s=V∗ps,w
s= a∗,ps,
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 947
This implies ha
¯
p
1−¯
p=ua∗,0
−ua†,0

ua∗,1
−ua†,1
.
Assuming p0>¯
p, unde ou policy, he p incipal ecommends he agen o ake a∗in he
i s pe iod and p omises o spli p0be ween 1 and ˜
pwi h p obabili y λin he second
pe iod, whe e (λ,˜
p)sol es
⎧
⎪
⎨
⎪
⎩
λ˜
p+(1−λ)1=p0
λua∗,˜
p+(1−λ)ua†,1
=w(p0)=ua†,p0−(1−δ)ua∗,p0
δ.
Replacing λ˜
p=λ−(1−p0)and λ(1−˜
p)=(1−p0)in o he second equa ion yields
λua∗,˜
p+(1−λ)ua†,1

=λua∗,1
−(1−p0)ua∗,1
+(1−p0)ua∗,0
+(1−λ)ua†,1

=ua∗,p0+(1−λ)ua†,1
−ua∗,1

=ua†,p0−(1−δ)ua∗,p0
δ
=⇒ λ=1−ua†,p0−ua∗,p0
δua†,1
−ua∗,1
=1−p0
δ−1−p0
δ
¯
p
1−¯
p.
Then i ollows ha he p incipal’s payo is
V=(1−δ) a∗,p0+δλ a∗,˜
p
=(1−δ)p0+δλ ˜
p a∗,1
+(1−δ)(1−p0)+δλ(1−˜
p) a∗,0

=p0−δ(1−λ) a∗,1
+(1−p0) a∗,0

= a∗,p0−δ(1−λ) a∗,1

= a∗,p0−p0+(1−p0)¯
p
1−¯
p a∗,1

=1−p0
1−¯
p a∗,¯
p,
which is exac ly he payo unde he KG policy, which spli s he ini ial belie p0in o ¯
p
wi h p obabili y 1−p0
1−¯
pand 1 wi h he complemen a y p obabili y.
Le VRbe he alue unc ion unde he andom ull-disclosu e policy. To show ha
ou policy τis also op imal when (a∗,0)
m(0)−u(a∗,0)= (a∗,1)
m(1)−u(a∗,1), we need o e i y ha
VR(p,w)=
ps∈supp(τ)
τ(ps)(1−δ) (as,ps)+δV R(ps,ws),∀(p,w)∈W.
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948 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
No e ha VR(p,w)=M(p)−w
M(p)−u(a∗,p) (a∗,p), since he p obabili y o ull disclosu e αsa -
is ies αM(p)+(1−α)u(a∗,p)=w.Hence,

ps∈supp(τ)
τ(ps)(1−δ) (as,ps)+δV R(ps,ws)
=λ·(1−δ) a∗,ˆ
p+δM(ˆ
p)−w(ˆ
p)
M(ˆ
p)−ua∗,ˆ
p a∗,ˆ
p
=λM(ˆ
p)−m(ˆ
p)
M(ˆ
p)−ua∗,ˆ
p a∗,ˆ
p,
whe e (λ,ˆ
p)sol es
!λˆ
p+(1−λ)1=p
λm(ˆ
p)+(1−λ)m(1)=w.
Since (a∗,0)
m(0)−u(a∗,0)= (a∗,1)
m(1)−u(a∗,1),weha e (a∗,ˆ
p)
M(ˆ
p)−u(a∗,ˆ
p)= (a∗,p)
M(p)−u(a∗,p)= (a∗,1)
M(1)−u(a∗,1).
The e o e, ecalling ha (a∗,1
)=0, we ha e
λM(ˆ
p)−m(ˆ
p)
M(ˆ
p)−ua∗,ˆ
p a∗,ˆ
p
=λM(ˆ
p)−m(ˆ
p)
M(ˆ
p)−ua∗,ˆ
p a∗,ˆ
p+(1−λ)M(1)−m(1)
M(1)−ua∗,1
 a∗,1

= a∗,p
M(p)−ua∗,pλM(ˆ
p)−m(ˆ
p)+(1−λ)M(1)−m(1)
= a∗,p
M(p)−ua∗,pM(p)−w=VR(p,w).
A.5 Theo em 1
To p o e Theo em 1, we i s in oduce he ollowing lemma.
Lemma 2. Conside any easible policy inducing he alue unc ion ˜
V.I ˜
Vis conca e in
bo h a gumen s, dec easing in wand sa is ies
˜
Vp,m(p)≥(1−δ) a∗,p+δ˜
Vp,w(p),
o all p∈Q1, hen he policy is op imal.
P oo . We a gue ha ˜
Vis he ixed poin o he ope a o T,hence ˜
V=V∗.Le
(λs,ps,ws,as)s∈Sbe a solu ion o he maximiza ion p oblem T(˜
V)(p,w).Wes a by
he ollowing obse a ion. Conside any ssuch ha as= a∗.Weha e
(1−δ) (as,ps)+δ˜
V(ps,ws)=δ˜
V(ps,ws)≤˜
V(ps,ws)≤˜
Vps,(1−δ)u(as,ps)+δws,
15557561, 2024, 2, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5056 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 949
whe e he las inequali y ollows om he ac ha ˜
Vis dec easing in wand m(ps)≤
(1−δ)u(as,ps)+δws≤(1−δ)m(ps)+δws≤ws.
Conside now any ssuch ha as=a∗. Since (λs,ps,ws,as)s∈Sis easible, we ha e
(1−δ)ua∗,ps+δws≥m(ps),
hence ps∈Q1and, he e o e,
˜
Vps,m(ps)≥(1−δ) a∗,ps+δ˜
Vps,−(1−δ)ua∗,ps+m(ps)
δ
 
