Innovation adoption by forward-looking social learners

F ick, Mi a; Ishii, Yuh a
A icle
Inno a ion adop ion by o wa d-looking social lea ne s
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: F ick, Mi a; Ishii, Yuh a (2024) : Inno a ion adop ion by o wa d-looking social
lea ne s, Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 19,
Iss. 4, pp. 1505-1541,
h ps://doi.o g/10.3982/TE4455
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Theo e ical Economics 19 (2024), 1505–1541 1555-7561/20241505
Inno a ion adop ion by o wa d-looking social lea ne s
Mi a F ick
Depa men o Economics, P ince on Uni e si y
Yuh a Ishii
Depa men o Economics, Pennsyl ania S a e Uni e si y
We build a model s udying he e ec o an economy’s po en ial o social lea n-
ing on he adop ion o inno a ions o unce ain quali y. Assuming consume s a e
o wa d-looking (i.e., ecognize he alue o wai ing o in o ma ion), we analyze
how quali a i e and quan i a i e ea u es o he lea ning en i onmen a ec equi-
lib ium adop ion dynamics, wel a e, and he speed o lea ning. Based on his, we
show how di e ences in he lea ning en i onmen ansla e in o obse able di -
e ences in adop ion dynamics, sugges ing a pu ely in o ma ional channel o wo
commonly documen ed adop ion pa e ns: S-shaped and conca e cu es. We
also iden i y en i onmen s ha a e subjec o a sa u a ion e ec : Inc eased op-
po uni ies o social lea ning can slow down adop ion and lea ning, and do no
inc ease consume wel a e, possibly e en being ha m ul.
Keywo ds. Inno a ion adop ion, social lea ning, in o ma ional ee- iding,
s a egic expe imen a ion, exponen ial bandi s.
JEL classi ica ion. D80, D83, O33.
1. In oduc ion
Suppose a new p oduc o unce ain quali y, such as a no el elec i e medical p oce-
du e (e.g., Lasik eye su ge y o ba ia ic weigh -loss su ge y) o a new mo ie, is eleased
in o he ma ke . In ecen yea s, he ise o online e iew si es, sea ch engines, ideo-
sha ing pla o ms, and social ne wo king si es has g ea ly inc eased he po en ial o
social lea ning in he economy: I o he pa ien s su e a se ious complica ion o many
iewe s enjoy he mo ie, his is mo e likely han e e o ind i s way in o he public do-
main; and he e a e mo e and mo e people who ha e access o his common pool o
consume -gene a ed in o ma ion.
Mi a F ick: [email p o ec ed]
Yuh a Ishii: [email p o ec ed]
This pape is a e ised e sion o chap e s o ou PhD disse a ions a Ha a d Uni e si y. We a e g a e-
ul o D ew Fudenbe g, A ila Amb us, E ic Maskin, and Tomasz S zalecki o gene ous ad ice and en-
cou agemen . Fo help ul commen s ha signi ican ly imp o ed he pape , we hank h ee anonymous
e e ees, as well as Nageeb Ali, Di k Be gemann, Aislinn Boh en, Yeon-Koo Che, Thomas Co e , Ma -
in C ipps, Ben Golub, Ma ina Halac, Johannes Hö ne , Ryo a Iijima, Boyan Jo ano ic, Daniel Kenis-
on, Da ia Kh omenko a, Sco Komine s, Da id Laibson, G eg Lewis, Chia a Ma ga ia, Césa Ma inelli,
S ephen Mo is, Giuseppe Mosca ini, Pauli Mu o, Aniko Ö y, Luciano Poma o, S en Rady, La y Samuel-
son, Hea he Scho ield, Jesse Shapi o, Ron Siegel, Andy Sk zypacz, Ca oline Thomas, Jean Ti ole, Ch is
Ud y, Juuso Välimäki, Leea Ya i , and nume ous con e ence and semina audiences.
©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/TE4455
1506 F ick and Ishii Theo e ical Economics 19 (2024)
This pape builds a model s udying he e ec o an economy’s po en ial o social
lea ning on he adop ion o inno a ions o unce ain quali y. A cen al ing edien o ou
model is ha consume s a e o wa d-looking social lea ne s: In choosing whe he o
adop an inno a ion, hey ecognize he alue o delaying hei decision o lea n om
o he adop e s’ consump ion expe iences.1We analyze how consume s’ delay incen-
i es depend on quali a i e and quan i a i e ea u es o he lea ning en i onmen , and
how his a ec s equilib ium adop ion, wel a e, and he speed o lea ning. Ou analysis
has wo main implica ions. Fi s , quali a i ely, we show how di e ences in he lea ning
en i onmen ansla e in o obse able di e ences in adop ion dynamics. This implies
a new, pu ely in o ma ional channel o wo o he mos commonly documen ed adop-
ion pa e ns: S-shaped and conca e cu es. Second, quan i a i ely, we sugges cau ion
in e alua ing he impac o inc eases in he po en ial o social lea ning. We iden i y
en i onmen s ha a e subjec o a sa u a ion e ec , whe eby beyond a ce ain le el, in-
c eased oppo uni ies o social lea ning can slow down adop ion and lea ning, and do
no imp o e consume wel a e (possibly e en being ha m ul).
In ou model (Sec ion 2), an inno a ion o ixed, bu unce ain quali y (be e o
wo se han he s a us quo) is in oduced o a la ge popula ion o o wa d-looking con-
sume s. Consume s a e (ex an e) iden ical, sha ing he same p io abou he quali y o
he inno a ion, he same discoun a e, and he same as es o good and bad quali y.
A each ins an ∈R+, consume s ecei e s ochas ic oppo uni ies o adop he inno-
a ion. A consume who ecei es an oppo uni y mus choose whe he o i e e sibly
adop he inno a ion o o delay his decision un il he nex oppo uni y. In equilib ium,
consume s op imally ade o he oppo uni y cos o delays agains he bene i o lea n-
ing mo e abou he quali y o he inno a ion.
Lea ning is summa ized by a public signal p ocess, ep esen ing news ha is ob-
ained endogenously—based on he expe iences o p e ious adop e s—and possibly
also om exogenous sou ces (e.g., wa chdog agencies, p o essional c i ics). To s udy he
impo ance o quan i a i e and quali a i e ea u es o he news en i onmen , we build
on he exponen ial-bandi amewo k widely used in he li e a u e on s a egic expe -
imen a ion (see Sec ion 1.1): Indi idual adop e s’ expe iences gene a e public signals
a a ixed Poisson a e ha we use o quan i y he po en ial o social lea ning. Quali-
a i ely, as we in e p e in Sec ion 2.2, he e is a na u al dis inc ion be ween bad news
ma ke s, whe e signal a i als (b eakdowns) indica e bad quali y and he absence o sig-
nals makes consume s mo e op imis ic abou he inno a ion, and good news ma ke s,
whe e signals (b eak h oughs) sugges good quali y and he absence o signals makes
consume s mo e pessimis ic.
Sec ion 3analyzes and con as s equilib ium adop ion dynamics in bad and good
news ma ke s. As in many applica ions o Poisson lea ning, we ocus on he s a k bu
ac able case o pe ec bad ( espec i ely, good) news, whe e a single signal a i al con-
clusi ely indica es bad ( espec i ely, good) quali y. Thus, incen i es a e non i ial only
absen signals. As a p elimina y s ep, Lemma 1shows ha equilib ium incen i es o e
1Fo wa d-looking social lea ning is well documen ed empi ically, e.g., in he de elopmen economics
li e a u e s udying he adop ion o ag icul u al inno a ions (see Sec ion 4.2).
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Theo e ical Economics 19 (2024) Inno a ion adop ion 1507
ime sa is y a single-c ossing p ope y: Absen signals, he e is a mos one ansi ion
om p e e ence o adop ion o p e e ence o wai ing, o ice e sa, wi h a possible
pe iod o indi e ence in be ween. Building on his, Theo ems 1and 2es ablish equi-
lib ium exis ence and uniqueness unde bad and good news.2Equilib ium adop ion
dynamics admi simple closed- o m desc ip ions ha a e Ma ko ian in cu en belie s
and in he mass o consume s who ha e no ye adop ed.
Unde bad news, he unique equilib ium is cha ac e ized by wo cu o imes 0 ≤
∗
1≤ ∗
2. Un il ∗
1, no adop ion akes place and consume s acqui e in o ma ion only om
exogenous sou ces; om ∗
2on, all consume s adop immedia ely when gi en a chance
(absen b eakdowns). I ∗
1<
∗
2, hen h oughou ( ∗
1, ∗
2) he e is pa ial adop ion:Only
some consume s adop when gi en a chance, wi h o he s ee- iding on he in o ma ion
gene a ed by he adop e s, whe e he low o adop e s on ( ∗
1, ∗
2)ensu es indi e ence
be ween adop ing and delaying h oughou his in e al. A pe iod o pa ial adop ion
a ises in economies wi h a la ge enough po en ial o social lea ning, and wi h su i-
cien ly pa ien and no oo op imis ic consume s; o he wise, he e is no pa ial adop-
ion. By con as , he unique good news equilib ium is always all-o -no hing, ea u -
ing immedia e adop ion up o some ime ∗and no adop ion om ∗on (absen b eak-
h oughs). Thus, ega dless o he po en ial o social lea ning, consume s’ discoun
a e, o p io belie s, he e is ne e any pa ial adop ion. This highligh s a new dis inc-
ion be ween he way in which bad and goods news lea ning a ec s consume s’ incen-
i es. Speci ically, as Sec ion 3.3 explains, sus aining pe iods o indi e ence be ween
immedia e adop ion and delay equi es he p ospec o ecei ing news ha makes con-
sume s (ins an aneously) go om being willing o adop o being unwilling o adop ;
b eakdowns ha e his e ec , bu b eak h oughs do no .
