Data and incentives

Liang, Annie; Madsen, E ik
A icle
Da a and incen i es
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Liang, Annie; Madsen, E ik (2024) : Da a and incen i es, Theo e ical Economics,
ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 19, Iss. 1, pp. 407-448,
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Theo e ical Economics 19 (2024), 407–448 1555-7561/20240407
Da a and incen i es
Annie Liang
Depa men o Economics, No hwes e n Uni e si y
E ik Madsen
Depa men o Economics, New Yo k Uni e si y
“Big da a” gi es ma ke s access o p e iously unmeasu ed cha ac e is ics o indi-
idual agen s. Policymake s mus decide whe he and how o egula e he use o
his da a. We s udy how new da a a ec s incen i es o agen s o exe e o in
se ings such as he labo ma ke , whe e an agen ’s quali y is ini ially unknown
bu is o ecas om an obse able ou come. We show ha measu emen o a new
co a ia e has a sys ema ic e ec on he a e age e o exe ed by agen s, wi h he
di ec ion o he e ec de e mined by whe he he co a ia e is in o ma i e abou
long- un quali y e sus a shock o sho - un ou comes. Fo a class o co a ia es
sa is ying a s a is ical p ope y ha we call s ong homoskedas ici y, hise ec is
uni o m ac oss agen s. Mo e gene ally, new measu emen s can impac agen s un-
equally, and we show ha hese dis ibu ional e ec s ha e a i s -o de impac on
social wel a e.
Keywo ds. Big da a, o ecas ing, e o incen i es, ca ee conce ns.
JEL classi ica ion. C72, D83, L51.
1. In oduc ion
Online pla o ms and da a b oke s ex ensi ely ack, eco d, and agg ega e consume
ac i i ies, p oducing measu emen s o e e y hing om he size o an indi idual’s so-
cial ne wo k,1 o how o en hey mo e esidences,2 o he amoun o ime hey spend
playing ideo games.3These new measu emen s a e inc easingly a ailable o i ms and
o ganiza ions, who may ind hem use ul as p edic o s o economic ou comes, such as
Annie Liang: [email p o ec ed]
E ik Madsen: [email p o ec ed]
We a e g a e ul o Edua do Aze edo, Cuimin Ba, Di k Be gemann, Alessand o Bona i, Syl ain Chassang,
Yash Deshpande, Ben Golub, Ma Jackson, Yizhou Jin, Na in Ka ik, Rishabh Ki palani, Alessand o Lizze i,
S e en Ma hews, Xiaosheng Mu, La y Samuelson, And zej Sk zypacz, Juuso Toikka, and Weijie Zhong o
use ul con e sa ions, and o Na ional Science Founda ion G an SES-1851629 o inancial suppo . We
hank Changhwa Lee o aluable esea ch assis ance on his p ojec .
1The inance s a up Lenddo e alua ed bo owe s on he basis o ac o s such as “how many iends o
ollowe s hey ha e on hei social ne wo ks” (Gage,2012).
2The al e na i e c edi sco ing company Zes Finance used bo owe s’ equency o esidence changes o
p edic hei c edi wo hiness (Lippe ,2014).
3China’s widely-publicized social c edi sco ing sys em epo edly plans o inco po a e da a on how
many ideo games a consume pu chases and how much ime hey spend playing hem (Canales and Mok,
2022).
©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/TE5289
408 Liang and Madsen Theo e ical Economics 19 (2024)
a wo ke ’s u u e p oduc i i y in a new job.4Regula ing such uses o pe sonal da a has
eme ged as an impo an policy issue,5bu ou unde s anding o when and how o do
so emains p elimina y. The use o new da a in such se ings may ha e a - anging so-
cial impac s beyond di ec p i acy conce ns, eshaping he c ea ion and dis ibu ion o
economic su plus.
In his pape , we s udy he impac o new da a on ma ke s in which mo al haza d
is an impo an conce n.6A mo i a ing applica ion is he labo ma ke , whe e wages
and job oppo uni ies a e commonly ied o ma ke o ecas s o a wo ke ’s p oduc i i y
based on pas ou pu . A well- ecognized consequence o his p ac ice is ha wo ke s a e
incen i ized o wo k ha d o imp o e he ma ke ’s o ecas . Since ou pu is ypically so-
cially aluable, new da a which impac s wo ke s’ e o incen i es may nega i ely a ec
labo ma ke p oduc i i y absen egula ion.
We p opose a simple model o epu a ional incen i es ha isola es he e ec o new
da a on mo al haza d. Ou model builds on he classic “ca ee conce ns” amewo k
o Holms öm (1999), in which an agen exe s e o o imp o e an ou come used by a
ma ke o o ecas his ype. Di e en om Holms öm (1999), we suppose ha he ma -
ke addi ionally bases i s o ecas on auxilia y da a consis ing o co a ia es desc ibing
he agen , which a e obse ed p io o his choice o e o .
We sepa a e co a ia es in o wo ca ego ies: Some co a ia es, which we call a -
ibu es, desc ibe he agen ’s ype; while o he s, which we call ci cums ances, a e in-
o ma i e abou a ansien shock o his ou come. Fo example, a wo ke ’s c ea i i y is
an a ibu e, while an illness o inju y is a ci cums ance. We model he acquisi ion o
new da a as an expansion o he se o co a ia es ha a e measu ed and may be used o
o ecas ing. Measu emen o new co a ia es upda es he ma ke ’s belie s abou a gi en
agen ’s ype and shock, eshaping incen i es o e o .
Ou main esul s cha ac e ize how measu emen o a new co a ia e impac s he
popula ion dis ibu ion o e o and agg ega e wel a e. Ou basic posi i e esul is ha
inco po a ing a new co a ia e in o he ma ke ’s ype o ecas leads o bo h a sys em-
a ic educ ion in unce ain y ac oss he popula ion and a edis ibu ion o unce ain y
be ween agen s. While he sys ema ic e ec mo es he e o o all agen s in he same
di ec ion, he edis ibu iona y e ec leads o he e ogeneous e o esponses, which
may di e e en di ec ionally ac oss agen s. Each o hese e ec s has a i s -o de e ec
on agg ega e wel a e, and we ind ha he edis ibu iona y e ec can oppose and e en
o e u n he wel a e impac o educing unce ain y.
4Employe s al eady widely use simila in o ma ion collec ed om in e ne sea ches o sc een po en ial
hi es on he basis o ac o s such as social media ac i i y (Ca ee Builde ,2018).
5Fo ins ance, p oposed Eu opean Union ules o a i icial in elligence ha e lagged au oma ed employ-
men sc eening sys ems as “high isk” applica ions subjec o s ic egula ion, in pa icula ega ding he
da a se s hey ely on (Eu opean Commission,2021).
6Ou app oach complemen s ecen wo k ocusing on how da a collec ion impac s ma ke s shaped by
asymme ic in o ma ion. See, o ins ance, Be gemann, Bona i, and Gan (2022), Ellio , Galeo i, and Koh
(2022), Yang (2022) o p ice disc imina ion; Ichihashi (2019), Hidi and Vellodi (2021), Gomes and Pa an
(2022) o ma ching on pla o ms; and B a e man and Chassang (2022), B unne meie , Lamba, and Segu a-
Rod iguez (2021) o insu ance p icing.
Theo e ical Economics 19 (2024) Da a and incen i es 409
We o malize ou posi i e indings h ough a pai o heo ems. Theo em 1shows
ha qui e gene ally, measu emen o a new a ibu e educes a e age e o in he popu-
la ion, while measu emen o a new ci cums ance inc eases a e age e o . Impo an ly,
his esul does no gua an ee ha all agen s change hei e o by he same amoun ,
o e en in he same di ec ion. Theo em 2shows ha any he e ogenei y in he e o
esponses o di e en agen s is en i ely a ibu able o a edis ibu ion o unce ain y
ac oss he popula ion.
Ou no ma i e indings a e summa ized in Theo em 3. I es ablishes ha , in he ab-
sence o edis ibu ion o unce ain y, he di ec ional e ec on wel a e o a newly mea-
su ed co a ia e is join ly de e mined by i s classi ica ion as an a ibu e o ci cums ance
along wi h he weigh ha agen s place on hei u u e epu a ions. I u he shows
ha g ea e edis ibu ion o unce ain y leads o educed wel a e gains om measu e-
men o a co a ia e, and ha his educ ion can be so ex eme ha he measu emen
o some co a ia es is ne e wel a e-enhancing, ega dless o he magni ude o agen s’
epu a ional conce ns.
Ou wo k con ibu es o an eme ging li e a u e s udying he use o pe sonal da a o
o ecas ing. Exis ing wo k has highligh ed incen i es o agen s o game o ecas s by dis-
o ing (Ball (2022), Bona i and Cis e nas (2020), F ankel and Ka ik (2022), Hagh alab,
Immo lica, Lucie , and Wang (2020), Hu, Immo lica, and Vaughan (2019)) o mis epo -
ing (Eliaz and Spiegle (2019,2022)) hei co a ia es. Such incen i es a e especially im-
po an when a small numbe o co a ia es shape o ecas s in a well-unde s ood way.
We s udy he complemen a y ques ion o how da a usage impac s incen i es o agen s
o di ec ly imp o e ou comes. These incen i es a e pa icula ly ele an when ou comes
a e a p ima y o ecas ing inpu , making hem a na u al a ge o manipula ion; o when
algo i hms used o inco po a e addi ional co a ia es in o o ecas s a e opaque, obscu -
ing e ec i e s a egies o gaming hem.
Addi ionally, ou model builds on he ca ee conce ns li e a u e. Compa ed o he
o iginal Holms öm (1999) model, we ocus on a wo-pe iod model which inco po a es
auxilia y signals and non-Gaussian in o ma ion s uc u es. We sha e hese modeling
ea u es wi h he closely ela ed wo k o Dewa ipon , Jewi , and Ti ole (1999)andRo-
dina (2018).7These pape s show ha unde ce ain condi ions, signals abou a single
agen ’s ype lowe his e o while signals abou he shock aise i .8In ou model wi h
a popula ion o he e ogeneous agen s, a simila esul holds on a e age ac oss agen s
7An adjacen li e a u e on ela i e pe o mance compa isons, e.g., Meye and Vicke s (1997), conside s
se ings in which he addi ional signal is no exogenous, bu is ins ead gene a ed by he ou come o ano he
agen wi h co ela ed unobse ables. See also Ti ole (2021), in which he addi ional signal is an ou come in
ano he domain in which he agen exe s e o .
