Kocou ek, Pa el; S eine , Jakub; S ewa , Colin
A icle
Boundedly a ional demand
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Kocou ek, Pa el; S eine , Jakub; S ewa , Colin (2024) : Boundedly a ional
demand, Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 19,
Iss. 4, pp. 1415-1442,
h ps://doi.o g/10.3982/TE5796
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Theo e ical Economics 19 (2024), 1415–1442 1555-7561/20241415
Boundedly a ional demand
Pa e l Kocou ek
Depa men o Mic oeconomics, Uni e si y o Duisbu g-Essen
Jakub S eine
Depa men o Economics, Uni e si y o Zu ich, CERGE-EI, and CTS UK
Colin S ewa
Depa men o Economics, Uni e si y o To on o
E idence sugges s ha consume s do no pe ec ly op imize, con a y o a c i ical
assump ion o classical consume heo y. We p opose a model in which consume
ypes can a y in bo h hei p e e ences and hei choice beha io . Gi en da a
on demand and he dis ibu ion o p ices, we iden i y he se o possible alues
o he consume su plus based on minimal a ionali y condi ions: e e y ype o
consume mus be no wo se o han i hey ei he always bough he good o ne e
did. We de elop a p ocedu e o na ow he se o su plus alues using iche da a
se s and p o ide bounds on coun e ac ual demands.
Keywo ds. Beha io al wel a e, bounded a ionali y, consume heo y, in o ma-
ion design, e ealed p e e ence.
JEL classi ica ion. D11, D6, D9.
1. In oduc ion
A key assump ion o he s anda d app oach o analyzing consume demand and wel a e
is ha consume s pe ec ly op imize. Ye i is clea om a numbe o empi ical s udies—
i no om in ospec ion alone— ha his assump ion does no gene ally hold in p ac-
ice. Fo example, simply changing he way ha p ices a e p esen ed o consume s
can ha e signi ican e ec s on demand (Che y, Looney, and K o (2009), Finkels ein
(2009)).1
Pa el Kocou ek: [email p o ec ed]
Jakub S eine : [email p o ec ed]
Colin S ewa : [email p o ec ed]
We ci cula ed an ea lie d a o his pape unde he i le “Demand in he Da k.” We ha e bene i ed om
he commen s o Sand o Ambuehl, Tibo Heumann, Zi Yang Kang, An on Kolo ilin, Nick Ne ze , Ma ek
Pycia, Andy Zapechelnyuk, a ious semina and wo kshop audiences, and se e al anonymous e e ees.
S ewa is g a e ul o PSE o hei hospi ali y. This wo k was suppo ed by ERC G an 770652, G an GAˇ
CR
24-10145S, and DFG G an 450175676, and by he Social Sciences and Humani ies Resea ch Council o
Canada.
1See also I o (2014) o empi ical e idence ha consume s do no co ec ly accoun o ma ginal elec-
ici y p icing and Feldman, Ka ušˇ
cák, and Kawano (2016) o ela ed e idence ega ding ma ginal ax a es.
Dickson and Sawye (1990) documen ha mo e han hal o he supe ma ke shoppe s hey su eyed we e
©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/TE5796
1416 Kocou ek, S eine , and S ewa Theo e ical Economics 19 (2024)
Failu es o op imize pe ec ly can occu o a a ie y o easons. The consume s may
no be ully a en i e o he p ice o a good hey buy, pe haps because i is a habi ual pu -
chase o because p ices in ol e complexi ies ha equi e e o o unde s and. Simila ly,
consume s may no always be awa e o he p ice o a good hey do no buy. Al e na i ely,
hey may simply make s ochas ic e o s, as is ypically assumed in andom u ili y mod-
els. Va ying salience o ce ain ea u es o he p oduc , such as whe he i is on sale, may
also in luence consume s’ choices.
How a e we o make in e ences abou a consume ’s p e e ences and wel a e in he
absence o op imal choices? We p opose an app oach g ounded in minimal assump-
ions abou he consume ’s a ionali y. We ocus on a se ing wi h uni demand in which
he da a consis o a demand cu e indica ing he p obabili y o pu chase a each p ice
oge he wi h a dis ibu ion o p ices. Ou a ionali y condi ions equi e only ha no
ypes o consume would be be e o i hey swi ched o ei he always buying o ne e
buying he good.2
We conside an analys who seeks o a ionalize he da a wi h a model desc ibing a
dis ibu ion o ypes o he consume , wi h each ype speci ying he consume s’ alue
oge he wi h hei demand cu e. A model a ionalizes he da a i , a each p ice, he
obse ed pu chase p obabili y is equal o he expec ed demand ac oss he ypes in he
model. We s udy wo main ques ions. Fi s , wha can he analys in e om he obse ed
da a abou he su plus he consume ecei es om pa icipa ing in he ma ke o his
good? Ou app oach can help o quan i y he unce ain y due o bounded a ionali y in
es ima ed measu es o consume su plus con ibu ed by new p oduc s.3Second, wha
can he analys p edic abou he coun e ac ual demand i he consume we e able o
ully op imize? This ques ion could be o in e es o a egula o o a monopolis consid-
e ing ixing he p ice in he ma ke , he eby elimina ing any e o s in choice a ising om
ina en ion o luc ua ions in p ices. Fo each o hese ques ions, we i s ob ain bounds
using simple da a se s and hen show how o na ow hese bounds wi h iche da a.
The e a e wo o mally equi alen in e p e a ions o ou amewo k. Unde one
in e p e a ion—which is he one we use in ou desc ip ion— he e is a single consume
wi h a s ochas ic ype. Unde he o he in e p e a ion, he e is a con inuum o con-
sume s, and each ype desc ibes an indi idual consume whose p e e ences and beha -
io a e ixed.
The consume ’s ype cap u es bo h luc ua ions in alue (as in a andom u ili y
model) and a ying a en ion ha could be co ela ed wi h he alue. Fo example, i
could be ha when he consume ’s alue is high, she pays li le a en ion o he p ice,
whe eas when i is low, she checks mo e ca e ully. Al e na i ely, he a en ion could a y
due o ex e nal ac o s such as ime p essu e. We hink o a ype’s demand as esul ing
unable o accu a ely epo he p ice o an i em immedia ely a e placing i in hei ca . Taubinsky and
Rees-Jones (2018) and Tipoe (2021) ind ha he e is signi ican he e ogenei y in a en ion o p ices.
2We e alua e expec ed u ili ies wi h espec o he dis ibu ion o p ices obse ed by he analys , making
an implici iden i ying assump ion ha his dis ibu ion ag ees wi h he consume ’s subjec i e belie o
expe ience.
3Fo example, Cohen, Hahn, Hall, Le i , and Me cal e (2016) es ima e he o al consume su plus due
o Ube , Goolsbee and Pe in (2004) es ima e ha due o di ec b oadcas sa elli es, and Pe in (2002)es i-
ma es ha due o he In oduc ion o he mini an.
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Theo e ical Economics 19 (2024) Boundedly a ional demand 1417
om a combina ion o he alue o he good and he a en ion (which is unobse ed
and unmodelled).