We highligh wo implica ions o ou analysis. Sec ion 4.1 shows ha , depending
on he in o ma ional en i onmen , ou model gene a es wo commonly documen ed
adop ion cu es (e.g., Hoye , MacInnis, and Pie e s (2012), Keillo (2007)). Bad news
equilib ia wi h ∗
1<
∗
2lead o he leading empi ical pa e n o S-shaped adop ion: Ab-
sen b eakdowns, he sha e o adop e s inc eases con exly h oughou he pa ial adop-
ion phase ( ∗
1, ∗
2), as con ex g ow h ensu es ha , despi e becoming inc easingly op i-
mis ic, consume s emain indi e en be ween adop ing and delaying; du ing he imme-
dia e adop ion phase om ∗
2on, adop ion is conca e, e lec ing he g adual deple ion
o he popula ion. In con as , he all-o -no hing s uc u e o good news equilib ia (o
bad news equilib ia wi h ∗
1= ∗
2) leads o pu ely conca e adop ion cu es.
Sec ion 4.2 conside s inc eases in he po en ial o social lea ning. P oposi ion 1es-
ablishes a sa u a ion e ec : I lea ning is ia bad news and he equilib ium ea u es
pa ial adop ion, hen such inc eases a e (ex an e) wel a e-neu al. Indeed, hey a e bal-
anced ou by an expansion o he pe iod ( ∗
1, ∗
2)o in o ma ional ee- iding, which slows
down he adop ion o (bo h good and bad) p oduc s and has a non-mono onic e ec on
he speed o lea ning. Mo e s ongly, wi h he e ogeneous consume s, inc eased oppo -
uni ies o social lea ning can be Pa e o-ha m ul (Rema k 1). By con as , in en i on-
men s whe e equilib ium is all-o -no hing, inc easing he po en ial o social lea ning is
(essen ially) always s ic ly bene icial and speeds up lea ning a all imes.
2Uniqueness is in e ms o agg ega e adop ion beha io .
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1508 F ick and Ishii Theo e ical Economics 19 (2024)
1.1 Rela ed li e a u e
We s udy a model o inno a ion adop ion wi h endogenous iming and social lea ning
om public in o ma ion. Rela ed in o ma ional ex e nali ies and s a egic delay incen-
i es a e analyzed in he li e a u e on obse a ional lea ning wi h endogenous iming;3
see, e.g., Chamley and Gale (1994) and, mo e closely ela ed, Mu o and Välimäki (2011),
whe e playe s p i a ely ob ain Poisson signals abou he quali y o a isky p ojec a a
ixed exogenous a e un il hey choose o i e e sibly exi o a sa e ou side op ion. A key
di e ence is ha in his li e a u e, playe s hold p i a e in o ma ion abou he s a e and
d aw in e ences om o he s’ ac ions, whe eas news in ou model is public and de i ed
om p e ious adop e s’ expe iences. In o ma ion agg ega es in andom bu s s in hese
models a he han smoo hly as in ou se ing, and he a o emen ioned pape s do no
de i e adop ion cu es o s udy how hey a e shaped by he in o ma ional en i onmen .
Ou public lea ning model builds on he amewo k o s a egic expe imen a ion
wi h exponen ial bandi s, o igina ing wi h Kelle , Rady, and C ipps (2005)andKelle
and Rady (2010,2015) ( o a su ey, see Hö ne and Sk zypacz (2017)). We depa in wo
main ways. Fi s , we s udy i e e sible adop ion (i.e., exi o he isky a m), a he han
allowing o con inuous back-and- o h swi ching. Second, we assume a con inuum
o agen s, who each ha e a negligible in luence on public in o ma ion. These depa -
u es en ail a quali a i e di e ence be ween bad and good news lea ning— he p esence
s. absence o pa ial adop ion egions— ha has obse able implica ions o adop ion
cu es and is absen in he a o emen ioned pape s, whe e he symme ic Ma ko equi-
lib ium ea u es a egion o pa ial adop ion/mixing unde bo h bad and good news.4
Ano he implica ion o hese depa u es is ha , unlike s a egic expe imen a ion, ou
se ing does no ea u e an “encou agemen e ec ,” i.e., an incen i e o inc ease cu en
expe imen a ion o d i e up belie s and induce mo e u u e expe imen a ion by o he s.
This yields new compa a i e s a ics ha isola e he impac o in o ma ional ee- iding:
Fo example, in he bad news en i onmen o Kelle and Rady (2015), an inc ease in he
numbe o playe s o signal in o ma i eness makes playe s mo e willing o expe imen
a pessimis ic belie s, whe eas he sa u a ion e ec in P oposi ion 1 elies on he oppo-
si e e ec . Mo e ecen ly, Laiho, Mu o, and Salmi (2024) s udy in o ma ional ee- iding
incen i es in a ela ed model o collec i e expe imen a ion wi h i e e sible adop ion
and a con inuum o (he e ogeneous) agen s, ocusing, howe e , on B ownian news and
lea ning om he s ock a he han he low o adop e s.5
A la ge li e a u e in economics, ma ke ing, and sociology seeks o explain why in-
no a ions di use g adually and why S-shaped (and o a lesse ex en conca e) adop ion
3A la ge li e a u e s udies obse a ional lea ning/inno a ion adop ion wi h exogenous iming (e.g.,
Bane jee (1992), Bikhchandani, Hi shlei e , and Welch (1992), Smi h and Sø ensen (2000), He e a and
Hö ne (2013), Boa d and Meye - e -Vehn (2021)). S a egic delay incen i es wi hou social lea ning a e
a he cen e o he li e a u e on wa s o a i ion (e.g., Mayna d Smi h (1974), Fudenbe g and Ti ole (1986),
Ande son, Smi h, and Pa k (2017)).
4Bona i and Hö ne (2017) s udy a di e en depa u e—unobse able ac ions—and ind ha his also
leads o he symme ic equilib ium unde bad s. good news being in mixed s. pu e s a egies.
5Fajgelbaum, Schaal, and Tasche eau-Dumouchel (2017) s udy s a egic in es men iming by a con in-
uum o agen s whose in es men p oduces Gaussian public signals abou an e ol ing s a e. They show ha
his gene a es sel - ein o cing episodes o high unce ain y and low in es men .
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Theo e ical Economics 19 (2024) Inno a ion adop ion 1509
pa e ns a e p e alen . We ela e o o he lea ning-based models o hese phenomena.6
Unlike exis ing wo k ha ocuses on social lea ning by myopic consume s (e.g., Young
(2009)) o o wa d-looking lea ning om exogenous signals (e.g., Jensen (1982)), we con-
side a model o o wa d-looking social lea ning. This allows us o p o ide a pu ely in-
o ma ional explana ion o obse ed adop ion pa e ns, whe eas models wi h myopic
consume s o exogenous signals equi e speci ic o ms o consume he e ogenei y o
gene a e S-shaped adop ion.7Ou sa u a ion e ec also hinges on he combina ion o
o wa d-looking incen i es and social lea ning, as unde myopic o exogenous lea ning,
a g ea e ease o in o ma ion ansmission is always bene icial.
Ou ocus on he in o ma ional de e minan s o inno a ion adop ion con as s wi h
wo k ha combines in o ma ional and payo ex e nali ies. Rob (1991)modelsen y
in o a new ma ke , whe e he cu en numbe o i ms in he ma ke in luences no only
en an s’ lea ning abou a demand pa ame e , bu also hei p o i s ia he ma ke p ice.
Rela ed o ou bad news equilib ium, equilib ium en y is pinned down by a ze o p o i s
condi ion and is lowe han socially op imal. He does no s udy how he in o ma ional
en i onmen a ec s en y dynamics o p o ide condi ions o S-shaped g ow h, bo h o
which would also depend on he in e se demand unc ion. Be gemann and Välimäki
(1997) ob ain S-shaped adop ion as a esul o duopolis ic compe i ion be ween an es-
ablished and a new selle in a model wi h e e sible adop ion and lea ning on bo h
he buye and he selle side. Ini ial adop ion o he new p oduc exceeds he social
op imum in hei model. Laiho and Salmi (2018) build on ou model by inco po a ing
monopoly p icing and consume he e ogenei y.
2. Model
2.1 The game
Time ∈R+is con inuous. A ime =0, an inno a ion o unknown quali y θ∈{G=
1, B=−1}and o unlimi ed supply is eleased o a con inuum popula ion o po en ial
consume s o mass N0∈R>0. Consume s a e ex an e iden ical. They ha e a common
p io p0∈(0, 1) ha θ=G, hey a e o wa d-looking wi h common discoun a e >0,
and hey ha e he same ac ions and payo s, as speci ied below.
A each ime , consume s ecei e s ochas ic oppo uni ies o adop he inno a ion.
Adop ion oppo uni ies a e gene a ed independen ly ac oss consume s and his o ies
acco ding o a Poisson p ocess wi h exogenous a i al a e ρ>0.8Gi en an adop ion
6Non-in o ma ional models ( o su eys, see Bap is a (1999), Ge oski (2000)) include “epidemic” models
(e.g., Mans ield (1961), Bass (1969)), “p obi ” models o he e ogeneously e ol ing bene i s o adop ion (e.g.,
Da ies (1979)), and models o pu e payo ex e nali ies (e.g., Jo ano ic and Lach (1989), Fa ell and Salone
(1986)). Woli zky (2018) con as s adop ion le els o cos -sa ing s. ou come-imp o ing inno a ions in a
model o lea ning om o he s’ ou comes. Che and Hö ne (2018) ake a mechanism design app oach o
incen i izing social lea ning abou an inno a ion.
7In hose models, agen s adop i and only i hei belie s exceed a cu o . This p ecludes egions o con-
ex adop ion wi h iden ical agen s, ins ead equi ing speci ic dis ibu ions o he e ogeneous p io s/ as es.