8Rodina (2018) addi ionally s udies whe he ga bling he ou come signal can imp o e agen incen i es,
a ques ion also examined by Hö ne and Lambe (2021) and Smolin (2021) in ela ed con ex s. By con as ,
we assume ha he agen ’s ou come is pe ec ly obse able and ocus on egula ion o access o addi ional
co a ia es.
410 Liang and Madsen Theo e ical Economics 19 (2024)
(Theo em 1). Bu addi ionally, in oduc ion o new da a po en ially changes he c oss-
sec ional dispe sion o e o in he popula ion. As we show in Sec ion 3.3, his dis ibu-
ional e ec has a i s -o de e ec on wel a e, and in some cases i can o e ide he wel-
a e e ec o an a e age change in e o . Ou esul s he e o e highligh he impo ance
o explici ly modeling he he e ogenei y ac oss agen s ha is p esen in applica ions.
The emainde o his pape p oceeds as ollows. Sec ion 2desc ibes ou model; Sec-
ion 3es ablishes ou main esul s abou he impac o new measu emen s on e o and
wel a e; Sec ion 4discusses ex ensions; and Sec ion 5concludes. Suppo ing analyses
and all p oo s a e collec ed in he Appendix.
2. Model
In Sec ion 2.1, we desc ibe ou basic model o epu a ional incen i es o e o , which
is a 2-pe iod e sion o he Holms öm (1999) ca ee conce ns model wi h gene al in-
o ma ion s uc u es. In Sec ion 2.2, we augmen he model by in oducing auxilia y
da a.
2.1 E o and wel a e
An agen pa icipa es in a ma ke ac oss wo pe iods =1, 2. He possesses a quali y ype
θ∼Fθ, which is pe sis en ac oss ime and unknown o himsel and he ma ke .
In pe iod 1, he agen p i a ely chooses an e o le el e∈R+a cos C(e)=1
2e2.(We
ex end ou esul s o gene al cos unc ions in Sec ion 4.3.) The agen ’s e o choice,
along wi h his quali y θand a ansien shock ε∼Fε, de e mine he ealiza ion o an
obse able ou come
Y=e+θ+ε.
We assume ha E(θ)=μ>0 while E(ε)=0. The agen ’s ewa d om his pe iod-1 in e -
ac ion is independen o Yandno malized obe0.
9His pe iod-1 payo is he e o e
U1=−1
2e2.
In pe iod 2, he agen ecei es a epu a ional payo s anding in o e u ns om
u u e pa icipa ion in he ma ke . This payo is equal o he ma ke ’s expec a ion o
his quali y condi ional on he ou come a iable Y.10 Since he agen ’s e o choice is
p i a e, he ma ke ’s o ecas is based on a conjec u ed le el o e o ˆ
e. Le ing Yˆ
e≡
ˆ
e+θ+εbe he ou come supposing ha he ma ke ’s e o conjec u e is co ec , he
agen ’s second-pe iod payo condi ional on he ealized ou come Y=yis
U2=Eˆ
e(θ|Y=y),
9This no maliza ion does no ule ou wage paymen s which depend on he ma ke ’s o ecas o pe iod-1
e o , as in Holms öm (1999). Such e ec s do no impac equilib ium e o o social su plus, and so we
do no explici ly model hem.
10None o ou esul s would change i he agen ’s epu a ional payo we e ins ead he ma ke ’s expec a-
ion o any s ic ly inc easing unc ion o θ. Ou model he e o e accommoda es a a ie y o in e p e a ions
o he sou ce o epu a ional e u ns om e o .

Theo e ical Economics 19 (2024) Da a and incen i es 411
whe e Eˆ
e(θ|Y=y)deno es he ma ke ’s (po en ially misspeci ied) expec a ion o θ,
upda ed based on he ealized ou come assuming ha Y=Yˆ
e.
The agen ’s ex pos payo om pa icipa ing in he ma ke in bo h pe iods is a
weigh ed sum o payo s ac oss he wo pe iods:
U=(1−β)·U1+β·U2,
whe e β∈(0, 1)is he epu a ion weigh , which deno es he impo ance o he agen
o u u e epu a ional ewa ds e sus cu en e o cos s. The agen ’s expec ed payo
unde e o le el eis he e o e
Ee(U)=β·EeEˆ
e(θ|Y)−(1−β)·e2
2,
whe e Eedeno es he expec a ion ope a o gi en he ue e o le el e.
In equilib ium, he agen mus ha e no incen i e o de ia e om he ma ke ’s con-
jec u ed le el o e o . Le e∗deno e equilib ium e o . Then in equilib ium he
ma ginal alue o e o (i.e., he equilib ium ma ginal impac o e o on he expec ed
epu a ional ewa d), discoun ed by i s ela i e weigh β/(1−β), equals he ma ginal
cos o e o :
β
1−β·∂
∂eEeEe∗(θ|Y)e=e∗
=e∗.
Because e o impac s he ou come addi i ely, he ma ginal alue o e o appea ing in
his i s -o de condi ion is independen o e∗and may be w i en as
MV ≡E0∂
∂Y E0(θ|Y),(1)
whe e E0deno es he expec a ion ope a o assuming ha he agen does no exe e o
o dis o he ou come (see Appendix D.1 o de ails). As (1) does no depend on e∗, he
unique e o le el sa is ying he i s -o de condi ion is hen e∗=β
1−β·MV . Th oughou
his pape , we will assume ha he i s -o de app oach is alid, so ha e∗cons i u es he
unique equilib ium e o choice.
We measu e wel a e using a s anda d c i e ion ha ea s bo h he ou come a iable
Yand he agen ’s e o cos C(e)as wel a e- ele an .11 An agen whose ype is θand
e o choice is e hus gene a es wel a e
w(θ,e)≡Ee(Y|θ)−C(e)=θ+e−1
2e2.(2)
This unc ion is s ic ly conca e in e o and maximized a he “ i s -bes ” e o le el
eFB =1 no ma e he agen ’s ype. Appendix Aex ends ou analysis o al e na i e wel-
a e speci ica ions inco po a ing lea ning-by-doing and unp oduc i e gaming.
11The assump ion ha Ycon ibu es o social wel a e is app op ia e o se ings such as he labo ma -
ke , in which he ou come cap u es p oduc i e ou pu o some o he socially aluable ac i i y. We do no
include he agen ’s equilib ium epu a ional payo in he wel a e calcula ion because on a e age ha pay-
o is ixed a Ee∗(Ee∗(θ|Y)) =μ, independen o he equilib ium e o le el. (This p ope y con inues o
hold when he model is augmen ed wi h da a.).
412 Liang and Madsen Theo e ical Economics 19 (2024)
2.2 Da a and belie s
We now augmen he basic model by supposing ha θand εa e p edic able om un-
de lying (and po en ially measu able) co a ia es, wi h da a e ealing a subse o hese
co a ia es. We e e o hose co a ia es which p edic θas a ibu es, deno ed by he
andom a iables (a1,,aJ), and hose co a ia es which p edic εas ci cums ances,
deno ed by he andom a iables (c1,,cK). Speci ically, he ype θand shock εsa is y
θ= 1(a1)+···+ J(aJ)+uθ
ε=g1(c1)+···+gK(cK)+uε
whe e each j,j∈{1, ,J},andgk,k∈{1, ,K}, is a de e minis ic and one- o-one
e ec size unc ion. (This speci ica ion nes s he s anda d linea eg ession model as a
special case when all e ec size unc ions a e a ine.) Fo con enience, we will de ine
he ype componen s θj≡ j(aj) o each j=1, ,Jand shock componen s εk≡gk(ck)
o each k=1, ,K.12
The idiosync a ic noise e ms uθand uεa e independen o one ano he and o all
co a ia es, and ha e ull suppo on he eals.13 We allow o co ela ion be ween a -
ibu es and be ween ci cums ances, bu assume ha he ec o o a ibu es is inde-
penden o he ec o o ci cums ances, i.e., (a1,,aJ)⊥⊥ (c1,,cK), implying in pa -
icula ha θ⊥⊥ ε. (We conside co a ia es which a e co ela ed wi h bo h he ype and
shockinSec ion4.1.)
Some co a ia es a e measu ed, making hem obse able o he agen and he ma -
ke . We use J⊆{1, ,J} o deno e he se o measu ed a ibu es and K⊆{1, ,K} o
deno e he se o measu ed ci cums ances. All measu ed co a ia es a e obse ed a he
ou se o he in e ac ion, leading he agen and ma ke o sha e a common belie ha he
agen ’s ype and shock ollow hei dis ibu ions condi ional on he agen ’s measu ed co-
a ia e alues. We iew symme ic unce ain y as a na u al concep ual benchma k ha
allows us o cleanly disen angle mo al haza d om issues o selec ion. In Sec ion 2.3,we
discuss how ou esul s would change i ei he side had addi ional p i a e in o ma ion.
The in e ac ion hen p oceeds as desc ibed in Sec ion 2.1, wi h app op ia e adjus -
men s o he calcula ion o equilib ium e o . Condi ioning on he measu ed co a ia es,
he agen ’s ma ginal alue o e o changes om (1) o he quan i y
MVJ,K≡E0∂
∂Y E0(θ|Y,aJ,cK)|aJ,cK(3)
and equilib ium e o becomes
e∗
J,K=β
1−β·MVJ,K.(4)
12In e ibili y o he e ec size unc ions implies ha obse a ion o a new co a ia e ajo ckis equi alen
o obse a ion o he co esponding ype o shock componen θjo εk. Some o ou esul s, pa icula ly
hose in ol ing s ong homoskedas ici y, do no depend on in e ibili y.
13The ull suppo assump ion ensu es ha he dis ibu ions o θand εcondi ional on any amily o mea-
su ed co a ia es ha e ull suppo , simpli ying ou p oo s. All o ou esul s con inue o hold in he absence
o ull suppo , and so we will make ee use o examples which do no ea u e ull-suppo idiosync a ic
noise e ms.
Theo e ical Economics 19 (2024) Da a and incen i es 413
No e ha bo h he ma ginal alue o e o and equilib ium e o may a y wi h he
alues o he agen ’s measu ed co a ia es, and hey a e he e o e bo h andom quan i-
ies. We in e p e his andomness om a popula ion pe spec i e, by supposing ha he
ma ke in e ac s wi h a con inuum o agen s possessing a ying a ibu es and ci cum-
s ances. F om his pe spec i e, andom a ia ion in e∗
J,Kco esponds o a dis ibu ion
o e o ac oss he popula ion o agen s.