One special case o ou model o pa icula in e es a ises when each ype o he
consume is Bayes a ional and impe ec ions in op imiza ion a e due o incomple e
in o ma ion abou p ices. In ha case, each ype is desc ibed by a alue o he good
oge he wi h an in o ma ion s uc u e abou p ices, wi h hei demand ollowing om
op imiza ion o hei expec ed payo . Each such ype o consume sa is ies ou minimal
a ionali y condi ions since he s a egies o always o ne e buying he good can be im-
plemen ed by igno ing any in o ma ion hey ha e abou p ices. One migh hink, hen,
ha his case would lead o na owe bounds on su plus and coun e ac ual demand
han he ones esul ing om ou gene al a ionali y condi ions. I u ns ou , howe e ,
ha he bounds a e iden ical: any model sa is ying ou gene al a ionali y condi ions
can be ob ained wi h Bayes a ional consume ypes o app op ia ely chosen in o ma-
ion s uc u es.
Rela i e o ou se ing, he s anda d assump ion o op imal choice (coupled wi h
quasilinea p e e ences) simpli ies he analysis in wo ways. Fi s , each ype has a
h eshold demand, buying he good i and only i he p ice is below ha ype’s alue
o he good, and hus he alue can be di ec ly in e ed om i s demand. Second, any
demand cu e admi s a unique decomposi ion in o h eshold demands o indi idual
consume ypes; hus, he dis ibu ion o alues can be di ec ly in e ed om he de-
mand obse ed in he da a. In con as , in ou se ing, ypes wi h he same alue may
di e in hei demand, co esponding o di e ences in a en ion o sophis ica ion. Thus
ypes’ alues canno be di ec ly in e ed om hei demands, and hei demands need
no ake a simple h eshold o m.
The e a e gene ally many di e en models ha can a ionalize he da a. Fi s , he
analys mus conside a ious decomposi ions o he o e all demand in o demands o
indi idual ypes. Second, o each ype, gi en i s demand, he e is a ange o incen i e-
compa ible alues, i.e., alues o which ou a ionali y condi ions a e sa is ied. In ligh
o his lexibili y, i is no possible o pin down he su plus exac ly. Fo ins ance, he
analys can assume pe ec op imiza ion and a ionalize he demand in he s anda d
way o ob ain he usual consume su plus. A he opposi e ex eme, he analys can
a ibu e all s ochas ici y in beha io o e o s by assuming a single ype whose demand
ma ches he obse ed demand. Al e na i ely, he analys can employ a iche model
wi h many ypes ha may be op imizing o a ying deg ees.
We cha ac e ize he le els o consume su plus (and coun e ac ual demands) ac oss
all a ionaliza ions o he da a. The le els o su plus consis en wi h he da a comp ise
an in e al anging om 0 o an uppe bound ha has a simple ma hema ical s uc u e
akin o ha o he s anda d consume su plus. Jus as he s anda d su plus is he a ea
be ween he p ice line and he in e se demand up o he quan i y demanded, he uppe
bound is he a ea be ween he p ice line and an “ele a ed” in e se demand up o he
quan i y demanded. As he name sugges s, his ele a ed demand, which depends on
bo h he obse ed demand and he p ice dis ibu ion, lies abo e he obse ed demand.
A i s blush, i may be su p ising ha he consume su plus could be highe han
he s anda d su plus unde pe ec op imiza ion. To see how his may happen, conside
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1418 Kocou ek, S eine , and S ewa Theo e ical Economics 19 (2024)
a consume who belie es ha he p ice o he good is no mally oo high o i o be
wo h pu chasing. She he e o e igno es i and checks he p ice only i she no ices ha
i is on sale. I sales a e announced a p ices below some h eshold ha is lowe han
he consume ’s alue o he good, an analys who ea s he consume as i she is ully
op imizing will in e ha he h eshold p ice o a sale is he consume ’s alue, he eby
unde es ima ing he su plus.
Each easible le el o su plus can be ob ained wi h a simple model: as in he s anda d
app oach, he obse ed demand is decomposed in o h eshold demands o indi idual
ypes. Howe e , he alue o a ype ha does no pe ec ly op imize need no be equal
o he p ice a he co esponding h eshold. Ou a ionali y condi ions imply bounds
on he alue ha a ype wi h a gi en demand h eshold could ha e: he alue can be no
g ea e han he expec ed p ice condi ional on no buying he good, and no lowe han
he expec ed p ice condi ional on buying he good. The ull ange o le els o consume
su plus consis en wi h he da a can be a ained by a ying he ypes’ alues wi hin hese
bounds.
I u ns ou o be use ul o ocus on he alue a andomly chosen ype assigns o he
good, which we e e o as he s ochas ic alue. (Thus a s ochas ic alue only pa ially
desc ibes a model in ha i does no speci y he demand o each ype.) Ou bounds
on su plus and coun e ac ual demand a e based on bounds on he s ochas ic alue
wi h espec o a ious s ochas ic o de s. Fo he uppe bound on consume su plus, we
make use o he inc easing con ex o de (ICX);4 o he lowe bound on su plus, we use
second-o de s ochas ic dominance (SOSD); o he bounds on coun e ac ual demand,
we use i s -o de s ochas ic dominance (FOSD).
Bounds on he s ochas ic alue a e pa icula ly use ul wi h iche da a. In Sec ion 6,
we conside da a se s comp ising wo o mo e ma ke egimes ha may di e in he
dis ibu ion o p ices and/o he consume ’s beha io a any gi en p ice. Fo example,
i could be ha , as in Che y, Looney, and K o (2009), sales axes a e included in he
pos ed p ice in one egime bu no included in he o he . The analys conside s all a-
ionaliza ions o he da a se s in which he alue o each ype is ixed ac oss egimes
( hough i s demand may a y, o example, due o changes in salience o a en ion); in
o he wo ds, he s ochas ic alue is held cons an ac oss egimes.
We p opose a simple p ocedu e o na owing he bounds on su plus o coun e ac-
ual demand wi hin each egime using he da a om he o he egimes. This p ocedu e
in ol es aking he collec ion o bounds on he s ochas ic alue ac oss egimes and com-
bining hem o ob ain a common igh e bound. (In pa icula , his p ocedu e can gi e
ise o a nonze o lowe bound.) To compu e his combined bound, we exploi a map-
ping be ween s ochas ic alues and con ex unc ions and iden i y a common bound on
hese con ex unc ions.
In a simila spi i o Be nheim and Rangel (2009), we p opose a e ealed-p e e ence
app oach o measu ing he wel a e o a decision-make who may no be pe ec ly a io-
4The ICX o de can be iewed as he analogue o second-o de s ochas ic dominance o a decision-
make who is isk lo ing ins ead o isk a e se. Thus, whe eas second-o de s ochas ic dominance a o s
highe means and smalle sp eads, ICX a o s highe means and la ge sp eads.