8In he con ex o ou mo i a ing examples, s ochas ic adop ion oppo uni ies may ep esen , e.g., con-
enien imes o ake o wo k o unde go an elec i e su ge y o a ee e ening o wa ch a mo ie. Sec ion 5
discusses he case when ρ→∞.
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1510 F ick and Ishii Theo e ical Economics 19 (2024)
oppo uni y, a consume mus choose whe he o adop he inno a ion (a =1) o o
wai (a =0). I a consume adop s, he ecei es an expec ed lump sum payo o E [θ],
condi ioned on in o ma ion a ailable up o ime , and d ops ou o he game.9I he
consume chooses o wai o does no ecei e an adop ion oppo uni y a , he ecei es
a low payo o 0 un il his nex adop ion oppo uni y, whe e he aces he same decision
again.
2.2 Lea ning
O e ime, consume s obse e public signals ha con ey in o ma ion abou he qual-
i y o he inno a ion. We employ a a ia ion o he Poisson lea ning models used in he
li e a u e on s a egic expe imen a ion. Le n deno e he low o o consume s newly
adop ing he inno a ion a ime , which we de ine mo e p ecisely in Sec ion 2.3.Con-
di ional on he quali y o he inno a ion being θ, public signals a i e acco ding o an
inhomogeneous Poisson p ocess wi h a i al a e εθ+λθn ,whe eλθ>0andεθ≥0a e
exogenous pa ame e s ha depend on θ.
The signal p ocess summa izes news e en s ha a e gene a ed om wo sou ces.
Fi s , he social lea ning e m λn ep esen s news gene a ed endogenously, based on
he expe iences o o he consume s. I cap u es a low n o new adop e s each gene a -
ing signals a a e λ.10 Thus, he g ea e he low o consume s adop ing he inno a ion
a , he mo e likely i is o a signal o a i e a ; hence, he absence o a signal a is
mo e in o ma i e he la ge is n . Second, we also allow o (bu do no equi e) signals
o a i e a a ixed exogenous a e ε, ep esen ing in o ma ion gene a ed independen ly
o consume s’ beha io (e.g., by wa chdog agencies o p o essional c i ics).
As in many applica ions o Poisson lea ning, we ocus o ac abili y on pe ec news
p ocesses, whe e a single signal p o ides conclusi e e idence o he quali y o he in-
no a ion. Quali a i ely, he e is a na u al dis inc ion be ween wo ypes o news en i-
onmen s. Lea ning is ia pe ec bad news ( o sho , bad news)i εG=λG=0and
εB=ε≥0, λB=λ>0; ha is, he a i al o a signal (a b eakdown)isconclusi ee i-
dence ha he inno a ion is bad. Lea ning is ia pe ec good news ( o sho , good news)
i εB=λB=0andεG=ε≥0, λG=λ>0; ha is, a signal a i al (a b eak h ough)iscon-
clusi e e idence ha he inno a ion is good. The na u e o he news en i onmen may
be in luenced by whe he a bad o good quali y inno a ion is mo e likely o gene a e
newswo hy (e.g., ex eme) payo ealiza ions. Fo example, an unsa e medical p oce-
du e may cause se ious complica ions ha a e widely epo ed, bu a sa e p ocedu e
ha pe o ms as in ended may no lead o newswo hy ou comes.11 Al e na i ely, he
9I e e sible adop ion is na u al o inno a ions such as medical p ocedu es o mo ies, o which “con-
sump ion” is ypically a one- ime e en , o o echnologies wi h la ge swi ching cos s.
10We ob ain quali a i ely simila esul s when he social lea ning componen a ime is aken o depend
on he s ock, 
0nsds, a he han he low o adop e s a .SeeSec ion5.
11Mo e gene ally, suppose payo s o he quali y θinno a ion a e d awn (independen ly ac oss con-
sume s) om cumula i e dis ibu ion unc ion Fθ,whe e∞
−∞ ξdF
θ(ξ)=θ. Suppose payo ealiza ions
ξa e newswo hy i and only i ξ≤ξo ξ≥ξ o some “ex eme” low and high payo s ξ < ξ, and ha
newswo hy payo s gene a e public signals a some a e. Bad news lea ning assumes FB(ξ)>0=FG(ξ)
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Theo e ical Economics 19 (2024) Inno a ion adop ion 1511
news en i onmen may e lec epo ing p ac ices o he a ailable social lea ning sys-
ems. Fo example, se e al mo ie e iew agg ega o and s eaming si es p o ide “bes
o ” lis s o new eleases wi h he highes use a ings, bu do no display “wo s o ” lis s.
Quan i a i ely, we use 0:=λN0as a simple measu e o he po en ial o social lea n-
ing in he economy, summa izing bo h he likelihood λwi h which indi idual adop e s’
expe iences ind hei way in o he public domain and he size N0o he popula ion ha
can con ibu e o and access he common pool o in o ma ion.
Unde bad news, consume s’ pos e io on θ=Gpe manen ly jumps o 0 a he i s
b eakdown, while unde good news, consume s’ pos e io on θ=Gpe manen ly jumps
o 1 a he i s b eak h ough. Le p deno e consume s’ no-news pos e io , i.e., he belie
a ha θ=Gcondi ional on no signals ha ing a i ed on [0, ). Gi en a low o adop e s
(n ), Bayesian upda ing implies12
p =⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
p0
p0+(1−p0)e−
0(ε+λns)ds unde bad news
p0e−
0(ε+λns)ds
p0e−
0(ε+λns)ds +(1−p0)
unde good news.
(1)
In pa icula , i (n )is con inuous in on some open in e al, hen on his in e al (p )
e ol es acco ding o he o dina y di e en ial equa ion (ODE)
˙
p =(ε+λn )p (1−p )unde bad news
−(ε+λn )p (1−p )unde good news.
No e ha he no-news pos e io is con inuous. Mo eo e , i is inc easing unde bad
news and dec easing unde good news.
2.3 Equilib ium
Ou in e es is in he agg ega e adop ion dynamics o he popula ion. Thus, ou equi-
lib ium concep akes as i s p imi i e he agg ega e low (n )o new adop e s and does
no explici ly model indi idual consume s’ beha io . Gi en ou ocus on pe ec news
p ocesses, incen i es a e non i ial only in he absence o signals: Unde bad news, no
consume s adop a e a b eakdown, while unde good news, all emaining consume s
adop a hei i s oppo uni y a e a b eak h ough. We hence o h deno e by n he
low o new adop e s a condi ional on no signals up o ime and we de ine equilib-
ium in e ms o his quan i y.
Cap u ing ha agg ega e adop ion is p edic able wi h espec o he public news
p ocess, we equi e (n ) o be a de e minis ic unc ion o ime. We conside all such
unc ions ha a e easible; ha is, (n )is igh -con inuous in and n ∈[0, ρN ] o all
∈R+,whe eN :=N0−
0nsds deno es he mass o consume s emaining in he game
and FB(ξ)=FG(ξ)=1, i.e., bad inno a ions some imes gene a e ex eme low payo s, bu nei he good
no bad inno a ions gene a e ex eme high payo s. Good news lea ning assumes FB(ξ)=FG(ξ)=0 and
FB(ξ)=1>F
G(ξ).
12Sec ion 2.3 imposes measu abili y on (n ),so heexp essionsin(1) a e well de ined.
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1512 F ick and Ishii Theo e ical Economics 19 (2024)
a ime . Imposing n ≤ρN ensu es ha a each ,n is consis en wi h all emaining
N consume s independen ly ecei ing adop ion oppo uni ies a a e ρ. Any easible
adop ion low (n )induces a no-news pos e io (p ) ia (1).
In equilib ium, we equi e ha a each ,n is consis en wi h op imal beha io
by he emaining N o wa d-looking consume s: Consume s who ecei e an adop-
ion oppo uni y a conside he expec ed payo o adop ing immedia ely, which is
u :=2p −1 absen news, and op imally ade his o agains he alue o wai ing, ak-
ing in o accoun ha u u e adop ion e ol es acco ding o p ocess (n ).
Fo mally, de ine he alue o wai ing (W )associa ed wi h p ocess (n ) o be he so-
lu ion o he ollowing Bellman equa ion a each .13 Unde bad news,
W =∞
ρe−( +ρ)(s− )p +(1−p )e−s
(ε+λnk)dk
 
p ob. o no b eakdown
in [ ,s)
max{us,Ws}ds;
ha is, W is he expec ed discoun ed payo o wai ing un il he nex s ochas ic adop ion
oppo uni y s, and hen adop ing a his oppo uni y i and only i (i) he e has been no
b eakdown and (ii) a he upda ed belie ps, he expec ed payo o adop ing usexceeds
henew alue owai ingWs.
Unde good news,
W =∞
ρe−( +ρ)(s− )1−p +p e−s
(ε+λnk)dk
 
p ob. o no b eak h ough
in [ ,s)
max{us,Ws}
+p 1−e−s
(ε+λnk)dk
 
p ob. o b eak h ough
in [ ,s)ds;
ha is, W is he expec ed discoun ed payo o wai ing un il he nex adop ion oppo -
uni y s, and adop ing a his oppo uni y i ei he (i) he e has been no b eak h ough
and, a he upda ed belie ps, he expec ed payo o adop ing usexceeds he new alue
o wai ing Ws, o (ii) he e has been a b eak h ough.
De ini ion 1. An equilib ium is a easible adop ion low (n )such ha
(i) W ≥u o all such ha n <ρN
(ii) W ≤u o all such ha 0 <n
.