Agg ega e wel a e gi en measu ed co a ia es (J,K)is he expec a ion o ealized
wel a e w(θ,e∗
J,K)as de ined in (2), a e aging o e a ia ion in he ype θand e o
e∗
J,Kac oss he popula ion:
W(J,K)≡Ewθ,e∗
J,K=μ+Ee∗
J,K−1
2e∗
J,K2.
(Recall ha μ≡E(θ)is he uncondi ional a e age quali y in he popula ion.) Agg ega e
wel a e is maximized when all agen s exe he i s -bes le el eFB =1.
Ou main esul s compa e e o and wel a e when he se o measu ed co a i-
a es changes om some baseline amily (J,K) o an expanded amily (J∪{j},K)o
(J,K∪{k})con aining one addi ional co a ia e. To simpli y exposi ion, h oughou
he main ex we de elop ou esul s assuming ha he baseline amily is (J,K)=
(∅,∅), while he expanded amily is ei he ({1},∅)o (∅,{1}), co esponding o measu e-
men o a ibu e 1 o ci cums ance 1. (Ou esul s ex end s aigh o wa dly o gene al
baselines—see Appendix B o de ails.) In his con ex , we de ine η≡j>1 j(aj)+uθ
and δ≡k>1gk(ck)+uεand decompose he ype and shock as
θ=θ1+η,ε=ε1+δ,
so ha he ype componen θ1and shock componen ε1summa ize he in o ma ion
e ealed by a new measu emen , while ηand δa e he esidual unknowns.
We impose a se o s anda d egula i y condi ions on he dis ibu ions o hese a i-
ables.
Assump ion 1 (Admissibili y). The andom a iables θand εha e log-conca e densi y
unc ions, and he condi ional andom a iables η|a1and δ|c1ha e log-conca e densi y
unc ions o e e y ealiza ion o a1and c1.
Assump ion 2(Diffe en iabili y). Fo e e y e o le el e, he de i a i e ∂
∂Y Ee(θ|Y)ex-
is s and is uni o mly bounded ac oss all ealiza ions o Y. Addi ionally, o e e y e o
le el eand ealiza ion o (a1,c1), he de i a i es ∂
∂Y Ee(θ|Y,a1)and ∂
∂Y Ee(θ|Y,c1)exis
and a e uni o mly bounded ac oss all ealiza ions o Y.
In addi i e s a is ical in e ence models, log-conca i y is a canonical assump ion en-
su ing ha be e ou comes co espond o imp o ed in e ences abou la en a iables.14
14Speci ically, i an analys obse es an ou come Zwhich is decomposable as Z=X+Y,whe ebo h
Xand Ya e unobse ed, and i Xand Ya e s a is ically independen and ha e log-conca e densi y unc-
ions, hen upon obse ing Zhis pos e io belie s abou Xand Ya e highe (in he i s -o de s ochas ic
dominance o de ) o la ge ealiza ions o Z(Milg om (1981))..
414 Liang and Madsen Theo e ical Economics 19 (2024)
Assump ion 1ensu es ha in bo h he baseline and he expanded en i onmen s, be e
(wo se) ealiza ions o Ylead o highe (lowe ) pos e io belie s abou bo h he ype and
shock. Assump ion 2ensu es ha condi ional expec a ions a e su icien ly smoo h ha
we can ake de i a i es and exchange de i a i es and expec a ions whe e equi ed.
2.3 Discussion o modeling choices
P i a e in o ma ion on he side o he ma ke Ou model assumes ha he ma ke does
no know mo e abou an agen ’s ype o shock han he agen does himsel . In some
applica ions, p i a e in o ma ion on he side o he ma ke is possible i he ma ke has
access o da a on pas ou comes o o he agen s wi h simila co a ia es. (This iew o
big da a is embedded in, o ins ance, he “in e se selec ion” model o B unne meie ,
Lamba, and Segu a-Rod iguez (2021).) We show in Sec ion 4.2 ha ou esul s con inue
o hold unde his so o in o ma ional asymme y, so long as measu emen o a new
co a ia e leads he agen o belie e ha he ma ke has gained new in o ma ion abou
his ype o shock.
P i a e in o ma ion on he side o he agen Ano he possibili y is ha he agen knows
mo e abou his ype and shock han he ma ke . This asymme y implies no only ha
he agen knows mo e abou his own co a ia es, bu also ha he can disce n how hese
co a ia es impac his ype and shock dis ibu ions ( ia he e ec size unc ions jand
gk), a demanding assump ion in many applica ions. Ne e heless, ou esul s con inue
o hold wi h p i a e in o ma ion on he agen ’s side whene e he ma ke ’s pos e io
ype expec a ion is linea in he ou come signal ( o ins ance, whene e (θ1,,θJ)and
(ε1,,εK) ollow ellip ical dis ibu ions). Beyond hese se ings, he agen ’s pe cei ed
ma ginal alue o e o may a y wi h p i a e in o ma ion abou his ype o shock, com-
plica ing he analysis, bu we conjec u e ha ou main esul s ex end mo e b oadly.
Exogenei y o co a ia es Ou model con as s co a ia es, which a e ixed cha ac e is ics
o he agen (a leas in he sho un); and he ou come Y, which is suscep ible o ma-
nipula ion by e o . We iew his dicho omy as a use ul one o se e al easons. Fi s , θ
and εmay be de e mined by he agg ega ion o many co a ia es, each o which indi id-
ually plays only a small ole. In such con ex s, ou exe cise can be iewed as ocusing on
he agen ’s incen i es o in luence he ela i ely in o ma i e ou come signal Y, while ab-
s ac ing om any cos ly dis o ion o less-in o ma i e indi idual co a ia es. Second, o
compu e he alue o manipula ing a co a ia e, he agen mus know he p ecise shape
o he e ec size unc ion desc ibing how ha co a ia e impac s he ou come, which is
mo e demanding han he knowledge equi emen s ha we impose.
3. Main esul s
Ou main esul s cha ac e ize he impac o measu ing a new co a ia e on popula ion
e o and wel a e. We i s show ha o a b oad class o co a ia es, measu ing a new
a ibu e dec eases popula ion e o on a e age, while measu ing a new ci cums ance
inc eases i (Sec ion 3.1). Howe e , ou side a na owe class o co a ia es, new mea-
su emen s may yield e o esponses o he e ogeneous magni ude and e en di ec ion
Theo e ical Economics 19 (2024) Da a and incen i es 421
(a1>0.05) alls by oughly 50%, o e∗∗
H≈0.077 ·β
1−β, while e o o all emaining wo k-
e s ises i e old o e∗∗
L≈0.82 ·β
1−β. In ui i ely, al hough esiden ial s abili y has only
a small di ec impac on job pe o mance, i has a la ge impac on he ma ke ’s unce -
ain y abou wo ke eliabili y. The ma ke ’s unce ain y abou wo ke s who mo e e y
equen ly ( esiden ial s abili y alls below he bo om 5% pe cen ile) inc eases, while he
ma ke ’s unce ain y abou he ype o all emaining wo ke s dec eases. Since, mo eo e ,
EMV 2
+A≈0.039 >MV2≈0.026,
Theo em 3implies ha agg ega e wel a e dec eases upon measu ing a ibu e 1 o any
epu a ion weigh β.
4. Ex ensions
We now analyze se e al ex ensions o ou amewo k. Sec ion 4.1 s udies he e ec o
co a ia es, which a e co ela ed wi h bo h he ype and shock. Sec ion 4.2 es ablishes
ha ou esul s a e obus o agen unce ain y abou he ma ke ’s belie s. Sec ion 4.3
elaxes he assump ion ha he agen ’s e o cos s a e quad a ic.
4.1 Gene al co a ia es
Ou main esul s ha e assumed ha indi idual co a ia es a e in o ma i e abou he
agen ’s ype θo shock ε, bu no bo h. In some applica ions, co a ia es may plausi-
bly lie somewhe e be ween hese wo ex emes. We now show how ou esul s can be
adap ed o accommoda e such co a ia es.
As in ou main esul s, we ocus on a baseline in which no co a ia es a e obse ed
and an expanded da a se consis ing o a single co a ia e, whose alue we will deno e
by he andom a iable X. We allow his co a ia e o be co ela ed wi h bo h θand
εin a e y gene al way, which we summa ize by i s e ec on he condi ional mean o
he ou come. Le Y0≡θ+εbe he baseline ou come igno ing he agen ’s e o , and
de ine he andom a iable ¯
Y0≡E(θ+ε|X) o be he condi ional mean o he baseline
ou come gi en he co a ia e. We main ain an in e ibili y assump ion ensu ing ha
measu ing Xis equi alen o obse ing he condi ional mean ou come ¯
Y0.
Assump ion 3 (Gene al in e ibili y). E(θ+ε|X=x)is a one- o-one unc ion o x.
This condi ion is analogous o he in e ibili y assump ions imposed on he e ec
size unc ions jand gkin he baseline model, and i se es he same pu pose.
We addi ionally impose admissibili y and di e en iabili y assump ions analogous o
Assump ions 1and 2in he baseline analysis.
Assump ion 4 (Gene al admissibili y). (¯
Y0,Y0)a e s a is ically a ilia ed.
Assump ion 5(Gene aldiffe en iabili y). Fo e e y e o le el e, he de i a i e ∂
∂Y Ee(θ|
Y)exis s and is uni o mly bounded ac oss all ealiza ions o Y. Fo e e y e o le el eand
ealiza ion o X, he de i a i e ∂
∂Y Ee(θ|Y,X)exis s and is uni o mly bounded ac oss all
ealiza ions o Y.

422 Liang and Madsen Theo e ical Economics 19 (2024)
Assump ion 4se es he same ole as Assump ion 1in ensu ing ha be e ou comes
co espond o imp o ed in e ences abou la en a iables. In pa icula , i he “ne esid-
ual” Y0≡Y0−¯
Y0is independen o he ealiza ion o X, a ilia ion o (¯
Y0,Y0) educes
o log-conca i y o he densi y unc ion o Y0. Meanwhile, Assump ion 5is a s aigh -
o wa d adap a ion o Assump ion 2.
Finally, we impose an assump ion ensu ing ha θand εa e co ela ed only h ough
he co a ia e X. We main ain i o ocus on he simples con ex in which co ela ion
be ween he ype and shock migh a ise.
Assump ion 6 (Gene al independence). (θ,ε)a e independen condi ional on X.