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Theo e ical Economics 19 (2024) Boundedly a ional demand 1419
nal. Empi ical s udies o beha io al wel a e ypically assume, ei he explici ly o implic-
i ly, ha ce ain obse ed choices e eal he decision-make ’s ue p e e ences.5These
p e e ences can hen be used o assess he wel a e associa ed wi h o he choices ha
could be subop imal. Fo example, Che y, Looney, and K o (2009)andTaubinsky and
Rees-Jones (2018) eco e ue p e e ences om consume choices when a ax is made
salien and use hese p e e ences o measu e he wel a e loss a ising om mis akes ha
occu when he axes a e no salien ; B onnenbe g, Dubé, Gen zkow, and Shapi o (2015)
iden i y he ue p e e ences om expe s’ choices and use hese p e e ences o e alua e
non-expe s’ wel a e. An al e na i e app oach, aken by G ube and Köszegi (2001), is o
use a s uc u al model ha ela es ue p e e ences o choice beha io . Rela i e o hese
empi ical s udies, we make much weake assump ions abou he ex en o which ue
p e e ences can be in e ed om he da a, equi ing only ha beha io sa is ies ce ain
minimal a ionali y condi ions.
In he special case o ou model in which e o s a e due o impe ec in o ma ion, ou
wo k can be iewed as combining e ealed p e e ence wi h in o ma ion design, whe e
he design has he goal o maximizing o minimizing he su plus o coun e ac ual de-
mand consis en wi h he obse ed da a.6Be gemann, B ooks, and Mo is (2022)iden-
i y bounds on coun e ac ual beha io in abs ac games. Theo em 4in he p esen
pape conce ns coun e ac ual beha io in a mo e speci ic se ing, bu unlike in Be ge-
mann, B ooks, and Mo is (2022), he dis ibu ion o p e e ences in ou model is no
known o he analys . Be gemann, B ooks, and Mo is (2015,2017) iden i y he ange o
su plus alues ha can be a ained o gi en p e e ences as in o ma ion a ies in a mo-
nopolis ic ma ke o a i s -p ice auc ion. Condo elli and Szen es (2020,2022)cha ac e -
ize he ange o su plus alues consis en wi h pa ial knowledge o demand in se ings
wi h ma ke powe on he supply side. Rega ding e ealed p e e ence, we a e closes o
he b anch o he li e a u e ha uses choice da a o join ly iden i y p e e ences and in o -
ma ion, as in Masa lioglu, Nakajima, and Ozbay (2012)andManzini and Ma io i (2014).
When conside ing bounds on su plus using da a om mul iple egimes, we ep e-
sen andom a iables as con ex unc ions o cons uc bounds wi h espec o he ICX
o SOSD o de . A simila echnique has been used in Bayesian pe suasion p oblems by
Gen zkow and Kamenica (2016) and Kolo ilin, Mylo ano , Zapechelnyuk, and Li (2017).
Mülle and Sca sini (2006) es ablish la ice p ope ies o hese o de s using he same
ans o ma ion. This echnique has a na u al in e p e a ion in ou con ex : he con ex
unc ion ha ep esen s a gi en s ochas ic alue maps each p ice o he consume su -
plus ha would a ise unde ha s ochas ic alue unde pe ec op imiza ion. Kleine ,
Moldo anu, and S ack (2021) cha ac e ize he ex eme poin s o a se o dis ibu ions
bounded by a andom a iable wi h espec o he ICX o SOSD o de .7In con as , we
iden i y andom a iables ha p o ide bounds wi h espec o hese o de s on a se sa -
is ying ce ain cons ain s.
5See Be nheim and Taubinsky (2018) o asu ey.
6While in o ma ion design p oblems ypically place no es ic ions on he in o ma ion s uc u e, we im-
pose an implici es ic ion o ensu e ha each ype has mono one demand.
7In ela ed wo k, Yang and Zen e is (2023) cha ac e ize ex eme poin s o a se o dis ibu ions bounded
wi h espec o he FOSD o de .
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1420 Kocou ek, S eine , and S ewa Theo e ical Economics 19 (2024)
Se e al o he pape s s udy complemen a y p oblems o iden i ying bounds on con-
sume su plus. Va ian (1985) conside s issues o measu emen due o gaps in obse ed
demand. Kang and Vasse man (2022) analyze how unc ional o m es ic ions na ow
hese bounds. Sandomi skiy and Ushche (2022) iden i y bounds on consume wel a e
ac oss possible disagg ega ions o an obse ed agg ega e demand cu e wi hou quasi-
linea u ili y. Allen and Rehbeck (2021) p o ide bounds on su plus om ini e da a se s
o consume s who app oxima ely op imize; in con as , we ocus on idealized in ini e
da a wi h di e en bounded a ionali y assump ions ha a e no nes ed wi h hei s.
2. Se up
An analys obse es da a (Q,F)desc ibing he s ochas ic pu chasing beha io o a con-
sume wi h uni demand oge he wi h he dis ibu ion o p ices. The demand unc-
ion Q:[p,p]−→ [0, 1], which we assume is non-inc easing, speci ies he p obabili y
Q(p)o pu chase a each p ice p;wedeno ebyP(q) he in e se demand associa ed
wi h Q(p).8P ices a e dis ibu ed acco ding o he con inuous dis ibu ion F(p)wi h
suppo [p,p],whe ep≥0. As is s anda d when measu ing consume wel a e, we as-
sume ha he analys obse es he choke p ice, i.e., Q(p)=0; simila ly, we assume ha
Q(p)=1.
The demand Qis an agg ega ion o many choices made by he consume ac oss
which bo h he alue o he good and he beha io may a y. In each such choice,
he consume aces a ake-i -o -lea e-i o e a a andom p ice pd awn acco ding o F.
(We deno e andom a iables in bold ace and hei ealiza ions wi h he co esponding
non-bold symbol; all p obabili ies and expec a ions a e e alua ed wi h espec o hese
bold a iables.) The consume has a s ochas ic ype iwi h suppo I⊂R.Each ypei
speci ies he consume ’s alue i o he good oge he wi h a non-inc easing demand
Qi(p).
We in e p e he da a as desc ibing he choices o a single indi idual whose alue o
he good may be changing and whose beha io also a ies due o unobse ed ac o s
such as a en ion o salience. We allow o he possibili y ha hese ac o s a e ela ed o
he alue since, o example, consume s may be mo e a en i e o he p ice when hei
alue is lowe . An al e na i e in e p e a ion o he da a is ha hey combine choices
made by a la ge popula ion o consume s, wi h each ype co esponding o a dis inc
indi idual whose alue and beha io a e ixed.
We assume ha he consume ’s ype is independen o he p ice. This assump ion
is na u al o indi idual-le el da a, such as scanne da a, whe e he indi idual is neg-
ligible om he pe spec i e o he selle . I consume s’ alues a y sys ema ically o e
ime, as, o ins ance, would be expec ed o a seasonal p oduc , hen he analys should
spli he da a in o pe iods wi hin which independence o alues and p ices is plausible.
(The ools de eloped in Sec ion 6can be applied o he sepa a ed da a.) I , howe e , he
analys ails o accoun o empo al a ia ion ha a ec s he selle ’s p icing s a egy,
8Tha is, P(q):=in {p:Q(p)≤q}. Analogously, gi en any in e se demand unc ion ˜
P, we de ine he
co esponding demand unc ion ˜
Q(p)=in {q:˜
P(q)≤p}.
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Theo e ical Economics 19 (2024) Boundedly a ional demand 1421
hen ou esul s may no apply. Simila ly, i he selle engages in hi d-deg ee p ice dis-
c imina ion o o he wise sc eens consume s by o e ing di e en dis ibu ions o p ices
depending on a ibu es co ela ed wi h he consume ’s alue, he analys should g oup
he da a acco ding o he same a ibu es; ou esul s may no apply o he o iginal com-
bined da a.