w(ps)
.
The conca i y o ˜
Vimplies ha
˜
Vps,(1−δ)ua∗,ps+δws−˜
Vps,m(ps)≥δ˜
V(ps,ws)−˜
Vps,w(ps),
whe eweuse heiden i y(1−δ)u(a∗,ps)+δws−m(ps)=δ(ws−w(ps)) and obse a-
ion (a) abou conca e unc ions in Sec ion A.1.
Combining he abo e wo inequali ies implies
˜
Vps,(1−δ)ua∗,ps+δws≥(1−δ) a∗,ps+δ˜
V(ps,ws).
I ollows ha
T(˜
V)(p,w)=
s∈S
λs(1−δ) (as,ps)+δ˜
V(ps,ws)
≤
s∈S
λs˜
Vps,(1−δ)u(as,ps)+δws
≤˜
V
s∈S
λsps,
s∈S
λs(1−δ)u(as,ps)+δws)
≤˜
V(p,w),
whe e he second inequali y ollows om he conca i y o ˜
Vand he hi d inequali y
om ˜
Vbeing dec easing in w.
Con e sely, since he policy inducing ˜
Vis easible, we mus ha e ha T(˜
V)(p,w)≥
˜
V(p,w) o all (p,w). This comple es he p oo .
In oking Lemma 2, we only need o p o e he ollowing p oposi ion o p o e Theo-
em 1.
P oposi ion 4. Le Vq∗be he alue unc ion induced by he policy τ∗,wi h
q∗=supp∈Q1:Vq1p,m(p)≥Vq1(p,w) o all w.
Then Vq∗is conca e in (p,w), dec easing in w, and sa is ies
Vq∗p,m(p)≥(1−δ) a∗,p+δVq∗p∗,w(p),
o all p∈Q1.
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950 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
Figu e 8. The unc ion mq.
P o ing P oposi ion 4 equi es o cons uc he alue unc ion Vqinduced by he pol-
icy τq. The cons uc ion is edious, and we pos pone i o Appendix B.In he es o his
sec ion, we only epo he p ope ies we need o p o e P oposi ion 4.
We s a wi h an impo an iden i y, which we will use h oughou . Fo any q∈
[q1,q1], de ine he unc ion mq:[0, 1]→Ras
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
1−p
q1m(0)+p
q1mq1i p∈0, q1,
m(p)i p∈(q1,q],
1−p
1−qm(q)+p−q
1−qm(1)i p∈(q,1
].
No e ha mqis con ex, mq(p)≥m(p) o all p∈[0, 1],mq(0)=m(0),andmq(1)=
m(1). Fo a g aphical illus a ion, see Figu e 8.
I is s aigh o wa d o check ha we ha e he ollowing iden i y:
Vq(p,w)=λ(p,w)Vq(ϕ(p,w),mqϕ(p,w),(9)
whe e he unc ions λand ϕa e de ined as in he main ex , bu wi h mqins ead o m;
see equa ion (5). This iden i y s a es ha knowing Vqon he se {(p,w)∈W:(p,w)=
(p,mq(p))}su ices o econs uc Vqa all poin s on i s domain. We now make wo
addi ional obse a ions.
Obse a ion A. Fo all q∈[q1,q1], we ha e he ollowing iden i y:
Vq(p,w)=1−p
1−pVqp,1−p
1−pw+p−p
1−pmq(1).
P oo o Obse a ion A. Le w=1−p
1−pw+p−p
1−pmq(1).
Assume ha w> mq(p). Since
λp,wϕp,w
mqϕp,w+1−λp,w1
mq(1)=p
w,
we ha e
1−p
1−pλp,wϕp,w
mqϕp,w+1−1−p
1−pλp,w1
mq(1)=p
w.
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 951
The e o e, λ(p,w)=1−p
1−pλ(p,w)and ϕ(p,w)=ϕ(p,w)since he solu ion (λ(p,w),
ϕ(p,w)) is unique when w>m
q(p). The s a emen hen ollows om equa ion (9).
Assume ha w=mq(p). F om he con exi y o mq, his equi es ha w=mq(p),so
ha mq(p)=1−p
1−pmq(p)+p−p
1−pmq(1). The esul ollows om con inui y as
Vqp,mq(p)=lim
w→mq(p)Vq(p,w),
=lim
w→mq(p)
1−p
1−pVqp,1−p
1−pw+p−p
1−pmq(1),
=1−p
1−pVqp,1−p
1−pmq(p)+p−p
1−pmq(1),
=1−p
1−pVqp,mqp.
No e ha his implies ha
Vqp,w(p)+c=λp,w(p)Vqϕp,w(p),mqϕp,w(p)+c
λp,w(p),
whe e cis a posi i e cons an .
Obse a ion B. The alue unc ion Vq1(p,·):[mq1(p),M(p)] →Ris conca e in w,
o each p. See Lemma 3in Sec ion B.2.
A.5.1 P oposi ion 4(a) We p o e ha Vq∗is dec easing in w. To s a wi h, ix p∈[0, 1]
and (w,w)∈[mq∗(p),M(p)] ×[mq∗(p),M(p)],wi hw>w.
Fi s , assume ha p≤q∗.I w=mq∗(p), henVq∗(p,w)≤Vq∗(p,w)by cons uc ion
o q∗.I w>mq∗(p),weha e ha
Vq∗p,w−Vq∗(p,w)
w−w=Vq1p,w−Vq1(p,w)
w−w
≤Vq1(p,w)−Vq1p,mq∗(p)
w−mq∗(p)
=Vq∗(p,w)−Vq∗p,mq∗(p)
w−mq∗(p)≤0,
whe e he inequali y ollows om he conca i y o Vq1wi h espec o w, o allw≥
mq1(p).(Recall ha mq∗(p)=mq1(p) o all p≤q∗.)
Second, assume ha p>q
∗. We show in de ail how o make use o Obse a ion A o
deduce he esul . We epea edly use simila compu a ions la e on. We ha e
Vq∗p,w=λp,wVq∗ϕp,w,mq∗ϕp,w
=λp,w1−ϕp,w
1−ϕ(p,w)Vq∗ϕ(p,w),1−ϕ(p,w)
1−ϕp,wmq∗ϕp,w
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952 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
+1−1−ϕ(p,w)
1−ϕp,wmq∗(1)
=λ(p,w)Vq∗ϕ(p,w),λp,w
λ(p,w)mq∗ϕp,w+1−λp,w
λ(p,w)mq∗(1)
=λ(p,w)Vq∗ϕ(p,w),mq∗ϕ(p,w)+w−w
λ(p,w),
whe e he i s line ollows om he cons uc ion o Vq∗, he second line om Obse a-
ion A, he hi d line om he de ini ion o he unc ions λand ϕ, and he las line om
he ollowing compu a ions:
λp,w
λ(p,w)mq∗ϕp,w+1−λp,w
λ(p,w)mq∗(1)
=1
λ(p,w)w+1−1
λ(p,w)mq∗(1)
=1
λ(p,w)w+1−1
λ(p,w)w−λ(p,w)mq∗ϕ(p,w)
1−λ(p,w)
=mq∗ϕ(p,w)+w−w
λ(p,w).
Thus, we a e able o exp ess Vq∗(p,w)as λ(p,w)Vq∗(ϕ(p,w),˜
w),wi h ˜
w he abo e ex-
p ession. Mo eo e , ϕ(p,w)≤q∗as w≥mq∗(p). We can use he (al eady es ablished)
conca i y o Vq∗in w o each p≤q∗ o deduce he desi ed esul . Mo e p ecisely, we
ha e ha
Vq∗p,w−Vq∗(p,w)
w−w
=
λ(p,w)Vq∗ϕ(p,w),mq∗ϕ(p,w)+w−w
λ(p,w)−Vq∗ϕ(p,w),mq∗ϕ(p,w)
w−w
≤0,
whe e he inequali y ollows om he conca i y o Vq∗in wa all p≤q∗.
Las ly, since Vq∗(p,w)=Vq∗(p,mq∗(p)) o all w∈[m(p),mq∗(p)], he esul imme-
dia ely ollows o all (w,w),wi hw∈[m(p),mq∗(p)].
A.5.2 P oposi ion 4(b) We p o e he conca i y o Vq∗wi h espec o bo h a gumen s
(p,w).
Le W={(p,w):w≥mq∗(p)}.Le
(p,w)∈W,(p,w)∈Wand α∈[0, 1].W i e
(pα,wα) o
αp
w+(1−α)p
w.
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 953
Wi hou loss o gene ali y, assume ha p≤p.Weha e ha
αVq∗(p,w)+(1−α)Vq∗p,w
=α1−p
1−pVq∗p,1−p
1−pw+p−p
1−pmq∗(1)
 