We now de i e condi ions unde which measu ing Xinc eases o dec eases e o ,
ex ending he esul s o Theo em 1 o his se ing.
P oposi ion 1. Suppose ha Assump ions 3–6hold.
1. I (¯
Y0,θ)and (¯
Y0,−ε)a e each s a is ically a ilia ed, hen measu ing X educes
a e age e o .
2. I (¯
Y0,−θ)and (¯
Y0,ε)a e each s a is ically a ilia ed, hen measu ing Xinc eases
a e age e o .
This esul es ablishes ha a co a ia e which is posi i ely associa ed wi h one com-
ponen o he ou come, and is simul aneously nega i ely associa ed wi h he emaining
componen , has an unambiguous impac on he expec ed ma ginal alue o e o . The
posi i e associa ion condi ion he e is a di ec analog o he a ilia ion condi ion in The-
o em 1, and is needed o he same eason. Meanwhile, he nega i e associa ion condi-
ion ules ou scena ios in which a good co a ia e ealiza ion implies bo h a high ype
and a high shock. Since hese in e ences ha e con lic ing e ec s on he ma ginal alue
o e o , he ne e ec o measu ing such a co a ia e is inhe en ly ambiguous. By con-
as , i a good co a ia e ealiza ion sugges s a high ype and a low shock, o ice e sa,
he wo e ec s ein o ce and he measu emen has an unambiguous impac on a e age
e o .
This esul can be s eng hened o ob ain a uni o m e ec on e o unde ho-
moskedas ici y condi ions simila o hose imposed in Theo em 2. In pa icula , le
θ ≡θ−E(θ|X)be he esidual unobse ed ype componen a e measu ing X.De-
ine ε simila ly wi h espec o he shock. Then i he join dis ibu ion o (θ,ε)
is independen o he ealiza ion o X, measu ing Xa ec s e o uni o mly ac oss all
agen s.
To illus a e hese o ces conc e ely, we analyze he e ec o measu ing a new co a i-
a e in a mul i a ia e Gaussian se ing. Suppose ha θand εa e decomposable as
θ=μ+b·X+Z,ε=d·X+W
Theo e ical Economics 19 (2024) Da a and incen i es 423
whe e X∼N(0, σ2
x),Z∼N(0, σ2
z),andW∼N(0, σ2
w)a e mu ually independen and
μ,b,andda e known cons an s. The ollowing lemma ensu es ha he egula i y as-
sump ions imposed in P oposi ion 1a e sa is ied in his se ing whene e b+d= 0, a
condi ion we will main ain going o wa d.26
Lemma 1. Assump ions 3–6a e sa is ied in a mul i a ia e Gaussian se ing wi h a gene al
co a ia e whene e b+d= 0.
We now check when he condi ions iden i ied in P oposi ion 1unde which mea-
su ing Xinc eases o dec eases e o a e sa is ied. (¯
Y0,θ)and (¯
Y0,ε)a e each join ly
Gaussian, and (¯
Y0,±θ)a e posi i ely co ela ed i and only i sign(b+d)=±sign(b).
Simila ly, (¯
Y0,±ε)a e posi i ely co ela ed i and only i sign(b+d)=±sign(d).Then
whene e b>0, P oposi ion 1implies ha measu ing X educes e o i b+d>0≥d,
i.e., d∈(−b,0
]. Simila ly, whene e d>0, measu ing Xinc eases e o i b+d>0≥b,
i.e., b∈(−d,0
].
The bounds d≤0andb≤0 illus a e he gene al poin made ea lie : Measu ing X
has an unambiguous e ec on e o only i i s in o ma i eness abou one componen o
he ou come is ein o ced a he han opposed by i s in o ma i eness abou he emain-
ing componen . The emaining condi ion b+d>0 ensu es ha be e ou comes co -
espond o imp o ed in e ences abou he ype o shock in he baseline, wi hou which
he expec ed di ec ional e ec o a measu emen can e e se.
We can e i y hese esul s by explici ly calcula ing MV and MV+, he ma ginal alue
o e o be o e and a e measu ing X. The ollowing esul summa izes he calcula ion.
P oposi ion 2. sign(MV −MV+)=sign((b+d)( b
σ2
z
−d
σ2
w)).
I b>0andd∈(−b,0
], his esul implies ha MV+<MV, in line wi h he p edic-
ion o P oposi ion 1. Simila ly, i d>0andb∈(−d,0
], henMV+>MV.Con e sely,
i bo h band da e posi i e, he sign o MV −MV+is ambiguous. Depending on he
sizes o hese coe icien s ela i e o he esidual unce ain y abou θand ε, measu ing
Xcould mo e he ma ginal alue o e o in ei he di ec ion.
4.2 Model unce ain y and misspeci ica ion
Suppose ha , con a y o ou assump ions in he baseline model, he agen is subjec-
i ely unce ain abou he ma ke ’s pe cei ed dis ibu ion o (θ,ε)gi en his measu ed
co a ia es. Such a si ua ion may a ise i he does no know which se o co a ia es he
ma ke obse es, o i he does no know how he ma ke maps his co a ia e alues in o
pe cei ed ype and shock dis ibu ions.
This subjec i e unce ain y can be modeled by supposing he agen possesses belie s
o e possible join dis ibu ions o (θ,ε) ha he ma ke migh hold when o ecas ing
he agen ’s ype. (I is no impo an ha he ma ke ’s ue model be con ained in he
26I b+d=0, hen Xdoes no impac Yand canno be es ima ed by obse ing he ou come. As a esul ,
measu ing i has no impac on he ma ginal alue o e o .
424 Liang and Madsen Theo e ical Economics 19 (2024)
suppo o he agen ’s belie s, so he agen may be misspeci ied.) We will con inue o
main ain he assump ion ha he agen is no asymme ically in o med abou his ype,
and so his own subjec i e belie abou he dis ibu ion o his ype and ou come is he
expec a ion o his belie abou he ma ke ’s dis ibu ion.
In his se ing, all o ou esul s ex end in he ollowing sense: I he agen becomes
con inced ha he ma ke ’s s a is ical model has become “be e -in o med” abou he
agen ’s ype o shock, his e o will mo e in he di ec ion p edic ed by ou esul s, so
long as he co esponding s a is ical assump ions hold o each model in he suppo
o he agen ’s belie s. Mo e p ecisely, an agen belie es he ma ke has become “be e -
in o med” i he hinks ha , ega dless o wha s a is ical model i is in ac using, he
ma ke has gained access o an addi ional a ibu e o addi ional co a ia e. In ha case,
he ma ginal alue o e o mo es in he same di ec ion o e e y model in he suppo
o he agen ’s belie s, and he expec ed ma ginal alue o e o he e o e mo es in his
di ec ion as well. Ou main esul s he e o e con inue o hold in his en i onmen .
4.3 Gene al con ex cos unc ions
We ha e es ablished ou main esul s unde he assump ion ha e o cos s ake he
o m C(e)=1
2e2. Unde his cos unc ion, equilib ium e o is iden ical o he ma ginal
alue o e o , allowing us o cha ac e ize he o me by analyzing he la e . Mo e gen-
e ally, when Cis a s ic ly con ex cos unc ion, equilib ium e o is a uniquely de e -
mined, s ic ly inc easing unc ion o he ma ginal alue o e o :
e∗
J,K=C−1β
1−β·MVJ,K,
whe e MVJ,Kis as de ined in (3). As a esul , unde such a cos unc ion, a de e minis ic
shi in he ma ginal alue o e o implies a change in e o in he same di ec ion. This
implies in pa icula ha he esul s o Theo em 2unde s ong homoskedas ici y ex end
immedia ely.
Theo em 1 o a ilia ed co a ia es ex ends so long as all agen s change hei e o in
he same di ec ion, and mo e gene ally unde a condi ion on he hi d de i a i e o he
e o cos unc ion.27 (In Appendix C, we p esen simila , bu mo e es ic i e, gene al-
iza ions o he wel a e esul s om Sec ion 3.3.)
P oposi ion 3. Suppose Assump ions 1–2hold.
(a) I a ibu e 1is a ilia ed, hen measu ing i educes a e age e o i C ≥0o all
agen s change hei e o in he same di ec ion.
(b) I ci cums ance 1is a ilia ed, hen measu ing i inc eases a e age e o i C ≤0
o all agen s change hei e o in he same di ec ion.
27The p oo o his esul is a s aigh o wa d applica ion o he p oo o Theo em 1,combinedwi h he
logic o he discussion ollowing he p oposi ion s a emen .
Theo e ical Economics 19 (2024) Da a and incen i es 425
The new o ce which a ises unde gene al cos unc ions is ha a e age e o may
espond o mean-p ese ing sp eads o he ma ginal alue o e o . To illus a e his
possibili y, conside any cos unc ion C(e)∝ek,whe ek>1. I k>2, hen unde
such a cos unc ion he ma ginal cos o e o is con ex, so equilib ium e o is a con-
ca e unc ion o he ma ginal alue o e o . Hence, any mean-p ese ing sp ead o he
ma ginal alue o e o educes a e age e o . Con e sely, i 2 >k>1, e o is a con ex
unc ion o he ma ginal alue o e o , and a mean-p ese ing sp ead o he ma ginal
alue o e o inc eases a e age e o .
Measu ing a new a ilia ed co a ia e has wo e ec s: I shi s he a e age ma ginal
alue o e o , and (whene e s ong homoskedas ici y ails) i may addi ionally in o-
duce a sp ead in he dis ibu ion o ma ginal alues. I he ma ginal cos o e o is con-
ex, his second e ec ends o educe equilib ium e o . Thus, when a new a ibu e
is measu ed, hese wo o ces wo k oge he o lowe a e age e o , and he esul s o
Theo em 1con inue o hold. A simila ou come holds when he ma ginal cos o e o
is conca e and a new ci cums ance is measu ed. When he wo o ces con lic , he ne
e ec on e o is ambiguous. In pa icula , i agen s change hei e o in di e en di-
ec ions, a e age e o could mo e in he opposi e di ec ion om he a e age ma ginal
alue o e o .
5. Conclusion
As i ms and go e nmen s mo e owa d collec ing la ge consume da a se s as inpu s
o decision-making, he ques ion o whe he and how o egula e he usage o pe sonal
da a has eme ged as an impo an policy ques ion. Recen egula ions, such as he Eu o-
pean Union’s Gene al Da a P o ec ion Regula ion, ha e ocused on p o ec ing consume
p i acy and imp o ing anspa ency ega ding wha kind o da a is being collec ed. An
impo an complemen a y conside a ion is how da a impac s economic ou comes. In
his pape , we ha e ocused on one such ac o — he e ec ha ma ke access o no el
co a ia es has on incen i es o hidden e o .