The analys assumes ha all ypes o he consume sa is y minimal a ionali y e-
s ic ions: we equi e ha no ype can be wo se o han she would be i she ne e
bough he good ( ega dless o i s ealized p ice), no can she be wo se o han i she
always bough he good. Le ing si=E[( i−p)Qi(p)] deno e ype i’s expec ed su plus,
his es ic ion co esponds o he pai o incen i e-compa ibili y cons ain s
si≥0(1)
si≥ i−E[p].(2)
This equi emen imposes discipline on he ela ionship be ween a ype’s alue and i s
demand: subs i u ing he de ini ion o siand isola ing iyields
Ep1−Qi(p)
E1−Qi(p)≥ i≥
EpQi(p)
EQi(p).(3)
The e a e se e al easons o expec ha a consume ’s demand may no pe ec ly e-
lec he alue. I could be ha she does no always check he p ice o he good o does
so only i she no ices ha i is on sale, he pos ed p ice may no include axes ha he
consume does no accu a ely compu e, o he consume may make andom e o s in
assessing he alue o he good, as in a andom u ili y model. In each o hese cases,
he consume ’s a en i eness and likelihood o making a mis ake could depend on he
cu en alue. The analys he e o e allows o ypes’ demands o a y along wi h hei
alues in a gene al way, imposing only ha no ype makes sys ema ic e o s such ha
hey would be be e o ei he always buying o ne e buying.
The analys seeks o explain he obse ed choices wi h a model ha consis s o a
dis ibu ion Mo ypes i∈I oge he wi h a speci ica ion ( i,Qi)i∈Io alues and non-
inc easing demand unc ions o each ype sa is ying (1)and(2). We say ha a gi en
model a ionalizes da a (Q,F)i Q(p)=E[Qi(p)] o all p. Gi en a model ha a io-
nalizes he da a, he (ex an e) consume su plus is s=E[si]. In gene al, da a can be a-
ionalized by many di e en models which in u n yield di e en alues o su plus. We
say ha su plus s∈Ris consis en wi h he da a i he e exis s a model ha a ionalizes
(Q,F)and gene a es su plus s.
A special case o ou en i onmen ha may be o pa icula in e es a ises when he
consume is Bayesian bu obse es only a noisy signal o he p ice o he good. The noise
in his signal could esul om ina en ion; o example, he consume may assume ha
he p ice o he good exceeds he willingness o pay unless she no ices ha i is on sale
(in which case she checks he p ice). In his case, a ype o he consume can be de-
sc ibed by a alue and an in o ma ion s uc u e. The ype’s demand is hen de e mined
by he condi ion ha she buys p ecisely when he alue exceeds he pos e io expec-
a ion o he p ice. Bayesian op imali y implies ha (1)and(2) a e sa is ied o each
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1422 Kocou ek, S eine , and S ewa Theo e ical Economics 19 (2024)
ype. Con e sely, o each ype ( i,Qi)sa is ying (1)and(2), he e exis s an in o ma ion
s uc u e o which a Bayesian ype wi h alue iwould ha e demand Qi. Namely, gi en
a ype( i,Qi), ake he bina y in o ma ion s uc u e ha gene a es a “buy” signal wi h
p obabili y Qi(p)and an “abs ain” signal o he wise. By (1)and(2), ollowing he ac ion
ecommended by he signal is incen i e compa ible. Consequen ly, all o ou esul s
apply as w i en o his special case; in pa icula , Bayesian op imali y does no na ow
he bounds on consume su plus ela i e o hose ob ained wi h ou minimal a ionali y
assump ions.
Example 1. The analys obse es he linea demand unc ion Q(p)=1−pand p ices
uni o mly dis ibu ed on [0, 1]. The e a e many possible models ha a ionalize his
da a. Fo ins ance, i could be ha , as in he s anda d analysis, he consume always
makes he op imal decision: he alue iis uni o mly dis ibu ed on [0, 1]and each
ype idemands he good p ecisely when p<
i. Fo any ealized p ice p, hisconsume
ecei es su plus (1−p)2/2 (co esponding o he a ea be ween he demand cu e and
he p ice). The expec ed consume su plus o his model is he e o e s=E[(1−p)2/2]=
1/6.
Al e na i ely, he da a can be a ionalized by a model wi h s ochas ic choices. Pe -
haps he simples such a ionaliza ion ea u es a consume wi h a single ype. Fo each
p ice ealiza ion p, he consume pu chases he good wi h p obabili y Q(p),which i -
ially gene a es he obse ed agg ega e demand. The inequali ies in (3) place limi s on
his ype’s alue, :i mus bea leas =E[pQ(p)]/E[Q(p)] =1/3 o ensu e ha she
does no p e e o abs ain om buying, and a mos =E[p(1−Q(p))]/E[1−Q(p)] =
2/3 o ensu e ha she does no p e e o always buy. Taking = leads o a su plus o
E[( −p)Q(p)] =1/6; aking = leads o a su plus o E[( −p)Q(p)] =0. Using alues
o in be ween hese wo ex emes, any su plus in [0, 1/6]can be ob ained.9
Mo e complex models can yield addi ional alues o he su plus. Conside wo
equally likely ypes, 1 and 2, wi h espec i e demands
Q1(p)=2(1−p)i p≥1/2
1o he wise
and Q2(p)=0i p≥1/2
1−2po he wise.
Since (Q1+Q2)/2=Q, hese ypes’ demands oge he gene a e he obse ed o al de-
mand. Gi en each ype’s demand Qi,(3) places es ic ions on he alues o he o m
i∈[ i, i]. Using he maximal alues gi es su plus 1
2i iE[Qi(p)] −E[pQ(p)] =2/9
(whe eas he minimal alues again gi e a su plus o 0).10 ♦
We see om his example ha i is possible o ob ain alues o he su plus consis-
en wi h he da a ha exceed he s anda d consume su plus. This obse a ion may
9Tha he uppe bound o 1/6 is equal o he s anda d su plus is a coincidence ha does no gene ally
hold ou side o his example. On he o he hand, 0 is a igh lowe bound ega dless o he da a as he e
a e always models in which each ype o he consume is indi e en be ween choosing acco ding o he
demand and ne e buying he good.
10Type 1 buys wi h p obabili y 3/4 and has maximal alue 1=5/6, and ype 2 buys wi h p obabili y 1/4
and has maximal alue 2=11/18, gi ing a su plus o 1/2·5/6·3/4+1/2·11/18 ·1/4−1/6=2/9.
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Theo e ical Economics 19 (2024) Boundedly a ional demand 1429
o which
s(p)=E( i−p)Qi(p).
Lemma 5. I s(p)is an in e im consume su plus consis en wi h he da a, hen s(p)≤
CS(Q,P;p).