≥mq∗(p)
+(1−α)Vq∗p,w
≤α1−p
1−p+(1−α)Vq∗⎛
⎜
⎜
⎝p,
α1−p
1−p1−p
1−pw+p−p
1−pmq∗(1)+(1−α)w
α1−p
1−p+(1−α)
⎞
⎟
⎟
⎠
=1−pα
1−pVq∗p,1−p
1−pα
wα+p−pα
1−pα
mq∗(1)
=Vq∗(pα,wα),
whe e he inequali y ollows om he conca i y o Vq1wi h espec o w o each pand
he p ope y ha Vq∗(p,w)=Vq1(p,w) o all (p,w)such ha w≥mq∗(p).No ice ha
we use wice Obse a ion A.
Finally, o all (p,w)∈W, o all(p,w)∈Wand o all α,weha e ha
αVq∗(p,w)+(1−α)Vq∗p,w
=αVq∗p,maxw,mq∗(p)+(1−α)Vq∗p,maxw,mq∗p
≤Vq∗pα,αmaxw,mq∗(p)+(1−α)maxw,mq∗p
≤Vq∗(pα,wα),
since αmax(w,mq∗(p))+(1−α)max(w,mq∗(p)) ≥wαand he ac ha Vq∗is dec easing
in w o all p. This comple es he p oo o conca i y.
A.5.3 P oposi ion 4(c) We p o e ha Vq∗(p,m(p)) ≥(1−δ) (a∗,p)+δVq∗(p,w(p))
o all p∈Q1.
The s a emen is ue o all p≤q∗by de ini ion since Vq∗(p,w)=Vq1(p,w) o all w.
Assume ha p>q
∗. F om Lemma 4, he eexis sqsuch ha ϕ(p,w(p)) ≥
ϕ(p,w(p)) o all p≥p≥q. Mo eo e , i ollows om A.6.3 and A.6.4 ha V(p,
m(p)) ≥V(p,w) o all w, o allp≤q. The e o e, we mus ha e ha q∗≥q.I ol-
lows ha ϕ(p,w(p)) <ϕ
(q∗,w(q∗)) ≤q∗,hencew(p)≥mq∗(p). We he e o e ha e ha
Vq∗(p,w(p)) =Vq1(p,w(p)).
Since Vq1(p,m(p)) =(1−δ) (a∗,p)+δVq1(p,w(p)) o all p∈Q1and Vq∗(p,
m(p)) =Vq∗(p,mq∗(p)) =Vq1(p,mq∗(p)), i is enough o p o e ha Vq1(p,mq∗(p)) ≥
Vq1(p,m(p)).
Clea ly, he e is no hing p o e i mq∗(p)=m(p) o all p∈Q1, ha is,i q∗=q1( e-
membe ha mq1(p)=m(p) o all p∈Q1).
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954 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
So, assume ha mq∗(p)>m
(p) o some p∈(q∗,q1),hencemq∗(p)>m
(p) o all
p∈(q∗,q1). We now a gue ha i Vq1(p,w)>V
q1(p,m(p)) o some w≥mq∗(p), hen
Vq1p,mp<1−p
1−pVq1(p,w),
o all p>p. To see his, obse e ha w>m
(p)and, acco dingly,
1−p
1−pw+p−p
1−pm(1)−mp>0,
since mis con ex. Hence,
0<Vq1(p,w)−Vq1p,m(p)
w−m(p)
=
1−p
1−pVq1p,1−p
1−pw+p−p
1−pm(1)−Vq1p,1−p
1−pm(p)+p−p
1−pm(1)
w−m(p)
≤
Vq1p,1−p
1−pw+p−p
1−pm(1)−Vq1p,mp
1−p
1−pw+p−p
1−pm(1)−mp,
whe e he equali y ollows Obse a ion A and he inequali y om he conca i y o Vq1in
w o each p. Since
Vq1(p,w)=1−p
1−pVq1p,1−p
1−pw+p−p
1−pm(1),
we ha e he desi ed esul .
Finally, om he de ini ion o q∗, o alln>0, he e exis pn∈(q∗,min(q∗+1
n,q1)]
and wn≥m(pn)such ha Vq1(pn,m(pn)) <V
q1(pn,wn). F om he conca i y o Vq1in w
o all p,Vq1(pn,m(pn)) <V
q1(pn,mq∗(pn)) o all n.
F om he abo e a gumen , o all p, o allnsu icien ly la ge, ha is, such ha pn<
p,weha e ha
Vq1p,m(p)<1−p
1−pn
Vq1pn,mq∗(pn).
Taking he limi as n→∞,weob ain ha
Vq1p,m(p)<1−p
1−q∗Vq1q∗,mq∗q∗=Vq1p,mq∗(p),
which comple es he p oo .
Appendix B: Cons uc ing he alue unc ion
This sec ion cha ac e izes he alue unc ion Vqinduced by he policy τq. As explained in
he ex , i su ices o cha ac e ize Vq1since Vq(p,w)=Vq1(p,w) o all (p,w)∈W W3
q
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 955
Figu e 9. Cons uc ion o he h esholds.
and Vq(p,w)=1−p
1−qVq1(q,m(q)) o all (p,w)∈W3
q. We i s s a wi h he de ini ion o
impo an subse s o [0, 1].
B.1 Cons uc ion o he se s Qk
Le Q0:=[0, 1]. We de ine induc i ely he se Qk⊆[0, 1],k≥0. We w i e qk( esp., qk)
o in Qk( esp., supQk). Fo any k≥0, de ine he unc ion Uk:[qk,1
]→R:
Uk(q):=1−q
1−qkmqk+q−qk