Condi ion (i) says ha i some consume s who ecei e an adop ion oppo uni y a
decide no o adop , hen he alue o wai ing W mus weakly exceed he expec ed
payo o immedia e adop ion u . Simila ly, (ii) equi es ha i some consume s adop
a ime , hen he alue o wai ing mus be weakly less han he payo o immedia e
13A unique solu ion exis s by s anda d a gumen s (e.g., Theo em 3.3 in S okey, Lucas, and P esco
(1989)).
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Theo e ical Economics 19 (2024) Inno a ion adop ion 1519
Figu e 2. Le : The S-shaped adop ion cu e unde bad news condi ional on no b eakdowns
( ∗
1=0). Righ : Conca e adop ion cu es unde good news (blue =b eak h ough be o e ∗;yel-
low =b eak h ough a e ∗;pink=bad quali y).
Co olla y 1. Bad News. In he unique equilib ium o Theo em 1,A =0 o 0≤ <
∗
1,
A is s ic ly inc easing and con ex in o ∗
1≤ <
∗
2,andA is s ic ly inc easing and
conca e in o ≥ ∗
2. I he i s b eakdown occu s a ime , adop ion ceases om hen
on.
Good News. In he unique equilib ium o Theo em 2,A =1−e−ρ is s ic ly inc eas-
ing and conca e o all <
∗. I he e is a b eak h ough p io o ∗, henA =1−e−ρ o
all . I he i s b eak h ough occu s a s>
∗(which equi es ε>0), hen adop ion comes
o a empo a y s ands ill be ween ∗and s, and o all ≥s,A is s ic ly inc easing and
conca e, and is gi en by 1−e−ρ( ∗+ −s).
Thus, in bad news ma ke s (Figu e 2, le ), he adop ion cu e exhibi s an S-shaped
(i.e., con ex–conca e) g ow h pa e n whene e ∗
1<
∗
2, whe e con ex g ow h coincides
wi h he pa ial adop ion egion ( ∗
1, ∗
2). By con as , in good news ma ke s (Figu e 2,
igh ), adop ion p oceeds in (up o wo) conca e bu s s. Conca e adop ion cu es also
a ise in bad news ma ke s wi h e y op imis ic and impa ien consume s o li le po en-
ial o social lea ning (so ha ∗
1= ∗
2by Lemma 2).
The ac ha hecon exg ow hpe iodo A unde bad news coincides wi h he
pa ial adop ion egion ( ∗
1, ∗
2)is ied o consume indi e ence in his egion. Absen
b eakdowns, consume s g ow inc easingly op imis ic abou he quali y o he inno a-
ion, which inc eases hei oppo uni y cos o delaying adop ion. To main ain indi -
e ence, he bene i o delaying adop ion mus hen also inc ease o e ime. This is
achie ed by inc easing he a i al a e o u u e b eakdowns, which imp o es he odds
ha wai ing will allow consume s o a oid he bad p oduc . Since he a i al a e o in-
o ma ion is inc easing in he low n o new adop e s, his means ha n mus s ic ly
inc ease h oughou ( ∗
1, ∗
2), i.e., ha A is con ex.25 By con as , he conca e g ow h
egions unde bo h bad and good news simply e lec he g adual deple ion o he pop-
ula ion when all consume s adop immedia ely upon an oppo uni y.26
25This a gumen o con ex g ow h does no ely on he linea i y o λn ; i emains alid as long as he
a e a which he bad p oduc gene a es b eakdowns a is inc easing in n .
26I he e is an in low o new consume s o i =γN a all (i.e., he popula ion size g ows exponen ially a
a e γabsen adop ion), hen i can be shown ha adop ion is e en ually conca e i and only i he g ow h
a e γis less han he a e ρo s ochas ic adop ion oppo uni ies.
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1520 F ick and Ishii Theo e ical Economics 19 (2024)
As discussed in he In oduc ion, S-shaped adop ion is documen ed o many in-
no a ions. Ou model complemen s exis ing explana ions (see Sec ion 1.1)byiden-
i ying a pu ely in o ma ional channel o his egula i y: I he e is a high enough
chance ha p e ious adop e s’ expe iences may e eal nega i e in o ma ion abou
he inno a ion and consume s a e o wa d-looking, hen S-shaped adop ion can a ise
due o some consume s s a egically delaying adop ion. This channel may be es-
pecially na u al o inno a ions whose in oduc ion is accompanied by subs an ial
sa e y conce ns, as may plausibly be he case o ou mo i a ing example o new
medical p ocedu es, whe e S-shaped adop ion pa e ns a e indeed commonly docu-
men ed.27
Though less p e alen han S-shaped cu es, conca e adop ion is ano he leading
pa e n documen ed in he ma ke ing li e a u e (e.g., Keillo (2007), pp. 51–61), wi h
leisu e-enhancing inno a ions such as mo ies, books, and games as examples. While
ou model abs ac s away om many impo an p oduc -speci ic o ces, Co olla y 1sug-
ges s some ac o s ha could con ibu e o conca e adop ion. In pa icula , high le els
o consume impa ience o op imism, o i social lea ning in hese ma ke s is p edomi-
nan ly ia good news signals o hei absence (as Sec ion 2.2 sugges ed could be d i en
by ea u es o he ele an e iew pla o ms).
4.2 The e ec o inc eased oppo uni ies o social lea ning
Nex , we conside an inc ease in he po en ial o social lea ning 0:=λN0, cap u ing
ei he a g ea e ease o in o ma ion ansmission (e.g., due o he in oduc ion o new
social ne wo king pla o ms) o a la ge communi y o consume s. We ask how his a -
ec s wel a e, lea ning, and adop ion dynamics. Again, in o ma ional ee- iding in he
o m o pa ial adop ion has impo an implica ions. Indeed, unde bad news, an econ-
omy’s abili y o ha ness i s po en ial o social lea ning is subjec o a sa u a ion e ec :I
he equilib ium ea u es pa ial adop ion, hen u he inc eases in he po en ial o so-
cial lea ning a e wel a e-neu al, cause lea ning o slow down o e ce ain pe iods, and
dec ease adop ion le els a all imes.
Fo mally, we ix all o he pa ame e s and s udy he e ec o inc easing 0on ex an e
equilib ium wel a e W0(0), no-news pos e io s p0
, and ex an e expec ed adop ion le -
els A (0,G)and A (0,B)condi ional on good and bad quali y, espec i ely. We as-
sume ha he o iginal po en ial o social lea ning 0is such ha he e is pa ial adop-
ion, i.e., ∗
1(0)<
∗
2(0); unde he condi ions in Lemma 2, his is he case whene e 0
is la ge enough.
P oposi ion 1. Conside lea ning ia bad news. Fix ,ρ,ε,andp0.I 0is such ha
∗
1(0)<
∗
2(0), hen an inc ease in he po en ial o social lea ning o ˆ
0>
0has he
ollowing e ec :
27See, e.g., he adop ion da a o ba ia ic su ge y in Buchwald and Oien (2009,2013).
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Theo e ical Economics 19 (2024) Inno a ion adop ion 1521
(i) Wel a e Neu ali y. We ha e W0(ˆ
0)=W0(0).
(ii) Non-Mono onici y o Lea ning. The e exis s >
∗
2(0)such ha
⎧
⎪
⎪
⎨
⎪
⎪
⎩
p0
=pˆ
0
i ≤ ∗
2(0)(lea ning is equally as unde 0and ˆ
0)
p0
>pˆ
0
i ∗
2(0)< < (lea ning is slowe unde ˆ
0)
p0
<pˆ
0
i > (lea ning is as e unde ˆ
0).
(iii) Slowdown o Adop ion. Fo all and θ=B,G, we ha e A (0,θ)≥A (ˆ
0,θ),
wi h s ic inequali y o all >
∗
1(0).
We p o e P oposi ion 1in Appendix A.6. The idea behind (i) is as ollows. Since
he equilib ium ea u es pa ial adop ion a 0, he same is ue when he po en ial o
social lea ning inc eases o ˆ
0. Mo eo e , bo h he ime ∗
1a which adop ion begins and
he pos e io p ∗
1a ∗
1a e he same unde 0and ˆ
0.28 Since consume s s ic ly p e e
o delay a all <
∗
1, and a e indi e en be ween delaying and adop ing a ∗
1,exan e
wel a e unde bo h 0and ˆ
0 hen co esponds o he expec ed payo o wai ing un il
∗
1and adop ing a ∗
1absen b eakdowns. Thus, W0(ˆ
0)=W0(0).29
This wel a e neu ali y esul con as s wi h he coope a i e benchma k whe e con-
sume s coo dina e on socially op imal adop ion le els. In he la e case, inc eased op-
po uni ies o social lea ning a e s ic ly bene icial and o any p0>1
2, he i s -bes
(comple e in o ma ion) payo o ρ
+ρp0can be app oxima ed in he limi as 0→∞.30
The esul also con as s wi h myopic social lea ning o o wa d-looking exogenous
lea ning, whe e wel a e necessa ily inc eases in esponse o mo e in o ma i e signals
(e en i consume s a e he e ogeneous).31
Poin s (ii) and (iii) u he illumina e he o ces behind wel a e neu ali y. By (ii), an
inc ease in 0a ec s lea ning dynamics in a non-mono onic manne . Thus, he im-
pac on a consume ’s expec ed payo a ies wi h he ime a which he ob ains his i s
adop ion oppo uni y. I ≤ ∗
2(0), his expec ed payo is he same unde 0and ˆ
0.I
∈( ∗
2(0), ), he is wo se o unde ˆ
0, because in case he inno a ion is bad, he is less
likely o ha e ound ou by hen han unde 0.32 Finally, i > , he is be e o unde
ˆ
0. Depending on ˆ
0, adjus s endogenously o balance ou he bene i s, which a i e
a imes a e , wi h he cos s incu ed a imes ( ∗
2(0), ).