P oo . Fi s no e ha , o a gi en s ochas ic alue , he consume can su e om wo
ypes o losses ela i e o op imal beha io : (i) he p obabili y o pu chase a a gi en
p ice may no be op imal, i.e., Q(p)may di e om Qs(p; ), and (ii) he se o ypes
pu chasing he good a a gi en p ice may no be hose wi h he highes alues. S a ing
om any model ha a ionalizes he da a, ealloca ing demands ac oss ypes o elimi-
na e his la e loss (igno ing incen i e compa ibili y) gi es an uppe bound CS(Q, ;p)
on he in e im su plus a each p o models wi h s ochas ic alue . The e o e, he su -
plus s(p)gene a ed by any such model sa is ies
s(p)≤CS(Q, ;p)≤CS(Q, ;p)=CS(Q,P;p),
whe e he middle inequali y ollows om Lemmas 2and 4.
Since he in e im su plus is bounded om abo e by CS(Q,P;p) o each p ice p,
he ex an e su plus is bounded by E[CS(Q,P;p)], as needed. This concludes he p oo
o Theo em 1.
6. Mul iple da a se s
The bounds on consume su plus can be na owed i he analys obse es he con-
sume ’s choices unde a ying ma ke condi ions, which we e e o as egimes.Weas-
sume ha he consume ’s ypes’ alues a e ixed ac oss egimes, bu he egimes may
di e in he dis ibu ion o p ices o in he pu chasing beha io o each ype a any
gi en p ice (o bo h). Fo example, one such egime may co espond o a publicly an-
nounced “sale” associa ed wi h a low dis ibu ion o p ices, while ano he co esponds
o he same ma ke in he absence o a sale; he sale announcemen may a ec he con-
sume ’s s ochas ic choice a each p ice h ough changes in a en ion o salience. Al e -
na i ely, he egimes may di e only in how p ices a e p esen ed o consume s, as in he
empi ical s udies o Che y, Looney, and K o (2009)andFinkels ein (2009).
The analys obse es a p o ile o da a se s (Qk,Fk),k=1, ,K,whe eQk(p)and
Fk(p)a e, espec i ely, he p obabili y ha he consume makes a pu chase a each
p ice pand he dis ibu ion o p ices in egime k, and each (Qk,Fk)sa is ies he as-
sump ions on da a made in Sec ion 2. The consume has a s ochas ic ype i, wi h each
ealiza ion ispeci ying he alue i o he good and he (non-inc easing) demand unc-
ion Qk
i(p)in each egime. The dis ibu ion o ypes and he alue o each ype a e
he same ac oss all egimes. A model o he analys consis s o a dis ibu ion o ypes
oge he wi h a speci ica ion o ( i,Q1
i,,QK
i) o each ype i.
We say ha a model a ionalizes he p o ile o da a se s (Qk,Fk)ki , o each egime
k, i a ionalizes da a se (Qk,Fk)when each ype ihas demand Qk
i. In pa icula , wi hin
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1430 Kocou ek, S eine , and S ewa Theo e ical Economics 19 (2024)
each egime, we impose he basic a ionali y assump ions desc ibed by (1)and(2).
A s ochas ic alue is consis en wi h he p o ile o da a se s (Qk,Fk)ki he e exis s a
model ha a ionalizes his p o ile and sa is ies d
= i.
Example 3. Conside wo egimes. The da a in egime 1 consis o he linea demand
Q1(p)=1−pand uni o m p ice dis ibu ion p∼U[0, 1],asinExample1.Theda ain
egime 2 consis o he s ep- unc ion demand Q2(p)=1p≤2/3and uni o m p ice dis i-
bu ion p∼U[2/3−ε,2/3+ε],whe e0<ε≤1/3. These wo egimes a e join ly a ional-
izable, o example, by a single ype wi h alue 2/3 and demand Qk(p) o k=1, 2. ♦
Gi en a model, he in e im consume su plus in egime ka p ice pis
sk(p)=EQk
i(p)( i−p).
We w i e sk=E[sk(p)] o he ex an e su plus in egime k(wi h p∼Fk). In e im con-
sume su plus sk(p)in egime kis consis en wi h he p o ile o da a se s (Qk,Fk)ki
he e exis s a model ha a ionalizes his p o ile and gene a es su plus sk(p)in egime k
(and analogously o he ex an e su plus).
6.1 Uppe bound
The nex esul p o ides an uppe bound on he su plus wi hin each egime ha gen-
e ally imp o es upon he bounds ha can be ob ained o each egime sepa a ely. The
basic idea is o de i e he uppe bounds on he s ochas ic alue wi h espec o he in-
c easing con ex o de when conside ing each egime sepa a ely and o combine hem
in such a way as o gene a e a igh e bound. The app oach he e o e equi es combin-
ing bounds on andom a iables wi h espec o ha s ochas ic o de . To do so, building
on ideas o Gen zkow and Kamenica (2016)andKolo ilin e al. (2017), we exploi he ep-
esen a ion o a andom a iable in e ms o a con ex unc ion desc ibed in Sec ion 4,in
his case, he s ochas ic alue in e ms o he s anda d consume su plus. Acco ding o
Lemma 1, compa isons o s ochas ic alues in he inc easing con ex o de co espond
o compa isons o he s anda d consume su plus. Using his connec ion, we ind he
la ges andom a iable ha sa is ies he bounds on he s ochas ic alue ac oss all o he
egimes by inding he la ges con ex unc ion lying below he co esponding bounds
on he s anda d consume su plus.
Le kbe he uppe bound on s ochas ic alues consis en wi h he da a o egime k
wi h espec o he inc easing con ex o de , as in Lemma 4.16 Fo each k, hebound
kco esponds o he con ex unc ion CSs(p; k). The uppe bound using da a ac oss
all egimes he e o e co esponds o he la ges con ex unc ion ha lies below each
CSs(p; k). Acco dingly, le CSs
∗(p)deno e he con ex closu e o he unc ion
minkCSs(p; k).17 We e e o CSs
∗as he con exi ica ion o minkCSs(p; k).
16Tha is, k=Pk(i)wi h i∼U[0, 1], whe e he ele a ed demand o egime kis Pk(i)=E[p|p≥Pk(i)]
wi h p∼Fkand Pkis he in e se demand o Qk.
17Recall ha he con ex closu e o a unc ion g(p)is he unc ion ha maps each p o in {s:(p,s)∈
co(g)},whe eco
(g)deno es he con ex hull o he g aph o he unc ion g. In he e minology o con ex
analysis, CSs
∗is he biconjuga e unc ion o minkCSs(p; k).
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Theo e ical Economics 19 (2024) Boundedly a ional demand 1431
Figu e 3. Con exi ica ion o he wo egimes desc ibed in Example 3wi h ε=1/3. No e ha
he g aphs depic only p ices p∈[1/2, 1]since he con exi ica ion is i ial o p<1/2. (a)
S anda d consume -su plus unc ions CSs(p; 1)(dashed) and CSs(p; 2)(do ed), and he con-
exi ica ion CSs
∗(p)( hick). (b) S anda d demands associa ed wi h s ochas ic alues Qs(p; 1)
(dashed), Qs(p; 2)(do ed), and Qs
∗(p)( hick).
To map CSs
∗back o a s ochas ic alue, ecall om Sec ion 4 ha CSs(p; )is he
in eg al o he uppe ail o he s anda d demand Qs(p; ), which is he complemen a y
dis ibu ion unc ion o . De ine he demand unc ion Qs
∗(p)=−∂−CSs
∗(p),whe e∂−
deno es he le de i a i e. No e ha 1 −Qs
∗is a dis ibu ion unc ion and le ∗be a
s ochas ic alue associa ed wi h his dis ibu ion.18 See Figu e 3 o an illus a ion.