1−qkm(1),
wi h he con en ion ha Uk≡m(1)i qk=1. No e ha U0(q)=M(q)and Uk(q)≥m(q)
o all k. We de ine Qk+1as ollows:
Qk+1=q∈Qk:(1−δ)ua∗,q+δUk(q)≥m(q).
Fo a g aphical illus a ion, see Figu e 9.
Few obse a ions a e wo h making. Fi s , we ha e ha P⊆Qk o all k.Second,
we ha e a dec easing sequence, ha is, Qk+1⊆Qk o all k.Thi d,i Qkand Pa e
nonemp y, hen hey a e closed in e als. Fou h, he limi Q∞=limk→∞ Qk=#kQk
exis s and includes P.Mo eo e ,i P= ∅, henq∞=p,whe ep:=in P.I P=∅, hen
Q∞=∅. Consequen ly, he e exis s k∗<∞such ha ∅=Qk∗+1⊂Qk∗= ∅.
The i s o he hi d obse a ions a e eadily p o ed, so we concen a e on he p oo
o he ou h obse a ion. The limi exis s as we ha e a dec easing sequence o se s.
We p o e ha i P=∅, henQ∞=∅. So, assume ha P=∅. We i s a gue ha i
canno be ha Qk=Qk−1= ∅ o some k≥0. To he con a y, assume ha Qk=Qk−1=
∅ o some k≥0, hence Qk=Qk−1 o all k≥k. F om he con exi y and con inui y o
mand he linea i y o u,Qk−1is he closed in e al [qk−1,qk−1], wi h he wo bounda y
poin s solu ion o
(1−δ)ua∗,q+δUk−2(q)=m(q).
The e o e, i (qk,qk)=(qk−1,qk−1),weha e ha
mqk−1=(1−δ)ua∗,qk−1+δmqk−1,
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956 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
mqk−1=(1−δ)ua∗,qk−1+δ1−qk−1
1−qk−1mqk−1+qk−1−qk−1
1−qk−1m(1),
≤(1−δ)ua∗,qk−1+δmqk−1.
This implies ha u(a∗,qk−1)=m(qk−1)and u(a∗,qk−1)=m(qk−1)and, he e o e, ∅=
Qk−1⊆P, a con adic ion.
We hus ha e an in ini e sequence o s ic ly dec easing nonemp y closed in e als.
Le ε:=minp∈[0,1]m(p)−u(a∗,p). Since P=∅,weha e ha ε>0. Fo all p∈Q∞, o
all k,
m(p)≤(1−δ)ua∗,p+δUk(p)
≤(1−δ)m(p)−ε+δUk(p).
Assume ha Q∞is nonemp y and le q∞i s g ea es lowe bound. Since q∞∈Qk o all
k,weha e ha Uk(q∞)≥m(q∞)+ε(1−δ)/δ o all k. Since limk→∞ Uk(q∞)=m(q∞),
we ha e ha m(q∞)≥m(q∞)+ε(1−δ)/δ, a con adic ion.
We now p o e ha i P= ∅, henq∞=p.F omabo e,weha e ha i Qk=Qk−1= ∅
o some k≥0, hence Qk=Qk−1 o all k≥k, henP=Qksince P⊆Qk.I weha ean
in ini e sequence o s ic ly dec easing se s, o all q∈Q∞,
(1−δ)ua∗,q+δ1−q
1−q∞mq∞+q−q∞
1−q∞m(1)≥m(q).
Taking he limi q↓q∞,weob ain ha u(a∗,q∞)=m(q∞), ha is,q∞∈P.Hence,q∞=
p.
B.1.1 De i a ion o Vq1We i s de i e Vq1 o all (p,w)∈W W2
q1.
To s a wi h, Vq1(1, m(1)) =0 since a∗is no op imal a p=1. Simila ly, Vq1(0,
m(0)) =0i a∗is no op imal a p=0, while Vq1(0, m(0)= (a∗,0
)i a∗is op imal a
p=0. Also, Vq1(q1,m(q1)) =(1−δ) (a∗,q1)i q1>0; while Vq1(0, m(0)) = (a∗,0
)i
q1=0, since a∗is hen op imal a p=0.
Wi h he unc ion Vq1de ined a hese h ee poin s, i is hen de ined a all poin s
(p,w)in W1
q1∪W4
q1. In pa icula , i is easy o show ha
Vq1q1,w=Mq1−w
Mq1−mq1(1−δ) a∗,q1=Mq1−w
Mq1−ua∗,q1 a∗,q1,
o all w∈[m(q1),M(q1)].
A all poin s (p,w)∈W3
q1,
Vq1(p,w)=1−p
1−q1Vq1q1,mq1.
The e o e, Vq1is well-de ined a all (p,w)∈W W2
q1.
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 963
Since ϕηis dec easing in η,weha eϕη≤ϕηwhen η>η, and hence ϕ(ϕη,w(ϕη)) ≤
ϕ(ϕη,w(ϕη)),asϕη≤ϕη≤p≤q. Simila ly, since ϕη<p≤q,weha e ha
ϕ(ϕη,w(ϕη)) ≤ϕ(p,w(p)) and, he e o e, η−(1−δ)(1−λη)
δ>0.
We now e u n o he compu a ion o he g adien . We ha e
=(1−δ) a∗,p+δVq1p,w(p)
−(1−δ)λη a∗,ϕη+δVq1p,w(p)+η−(1−δ)(1−λη)
δm(1)−ua∗,1