28Indeed, as we saw in Sec ion 3.2, ∗
1is he i s ime a which he pos e io exceeds he h eshold p=
ε+
ε+2 and lea ning up o ∗
1is pu ely ia he exogenous news sou ce.
29Rela ed wel a e neu ali y esul s can a ise in mixed equilib ia in o he games; e.g., in ce ain s a ic
public goods p o ision games, he equilib ium wel a e/p o ision p obabili y o he public good can be
independen o he numbe o playe s.
30F ick and Ishii (2023) (Supplemen C) show he coope a i e benchma k is all-o -no hing, wi h no ( esp.
immedia e) adop ion below ( esp. abo e) a cu o belie pSO. Equilib ium adop ion displays wo ine icien-
cies: (i) i s a s oo la e (pSO <p
∗
1); (ii) once i s a s i is ini ially oo low (i ∗
1<
∗
2).
31To de ine ex an e wel a e wi h myopic consume s, assume ha consume s’ payo s a e discoun ed a
some a bi a y a e >0, bu consume s beha e myopically.
32The ac ha lea ning on ( ∗
2(0), )is slowe unde ˆ
0 han 0 e lec s ha he low o adop e s unde
0jumps up a ∗
2(0)(due o he ansi ion om he pa ial adop ion o immedia e adop ion egions),
whe eas unde ˆ
0, pa ial adop ion con inues un il ∗
2(ˆ
0)>
∗
2(0).
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1522 F ick and Ishii Theo e ical Economics 19 (2024)
Figu e 3. The e ec o inc eased oppo uni ies o social lea ning on he adop ion o a good
p oduc unde bad news ( ˆ
0>
0).
Simila ly, by (iii), an inc ease in 0s ic ly dec eases he adop ion A (0,G)o good
p oduc s (which is ha m ul), bu also dec eases he ex an e expec ed adop ion A (0,B)
o bad p oduc s (which is bene icial), and wel a e neu ali y ob ains because hese o ces
balance ou in equilib ium. Figu e 3illus a es ha he s ic slowdown in he adop-
ion o good p oduc s is due o wo e ec s: On he ex ensi e ma gin, he inc ease in
0pushes ou ∗
2, i.e., p olongs ee- iding; on he in ensi e ma gin, he inc ease d i es
down he g ow h a e o A a all <
∗
2(0).
Poin (iii) yields new es able implica ions ela i e o exis ing models o inno a ion
adop ion, sugges ing, o example, ha he ac ion o adop e s may g ow mo e slowly
in la ge communi ies. B oadly consis en wi h his, Bandie a and Rasul (2006)s udy
he adop ion o a new c op by a me s in Mozambique and ind ha a me s whose
ne wo k includes many adop e s may be less likely o adop ini ially hemsel es; hus, in
equilib ium, la ge ne wo ks o a me s should ea u e lowe pe cen ages o adop ion.33
Finally, he logic behind he sa u a ion e ec elies c ucially on pa ial adop-
ion/in o ma ional ee- iding. I unde bad news, 0is so low ha he e is no pa ial
adop ion in equilib ium, hen inc easing 0is s ic ly bene icial (see F ick and Ishii
(2023), Supplemen B.1). Likewise, he e is no sa u a ion e ec unde good news (see
F ick and Ishii (2023), Supplemen B.2): Since equilib ium adop ion is all-o -no hing,
inc easing he po en ial o social lea ning speeds up lea ning a all imes, which s ic ly
imp o es wel a e (p o ided ε>0).34
Rema k 1. P oposi ion 1shows ha inc easing 0is wel a e-neu al unde bad news.
Mo e s ongly, i consume s ha e he e ogeneous discoun a es, hen inc easing he po-
en ial o social lea ning can lead o Pa e o dec eases in ex an e wel a e. To illus a e,
33In ela ed wo k, Munshi (2004) inds ha in ice-g owing egions in India, whe e (due o mo e he -
e ogeneous plo condi ions) social lea ning is less easible han in whea -g owing a eas, a me s a e mo e
likely o expe imen wi h a new c op han hei coun e pa s in whea -g owing a eas.
34E en unde good news, inc easing 0inc eases wel a e only i his a ec s agen s’ p e e ence o adop-
ion s. delay a some his o ies. I ε=0, agen s weakly p e e o adop a all his o ies (no e u =W o all
≥ ∗as p =ps=1
2 o all ≥ ∗); hence, W0(0)=ρ
+ρ(2p0−1)is independen o 0.I ε>0, inc easing
0imp o es wel a e by leading mo e agen s o adop only a e a b eak h ough (u <W
o all >
∗and ∗
is dec easing in 0).
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Theo e ical Economics 19 (2024) Inno a ion adop ion 1523
suppose ε=0 and in oduce a single (mass 0) impa ien agen wi h discoun a e i>
in o he popula ion.35 Then, unde he assump ions in P oposi ion 1, inc easing λ o
ˆ
λis wel a e-neu al o he o iginal popula ion, bu makes his impa ien agen s ic ly
wo se o . Indeed, since he pa ien agen s a e ini ially indi e en be ween adop ing and
delaying, he impa ien agen adop s upon his i s oppo uni y absen b eakdowns in
bo h en i onmen s. By he non-mono onici y o lea ning in P oposi ion 1, he eexis s
some ime >
∗:= ∗
2(λ)such ha lea ning is s ic ly slowe unde ˆ
λbe ween ∗and ,
bu as e om ime on (and lea ning is equally as unde λ,ˆ
λup o ∗). Fo pa ien
agen s, he cos s o he ea ly decele a ion in lea ning and he bene i s o he la e accel-
e a ion exac ly balance ou . Howe e , he impa ien agen is hu , because ela i e o a
pa ien agen , he weigh s he ea ly cos s mo e hea ily han he la e bene i s. 
5. Concluding ema ks
This pape de elops a model o inno a ion adop ion when consume s a e o wa d-
looking and lea ning is social. Ou analysis isola es he e ec o pu ely in o ma ional
incen i es on agg ega e adop ion dynamics, lea ning, and wel a e. We highligh how
quali a i e and quan i a i e ea u es o he lea ning en i onmen shape hese incen-
i es, mos impo an ly by de e mining whe he o no he e is in o ma ional ee- iding
in he o m o pa ial adop ion. The p esence o absence o pa ial adop ion has obse -
able implica ions, sugges ing a no el channel o wo widesp ead adop ion pa e ns:
S-shaped and conca e cu es. Mo eo e , pa ial adop ion has impo an wel a e impli-
ca ions, en ailing ha inc eased oppo uni ies o social lea ning need no bene i con-
sume s and can be s ic ly ha m ul. Below, we b ie ly commen on some modi ica ions
and ex ensions o ou model.
Adop ion oppo uni ies. We assumed ha consume s ecei e adop ion oppo u-
ni ies a an a bi a ily la ge bu ini e Poisson a e ρ. This a oided echnical issues e-
la ed o de ining s a egies and con inua ion payo s when agen s can mo e con inu-
ously and adop ion p ocesses can ea u e mass poin s. The key quali a i e implica ion
o a ini e ρin bo h he bad and good news equilib ium is o gene a e conca e adop-
ion egions. To illus a e wha happens as ρ→∞, suppose ε=0andp0>1
2.Unde
bad news, he immedia e (i.e., conca e) adop ion phase disappea s as ρ→∞.InFig-
u e 1,limρ→∞ N∗(p)=0 o allp<1, so egion I anishes.36 Thus, by Theo em 1, he e
is an ini ial pa ial adop ion phase wi h low o adop e s n = (2p −1)
λ(1−p )and, in he limi as
ρ→∞, his phase con inues all he way un il he ini e ime ∗
2a which he popula ion
is ully deple ed.37 Unde good news, Theo em 2implies ha o any ini e ρ, equilib-
ium is all-o -no hing wi h cu o pos e io ps=1
2,bu asρ→∞, he ime ∗i akes o
35Sec ion 4.3 o F ick and Ishii (2015) ins ead conside ed a small mass o impa ien consume s.
36In ui i ely, i he e is any posi i e mass M o immedia e adop e s, hen i is s ic ly bene icial o wai an
ins an , as he cos o delaying he decision by an ins an is negligible (o o de d ) ela i e o he p obabili y
(1−p )(1−e−λM )o obse ing a b eakdown and a oiding he bad p oduc .
37To see why ∗
2is ini e, no e ha he ODE o pa ial adop ion implies n =
λ
2p0−1
e− p0−(2p0−1),which ends
o ∞by he ini e ime =1
ln p0
2p0−1. We also no e ha pa s (i) and (iii) o P oposi ion 1 emain alid as
ρ→∞, bu he non-mono onici y o lea ning in pa (ii) no longe a ises in he limi , because he accele -
a ion/decele a ion in lea ning occu s du ing he immedia e adop ion phase.