The ollowing esul is he main s ep unde lying he uppe bound o mul iple e-
gimes.
Lemma 6. I a s ochas ic alue is consis en wi h he p o ile o da a se s, hen ∗icx .
The lemma ollows om Theo em 3.2 o Mülle and Sca sini (2006).
A di ec a gumen is as ollows. I is consis en wi h he p o ile o da a se s, hen i is
consis en wi h each da a se sepa a ely; hus, kicx o each egime k. By Lemma 1,
minkCSs(p; k)≥CSs(p; ). Since CSs(p; )is con ex in p,CSs(p; )is no g ea e han
he con exi ica ion o minkCSs(p; k). Finally, again by Lemma 1, ∗icx .
Combining Lemmas 2and 6leads o he ollowing uppe bound on he consume
su plus wi hin each egime.
Theo em 2. I he in e im consume su plus sk(p)in egime kis consis en wi h he
p o ile o da a se s, hen sk(p)≤CS(Qk, ∗;p).
As an immedia e consequence, he ex an e consume su plus in egime kconsis-
en wi h he p o ile o da a se s is bounded om abo e by E[CS(Qk, ∗;p)],whe e he
expec a ion is wi h espec o he dis ibu ion o p ices in egime k.
18Since CSs
∗is con ex, i s le de i a i e exis s, and Qs
∗is non-inc easing and le -con inuous. Ad-
di ionally, CSs
∗(p)=0 o p>pand CSs
∗(p)has slope −1 o p<p
;hence,limp→−∞ Qs
∗(p)=1 and
limp→+∞ Qs
∗(p)=0. Thus, 1 −Qs
∗is a dis ibu ion unc ion.
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1432 Kocou ek, S eine , and S ewa Theo e ical Economics 19 (2024)
P oo o Theo em 2. Gi en a s ochas ic alue , he in e im consume su plus sk(p)
in egime kis a mos CS(Qk, ;p)because his is he su plus associa ed wi h ha ing he
measu e Qk(p)o ypes wi h he highes alues buy a p. By Lemma 2,CS(Qk, ;p)is
non-dec easing in wi h espec o he inc easing con ex o de . Finally, by Lemma 6,
any s ochas ic alue consis en wi h he p o ile o da a se s is bounded by ∗in he
inc easing con ex o de .
Lemma 6, when combined wi h Lemma 1, also p o ides an uppe bound on he
coun e ac ual consume su plus ha would a ise i he consume pe ec ly op imized.
Co olla y 1. Gi en a p o ile o da a se s (Qk,Fk)k, he consume su plus ha would
a ise a p ice pi he consume chose op imally is no g ea e han CSs
∗(p).
6.2 Lowe bound
An analogous cons uc ion o ha o he uppe bound can be used o ob ain a non i ial
lowe bound on su plus using da a om mul iple egimes. Gi en a s ochas ic alue and
demand Q(p), we can compu e a lowe bound on su plus by supposing ha he measu e
Q(p)o he lowes ypes pu chase he good a each p(as opposed o he highes ypes
we used o he uppe bound). Unde his assignmen , oughly speaking, lowe means
and g ea e sp eads o he s ochas ic alue bo h educe he lowe bound on su plus.
Consequen ly, he ele an o de ing o s ochas ic alues is sosd (as opposed o icx o
he uppe bound).
Jus as is he highes and he mos sp ead ou s ochas ic alue consis en wi h he
da a in a single egime, (as de ined in Sec ion 5) is he lowes and he mos sp ead ou
such s ochas ic alue. Mo e p ecisely, is a lowe bound wi h espec o sosd on all
consis en wi h he da a. While he cen al s ep o he p oo o Lemma 4was o show ha
a decomposi ion in o h eshold demands induces a mean-inc easing sp ead o he non-
buying p ice expec a ions, a symme ic a gumen implies ha he same decomposi ion
induces a mean-dec easing sp ead o he buying p ice expec a ions.
To ep esen he second-o de s ochas ic dominance o de , we de ine he comple-
men a y s anda d consume su plus
CSs(p; ):=p
−∞1−Qsp; dp
and no e ha i is non-dec easing and con ex in p. By he well known cha ac e iza ion
o Hada and Russell (1969)andRo hschild and S igli z (1970), he anking o s ochas ic
alues wi h espec o sosd implies he opposi e anking o
CSs(p; ), and he con e se
also ob ains p o ided he la e anking holds uni o mly ac oss all p.
Wi h mul iple egimes, ollowing he analogous cons uc ion o ha o he uppe
bound, mink
CSs(p; k)is an uppe bound on
CSs(p; ),whe e is a s ochas ic alue
consis en wi h he da a in each egime and, o each k, kis he lowe bound on s ochas-
ic alues wi h espec o sosd consis en wi h he da a se (Qk,Fk). Since mink
CSs(p;
k)is no gene ally con ex, i may no co espond o any s ochas ic alue; acco dingly,
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Theo e ical Economics 19 (2024) Boundedly a ional demand 1433
le
CSs
∗(p)deno e i s con exi ica ion. Le ∗be he s ochas ic alue associa ed wi h
CSs
∗.19 Along he same lines as in Lemma 6, ∗is a lowe bound wi h espec o sosd
on s ochas ic alues consis en wi h he p o ile o da a se s.
Le
P(q; ):=Ps(1−q; )deno e he q h lowes quan ile o .
Theo em 3. I he in e im consume su plus sk(p)in egime kis consis en wi h he
p o ile o da a se s, hen sk(p)≥CS(Qk,
P(·, ∗);p).
Once again, aking expec a ions wi h espec o he p ice in each egime gi es a lowe
bound on he ex an e su plus in ha egime.
To unde s and his esul , conside a consume wi h s ochas ic alue . Acco ding o
he da a o egime k, a measu e Qk(p)o ypes buy a each p ice p. Selec ing he ypes
wi h he lowes alues gene a es su plus CS(Qk,
P(; );p)in egime k; hislowe bound
is non-dec easing in wi h espec o second-o de s ochas ic dominance. Finally, be-
cause s ochas ic alues consis en wi h he p o ile o da a se s a e bounded om below
wi h espec o sosd by ∗, heboundonsk(p) om he heo em applies.
Example 4. To illus a e he lowe bound, conside he egimes om Example 3wi h
ε=1/3. In his case, 1is uni o mly dis ibu ed on [0, 1/2]and 2is almos su ely equal
o 1/2. Thus 2second-o de s ochas ically domina es 1, making he con exi ica ion
i ial wi h ∗= 2. The lowe bound om Theo em 3on he ex an e consume su plus
in egime 1 is he e o e 1/2·1/2−1/6=1/12 and he lowe bound in egime 2 is 0. ♦
6.3 Tigh ness o he bounds
Theo em 1p o ides igh bounds on consume su plus o da a om a single ma ke
egime; o each alue wi hin he bounds, we ha e cons uc ed a model o which he
su plus is equal o ha alue. While he bounds on su plus in Theo ems 2and 3a e
gene ally igh e wi hin each egime han he bounds ob ained om he da a in ha
egime alone, hey a e no hemsel es igh bounds.