×η−1
=(1−δ)
η a∗,p−λη a∗,ϕη
+δ
ηVq1p,w(p)−Vq1p,w(p)+η−(1−δ)(1−λη)
δm(1)−ua∗,1

=(1−δ)
η(1−λη) a∗,1

+δ
ηVq1p,w(p)−Vq1p,w(p)+η−(1−δ)(1−λη)
δm(1)−ua∗,1
. (10)
We u he de elop he abo e exp ession. To ease no a ion, we w i e (ϕ(p),λ(p))
o (ϕ(p,w(p)),λ(p,w(p))).No e ha ϕ(p)∈(qk−1,qk], since p∈(qk,qk+1].As
η−(1−δ)(1−λη)
δ>0, we ha e ha
=(1−δ)
η(1−λη) a∗,1

+δ
ηVq1p,w(p)−Vq1p,w(p)+η−(1−δ)(1−λη)
δm(1)−ua∗,1

=(1−δ)
η(1−λη) a∗,1
+δ
η
η−(1−δ)(1−λη)
δ
×
Vq1p,w(p)−Vq1p,w(p)+η−(1−δ)(1−λη)
δm(1)−ua∗,1

η−(1−δ)(1−λη)
δ
=(1−δ)
η(1−λη) a∗,1
+1−(1−δ)(1−λη)
η
×λ(p)Vq1ϕ(p),mq1ϕ(p)
−Vq1ϕ(p),mq1ϕ(p)+η−(1−δ)(1−λη)
δλ(p)m(1)−ua∗,1

×η−(1−δ)(1−λη)
δ−1
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964 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
=(1−δ)
η(1−λη) a∗,1
+1−(1−δ)(1−λη)
η
×Vq1ϕ(p),mq1ϕ(p)
−Vq1ϕ(p),mq1ϕ(p)+η−(1−δ)(1−λη)
δλ(p)m(1)−ua∗,1

×η−(1−δ)(1−λη)
δλ(p)−1
≥(1−δ)
η(1−λη) a∗,1
+1−(1−δ)(1−λη)
η a∗,1
= a∗,1
,
whe e we use Obse a ion A and he induc ion s ep.
We now show ha he g adien is inc easing in η. To s a wi h, no e ha η−(1−δ)(1−λη)
δ
is inc easing in ηsince 1−λη
ηis dec easing in η(see Lemma 5). Fo any η>η
,weha e
he ollowing:
Vq1p,w(p)−Vq1p,w(p)+η−(1−δ)(1−λη)
δmq1(1)−ua∗,1

η−(1−δ)(1−λη)
δ
=λ(p)Vq1ϕ(p),mq1ϕ(p)
−λ(p)Vq1ϕ(p),mq1ϕ(p)+η−(1−δ)(1−λη)
δλ(p)mq1(1)−ua∗,1

×η−(1−δ)(1−λ)
δ−1
=Vq1ϕ(p),mq1ϕ(p)
−Vq1ϕ(p),mq1ϕ(p)+η−(1−δ)(1−λη)
δλ(p)mq1(1)−ua∗,1

×η−(1−δ)(1−λ)
δλ(p)−1
≥Vq1ϕ(p),mq1ϕ(p)
−Vq1ϕ(p),mq1ϕ(p)+η−(1−δ)(1−λη)
δλ(p)mq1(1)−ua∗,1