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1524 F ick and Ishii Theo e ical Economics 19 (2024)
each psabsen news ends o 0. Thus, he ini ial conca e adop ion egion app oxima es
a single mass poin o M0=1
λln p0
1−p0adop e s, whe e M0is such ha absen news, he
belie jumps down o ps.Hence,asρ→∞, he good news equilib ium app oxima es an
ini ial bu s o pa ial adop ion ( ollowed by a second bu s i he e is a b eak h ough),
bu a d awn-ou egion o pa ial adop ion can s ill only a ise unde bad news.38
Lea ning om he s ock o adop e s. In ou model, he social lea ning componen
o he signal a i al a e a ime ,λn , depends only on he low n o new adop e s. This
e ec i ely assumes ha adop e s can gene a e signals only once, a he ime o adop-
ion, app oxima ing se ings whe e he p obabili y o ecei ing signals abou he qual-
i y o he inno a ion (e.g., complica ions om a new medical p ocedu e) dep ecia es
apidly om he ime o adop ion. In con as , o some du able goods, i may be mo e
na u al o le signals a a i e a a e λS ,whe eS :=
0nsds ep esen s he s ock o
adop e s, cap u ing ha adop e s can gene a e signals epea edly o e ime. This would
p oduce simila esul s. Speci ically, simila a gumen s yield he exis ence and unique-
ness o equilib ium unde bo h bad and good news. The good news equilib ium is again
all-o -no hing, while, o app op ia e pa ame e s, he bad news equilib ium again ea-
u es a pa ial adop ion egion wi h beha io pinned down by he indi e ence condi ion
S = (2p −1)
λ(1−p )−ε
λ. Finally, he pa ial adop ion egion again exhibi s con ex g ow h in
adop ion le els.39
Mo e gene al signal p ocesses. As in many applica ions o Poisson lea ning, we ha e
ocused o ac abili y on conclusi e bad o good news signals. While a ca e ul in es i-
ga ion o mo e gene al signal p ocesses is beyond he scope o his pape , he analysis
ex ends eadily o hyb id en i onmen s wi h wo ypes o conclusi e Poisson signals:
bad news and good news signals wi h espec i e a i al a es λBn and λGn .Inpa ic-
ula , i λB>λ
G, he equilib ium is analogous o Theo em 1. Some o ou insigh s also
ex end beyond en i onmen s wi h conclusi e signals. Fo example, we no e ha pa ial
adop ion elies c ucially on he possibili y o news e en s ha igge disc e e down-
wa d jumps in belie s (al hough such e en s need no conclusi ely signal bad quali y).
Wi hou such e en s (e.g., when lea ning is based on inconclusi e good news Poisson
38I , ins ead, each consume ’s i s adop ion oppo uni y a i es a a e ρ<∞, bu subsequen adop ion
oppo uni ies a i e con inuously, he good news equilib ium is s ill all-o -no hing as in Theo em 2,excep
ha he cu o belie limρ→∞ ps=ε+
ε+2 is g ea e han ps(i ε>0). The bad news equilib ium is quali a i ely
unchanged: Unde sui able pa ame e s, he e is an ini ial pa ial adop ion egion wi h con ex adop ion
g ow h (which con inues un il he s ock o consume s who ha e ecei ed a i s adop ion oppo uni y is
deple ed); om hen on, he emaining consume s adop immedia ely a hei i s oppo uni y (leading o
conca e g ow h). Howe e , he non-mono onici y o lea ning in P oposi ion 1(ii) no longe a ises, as he
low o adop e s now ea u es a downwa d jump a he ansi ion om pa ial o immedia e adop ion.
39Indeed, as in Sec ion 3.2, indi e ence equi es he bene i o a oiding a bad p oduc when a b eakdown
occu s ((1−p )(λS +ε)) o equal he cos o delaying adop ion absen news ( (2p −1)). Since consume s
g ow mo e op imis ic absen news, his has wo implica ions h oughou he indi e ence egion: (i) belie s
p inc ease con exly, as he g ow h a e o p equals he ins an aneous p obabili y o a b eakdown ( ˙
p
p =
(1−p )(λS +ε)), which mus inc ease o e ime o balance ou he inc easing cos o delay; (ii) he s ock
o adop e s S =S(p )inc eases con exly as a unc ion o p , o ensu e ha b eakdowns a i e a a a e
ha coun e balances he con ex g ow h (wi h espec o p )o he a io (2p −1)
(1−p )be ween he cos o delay
and he p obabili y o acing a bad p oduc . Combining (i) and (ii), i ollows ha S —and,hence,adop ion
le els—inc eases con exly o e ime.
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Theo e ical Economics 19 (2024) Inno a ion adop ion 1525
o B ownian signals), a simila logic as in Sec ion 3.3 implies ha he e canno be con-
inuous egions o pa ial adop ion, because a consume who is willing o adop canno
ins an aneously acqui e decision- ele an in o ma ion (see F ick and Ishii (2023), Sup-
plemen D).40
Appendix:P oo s
A.1 P elimina y lemmas
The ollowing i e lemmas will be used h oughou he Appendix. Fo any easible adop-
ion low (n ),wedeno eby(W ) he co esponding no-news alue o wai ing and deno e
by (p ) he no-news pos e io , wi hou making explici he dependency on (n ).
Lemma A.1. Fo any easible adop ion low (n ), heco esponding(W )and (p )a e con-
inuous in .
The p oo is immedia e om he de ini ions o p and W in Sec ions 2.2 and 2.3.
Lemma A.2. Suppose ha (ns)is an equilib ium and ha W <2p −1 o some >0.
Then he e exis s ν>0such ha (Wτ)is con inuously di e en iable in τon he in e al
( −ν, +ν)and o all τ∈( −ν, +ν),
˙
Wτ= +ρ+(εG+λGρNτ)pτ+(εB+λBρNτ)(1−pτ)Wτ
−ρ(2pτ−1)−pτ(εG+λGρNτ)ρ
ρ+ .
P oo . Suppose W <2p −1 o some >0. Since (Wτ)and (pτ)a e con inuous in τ
(Lemma A.1), he e exis s ν>0such ha Wτ<2pτ−1 o allτ∈( −ν, +ν). Because
(ns)is an equilib ium, his implies ha nτ=ρNτ o all τ∈( −ν, +ν).Thus,nτis
con inuous a all τ∈( −ν, +ν).ThenWτis con inuously di e en iable in τ o all
τ∈( −ν, +ν),as
Wτ=
+ν
τ
ρe−(ρ+ )(s−τ)pτe−s
τ(εG+λGnx)dx −(1−pτ)e−s
τ(εB+λBnx)dxds
+e−( +ρ)( +ν−τ)pτe− +ν
τ(εG+λGnx)dx +(1−pτ)e− +ν
τ(εB+λBnx)dxW +ν
+
+ν
τ
ρe−(ρ+ )(s−τ)pτ1−e−s
τ(εG+λGnx)dxds
+e−( +ρ)( +ν−τ)pτ1−e− +ν
τ(εG+λGnx)dxρ
ρ+ .
40In con as , Laiho, Mu o, and Salmi (2024) ob ain pa ial adop ion/g adualism in a model wi h B ow-
nian lea ning om he s ock o adop e s and con inuous adop ion oppo uni ies.
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1526 F ick and Ishii Theo e ical Economics 19 (2024)
The de i a i e o Wτcan be compu ed using I o’s lemma o p ocesses wi h jumps. Gi en
pe ec Poisson lea ning, he de i a ion is simple and we p o ide i o comple eness. As
abo e, o any ∈(0, +ν−τ),wecan ew i eWτas
Wτ=
τ+
τ
ρe−(ρ+ )(s−τ)pτe−s
τ(εG+λGnx)dx −(1−pτ)e−s
τ(εB+λBnx)dxds
+e−( +ρ)pτe−τ+
τ(εG+λGnx)dx +(1−pτ)e−τ+
τ(εB+λBnx)dxWτ+
+
τ+
τ
ρe−( +ρ)(s−τ)pτ1−e−s
τ(εG+λGnx)dxds
+e−( +ρ)pτ1−e−τ+
τ(εG+λGnx)dxρ
ρ+ .
Since hisis ue o all∈(0, +ν−τ), he igh -hand side o his iden i y, which we
deno e R, is con inuously di e en iable wi h espec o and sa is ies d
dR≡0. Tak-
ing he limi as →0 and since ˙
Wτ=lim→0d
dτ Wτ+by con inuous di e en iabili y, we
hen ob ain
˙
Wτ= +ρ+(εG+λGnτ)pτ+(εB+λBnτ)(1−pτ)Wτ
−ρ(2pτ−1)−pτ(εG+λGnτ)ρ
ρ+ .
Plugging in nτ=ρNτyields he desi ed exp ession.
Lemma A.3. Suppose ha (nτ)is an equilib ium and ha W >2p −1 o some >0.
Then he e exis s ν>0such ha (Wτ)is con inuously di e en iable in τon he in e al
( −ν, +ν)and o all τ∈( −ν, +ν),
˙
Wτ= +pτεG+(1−pτ)εBWτ−pτεG
ρ
ρ+ .
P oo . The p oo ollows he same lines as ha o Lemma A.2. Lemma A.1 again
implies ha i W >2p −1, hen he e exis s ν>0such ha Wτ>2pτ−1 o all
τ∈( −ν, +ν). By he de ini ion o equilib ium, nτ=0 o allτ∈( −ν, +ν).
Hence, Wτsa is ies
Wτ=e− ( +ν−τ)pτe−εG( +ν−τ)+(1−pτ)e−εB( +ν−τ)W +ν
+pτ
+ν
τ
εGe−(εG+ )sρ
ρ+ ds
and, hus, is con inuously di e en iable in τ.
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Theo e ical Economics 19 (2024) Inno a ion adop ion 1527
To compu e he de i a i e, no e again ha o any ∈(0, +ν−τ),
Wτ=e− pτe−εG+(1−pτ)e−εBW ++pτ
τ+
τ
εGe−(εG+ )sρ
ρ+ ds.
Di e en ia ing bo h sides wi h espec o and aking he limi as →0,
˙
Wτ= +pτεG+(1−pτ)εBWτ−pτεG
ρ
ρ+ ,
as claimed.
Lemma A.4. Suppose (n )is an equilib ium unde bad news. Suppose ε>0o p0>1
2.
Then lim →∞ p =μ(ε,0,p0)and lim →∞ W =ρ
ρ+ (2μ(ε,0,p0)−1),whe e
μ(ε,0,p0):=⎧
⎨
⎩
1i ε>0
p0
p0+(1−p0)e−0i ε=0.
P oo . Suppose i s ha ε>0. Then i ially p →1as →∞. Since o any ,
ρ
ρ+ (2p −1)≤W ≤ρ
ρ+ , his implies ha lim →∞ W =ρ
ρ+ ,asclaimed.