Ou esul s indica e ha o ecas ing om da a on endu ing pe sonal a ibu es de-
c eases a e age e o ac oss he popula ion, while con e sely da a e lec ing sho -li ed
ci cums ances boos s e o . I is he e o e impo an o dis inguish be ween hese wo
classes o da a when egula ing da a usage. Fu he , new da a may lead o inc eased
a ia ion in e o ac oss wo ke s, an ou come which has a i s -o de impac on wel a e.
This inding sugges s ha egula o s should also ake in o accoun he dis ibu ional
e ec s o new da a when deciding whe he o pe mi i s use in pa icula ma ke s.
One way o in e p e he a ibu es and ci cums ances in ou model is as s and-
ins o co a ia es wi h di e en le els o pe sis ence in a dynamic model, whe e he
agen exe s e o o e mul iple pe iods and his ype e ol es o e ime. Gene alizing
ou esul s o a many-pe iod se ing is echnically challenging unde non-Gaussian in-
o ma ion s uc u es, since e o de ia ions oday may dis o u u e e u ns o e o .
None heless, doing so would pe mi a iche s udy o he wel a e implica ions o o e-
cas ing om da a wi h a ying pe sis ence, making i an impo an a enue o u u e
esea ch.
426 Liang and Madsen Theo e ical Economics 19 (2024)
Appendix A: Al e na i e wel a e speci ica ions
In his appendix, we ex end ou wel a e analysis o conside al e na i e en i onmen s
in which e o imp o es u u e as well as cu en ou comes (“lea ning-by-doing”) o is
pa ially dissipa i e (“gaming” e o ).
A.1 Lea ning-by-doing
In some applica ions, e o may imp o e u u e as well as cu en ou comes, o in-
s ance in labo ma ke se ings ea u ing lea ning-by-doing. In ha case, he agen ’s
ype is no cons an o e ime bu ins ead imp o es wi h pas e o , and e o has so-
cially bene icial e ec s o e mul iple pe iods.
Ou model can be modi ied o accommoda e his ea u e by allowing he agen ’s
ype θ( ), which de e mines he a e age ou come in pe iod , o be ime-dependen .
Conc e ely, we will suppose ha θ(2)=θ(1)+γ·e,whe eγ>0 is a lea ning-by-doing
pa ame e . Pe iod-1 ou pu is
Y=e+θ(1)+ε,
while he agen ’s pe iod-2 epu a ional ewa d is E(θ(2)|Y).
The p esence o lea ning by doing does no a ec equilib ium e o , because he
agen ’s epu a ional ewa d is based on he ma ke ’s o ecas o his e o (which is ixed)
a he han his ue e o . This expec a ion is
Ee∗θ(2)|Y=(1+γ)·e∗+Ee∗θ(1)|Y.
Exe ing addi ional e o is he e o e aluable o he agen only inso a as i imp o es
he ma ke ’s o ecas o θ(1), exac ly as in ou main model. Thus, equa ion (4) con inues
o cha ac e ize equilib ium e o .
The socially op imal e o le el, howe e , becomes eFB =1+γin his model. Equi-
lib ium e o he e o e alls below he i s -bes le el o a b oade ange o epu a ion
weigh s βas he lea ning-by-doing pa ame e γinc eases. An analogue o Theo em 3
con inues o hold, whe e he h eshold epu a ion weigh s β∗and β∗a e inc easing in
γ. In o he wo ds, inc eased lea ning-by-doing makes ci cums ances (which boos e -
o ) mo e a ac i e and a ibu es (which educe i ) less so a any gi en epu a ional
weigh .
A.2 “Gaming” e o
In o he applica ions, e o may be dissipa i e and se e o dis o a signal o quali y
wi hou p oducing social alue. This possibili y may a ise, o ins ance, in labo ma ke
se ings in which a wo ke can spend ime pe o ming “in luence ac i i ies” o inc ease
he isibili y o his accomplishmen s (as in Milg om and Robe s (1988)). I may also
a ise in educa ional se ings whe e he ou come a iable is a es sco e ha can be im-
p o ed by es p ep wi h no u he educa ional alue (as in F ankel and Ka ik (2022)).

Theo e ical Economics 19 (2024) Da a and incen i es 427
To accommoda e his possibili y, ou wel a e c i e ion can be modi ied o discoun
he wel a e bene i s o e o :
w(θ,e)=θ+δ·e−1
2e2,
whe e δ∈[0, 1)measu es he p opo ion o e o which is socially bene icial. When
δ=0, e o is o ally unp oduc i e, while δ∈(0, 1)cap u es si ua ions in which some
ac ion o e o con ibu es social alue.
The dissipa i e na u e o e o has no impac on equilib ium e o , bu educes
he i s -bes e o le el o eFB =δ. Equilib ium e o will he e o e exceed he i s -
bes le el o a b oade ange o epu a ion weigh s βas e o becomes inc easingly
dissipa i e. A esul analogous o Theo em 3can be es ablished in his se ing, wi h he
h eshold epu a ion weigh s β∗and β∗inc easing in δ. One in e es ing case is δ=0, in
which e o is ully dissipa i e e o . In ha case, measu ing new a ibu es imp o es
wel a e while measu ing new ci cums ances diminishes i , ega dless o he epu a ion
weigh β. (The one excep ion is o an a ibu e wi h signi ican dispa a e impac , which
may s ill be wel a e- educing o all β.)
Appendix B: Resul s o a gene al baseline
The esul s o Sec ion 3can be s aigh o wa dly gene alized o accommoda e se -
ings in which some co a ia es a e ini ially measu ed by he ma ke . Gi en any se s
J⊆{1, ,J}o measu ed a ibu es and K⊆{1, ,K}o measu ed ci cums ances,
de ine
ηJ≡
j/∈J
θj+uθ,δK≡
k/∈K
εk+uε
o be he sums o all unmeasu ed componen s o he agen ’s ype and shock.
Fix a baseline amily (J,K)o measu ed co a ia es. A ilia ion and s ong ho-
moskedas ici y may be gene alized o his en i onmen as ollows.
De ini ion B.1 (Affilia ion). The a ibu e j/∈Jis J-a ilia ed i (θj,ηJ∪{j})is a ili-
a ed condi ional on aJ. The ci cums ance k/∈Kis K-a ilia ed i (εk,δK∪{k})is a ili-
a ed condi ional on cK.
De ini ion B.2 (S ong homoskedas ici y). The a ibu e j/∈Jsa is ies J-s ong ho-
moskedas ici y i ηJ∪{j}−E(ηJ∪{j}|aJ∪{j})is independen o ajcondi ional on aJ.
The ci cums ance k/∈Ksa is ies K-s ong homoskedas ici y i δK∪{k}−E(δK∪{k}|
cK∪{k})is independen o ckcondi ional on cK.
As o mula ed, hese de ini ions apply ac oss all (J,K)-subpopula ions o agen s,
whe e each subpopula ion consis s o all agen s sha ing a pa icula ealiza ion o
(aJ,cK). They could al e na i ely be o mula ed mo e na owly o apply only o a pa -
icula se o ealized co a ia es, i he analys is p ima ily in e es ed in he impac o a
new co a ia e on a pa icula subpopula ion o agen s.
428 Liang and Madsen Theo e ical Economics 19 (2024)
Assump ions 1and 2, which imposed log-conca i y on la en a iables and bound-
edness o de i a i es o condi ional expec a ions, mus also be ex ended o a gene al
se o baseline measu ed co a ia es. We spli hese condi ions in o an assump ion we
main ain in he baseline en i onmen , and a condi ion imposed on newly measu ed
co a ia es.
Assump ion B.1 (Baseline admissibili y). The condi ional dis ibu ions θ|aJand ε|
cKha e log-conca e densi y unc ions, and o e e y e o le el eand ealiza ion o
(aJ,cK), he de i a i e ∂
∂Y Ee(θ|Y,aJ,cK)exis s and is uni o mly bounded ac oss all
ealiza ions o Y.
De ini ion B.3 (Admissible co a ia es). An a ibu e j/∈Jis J-admissible i : (1)
ηJ∪{j}|aJ∪{j}has a log-conca e densi y unc ion o e e y ealiza ion o aJ∪{j};and(2)
o e e y e o le el eand ealiza ion o co a ia es (aJ∪{j},cK), he de i a i e ∂
∂Y Ee(θ|
Y,aJ∪{j},cK)exis s and is uni o mly bounded ac oss all ealiza ions o Y.
A ci cums ance k/∈Kis K-admissible i : (1) δK∪{k}|cK∪{k}has a log-conca e den-
si y unc ion o e e y ealiza ion o cK∪{k}; and (2) o e e y e e y e o le el eand
ealiza ion o co a ia es (aJ,cK∪{k}), he de i a i e ∂
∂Y Ee(θ|Y,aJ,cK∪{k})exis s and is
uni o mly bounded ac oss all ealiza ions o Y.
Wi h hese concep s, we can gene alize Theo ems 1and 2as ollows.
Theo em B.1. Suppose Assump ion B.1 holds.
(a) I a ibu e jis J-admissible and sa is ies J-a ilia ion, hen measu ing i weakly
educes a e age e o wi hin each (J,K)-subpopula ion.
(b) I ci cums ance kis K-admissible and sa is ies K-a ilia ion, hen measu ing i
weakly inc eases a e age e o wi hin each (J,K)-subpopula ion.
Theo em B.2. Suppose Assump ion B.1 holds.
(a) I a ibu e jis J-admissible and sa is ies J-s ong homoskedas ici y, hen mea-
su ing i weakly educes e e y agen ’s e o . Fu he , he magni ude o he e o
change is he same o e e y agen in each (J,K)-subpopula ion.
(b) I ci cums ance kis K-admissible and sa is ies K-s ong homoskedas ici y, hen
measu ing i weakly inc eases e e y agen ’s e o . Fu he , he magni ude o he
e o change is he same o e e y agen in each (J,K)-subpopula ion.
These esul s can be applied epea edly o assess he impac o measu ing mul iple
co a ia es, so long as admissibili y and he co esponding s a is ical condi ion (a ilia-
ion o s ong homoskedas ici y) holds o each o he measu ed co a ia es ela i e o
i s espec i e baseline. No e in pa icula ha he exponen ial and mul i a ia e no mal
se ings o Examples 2and 3sa is y a ilia ion and s ong homoskedas ici y, espec i ely,
o any baseline and newly measu ed co a ia e.