×η−(1−δ)(1−λη)
δλ(p)−1
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 965
=
Vq1p,w(p)−Vq1p,w(p)+η−(1−δ)(1−λη)
δmq1(1)−ua∗,1

η−(1−δ)(1−λη)
δ
,
whe e he inequali y ollows om he ac ha ϕ(p)∈(qk−1,qk]and, he e o e, he g a-
dien G(ϕ(p);η)being inc easing in ηby he induc ion hypo hesis.
Finally, we ha e ha
1
ηVq1p,mq1(p)−Vq1p,w(p;η)
=(1−δ)(1−λη)
η a∗,1
+1−(1−δ)(1−λη)
η
×
Vq1p,w(p)−Vq1p,w(p)+η−(1−δ)(1−λη)
δm(1)−ua∗,1

η−(1−δ)(1−λη)
δ
≥(1−δ)(1−λη)
η a∗,1
+1−(1−δ)(1−λη)
η
×
Vq1p,w(p)−Vq1p,w(p)+η−(1−δ)(1−λη)
δmq1(1)−ua∗,1

η−(1−δ)(1−λη)
δ
=(1−δ)(1−λη)
η a∗,1
+1−(1−δ)(1−λη)
η
×
Vq1p,w(p)−Vq1p,w(p)+η−(1−δ)(1−λη)
δmq1(1)−ua∗,1

η−(1−δ)(1−λη)
δ
+(1−δ)(1−λη)
η−(1−δ)(1−λη)
η
×Vq1p,w(p)−Vq1p,w(p)+η−(1−δ)(1−λη)
δmq1(1)−ua∗,1

η−(1−δ)(1−λη)
δ
− a∗,1

≥1
ηVq1p,mq1(p)−Vq1p,wp;η
+(1−δ)(1−λη)
η−(1−δ)(1−λη)
η
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966 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
×Vq1ϕ(p),mq1ϕ(p)
−Vq1ϕ(p),mq1ϕ(p)+η−(1−δ)(1−λη)
δλ(p)mq1(1)−ua∗,1

×η−(1−δ)(1−λη)
δλ(p)−1
− a∗,1

≥1
ηVq1p,mq1(p)−Vq1p,wp;η.
The las inequali y ollows om he ac ha he g adien in he second b acke is
weakly la ge han (a∗,1
)by he induc ion hypo hesis and he ac ha 1−λη
η<1−λη
η
(Lemma 5).
Since limk→∞ qk=pwhen P= ∅, his comple es he p oo ha he g adien is
g ea e han (a∗,1
) o all p∈[0, p].
Fac 2: Fo all p∈I2,G(p;η)is inc easing in η.We i s ea he case P= ∅. Recall ha
o all p∈(p,q∞], we ha e an explici de ini ion o he alue unc ion Vq1(p,mq1(p)) as
a∗,p−mq1(p)−ua∗,p
mq1(1)−ua∗,1
 a∗,1
.
De ine ¯η(p)as he solu ion o ϕ¯η(p)=ϕ(p,w(p;¯η(p))) =p. No e ha o any p∈
(p,q∞], o anyη≤¯η,ϕη∈[p,q∞]. The e o e,
Vq1p,w(p;η)=ληVq1ϕη,mq1(ϕη)=λη a∗,ϕη−mq1(ϕη)−ua∗,ϕη
mq1(1)−ua∗,1
 a∗,1

= a∗,p−w(p;η)−ua∗,p
mq1(1)−ua∗,1
 a∗,1
.
I ollows ha he g adien is equal o (a∗,1
) o all p∈(p,p∗], o allη≤¯η.
Conside now η> ¯η. We ew i e he g adien G(p;η)as ollows:
Vq1p,mq1(p)−Vq1p,w(p;η)
η
=Vq1p,mq1(p)−Vq1p,wp;η1(p)
η
+Vq1p,wp;η1(p)−Vq1p,w(p;η)
η
=η1(p)
η
Vq1p,mq1(p)−Vq1p,wp,η1(p)
η1(p)
+η−η1(p)
η
Vq1p,wp;η1(p)−Vq1p,w(p;η)
η−η1(p)
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 967
=η1(p)
η a∗,1
+η−η1(p)
η
1−p
1−pVq1p,mq1(p)−Vq1p,wp;η−η1(p)
1−p
1−p
η−η1(p)
=η1(p)
η a∗,1
+η−η1(p)
ηGp;η−η1(p)
1−p
1−p.
Since we ha e al eady shown ha G(p;η)is inc easing in ηand weakly la ge han
(a∗,1
), we ha e ha he g adien G(p;η)is also weakly inc easing in η(and g ea e
han (a∗,1
)).
We now ea he case P=∅. De ine ¯η(p)as he solu ion o ϕ¯η(p)=ϕ(p,w(p;
¯η(p))) =q. No e ha o any p∈[q,q], o anyη≤¯η,ϕη∈[q,q]. The e o e, o all η≤¯η,
η=(1−δ)(1−λη)since he a io mq1(1)−w(ϕη)
1−ϕηis cons an in ηand so is ϕ(ϕη,w(ϕη)).
(Recall ha we a y ηa a ixed p.) I ollows hen om equa ion (10) ha
G(p;η)=(1−δ)
η(1−λη) a∗,1