Now suppose ε=0andp0>1/2. No e ha W ≤2p −1 o all . Indeed, suppose
W >2p −1 o some .I Ws>2ps−1 o alls≥ , henW =0, con adic ing W >2p −
1≥2p0−1>0. Thus, we can ind s> such ha Ws=2ps−1andWs>2ps−1 o all
s∈( ,s). This implies ns=0 o alls,and,hence,W =e− (s− )Ws=e− (s− )(2ps−1)=
e− (s− )(2p −1), again con adic ing W >2p −1>0.
Le N∗:=lim →∞ 
0nsds =sup 
0nsds ≤N0.Le p∗:=lim →∞ p =sup p .Fo any
ν>0, we can ind ∗such ha whene e >
∗, hene−λ
∗nsds >1−ν. Because 2p −1≥
W o all , we can hen w i e he alue o wai ing a all >
∗as
W =
∞

ρe−( +ρ)τp −(1−p )e−λτ
nsdsdτ
≤ρ
+ρp −(1−p )(1−ν).
By op imali y, W ≥ρ
ρ+ (2p −1) o all , so by combining, we ha e
ρ
ρ+ 2p∗−1≤lim
→∞ in W ≤lim
→∞ supW ≤ρ
+ρp∗−1−p∗(1−ν).
Since hisis ue o allν>0, i ollows ha
lim
→∞ W =ρ
+ρ2p∗−1,
which is s ic ly less han 2p∗−1, so o all su icien ly la ge we mus ha e 2p −1>
W .Then o all su icien ly la ge, we ha e n =ρN .Thus,N∗=N0and, he e o e,
p∗=μ(ε,0,p0).
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1528 F ick and Ishii Theo e ical Economics 19 (2024)
Lemma A.5. Suppose ha lea ning is ia bad news. Suppose ha ε=0and p0≤1
2.Then
he unique equilib ium sa is ies n =0 o all .
P oo . Suppose ha (ns)is an equilib ium and suppose, o a con adic ion, ha ∗
1:=
in { :n >0}<∞.Pick ≥ ∗
1such ha n >0. By igh -con inui y o (ns),weha enτ>0
o all τ> su icien ly close o . This implies
∞
∗
1
ρe−( +ρ)(s− )p ∗
1−(1−p ∗
1)e−s
∗
1λnkdkds > ρ
+ρ(2p ∗
1−1)≥2p ∗
1−1, (6)
whe e he second inequali y holds because p ∗
1=p0≤1
2. The in eg al on he le -hand
side is he expec ed payo a ime ∗
1 o adop ing a he i s oppo uni y in he u u e,
condi ional on no b eakdown ha ing occu ed p io o his oppo uni y. By op imali y
o he alue o wai ing, his is weakly less han W ∗
1.Hence,(6)impliesW ∗
1>2p ∗
1−1.
By con inui y o (Ws)and (ps), i ollows ha o all s≥ ∗
1su icien ly close o ∗
1,Ws>
2ps−1and,hence,ns=0, con adic ing he de ini ion o ∗
1.
This lea es n =0 o all as he only candida e equilib ium. In his case, W =0≥
2p0−1=2p −1 o all , so his is indeed an equilib ium.
A.2 P oo o Lemma 1
Good News. Suppose i s ha lea ning is ia good news.
S ep 1: W =2p −1=⇒ Wτ≥2pτ−1 o allτ≥ . Suppose W =2p −1a some
ime and suppose, o a con adic ion, ha a some ime s> ,weha eWs<2ps−1.
Le s∗:=sup{s<s
:Ws=2ps−1}.
By con inui y, s∗<s
,Ws∗=2ps∗−1, and Ws<2ps−1 o alls∈(s∗,s).Thenby
Lemma A.2, he igh -hand de i a i e o Ws−(2ps−1)a s∗exis s and sa is ies
lim
s↓s∗
˙
Ws−2˙
ps= (2ps∗−1)+ps∗(ε+λρNs∗)
ρ+ >0.
This implies ha o some s∈(s∗,s)su icien ly close o s∗,weha eWs>2ps−1, which
is a con adic ion.
S ep 2: W >2p −1=⇒ Wτ>2pτ−1 o allτ> . Suppose, o a con adic ion, ha
he e exis s s> such ha Ws=2ps−1. Le s∗:=in {s> :Ws=2ps−1}. By con inui y,
s∗> ,Ws∗=2ps∗−1, and Ws>2ps−1 o alls∈( ,s∗).No e ha ps∗≥1
2, because Ws∗is
bounded below by 0. Mo eo e , by Lemma A.3, he le -hand de i a i e o Ws−(2ps−1)
a s∗exis s and is gi en by
lim
s↑s∗
˙
Ws−2˙
ps= (2ps∗−1)+ps∗
ρ+ ε.
I ε>0, his is s ic ly posi i e, implying ha o some s∈( ,s∗)su icien ly close o s∗,
we ha e Ws<2ps−1, which is a con adic ion. I ε=0, hen o all s∈( ,s∗),weha e
ps∗=psand Ws=e− (s∗−s)Ws∗=e− (s∗−s)(2ps∗−1)≤2ps∗−1. Thus, Ws≤2ps−1, again
con adic ing Ws>2ps−1.
15557561, 2024, 4, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE4455 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 19 (2024) Inno a ion adop ion 1535
P oo . De ine
H :=p ∞
0(ε+λn +τ)e−(ετ+ +τ
λnsds)ρ
+ρe− τ dτ.
Thus, H ep esen s a consume ’s expec ed payo o wai ing a ime gi en ha om
on he adop s only i he e has been a b eak h ough and gi en ha he popula ion’s low
o adop ion ollows (ns). By op imali y o W ,wemus ha eH ≤W o all .Fo any
pos e io p∈(0, 1),le
H(p,0
):=p∞
0
εe−ετ ρ
+ρe− τ dτ =pερ
(ε+ )( +ρ).
Tha is, H(p,0
) ep esen s a consume ’s expec ed payo o wai ing a pos e io p,gi en
ha he adop s only once he e has been a b eak h ough and gi en ha b eak h oughs
a e only gene a ed exogenously.
No e ha by de ini ion o ∗,n >0 i and only i <
∗. This implies ha H(p ,0
)<
H i <
∗and H(p ,0
)=H =W i ≥ ∗;mo eo e ,2p −1≥W i <
∗and 2p −1≤
W i ≥ ∗. Finally, no e ha ps:=(ε+ )( +ρ)
2(ε+ )( +ρ)−ερ has he p ope y ha 2p−1≤H(p,0
)
i and only i p≤ps.
Combining hese obse a ions, i <
∗, hen2p −1≥W ≥H >H
(p ,0
),sop >
ps.I ≥ ∗, hen2p −1≤W =H(p ,0
),sop ≤ps,asclaimed.
A.6 P oo o P oposi ion 1
Fix ,ρ,ε,p0. Suppose 0is such ha ∗
1(0)<
∗
2(0). By he p oo s o Theo em 1
and Lemma 2, hismeans ha Condi ion1is sa is ied, p0<p
,and0> 0,whe e
0:=max{∗(p0),∗(p)}as in he p oo o Lemma 2.Conside anyˆ
0>
0.
A.6.1 P oo o pa (i) (wel a e neu ali y) W i e 1
0:=0and 2
0:=ˆ
0,wi hco e-
sponding cu o imes i
1and i
2, alue o wai ing Wi
, and no-news pos e io s pi
o
i=1, 2 (by he p oo o Theo em 1, hese quan i ies depend on λi,Ni
0only h ough i
0).
Since 1
1<
1
2and 2
0>
1
0> 0, Lemma 2implies 2
1<
2
2. Mo eo e , by he p oo o
Lemma A.7,weha emax{p0,p}=p1
1
1
=p2
2
1
. Because ni
=0 o all <
i
1 o bo h i=1, 2,
his implies ha 1
1= 2
1= 1.ThenW2
1=2p2
1−1=2p1
1−1=W1
1. Since he e is no adop-
ion un il 1,weha eWi
0=e− 1p 1
p0Wi
1 o i=1, 2, whence W1
0=W2
0,asclaimed.
A.6.2 P oo o pa (ii) (non-mono onici y o lea ning) We i s p o e he ollowing
lemma.
Lemma A.11. Suppose ha ˆ
0=ˆ
λˆ
N0>
0=λN0> 0, wi h co esponding equilib ium
lows o adop ion (ˆ
n )and (n ).Then
(i) ∗
1(0)= ∗
1(ˆ
0)
(ii) 0<
∗
2(0)<
∗
2(ˆ
0)
(iii) o all <
∗
2(0),λn =ˆ
λˆ
n .
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1536 F ick and Ishii Theo e ical Economics 19 (2024)
P oo . Fo (i), no e ha by he p oo o Lemma A.7, ime ∗
1unde bo h 0and ˆ
0is
pinned down by he condi ion max{p0,p}=p0
∗
1(0)=pˆ
0
∗
1(ˆ
0). Because up o ime ∗
1,
lea ning is pu ely exogenous unde bo h 0and ˆ
0, his implies ∗
1(0)= ∗
1(ˆ
0).
Fo (ii) and (iii), no e i s ha by Lemma 2,weha e ∗
2(ˆ
0), ∗
2(0)>0. Le ∗
2=
min{ ∗
2(ˆ
0), ∗
2(0)}. Then because ∗
1(0)= ∗
1(ˆ
0), he ODE in Co olla y A.1 implies ha
a all imes <
∗
2,weha ep0
=pˆ
0
=p . By Lemma A.6, his implies ha o all <
∗
2,
λn =ˆ
λˆ
n . (11)
No e ha (11) implies ha
 ∗
2=0− ∗
2
0
λn d < ˆ
0− ∗
2
0
ˆ
λˆ
n d =ˆ
 ∗
2.