Ou wel a e esul s also hold unde gene al baselines using app op ia e no ions o
egula i y and s ic egula i y.
Theo e ical Economics 19 (2024) Da a and incen i es 429
De ini ion B.4. Fix a baseline amily o measu ed co a ia es (J,K).Then:
•An a ibu e j/∈Jis (J,K)- egula i , condi ional on any ealiza ion o (aJ,cK),
measu ing jweakly educes he ma ginal alue o e o on a e age. I is s ic ly
(J,K)- egula i he educ ion is s ic o a posi i e ac ion o ealiza ions o
(aJ,cK).
•A ci cums ance k/∈Kis (J,K)- egula i , condi ional on any ealiza ion o
(aJ,cK), measu ing kweakly inc eases he ma ginal alue o e o on a e age. I
is s ic ly (J,K)- egula i he inc ease is s ic o a posi i e ac ion o ealiza ions
o (aJ,cK).
Weak egula i y imposes mono onici y sepa a ely on each subpopula ion o agen s.
S ic egula i y imposes he s onge equi emen o s ic mono onici y o a posi i e
ac ion o agen s. (In he special case o a baseline wi h no obse ed co a ia es, s ic
egula i y i ially implies s ic mono onici y o all agen s, co esponding o De ini-
ion 3.)
The ollowing esul gene alizes Theo em 3 o gene al baselines.
Theo em B.3. Fix a baseline amily o measu ed co a ia es (J,K).
(a) Fo e e y (J,K)- egula a ibu e j/∈J, he e exis s a h eshold epu a ion weigh
β∗∈(0, 1]such ha measu ing jis wel a e-imp o ing i and only i β>β
∗.Mo e-
o e , β∗<1i and only i
EMV 2
J∪{j},K<EMV 2
J,K(B.1)
whe e MVJ,Kis as de ined in (3).
(b) Fo e e y (J,K)- egula ci cums ance k/∈K, he e exis s a h eshold epu a ion
weigh β∗∈[0, 1)such ha measu ing kis wel a e-imp o ing i and only i β<β
∗.
Mo eo e , β∗>0i and only i kis s ic ly (J,K)- egula .
Appendix C: Wel a e unde gene al con ex cos s
Theo em 3, ou main wel a e esul , can be ex ended o nonquad a ic e o cos unc-
ions unde he same condi ions as P oposi ion 3, assuming ha e o cos s ollow a
powe law. We s a e and p o e his esul o gene al baselines, as in he analysis o
Appendix B.
P oposi ion C.1. Suppose ha C(e)=Aek o some A>0and k>1. Fix a baseline
amily o measu ed co a ia es (J,K).
(a) Suppose ha he ma ke measu es he addi ional egula a ibu e j/∈J.I ei he
k≥2o else all agen s change hei e o in he same di ec ion, hen he e exis s
a h eshold epu a ion weigh β∗∈(0, 1]such ha he measu emen is wel a e-
imp o ing i and only i β>β
∗.I jis s ic ly egula and all agen s change hei
e o in he same di ec ion, hen β∗<1.
430 Liang and Madsen Theo e ical Economics 19 (2024)
(b) Suppose ha he ma ke measu es he addi ional egula ci cums ance k/∈K.I
ei he k≤2o else all agen s change hei e o in he same di ec ion, hen he e ex-
is s a h eshold epu a ion weigh β∗∈[0, 1)such ha he measu emen is wel a e-
imp o ing i and only i β<β
∗.I kis s ic ly egula , hen β∗>0.
Fo gene al cos unc ions, ully cha ac e izing how agg ega e wel a e changes wi h β
becomes in ac able. Howe e , i can be shown ha i he e o cos unc ion is app oxi-
ma ely quad a ic nea ze o, hen when βis small, newly measu ed egula a ibu es e-
duce agg ega e wel a e and newly measu ed egula ci cums ances inc ease hem; while
o la ge β, hese e ec s e e se.
P oo o P oposi ion C.1. I a newly measu ed co a ia e is egula bu no s ic ly
egula , hen expec ed e o is unchanged while he dis ibu ion o e o in each sub-
popula ion unde goes a mean-p ese ing sp ead unde he measu emen . Then since
e o cos s a e s ic ly con ex, agg ega e wel a e mus a leas weakly dec ease no ma e
he alue o β, co esponding o β∗=1 o an a ibu e and β∗=0 o a ci cums ance.
Fo he emainde o he p oo , we assume ha he newly measu ed a ibu e is s ic ly
egula .
Le δ≡β/(1−β).No e ha e∗
J,Kdepends on βonly h ough δ, and we will w i e
e∗
J,K(δ) o make his dependence explici .
We i s conside he case in which he ma ke measu es a new a ibu e j.Allno a-
ion is as in he p oo o Theo em B.3. De ine
W(δ)≡Ew0e∗
J,K(δ)−w0e∗
J,K(δ),
whe e
w0(e)≡e−C(e).
Recall ha gi en any amily o measu ed co a ia es, equilib ium e o in a gi en sub-
popula ion sa is ies e∗=(C)−1(δ·MV ),whe eMV is he co esponding subpopula ion
ma ginal alue o e o . Thus,
∂e∗
∂δ =MV
CC−1(δ·MV )
and
∂
∂δwe∗=1−Ce∗∂e∗
∂δ =(1−δ·MV )·MV ·1
CC−1(δ·MV ).
When C(e)=Aek,weha e
1
CC−1(δ·MV )=A·(δ·MV )1
k−1−1,
whe e A≡1/((k−1)(Ak)1/(k−1))>0. Hence,
∂
∂δwe∗=A·δ1
k−1−1·(1−δ·MV )·MV 1
k−1.
Theo e ical Economics 19 (2024) Da a and incen i es 437
P oo . This is es ablished along e y simila lines o he p oo o Lemma D.2.Fix e-
aliza ions o (aJ,cK), condi ion all dis ibu ions on hei alues, and supp ess explici
condi ioning. Le ρη(u)be he densi y o ηJand ρδ|ε(x|z)be he condi ional densi y o
δ−K|εk. The condi ions equi ed o he s eps o he p oo o Lemma D.2 o go h ough
a e ha ρη(u)is log-conca e, ρδ|ε(x|z)is log-conca e in x o all z,and(δK,εk)a e
a ilia ed. The i s wo p ope ies ollow om Assump ion B.1, while he inal p ope y
holds by K-a ilia ion o ci cums ance k.
We compa e MVJ,Kwi h MVJ,Kin a manne e y simila o he case o an addi-
ional a ibu e. Fix ealiza ions o (aJ,cK), and de ine
Fε(z|y)≡P εk≤z|Y0=y,aJ,cK
o be he condi ional CDF o εkgi en he ou come Y0. Decompose Y0as
Y0=μ(J,K)+ηJ+δK.
Taking expec a ions o each side condi ional on (Y0,εk,aJ,cK)yields
Y0=μ(J,K)+EηJ|Y0,εk,aJ,cK+EδK|Y0,εk,aJ,cK.
Hence,
dFε(z|y)EηJ|Y0=y,εk=z,aJ,cK
=y−μ(J,K)−dFε(z|y)EδK|Y0=y,εk=z,aJ,cK.(D.2)
Lemma D.3 di ec ly implies ha
dFε(z|y)EδK|Y0=y,εk=z,aJ,cK
is weakly inc easing in y,so(D.2) is weakly dec easing in y.
Following he same logic as in he a ibu es case, mono onici y o (D.2) implies ha
∂
∂Y0EηJ|Y0,aJ,cK≤E∂
∂Y0EηJ|Y0,εk,aJ,cKY0,aJ,cK,
and i ollows ha
MVJ,K≤E[MVJ,K|aJ,cK].
Thus, he ma ginal alue o e o in each subpopula ion in he baseline is weakly lowe
han he expec ed ma ginal alue o e o when he ci cums ance kis addi ionally mea-
su ed.
D.3 P oo s o Theo ems 2and B.2
We p o e Theo em B.2, omwhichTheo em2 ollows immedia ely as a co olla y.

438 Liang and Madsen Theo e ical Economics 19 (2024)
D.3.1 Pa (a) Fix a baseline amily o measu ed co a ia es (J,K). As es ablished in
Lemma D.1, he ma ginal alue o e o is
MV (J,K)=E∂
∂Y0EηJ|Y0,aJ,cKaJ,cK,
whe e Y0≡θ+εis he baseline alue o he ou come a e sub ac ing ou he agen ’s
e o .
Now suppose he ma ke addi ionally obse es he a ibu e j/∈J,andle J≡J∪
{j}. Unde he expanded amily o measu ed co a ia es, he ma ginal alue o e o
becomes
MVJ,K=E∂
∂Y0EηJ|Y0,aJ,cKaJ,cK,
whe e, condi ional on (aJ,cK),MVJ,Kis a andom a iable whose alue is a unc ion
o he ealiza ion o aj.
The ou come Y0may be decomposed as
Y0=μ(J,K)+ηJ+δK,(D.3)
whe e
μ(J,K)≡
j∈J
θj+
k∈K
εk
is cons an condi ional on (aJ,cK). The esidual ype componen ηJmay be u he
decomposed as
ηJ=¯ηJ|j+ηJ,
whe e
¯ηJ|j≡E[ηJ|aJ],ηJ≡θ−E[θ|aJ].
We may he e o e ew i e (D.3)as
Y0=μ(J,K)+¯ηJ|j+ηJ+δK.
Now, no e ha
ηJ−E[ηJ|aJ]=
j∈J
θj+ηJ−E
j∈J
θj+ηJ|aJ=ηJ.
Hence, J-s ong homoskedas ici y o a ibu e jis equi alen o he assump ion ha
ηJis independen o ajcondi ional on (aJ,cK). The e o e, unde J-s ong ho-
moskedas ici y, Y0depends on ajonly h ough ¯ηJ|j. I ollows ha unde J-s ong
homoskedas ici y, E[ηJ|Y0,aJ,cK]=E[ηJ|Y0,¯ηJ|j,aJ,cK], and he la e ex-
pec a ion depends on ajonly h ough ¯ηJ|j.