Example 5. To illus a e, conside he uppe bound o he wo egimes om Example 3
wi h ε≤1/6. In his case, one can show ha CSs(p; 2)≤CSs(p; 1) o all p, making
he con exi ica ion i ial: CSs
∗(p)≡CSs(p; 2). The e o e, ∗d
= 2almos su ely akes
on he alue 2/3+ε/2, which is he non-buying p ice expec a ion o egime 2. How-
e e , his alue is no consis en wi h he da a o egime 1 since he non-buying p ice
expec a ion o a leas some ypes mus be no mo e han 2/3 ( he non-buying p ice ex-
pec a ion o demand Q1(p)). Thus, he uppe bound on consume su plus in egime 1
cons uc ed in Theo em 2is no a ainable in his case. ♦
I he analys obse es only one ma ke egime, hen o de e mine he ange o al-
ues o consume su plus, i su ices o conside ypes wi h simple h eshold demands.
19Tha is, le 1 −Qs(·; ∗)be he igh de i a i e o
CSs, obse e ha i is a dis ibu ion unc ion, and le
∗be a andom a iable wi h his dis ibu ion.
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1434 Kocou ek, S eine , and S ewa Theo e ical Economics 19 (2024)
Wi h mul iple egimes, decomposi ions in o h eshold demands a e no gene ally su -
icien ; i can happen ha he egimes a e no join ly a ionalizable by any model wi h
h eshold demands (bu can be a ionalized by o he models).20 The cons uc ions in
Theo ems 2and 3ci cum en his complica ion by using a bound on su plus in each
egime based on h eshold demands. The upside o his app oach is ha he combined
bound is simple. The downside is ha he combined bound need no co espond o a
model ha a ionalizes he p o ile o da a se s and. hence, he bound is no gene ally
igh .
7. Bounds on coun e ac ual demand
Re u ning o he o iginal model in which he analys obse es a single da a se (Q,F)
ha may esul om impe ec op imiza ion, we now conside he coun e ac ual de-
mand ha would a ise i ins ead he consume we e o pe ec ly op imize and pu chase
p ecisely when he alue iexceeds he p ice p. These bounds apply equally o a coun-
e ac ual ma ke wi h a ixed, de e minis ic p ice whe e ou minimal a ionali y condi-
ions imply ha he consume would choose op imally.
As o consume su plus, bounds on coun e ac ual demand co espond o bounds
on he consume ’s s ochas ic alue, albei wi h espec o a di e en s ochas ic o de :
while he inc easing con ex o de and second-o de s ochas ic dominance p o ide he
ele an bounds o consume su plus, he bounds o coun e ac ual demand co e-
spond o i s -o de s ochas ic dominance.
To s a e hese bounds, de ine he doubly ele a ed and doubly lowe ed in e se de-
mands, espec i ely, by
P(q):=Ep|p≥P(q),q≤q
P(q):=Ep|p≤P(q),q≥q,
whe e q∼U[0, 1]and p∼F. Bo h unc ions a e non-inc easing. Rela i e o he ele a ed
and lowe ed in e se demands Pand P, hese in e se demands a e u he ele a ed and
lowe ed, i.e., P(q)≥P(q)and P(q)≤P(q) o all q. To see his, obse e ha P(q)is
a con ex combina ion o P(q)ac oss q∈[0, q]and Pis non-inc easing; a symme ic
a gumen shows ha P(q)≤P(q). See Figu e 4 o an illus a ion.
Unlike Pand P, he doubly ele a ed and lowe ed in e se demands Pand Pcould
no a ise unde he coun e ac ual o pe ec op imiza ion: he co esponding s ochas ic
alues a e inconsis en wi h he obse ed da a. Howe e , o each q, he e exis models
a ionalizing he da a o which he coun e ac ual in e se demands a e P(q)and P(q),
espec i ely.
20Example 3p o ides one such example when ε<1/6. I he demand om egime 1 is decomposed in o
h eshold demands, hen a nonze o mass o ypes mus ha e h esholds below 1/3−2ε. The non-buying
p ice expec a ion o such ypes is less han 2/3−ε. The e o e, hese ypes would no buy a any p ice ha
occu s in egime 2, con adic ing ha Q2(p)=1 o p≤2/3.
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Theo e ical Economics 19 (2024) Boundedly a ional demand 1435
Figu e 4. Doubly ele a ed and doubly lowe ed in e se demands P( hick g ey) and P( hick
dashed) o gi en in e se demand P(black). Fo each q,P(q)is he expec ed p ice condi ional
on pand qlying in he uppe -le g ey a ea. Simila ly, P(q)is he expec ed p ice condi ional on
he lowe - igh g ey a ea. Fo compa ison, he hin g ey and hin dashed cu es depic Pand P,
espec i ely.
Theo em 4. Fo e e y s ochas ic alue consis en wi h da a (Q,F), he s anda d in-
e se demand unc ion sa is ies
P(q)≤Ps(q; )≤P(q)
o all q. These bounds a e igh in he sense ha o each q, he e exis s a s ochas ic alue
consis en wi h he da a such ha Ps(q; )=P(q), and simila ly o P(q).
We ske ch he a gumen o he uppe bound; he a gumen o he lowe bound
is analogous. Fo each q, gi en any s ochas ic alue , he s anda d in e se demand
Ps(q; )is a pa icula quan ile o (namely, he (1−q) h quan ile). The model ha
maximizes he coun e ac ual in e se demand a qamong hose a ionalizing he da a
is he e o e he one ha maximizes his quan ile. Acco dingly, bounds on coun e ac ual
demand co espond o bounds on s ochas ic alues wi h espec o i s -o de s ochas ic
dominance.
How can we maximize a gi en quan ile o (among s ochas ic alues consis en wi h
he da a)? Recall ha he highes alue compa ible wi h a ype’s demand is i s non-
buying p ice expec a ion. I u ns ou ha his non-buying p ice expec a ion is maxi-
mized when no o he ype has a highe alue and he demand o his ype is as la ge as
possible. Acco dingly, o maximize he alue a he (1−q) h quan ile, we use a model
in which he ype wi h he highes alue has measu e qand demand min{Q(p)/q,1
}.By
cons uc ion, he non-buying p ice expec a ion o his ype is exac ly P(q). To see ha
he bound is igh , no e ha such a ype can be pa o a model ha a ionalizes he da a
(in which he emaining measu e 1 −qo ypes gene a e he esidual demand).
As wi h consume su plus, da a om mul iple ma ke egimes can be used o igh en
he bounds on coun e ac ual demand. Assuming, as in Sec ion 6, ha p e e ences a e
s able ac oss egimes, a igh e bound can be ob ained by simply aking he minimum
and maximum, espec i ely, o he uppe and he lowe bounds om Theo em 4ac oss
all o he egimes.
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1436 Kocou ek, S eine , and S ewa Theo e ical Economics 19 (2024)
8. Discussion
I he analys does no know whe he he consume engages in op imal choice beha io ,
he consume su plus canno be poin -iden i ied om p ice and demand da a. None-
heless, weak a ionali y assump ions impose signi ican es ic ions on he le els o su -
plus consis en wi h he da a. Iden i ica ion o he consume su plus can be u he
sha pened by combining da a om ma ke egimes wi h a ying p io s o consume
demands.