Because p0
∗
2=pˆ
0
∗
2, Lemma A.7 implies ha ∗
2= ∗
2(0)<
∗
2(ˆ
0).F om hisand(11), i
is hen immedia e ha λn =ˆ
λˆ
n o all <
∗
2(0).
Now we p o e pa (ii) o P oposi ion 1. By Lemma A.11, ∗:= ∗
2(0)<
∗
2(ˆ
0),λn =
ˆ
λˆ
n ,andp0
=pˆ
0
o all ≤ ∗, which p o es he i s claim o pa (ii).
Fo he second claim o pa (ii), we no e ha he e exis s some ν>0such ha a all
imes ∈( ∗, ∗+ν),weha ep0
>pˆ
0
. To see his, we p o e he ollowing inequali y
o he equilib ium co esponding o 0:
lim
↑ ∗λn <lim
↓ ∗λn . (12)
Tha is, he e is a discon inui y in he equilib ium low o adop ion a ime ∗. Indeed,
because n =ρN o all ≥ ∗and by con inui y o N , easibili y implies ha lim ↑ ∗λn ≤
lim ↓ ∗λn . Suppose o a con adic ion ha lim ↑ ∗λn =lim ↓ ∗λn :=λn ∗.Thenλn ∗=
ˆ
λˆ
n ∗.Mo eo e , o all >
∗,weha eλn =ρ ∗e−ρ( − ∗), which is s ic ly dec easing in
. On he o he hand, ˆ
λˆ
n sa is ies
ˆ
λˆ
n =⎧
⎪
⎨
⎪
⎩
(2ˆ
p −1)
(1−ˆ
p )−εi ∈[ ∗, ∗
2(ˆ
0))
ρ ∗
2(ˆ
0)e−ρ( − ∗
2(ˆ
0)) i ≥ ∗
2(ˆ
0).
Thus, o ∈[ ∗, ∗
2(ˆ
0)),ˆ
λˆ
n is s ic ly inc easing in . This implies ha ˆ
λˆ
n >λn
o all
∈[ ∗, ∗
2(ˆ
0)).Hence,by(1), pˆ
0
∗
2(ˆ
0)>p
0
∗
2(ˆ
0), which by Lemma A.7 implies
ˆ
 ∗
2(ˆ
0)=∗pˆ
0
∗
2(ˆ
0)>
∗p0
∗
2(ˆ
0)>
∗
2(ˆ
0).
This yields ha o all ≥ ∗
2(ˆ
0),
ˆ
λˆ
n =ρe−ρ( − ∗
2(ˆ
0)ˆ
 ∗
2(ˆ
0)>ρe
−ρ( − ∗
2(ˆ
0) ∗
2(ˆ
0)=λn .
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Theo e ical Economics 19 (2024) Inno a ion adop ion 1537
Thus, ˆ
λˆ
n >λn
o all >
∗and, hence, pˆ
0
>p
0
o all >
∗. This implies Wˆ
0
∗>W0
∗,
which is a con adic ion, because we ha e
Wˆ
0
∗=2pˆ
0
∗−1=2p0
∗−1=W0
∗.
This p o es ha lim ↑ ∗λn <lim ↓ ∗λn .Hence,
lim
↓ ∗
ˆ
λˆ
n =lim
↑ ∗
ˆ
λˆ
n =lim
↑ ∗λn <lim
↓ ∗λn .
Thus, he e exis s some ν>0such ha ˆ
λˆ
n <λn
o all ∈[ ∗, ∗+ν). Toge he wi h he
ac ha p0
∗=pˆ
0
∗, his implies ha p0
>pˆ
0
o all ∈( ∗, ∗+ν), p o ing he second
claim.
Finally, o he hi d claim o pa (ii), obse e i s ha he e exis s some >
∗such
ha p0
=pˆ
0
. I no , hen by con inui y o belie s, p0
>pˆ
0
o all >
∗and we ha e
Wˆ
0
∗<W0
∗, again con adic ing Wˆ
0
∗=W0
∗=2p ∗−1. Then :=sup{s∈( ∗, ):p0
s>
pˆ
0
s}exis s, wi h >
∗by he second claim. Fu he , by con inui y, p0
=pˆ
0
,which
implies 
0λnsds =
0ˆ
λˆ
nsds. This yields  <ˆ
 , which implies ha ˆ
λˆ
n >λn
o all
> . Indeed, i ≥ ∗
2(ˆ
0), hisisob ious.I ∈( ∗, ∗
2(ˆ
0)), henwemus ha eλns<ˆ
λˆ
ns
o some s< , which implies ha λns<ˆ
λˆ
ns o all s∈(s, ∗
2(ˆ
0)), because Nis s ic ly
dec easing and ˆ
nis s ic ly inc easing on his domain. To see ha we also ha e λns<
ˆ
λˆ
ns o all s≥ ∗
2(ˆ
0), no e ha om he abo e, pˆ
0
∗
2(ˆ
0)>p
0
∗
2(ˆ
0), which as abo e implies
ha
ˆ
 ∗
2(ˆ
0)=∗pˆ
0
∗
2(ˆ
0)>
∗p0
∗
2(ˆ
0)>
∗
2(ˆ
0).
Hence, ˆ
λˆ
n >λn
o all > . Thus, in ei he case, pˆ
0
>p
0
o all > .
A.6.3 P oo o pa (iii) (slowdown o adop ion) Adop ion o Good P oduc s. By
Lemma A.11, ∗
1(0)= ∗
1(ˆ
0)=: ∗
1and λn =ˆ
λˆ
n o all ∈( ∗
1, ∗),whe e ∗:= ∗
2(0).
Then o all <
∗,
n
N0
=λn
0
=ˆ
λˆ
n
0
≥ˆ
λˆ
n
ˆ
0
=ˆ
n
ˆ
N0
,
wi h s ic inequali y o all ∈( ∗
1, ∗). The e o e, A (0,G)≥A (ˆ
0,G) o all <
∗,
wi h s ic inequali y o all ∈( ∗
1, ∗).
Finally no e ha o all ≥ ∗,n =ρN and so
A (0,G)=A ∗(0,G)+1−e−ρ( − ∗)1−A ∗(0,G)
A (ˆ
0,G)≤A ∗(ˆ
0,G)+1−e−ρ( − ∗)1−A ∗(ˆ
0,G),
whe e he second inequali y ollows om easibili y. Because A ∗(0,G)>A
∗(ˆ
0,G),
i ollows ha A (0,G)>A
(ˆ
0,G) o all >
∗
1,asclaimed.
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1538 F ick and Ishii Theo e ical Economics 19 (2024)
Adop ion o Bad P oduc s. Recall ha A (λ,N0,B)deno es he expec ed p opo ion
o adop e s a ime condi ional on θ=B. Tha is, le ing (n )deno e he associa ed
equilib ium, we ha e
A (λ,N0,B):=
0
(ε+λnτ)e−τ
0(ε+λns)dsτ
0
ns
N0dsdτ +e−
0(ε+λns)ds
0
ns
N0ds
=
0
nτ
N0e−τ
0(ε+λns)ds dτ,
whe e he inal equali y ollows om in eg a ion by pa s. Mo eo e , om he Ma ko-
ian desc ip ion o equilib ium in he p oo o Theo em 1, i is easy o see ha his
exp ession depends on λand N0only h ough 0=λN0, so we can deno e i by
A (0,B). Then we can assume wi hou loss o gene ali y ha 0and ˆ
0a e o he
o m 0=λN0and ˆ
0=ˆ
λN0, i.e., ha he wo en i onmen s ha e he same popula ion
size N0.
Le (n )and (ˆ
n )be he equilib ium unde λand ˆ
λ, espec i ely.Gi enana bi a y
s ic ly posi i e adop ion low (ms)and >0, no e ha he map
λ→
0
mτe−τ
0(ε+λms)ds dτ
is s ic ly dec easing in λ. Since ˆ
0>
0> 0,weha e ∗
1(0)= ∗
1(ˆ
0)=: ∗
1,andsowe
ge ha o all >0,
0
nτe−τ
0(ε+λns)ds dτ ≥
0
nτe−τ
0(ε+ˆ
λns)ds dτ, (13)
wi h s ic inequali y o all >
∗
1. We now show ha
0
nτe−τ
0(ε+ˆ
λns)ds dτ ≥
0
ˆ
nτe−τ
0(ε+ˆ
λˆ
ns)ds dτ.
Toge he wi h (13), his implies he desi ed conclusion ha A (ˆ
λN0,B)≤A (λN0,B) o
all >0, wi h s ic inequali y o all >
∗
1.
To p o e his, suppose o a con adic ion ha he e exis s some >0such ha
0
nτe−τ
0(ε+ˆ
λns)ds dτ <
0
ˆ
nτe−τ
0(ε+ˆ
λˆ
ns)ds dτ. (14)
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Theo e ical Economics 19 (2024) Inno a ion adop ion 1539
No e ha by he abo e esul o good p oduc s, N0Aτ(λ,G)=τ
0nsds ≥τ
0ˆ
nsds =
N0Aτ(ˆ
λ,G) o all τ≥0andso, o all ≥0,
0
εe−τ
0(ε+ˆ
λns)ds dτ ≤
0
εe−τ
0(ε+ˆ
λˆ
ns)ds dτ. (15)
Inequali ies (14)and(15) oge he imply
0
(ε+ˆ
λnτ)e−τ
0(ε+ˆ
λns)ds dτ <
0
(ε+ˆ
λˆ
nτ)e−τ
0(ε+ˆ
λˆ
ns)ds dτ.
This is equi alen o
1−e−
0(ε+ˆ
λns)ds<1−e−
0(ε+ˆ
λˆ
ns)ds,
which con adic s 
0nsds ≥
0ˆ
nsds.
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