Using his ac , we may w i e
EηJ|Y0,aJ,cK=¯ηJ|j+EηJ|Y0,¯ηJ|j,aJ,cK
Theo e ical Economics 19 (2024) Da a and incen i es 439
and
MVJ,K=E∂
∂Y0EηJ|Y0,¯ηJ|j,aJ,cK¯ηJ|j,aJ,cK.
The heo em holds i can we show ha he condi ional expec a ion o ηJis less
esponsi e o he ealiza ion o he ou come Y han he condi ional expec a ion o he
o iginal esidual ηJ.No e ha ηJis he sum o he (condi ionally) independen a i-
ables ¯ηJ|jand ηJ, so unce ain y abou ηJis mechanically lowe han unce ain y
abou ηJ. Bu his does no di ec ly ansla e in o a s a emen ha he pos e io ex-
pec a ion o ηJis less sensi i e o he ealiza ion o Y. In gene al, we a e no e en
gua an eed ha highe ealiza ions o Ylead o highe in e ences abou θJonce we
ha e condi ioned on he ealiza ion o ¯ηJ|j.29 We nex p o e a key echnical lemma,
which will imply an analogue o admissibili y o ou ans o med en i onmen .
Lemma D.4. (ηJ,¯ηJ|j,Y0)a e a ilia ed condi ional on (aJ,cK).
P oo . Fix a se o ealiza ions o (aJ,cK), and condi ion all dis ibu ions on hese
alues. To economize on no a ion, we supp ess explici condi ioning on hese co-
a ia es h oughou his p oo . Le ˜ρη,θ,Y(u, ,y)be he condi ional join densi y o
(ηJ,¯ηJ|j,Y0). We will show ha ˜ρη,θ,Yis log-supe modula .
Use ˜ρθ( ) o deno e he densi y o ¯ηJ|j,˜ρη|θ(u| ) o deno e he condi ional densi y
o ηJ|¯ηJ|j,and ˜ρY|η(y|u) o deno e he condi ional densi y o Y0|ηJ.No e ha Y0
is independen o ¯ηJ|jcondi ional on ηJ.So, ˜ρη,θ,Ymay be decomposed as
˜ρη,θ,Y(u, ,y)=˜ρθ( )˜ρη|θ(u| )˜ρY|η(y|u).
I is he e o e su icien o show ha ˜ρY|ηand ˜ρη|θa e log-supe modula .
Fi s , conside ˜ρY|η. Decompose Y0as
Y0=μ(J,K)+ηJ+δK.
Le ρδ(z)be he densi y o δK.Then
˜ρY|η(y|u)=ρδ(y−μ(J,K)−u).
Unde Assump ion B.1,ρδis log-conca e, meaning ˜ρY|ηis log-supe modula .
As o ˜ρη|θ,le ˜ρη(w)be he densi y o ηJ. Decompose ηJas
ηJ=¯ηJ|j+ηJ,
and ecall ha i jis J-s ongly homoskedas ic, hen ηJis independen o ajand
hence ¯ηJ|j. I ollows ha
˜ρη|θ(u| )=˜ρη(u− ),
29Recall ha ou admissibili y assump ions a e imposed on he o iginal ype componen θj, and no on
he cons uc ed ¯ηJ|j.
440 Liang and Madsen Theo e ical Economics 19 (2024)
and hence,
∂2
∂u∂ log ˜ρη|θ(u| )=− ∂2
∂w2log ˜ρη(w)w=u−
=− ∂2
∂u2log ˜ρη|θ(u| ).
Now, le ρη|a(u|α)deno e he condi ional densi y o ηJ|aj. De ine
ζ(α)≡ j(α)+E[ηJ|aj=α],
so ha
ηJ=ζ(aj)+ηJ.
S ong homoskedas ici y implies ha
ρη|a(u|α)=˜ρηu−ζ(α)=˜ρη|θu|ζ(α).
Le ˜
≡{ :ζ(α)= o some α∈Aj}deno e he suppo o ¯ηJ|j.Fixany ∈˜
.Then
o all uand e e y α∈Ajsuch ha ζ(α)= ,
∂2
∂u2log ˜ρη|θ(u| )=∂2
∂u2logρη|a(u|α).
Le ρη|adeno e he condi ional densi y o ηJ|aj.Thenρη|a(u|α)=ρη|a(u− j(α)|
α),so ha
∂2
∂u2log ˜ρη|θ(u| )=∂2
∂w2log ρη|a(w|α)w=u− j(α)
.
Unde Assump ion B.1,ρη|ais log-conca e and his inal de i a i e is nonposi i e,
meaning
∂2
∂u∂ log ˜ρη|θ(u| )=− ∂2
∂u2log ˜ρη|θ(u| )≥0
o e e y uand ∈˜
.Hence, ˜ρη|θis log-supe modula , as desi ed.
Following a gumen s iden ical o hose used o he p oo o Theo em B.1 (wi h ηJ
and ¯ηJ|jplaying he oles o ηJand θj, espec i ely), Lemma D.4 implies
MVJ,K≥E[MVJ,K|aJ,cK].
Thus, he ma ginal alue o e o in each subpopula ion in he baseline se ing is weakly
highe han he expec ed ma ginal alue once a ibu e jis addi ionally measu ed, o
any ealiza ions o (aJ,cK).
To comple e he p oo , we mus es ablish ha mono onici y holds uni o mly ac oss
ealiza ions o aj, and no jus on a e age. This ollows immedia ely om he ac ha
MVJ,Kis independen o ajcondi ional on (aJ,cK). To see his, decompose Yas
Y=e+μ(J,K)+¯ηJ|j+ηJ+δK.
Theo e ical Economics 19 (2024) Da a and incen i es 441
S ong homoskedas ici y implies ha ηJis independen o ajcondi ional on (aJ,
cK).Hence,ajen e s he ma ke ’s in e ence p oblem only ia a known addi i e shi
¯ηJ|j o he agen ’s ype dis ibu ion and, i s alue he e o e does no impac incen i es
o e o . So, hese incen i es mus be independen o aj,asclaimed.
D.3.2 Pa (b) Suppose he ma ke obse es he addi ional ci cums ance k/∈K.Le
K≡K∪{k}. Unde he expanded amily o measu ed co a ia es, he ma ginal alue o
e o is
MVJ,K=E∂
∂Y0EηJ|Y0,aJ,cKaJ,cK.
De ine
¯
δK,k≡E[δK|cK],δK≡ε−E[ε|cK].
Then Y0may be decomposed as
Y0=μ(J,K)+ηJ+¯
δK,k+δK.
Unde s ong homoskedas ici y, δKis independen o ckcondi ional on (aJ,cK),and
so he dis ibu ion o Y0depends on ckonly h ough ¯
δK,k.Thus,
EηJ|Y0,aJ,cK=EηJ|Y0,¯
δK,k,aJ,cK,
and he la e andom a iable depends on ckonly h ough ¯
δK,k. The e o e, in a man-
ne analogous o he a ibu e case, he ma ginal alue o e o a e measu ing jde-
pends on ckonly h ough ¯
δK,kand may be w i en
MVJ,K=E∂
∂Y0EηJ|Y0,¯
δK,k,aJ,cK¯
δK,k,aJ,cK.
Lemma D.5. (¯
δK,k,δK,Y0)a e a ilia ed condi ional on (aJ,cK).
P oo . This p oo ollows along e y simila lines o he p oo o Lemma D.4.Fix
ealiza ions o (aJ,cK), condi ion all dis ibu ions on hei alues, and supp ess ex-
plici condi ioning on hese co a ia es. Le ρη(u)be he condi ional densi y o ηJand
ρδ|c(x|z)be he condi ional densi y o δK|ck. The condi ions equi ed o he s eps
o he p oo o Lemma D.4 o go h ough a e ha ρη(u)is log-conca e, ρδ|c(x|z)is
log-conca e in x o all z,andδKis independen o ¯
δK|k. The i s wo p ope ies a e
ensu ed by Assump ion B.1, while he inal p ope y holds unde s ong homoskedas ic-
i y.
We compa e MVJ,Kand MVJ,Kin a manne e y simila o he a ibu e case. Fix
ealiza ions o (aJ,cK). De ine

Fε(z|y)≡P ¯
δK,k≤z|Y0=y,aJ,cK
442 Liang and Madsen Theo e ical Economics 19 (2024)
o be he dis ibu ion unc ion o ¯
δK,kcondi ional on he ou come Y0. Decompose Y0
as
Y0=μ(J,K)+ηJ+δK.
Taking expec a ions o each side condi ional on (Y0,¯
δK,k,aJ,cK)yield
Y0=μ(J,K)+EηJ|Y0,¯
δK,k,aJ,cK+EδK|Y0,¯
δK,k,aJ,cK.
Hence,
d
Fεz|yEηJ|Y0=y,¯
δK,k=z,aJ,cK
=y−μ(J,K)−d
Fεz|yEδK|Y0=y,¯
δK,k=z,aJ,cK.(D.4)
Lemma D.5 di ec ly implies ha
d
Fε(z|y)EδK|Y0=y,¯
δK,k=z,aJ,cK
is nondec easing in y,so(D.4) is noninc easing in y.
Following he same logic as in he a ibu es case, mono onici y o (D.4) implies ha
∂
∂Y0EηJ|Y0,aJ,cK≤E∂
∂Y0EηJ|Y0,¯
δK,k,aJ,cKY0,aJ,cK,
and i ollows ha
MVJ,K≤E[MVJ,K|aJ,cK].
Thus, he ma ginal alue o e o in each subpopula ion in he baseline is weakly lowe
han he expec ed ma ginal alue o e o when he ci cums ance kis addi ionally mea-
su ed.
The inal s ep in he p oo is o es ablish ha mono onici y holds uni o mly ac oss
ealiza ions o he addi ional ci cums ance, and no jus on a e age. This ollows om
a gumen s nea ly iden ical o hose used o he a ibu es case.
D.4 P oo o Theo em 3
De ine δ≡β/(1−β). Then gi en any se o measu ed co a ia es, equilib ium e o can
be w i en as e∗
J,K=δ·MVJ,K. We will cha ac e ize he agg ega e wel a e change om
measu ing a new co a ia e as a unc ion o δ, which can be mapped on o an equi alen
cha ac e iza ion in e ms o β.
Conside i s he case in which he ma ke obse es he new a ibu e j.Le J≡
J∪{j}. The agg ega e change in wel a e om measu ing j, as a unc ion o δ,is
W(δ)=Ew0(δ·MVJ,K)−w0(δ·MVJ,K),

Theo e ical Economics 19 (2024) Da a and incen i es 443
whe e
w0(e)≡e−1
2e2.