Two ele an ques ions ela ed o his p ojec emain open. Fi s , he bounds we p o-
ide unde mul iple egimes a e no igh ; ou bounds ely on sepa a e a ionaliza ions
o each egime, whe eas, in p inciple, iden i ica ion o he su plus can be igh ened by
simul aneously a ionalizing he p o ile o da a se s. Second, in he in e es o gene -
ali y, we ha e imposed minimal s uc u e on he ela ionship be ween he consume ’s
alue and he demand. Depending on he con ex , he e may be addi ional s uc u e
ha could be used o na ow he bounds on su plus o coun e ac ual demand.
Appendix:P oo s
P oo o Lemma 2.No e ha
CS(˜
Q, ;p)=CS˜
Q,Ps(·; );p=˜
Q(p)
0
Ps(q; )dq −˜
Q(p)p.
Conside any and such ha icx . Since he expendi u e ˜
Q(p)pdoes no depend
on he s ochas ic alue, i su ices o p o e ha q∗
0Ps(q; )dq ≥q∗
0Ps(q; )dq o each
q∗∈[0, 1].Fixq∗.Fo p=Ps(q∗; ),
q∗
0
Psq; dq =CSsp; +pq∗
≥CSs(p; )+pq∗
≥q∗
0Ps(q; )−pdq +pq∗
=q∗
0
Ps(q; )dq.
The i s inequali y ollows om Lemma 1, while he second ollows om he obse a-
ion ha CSs(p; )=maxqq
0(Ps(q; )−p)dq.
P oo o Lemma 4.S ep 1.Conside amodelsuch ha each ypeihas alue iand de-
mand Qi(p).Le = ibe he associa ed s ochas ic alue. Le = i,whe e i=E[p|q≥
Qi(p)] o q∼U[0, 1]deno es he non-buying p ice expec a ion associa ed wi h de-
mand Qi. By Lemma 3, i≥ i o each i.Thus, icx (because i s -o de s ochas i-
cally domina es ).
S ep 2. Fo each ype i, de ine a andom a iable ias ollows. Le Pi(q)be he in e se
demand o demand Qi,le Pi(q)=E[p|p≥Pi(q)] be he ele a ed demand o ype i,and
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Theo e ical Economics 19 (2024) Boundedly a ional demand 1437
de ine he s ochas ic alue i=Pi(q) o q∼U[0, 1]. Finally, le = i; hus is a
sp ead o ha eplaces
i= iwi h i o each i.
We will show ha icx (and, hence, icx ). I su ices o show ha i≤E[ i]
o each i. Indeed, o q∼U[0, 1], by he law o i e a ed expec a ions,
i=Ep|q≥Qi(p)
=Ep|p≥Pi(q)
=EEp|p≥Pi(q),qp≥Pi(q)
=EPi(q)|p≥Pi(q)
=EPi(q)|q≥Qi(p).
Since qcondi ional on q≥Qi(p) i s -o de s ochas ically domina es qi sel and Pi(q)
is non-inc easing, i ollows ha
i≤EPi(q)=E[ i],
as needed.
S ep 3. We conclude by p o ing ha d
= .Conside anypa which Qis con inuous
and le ˜
=E[p|p≥p].Fo anyj∈[0, 1],
j=P ( ≥˜
)=⇒ P(j)=˜
=⇒ P(j)=p=⇒ j=Q(p).
Hence, P ( ≥˜
)=Q(p). Likewise, P ( i≥˜
)=Qi(p) o almos all i(i.e., o all iexcep
hose o which Qiis discon inuous a p), and, hus,
P ≥˜
=P ( i≥˜
)=EQi(p)=Q(p)=P ( ≥˜
)
o all ˜
om a dense subse o he suppo o and , as needed.
P oo o Theo em 3. Conside a model consis en wi h he p o ile o da a se s and le
be i s associa ed s ochas ic alue. Recall ha
P(q; )=Ps(1−q; )is he q h lowes
quan ile o .No e ha
sk(p)≥CSQk,
P(·; );p
o each ksince he igh -hand side is he expec ed consume su plus i he measu e
Qk(p)o ypes wi h he lowes alues buy a p ice p.
Fo any p ice pand any wo s ochas ic alues and such ha sosd ,andany
demand unc ion ˜
Q,weclaim ha
CS˜
Q,
P(·; );p≥CS˜
Q,
P·; ;p.
The p oo o his claim is analogous o ha o Lemma 2. In pa icula , we may dis ega d
expendi u es since hey depend only on he i s and he las a gumen s o CS.I su ices
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1438 Kocou ek, S eine , and S ewa Theo e ical Economics 19 (2024)
o p o e ha i sosd , henq∗
0
P(q; )dq ≥q∗
0
P(q; )dq o e e y q∗. Fixing q∗and
le ing p=
P(q∗; ),weha e
q∗
0
Pq; dq =q∗p−
CSsp;
≤q∗p−
CSs(p; )
≤q∗p−q∗
0p−
P(q; )dq
=q∗
0
P(q; )dq.
The i s inequali y ollows om he in eg al condi ion o sosd and he second om
he ac ha
CSs(p; )=maxˆ
qˆ
q
0(p−
P(q; ))dq.
The e o e, i a s ochas ic alue is consis en wi h he p o ile o da a se s, hen
sk(p)≥CSQk,
P(·; );p≥CSQk,
P(·; ∗);p
since sosd ∗.
P oo o Theo em 4. We p o e only he uppe bound; he a gumen o he lowe
bound is analogous.
Fo any q∈(0, 1], conside a demand unc ion ˜
Q ha a ains alues in [0, q], i.e., a
non-inc easing unc ion om [p,p]on o [0, q].Le
˜
(˜
Q;q):=Ep|q≥˜
Q(p)
w(˜
Q;q):=qP q≥˜
Q(p)
o q∼U[0, q]and p∼F. To in e p e hese wo unc ions, conside a model wi h
ype dis ibu ion Mand a subse I⊆Io ypes such ha P (i∈I)=qand ˜
Q(p)=
i∈IQi(p)dM(i).Then
˜
(˜
Q;q)is he expec ed p ice condi ional on a ype andomly
d awn om Ino making a pu chase, and w(˜
Q;q)is he p obabili y ha a andomly
d awn ype lies in Iand does no buy.
No e he ollowing iden i y. Fo any qa,qb∈(0, q]such ha qa+qb=q,andany
wo demands Qaand Qb ha a ain alues in [0, qa]and [0, qb], espec i ely, such ha
Qa+Qb=˜
Q,
˜
(˜
Q;q)=w(Qa;qa)˜
(Qa;qa)+w(Qb;qb)˜
(Qb;qb)
w(Qa;qa)+w(Qb;qb).(6)
Gi en any model and a subse Io ypes such ha P (i∈I)=q,le ∗:=in i∈I i.To
es ablish he uppe bound, i su ices o show o each q ha he sup emum o ∗ac oss
all models ha a ionalize he da a and subse s Isuch ha P (i∈I)=qis a mos P(q).
Fix a model wi h ype dis ibu ion Mon Iand ypes ( i,Qi) ha a ionalizes he
da a. Fix a se Io ypes such ha P (i∈I)=q.Le ˜
Q(p)=i∈IQi(p)dM(i)be he
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