+δ
ηVq1p,w(p)−Vq1p,w(p)+η−(1−δ)(1−λη)
δm(1)−ua∗,1

=(1−δ)
η(1−λη) a∗,1
= a∗,1
.
We ha e ha he g adien G(p;η)is equal o (a∗,1
) o all p∈(q,q], o allη≤¯η.Fi-
nally, when η> ¯η, he same decomposi ion as in he case P= ∅ comple es he p oo .
Fac 3: Fo all p∈I3, heg adien G(p;η)is inc easing in η.We only ea he case
P= ∅.(ThecaseP=∅is ea ed analogously.) De ine ¯η(p)as he solu ion o ϕ¯η(p)=
ϕ(p,w(p;¯η(p))) =q∞. By cons uc ion, o all p∈(q∞,1
], o allη≤¯η(p),weha e ha
ϕη∈(q∞,1
]. The e o e, ϕη> q.
Choose ¯η(p)≤η≤η.Weha e ha ϕη≥ϕη≥qsince q∞≥qand, he e o e,
ϕp,w(p)+η−(1−δ)(1−λη)
δmq1(1)−ua∗,1

=ϕϕη,w(ϕη)
≥ϕ(ϕη,w(ϕη)=ϕp,w(p)+η−(1−δ)(1−λη)
δmq1(1)−ua∗,1
.
Also, since q≤ϕη≤p,weha e ha ϕ(ϕη,w(ϕη)) ≥ϕ(p,w(p)) and, he e o e,
η−(1−δ)(1−λη)
δ≤0.Thesameapplies oη. Finally, as al eady shown,
η−(1−δ)(1−λη)
δ<η−(1−δ)(1−λη)
δ.
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968 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
To ease no a ion, de ine (˜
λη,˜ϕη)as ollows:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
˜
λη=λp,w(p)−(1−δ)(1−λη)−η
δm(1)−ua∗,1

˜ϕη=ϕp,w(p)−(1−δ)(1−λη)−η
δm(1)−ua∗,1
(11)
No ice ha ˜ϕη=ϕ(ϕη,w(ϕη)) ∈I1since ϕη> q∞.
The es o he p oo is pu ely algeb aic and mi o s he case p∈I1.Fi s ,weha e
he ollowing:
Vq1p,w(p)−Vq1p,w(p)−(1−δ)(1−λη)−η
δmq1(1)−ua∗,1

(1−δ)(1−λη)−η
δ
=˜
ληVq1˜ϕη,mq1(˜ϕη)+(1−δ)(1−λη)−η
δ˜
ληmq1(1)−ua∗,1

−˜
ληVq1˜ϕη,mq1(˜ϕη)(1−δ)(1−λη)−η
δ−1
=
Vq1˜ϕη,w˜ϕη;(1−δ)(1−λη)−η
δ˜
λη−Vq1˜ϕη,mq1(˜ϕη)
(1−δ)(1−λη)−η
δ˜
λη
,
whe e we again use Obse a ion A. Simila ly, we ha e
Vq1p,w(p)−Vq1p,w(p)−(1−δ)(1−λη)−η
δmq1(1)−ua∗,1

(1−δ)(1−λη)−η
δ
=˜
ληVq1˜ϕη,w˜ϕη;(1−δ)(1−λη)−η
δ˜
λη
−˜
ληVq1˜ϕη,w˜ϕη;(1−δ)(1−λη)−η
δ˜
λη
−(1−δ)(1−λη)−η
δ˜
λη
×(1−δ)(1−λη)−η
δ−1
=Vq1˜ϕη,w˜ϕη;(1−δ)(1−λη)−η
δ˜
λη
−Vq1˜ϕη,w˜ϕη;(1−δ)(1−λη)−η
δ˜
λη
−(1−δ)(1−λη)−η
δ˜
λη
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 969
×(1−δ)(1−λη)−η
δ˜
λη−1
,
whe e again we use Obse a ion A and he ac
(1−δ)(1−λη)−η
δ˜
λη
>(1−δ)(1−λη)−η
δ˜
λη
.
Since ˜ϕη∈I1,weha e ha
Vq1˜ϕη,w˜ϕη;(1−δ)(1−λη)−η
δ˜
λη
−Vq1˜ϕη,w˜ϕη;(1−δ)(1−λη)−η
δ˜
λη
−(1−δ)(1−λη)−η
δ˜
λη
×(1−δ)(1−λη)−η
δ˜
λη−1
≤
Vq1˜ϕη,w˜ϕη;(1−δ)(1−λη)−η
δ˜
λη−Vq1˜ϕη,mq1(˜ϕη)
(1−δ)(1−λη)−η
δ˜
λη
,
whe e he inequali y ollows om ou p e ious a gumen on he in e al I1.
I ollows ha
Vq1p,w(p)−Vq1p,w(p)−(1−δ)(1−λη)−η
δmq1(1)−ua∗,1

(1−δ)(1−λη)−η
δ
≤
Vq1p,w(p)−Vq1p,w(p)−(1−δ)(1−λη)−η
δmq1(1)−ua∗,1

(1−δ)(1−λη)−η
δ
.
F om equa ion (10), we hen ha e ha
1
ηVq1p,mq1(p)−Vq1p,w(p;η)
=(1−δ)(1−λη)
η a∗,1
+(1−δ)(1−λη)
η−1
×
Vq1p,w(p)−Vq1p,w(p)−(1−δ)(1−λη)−η
δm(1)−ua∗,1

(1−δ)(1−λη)−η
δ
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970 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
≥(1−δ)(1−λη)
η a∗,1
+(1−δ)(1−λη)
η−1
×
Vq1p,w(p)−Vq1p,w(p)−(1−δ)(1−λη)−η
δmq1(1)−ua∗,1

(1−δ)(1−λη)−η
δ
=(1−δ)(1−λη)
η a∗,1
+1−(1−δ)(1−λη)
η
×
Vq1p,w(p)−(1−δ)(1−λη)−η
δmq1(1)−ua∗,1
−Vq1p,w(p)
(1−δ)(1−λη)−η
δ
=(1−δ)(1−λη)
η a∗,1
+1−(1−δ)(1−λη)
η
×
Vq1p,w(p)−(1−δ)(1−λη)−η
δmq1(1)−ua∗,1
−Vq1p,w(p)
(1−δ)(1−λη)−η
δ
+(1−δ)(1−λη)
η−(1−δ)(1−λη)
η
×%Vq1p,w(p)−(1−δ)(1−λη)−η
δmq1(1)−ua∗,1
−Vq1p,w(p)
(1−δ)(1−λη)−η
δ
− a∗,1