The agg ega e change in wel a e is a quad a ic unc ion o δ, whose i s de i a i e is
d
dδW(δ)=E[MVJ,K−MVJ,K]−δ·EMV 2
J,K−MV 2
J,K.
Obse e ha W(0)=0and
d
dδW(δ)|δ=0=E[MVJ,K−MVJ,K≤0
wi h he inequali y implied by egula i y o j.So,W (δ)is a quad a ic unc ion ha
anishes a δ=0 and is noninc easing he e. Suppose i s ha jis s ic ly egula , so
ha Wis s ic ly dec easing a 0. Then i s shape mus hen sa is y ei he o :
•W(δ)is s ic ly con ex and in e sec s ze o exac ly once o δ>0,
•W(δ)is weakly conca e and does no in e sec ze o o any δ>0.
De ine δ∗≡in {δ>0:W (δ)≥0}. This h eshold lies in (0, ∞), and has he p ope y
ha W(δ)>0 o δ>δ
∗,andW (δ)≤0 o δ∈(0, δ∗].Fu he ,δ∗<∞i and only i
Wis s ic ly con ex. Con exi y is de e mined by he sign o
d2
dδ2W (δ)=EMV 2
J,K−MV 2
J,K.
Hence, δ∗<∞i and only i E[MV 2
J,K]>E[MV 2
J,K]. Le ing β∗≡δ∗/(1+δ∗) he e o e
yields a epu a ion weigh h eshold wi h he desi ed p ope ies.
Suppose ins ead ha jis weakly bu no s ic ly egula . W i e
EMV 2
J,K|aJ,cK
=EMVJ,K−E[MVJ,K|aJ,cK]2|aJ,cK+E[MVJ,K|aJ,cK]2
=Va (MVJ,K|aJ,cK)+E[MVJ,K|aJ,cK]2.
Hence,
EMV 2
J,K|aJ,cK−MV 2
J,K
=E[MVJ,K|aJ,cK]2−MV 2
J,K+Va (MVJ,K|aJ,cK).
Weak bu no s ic egula i y implies ha MVJ,K=E[MVJ,K|aJ,cK]wi h p obabili y
1, so ha
EMV 2
J,K|aJ,cK−MV 2
J,K=Va (MVJ,K|aJ,cK)≥0
wi h p obabili y 1. The e o e, Wis a leas weakly conca e. Weak bu no s ic
egula i y addi ionally implies ha Whas ze o slope a δ=0. Hence, in his case
444 Liang and Madsen Theo e ical Economics 19 (2024)
W (δ)≤0 o allδ>0, and we se δ∗=∞and β∗=∞. We may summa ize he wo k
o he s ic ly and weakly egula cases wi h he conclusion ha β∗<∞i and only i
E[MV 2
J,K]>E[MV 2
J,K]. Indeed, in he s ic ly egula case, his equi alence was di-
ec ly es ablished, while in he weakly egula case, i is ue bo h ha β∗=∞and ha
he la e inequali y canno hold.
Now suppose he ma ke obse es he new ci cums ance k. Calcula ions e y simi-
la o hose o he a ibu e case show ha Wis a quad a ic unc ion o δwhich an-
ishes a δ=0 and is weakly inc easing he e, and s ic ly inc easing i kis s ic ly egula .
I is conca e i and only i E[MV 2
J,K]≥[MV 2
J,K], wi h he conca i y s ic i and only i
he inequali y is.
I kis only weakly egula , hen he calcula ions o he a ibu e case show ha
E[MV 2
J,K]≥E[MV 2
J,K], meaning ha Wmus be a leas weakly conca e, in which
case W(δ)≤0 o allδ>0. In his case, we se δ∗=0andβ∗=δ∗/(1+δ∗)=0.
Going o wa d, suppose ha kis s ic ly egula . De ining δ∗≡in {δ>0:W (δ)≤
0}yields a h eshold in (0, ∞]wi h he p ope y ha W(δ)<0 o allδ>δ
∗and
W(δ)>0 o allδ∈(0, δ∗). This h eshold is ini e i and only i Wis a s ic ly conca e
unc ion. Calcula ions e y simila o he a ibu e case imply ha
EMV 2
J,K|aJ,cK−MV 2
J,K
=E[MVJ,K|aJ,cK]2−MV 2
J,K+Va (MVJ,K|aJ,cK).
S ic egula i y implies ha he di e ence be ween he i s wo e ms on he hs is non-
nega i e o e e y ealiza ion o (aJ,cK), and posi i e wi h posi i e p obabili y. Fu he ,
he condi ional a iance o MVJ,Kmus be nonnega i e. The e o e,
EMV 2
J,K|aJ,cK≥MV 2
J,K,
wi h he inequali y s ic wi h posi i e p obabili y. Taking he uncondi ional expec a ion
o bo h sides yields he desi ed s ic conca i y o W, implying δ∗<∞. Le ing β∗≡
δ∗/(1+δ∗)yields a epu a ion weigh wi h he desi ed p ope ies.
D.5 P oo o P oposi ion 1
We p o e he i s pa o he esul , wi h he second ollowing om nea ly iden ical a -
gumen s. To s eamline no a ion, h oughou his p oo we will d op supe sc ip s on Y0
and ¯
Y0.
Unde Assump ion 3, condi ioning on he alue o Xis equi alen o condi ioning
on he alue o ¯
Y. The desi ed esul can he e o e be es ablished using a echnique
e y simila o he one employed in he p oo o Theo em B.1(a)subsequen oLemma
D.2,wi h ¯
Yplaying he ole o θj. Tha a gumen equi es wo condi ions: (A) (¯
Y,Y)
a e s a is ically a ilia ed, and (B) (¯
Y,θ)a e s a is ically a ilia ed condi ional on Y.We
main ain condi ion A by hypo hesis, so i emains only o es ablish condi ion B.
Le ρY|¯
Y,θ(y|y, )be he condi ional densi y o Ygi en (¯
Y,θ),ρ¯
Y,θbe he join
densi y o (¯
Y,θ),andρYbe he ma ginal densi y o Y. Then by Bayes’ ule,
ρ¯
Y,θ|Yy, |y=ρY|¯
Y,θy|y, ρ¯
Y,θy, 
ρY(y).
Theo e ical Economics 19 (2024) Da a and incen i es 445
Because (¯
Y,θ)a e a ilia ed, he densi y ρ¯
Y,θis log-supe modula . To es ablish condi-
ion B, i is he e o e su icien o show ha ρY|¯
Y,θ(y|y, )is log-supe modula in (y, ),
holding y ixed.
Le ρε|¯
Y,θ(e|y, )and ρε|¯
Y(e|y)be he condi ional densi ies o εgi en (¯
Y,θ)and
¯
Y, espec i ely. Gi en he condi ional independence o θand εgi en Xand he in e -
ibili y o E(¯
Y|X), i ollows ha θand εa e independen condi ional on ¯
Y. The e o e,
ρε|¯
Y,θe|y, =ρε|¯
Ye|y
o all , allowing us o w i e
ρY|¯
Y,θy|y, =ρε|¯
Y,θy− |y, =ρε|¯
Yy− |y.
Le ρε,¯
Y(e,y)be he join densi y o (ε,¯
Y)and ρ¯
Y(y)be he ma ginal densi y o ¯
Y.
Then using Bayes’ ule, we ha e
ρY|¯
Y,θy|y, =ρε,¯
Yy− ,y
ρ¯
Yy.
Condi ion B he e o e ollows i ρε,¯
Yis log-submodula . Equi alen ly, ρ−ε,¯
Ymus be log-
supe modula , whe e ρ−ε,¯
Yis he join densi y o (−ε,¯
Y). This condi ion is equi alen
o a ilia ion o (−ε,¯
Y),ashypo hesized.
D.6 P oo o Lemma 1
In a mul i a ia e Gaussian en i onmen , he condi ional mean o ou pu is
¯
Y0=μ+(b+d)X,
sa is ying Assump ion 3whene e b+d= 0. Since (X,Z,W)a e join ly Gaussian and
¯
Y0and Y0a e each linea combina ions o (X,Z,W), hepai (¯
Y0,Y0)a e also join ly
Gaussian. Addi ionally,
Y0=¯
Y0+Z+W,
and since ¯
Y0is independen o Z+Wi ollows ha Y0and ¯
Y0a e posi i ely co e-
la ed. The e o e, (¯
Y0,Y0)a e a ilia ed, sa is ying Assump ion 4. Mo eo e , condi ional
expec a ions in mul i a ia e Gaussian en i onmen s a e linea in he condi ioning a i-
able, ensu ing ha Assump ion 5is sa is ied. Finally, mu ual independence o X,Z,and
Wimplies ha (θ,ε)a e independen condi ional on X, sa is ying Assump ion 6.
D.7 P oo o P oposi ion 2
P io o he measu emen , (θ,Y0)ha e join dis ibu ion
θ
Y0∼Nμ
μ,σ2
θσθ,Y
σθ,Yσ2
Y
446 Liang and Madsen Theo e ical Economics 19 (2024)
whe e
σ2
θ=b2σ2
x+σ2
z,σθ,Y=b(b+d)σ2
x+σ2
z,σ2
Y=(b+d)2σ2
x+σ2
z+σ2
w.
The ma ke ’s ype o ecas gi en equilib ium e o e∗is he e o e
Ee∗(θ|Y)=Eθ|Y0=Y−e∗=μ+σθ,Y
σ2
Y
·Y−μ−e∗,
implying ha he baseline ma ginal alue o e o is
MV =σθ,Y
σ2
Y
=b(b+d)σ2
x+σ2
z
(b+d)2σ2
x+σ2
z+σ2
w
.
Meanwhile, a e measu ing X, he condi ional join dis ibu ion o (θ,Y0)is
θ
Y0X∼N μ+bX
μ+(b+d)X,σ2
zσ2
z
σ2
zσ2
z+σ2
w.
Gi en equilib ium e o e∗∗, he ma ke ’s pos e io ype o ecas a e measu ing Xis
Ee∗∗ (θ|X,Y)=Eθ|X,Y0=Y−e∗∗=μ+bX +σ2
z
σ2
z+σ2
wY−e∗∗ −μ−(b+d)X.
The agen ’s ma ginal alue o e o unde he expanded da a se is he e o e
MV+=σ2
z
σ2
z+σ2
w
.
Compa ing he exp essions o MV and MV+jus de i ed yields he iden i y in he
p oposi ion s a emen .
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