≥1
ηVq1p,mq1(p)−Vq1p,wp;η,
whe e he las inequali y ollows om
Vq1p,w(p)−(1−δ)(1−λη)−η
δmq1(1)−ua∗,1
−Vq1p,w(p)
(1−δ)(1−λη)−η
δ
=
˜
ληVq1˜ϕη,mq1(˜ϕη)−˜
ληVq1˜ϕη,w˜ϕη;(1−δ)(1−λη)−η
δ˜
λη
(1−δ)(1−λη)−η
δ
≥ a∗,1
.
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Theo e ical Economics 19 (2024) Con ac ing o e pe sis en in o ma ion 971
We now show ha he he g adien G(p;η)is smalle han (a∗,1
) o any η≤¯η(p).
F om equa ion (10), we ha e ha
1
ηVq1p,mq1(p)−Vq1p,w(p;η)
=(1−δ)(1−λη)
η a∗,1
−(1−δ)(1−λη)
η−1
×
Vq1p,w(p)−(1−δ)(1−λη)−η
δm(1)−ua∗,1
−Vq1p,w(p)
(1−δ)(1−λη)−η
δ
= a∗,1
−(1−δ)(1−λη)
η−1
×%Vq1p,w(p)−(1−δ)(1−λη)−η
δm(1)−ua∗,1
−Vq1p,w(p)
(1−δ)(1−λη)−η
δ
− a∗,1

= a∗,1
−(1−δ)(1−λη)
η−1
×%˜
ληVq1˜ϕη,mq1(˜ϕη)−˜
ληVq1˜ϕη,w˜ϕη;(1−δ)(1−λη)−η
δ˜
λη
(1−δ)(1−λη)−η
δ
− a∗,1
&
= a∗,1
−(1−δ)(1−λη)
η−1
 
≥0
×%Vq1˜ϕη,mq1(˜ϕη)−Vq1˜ϕη,w˜ϕη;(1−δ)(1−λη)−η
δ˜
λη
(1−δ)(1−λη)−η
δ˜
λη
− a∗,1
&
 
≥0
≤ a∗,1
,
whe e he inequali y ollows om he ac ha ˜ϕη≤p( he e o e, om ou a gumen s
on he in e al I1, whe e we show ha he g adien is la ge han (a∗,1
)).
Finally, we can use a simila decomposi ion as in he case p∈I2 o p o e ha he
g adien is inc easing o all η.
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972 Zhao, Mezze i, Renou, and Tomala Theo e ical Economics 19 (2024)
Appendix C
C.1 Recu si e o mula ion: A p oo
Ely (2015) p o es ha he p incipal’s maximal payo is maxw∈[m(p0),M(p0)] 
V∗(p0,w),
wi h 
V∗ he unique ixed poin o he con ac ion 
T, wi h he ope a o 
Tdi e ing om
he ope a o Tin ha he p omise-keeping cons ain is w i en as an equali y in all
maximiza ion p oblems 
T(V)(p,w); all o he cons ain s a e he same. No e ha , like
T, he ope a o 
Tis mono one.
Fo any (p,w)∈W,le "
V∗(p,w):=max
w∈[w,M(p)] 
V∗(p,
w)and "
w∗(p,w)amaxi-
mize . (I he e a e mul iple maximize s, choose an a bi a y one.)
We p o e ha V∗="
V∗.Todoso,wep o e ha T("
V∗)="
V∗. Since Tis a con ac ion,
hence has a unique ixed poin , i ollows ha V∗="
V∗. (No e ha we a e no a guing
ha T=
T.)
We s a wi h wo simple obse a ions: (i) T(V)(p,w)≥
T(V)(p,w) o all (p,w)∈
W, o allV, and (ii) "
V∗(p,w)≥
V∗(p,w) o all (p,w)∈W. The i s obse a ion ol-
lows om he ac he p omised-keeping cons ain is an equali y in 
T(V)(p,w), while
i is an inequali y in T(V)(p,w). The second obse a ion ollows immedia ely om he
de ini ion o "
V∗.
We now p o e ha T(˜
V∗)≥˜
V∗.Fo all(p,w)∈W,weha e
"
V∗(p,w)=
V∗p,"
w∗(p,w)=
T
V∗p,"
w∗(p,w),
≤T
V∗p,"
w∗(p,w),
≤T
V∗(p,w),
≤T"
V∗(p,w),
whe e he i s line ollows om he de ini ions o "
V∗,
V∗,and
T,and he ac ha 
V∗=

T(
V∗); he second line om obse a ion (i); he hi d line om he ac ha "
w∗(p,w)≥
w, so ha all easible solu ions o T(
V∗)(p,"
w∗(p,w)) a e also easible o T(
V∗)(p,w);
and he ou h line om obse a ion (ii) and he de ini ion o T(V)(p,w),V=
V∗,"
V∗.
We nex p o e ha T("
V∗)≤"
V∗. By con adic ion, suppose ha he e exis s (p,w)∈
Wand a easible policy (λs,ps,as,ws)s∈Ssuch ha
"
V∗(p,w)<
s∈S
λs(1−δ) (as,ps)+δ"
V∗(ps,ws).
Mo eo e , we ha e ha

s∈S
λs(1−δ) (as,ps)+δ"
V∗(ps,ws)=
s∈S
λs(1−δ) (as,ps)+δ
V∗ps,"
w∗(ps,ws)
≤
V∗p,
s∈S(1−δ)u(as,ps)+δ"
w∗(ps,ws)
≤"
V∗(p,w),
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