Kang, Kee-Youn
A icle
Digi al cu ency and p i acy
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Kang, Kee-Youn (2024) : Digi al cu ency and p i acy, Theo e ical Economics,
ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 19, Iss. 1, pp. 131-167,
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Theo e ical Economics 19 (2024), 131–167 1555-7561/20240131
Digi al cu ency and p i acy
Kee-Youn Kang
School o Business, Yonsei Uni e si y
We de elop a mone a y model in which a p i a e company issues digi al cu ency
and uses paymen da a o es ima e consume s’ p e e ences. Selle s pu chase p e -
e ence in o ma ion o p oduce goods ha be e ma ch consume s’ p e e ences.
A monopoly a ises in he digi al cu ency indus y, and digi al cu ency is no is-
sued i he in la ion a e is su icien ly high. Due o ein o cing in e ac ions be-
ween he alue o p e e ence in o ma ion and ade olume, mul iple equilib ia
(wi h and wi hou digi al cu ency) can exis , depending on ma ke s uc u es o
mone a y exchanges. When le o ma ke o ces alone, socially e icien uses o
paymen da a may no occu .
Keywo ds. Digi al cu ency, p i acy, ansac ion da a, p e e ence in o ma ion,
s a egic complemen a i ies.
JEL classi ica ion. E12, E40, E50, G10.
1. In oduc ion
As ou economy has become mo e digi alized, elec onic paymen s ha e s eadily in-
c eased o e ecen decades (see S a ins (2017)). Al hough elec onic means o pay-
men (hence o h, E-money), such as debi ca ds, Alipay, and PayPal, di e om a-
di ional cash in many espec s, one impo an di e ence is p i acy: Cash e ains use
p i acy, while digi al cu ency ansac ions a e collec ed by he company ha ope a es
he elec onic paymen sys em. Paymen his o ies can indica e indi idual p e e ences
o ce ain i ems, and his p e e ence in o ma ion, combined wi h use s’ pe sonal in o -
ma ion, can be used o ma ke ing pu poses and o design be e goods ha a e mo e
ailo ed o consume s’ p e e ences. Thus, paymen his o y da a ha e comme cial alue
and hei impo ance has inc eased wi h ad ances in analy ical echnologies such as
machine lea ning.
Al hough economic s udies on digi al cu ency ha e eme ged ecen ly since a su ge
in he Bi coin p ice, he p ac ice o using paymen da a o digi al cu ency has ecei ed
ela i ely li le a en ion in academic a eas. The ollowing ques ions s ill need o be ad-
d essed: Unde which condi ions does he E-money business—issuing digi al cu ency
Kee-Youn Kang: [email p o ec ed]
I would like o hank Jona han Chiu, Kim-Sau Chung, Rod Ga a , Ped o Gomis-Po que as, Inkee Jang,
Seungduck Lee, Cy il Monne , Pe e No man, Xi Wang, S ephen Williamson, Randall W igh , Syl ia Xiaolin
Xiao, and all semina pa icipan s in 2021 SaMMF Mone a y Economics Con e ence, 2021 KER, and 2022
Annual Mee ing o he Cen al Bank Resea ch Associa ion o help ul commen s and discussions. Finally,
I hank h ee anonymous e e ees o cons uc i e commen s and sugges ions ha ha e g ea ly imp o ed
he pape .
©2024 The Au ho . 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/TE5081
132 Kee-Youn Kang Theo e ical Economics 19 (2024)
and comme cially using paymen da a—exis in equilib ium, and is i good o bad o a
socie y? How does mone a y policy a ec he E-money business? How do equilib ium
ou comes depend on ma ke s uc u es? Wha a e he e ec s o go e nmen ’s ma ke
in e en ions on eal alloca ions and wel a e?
In his pape , we cons uc a money sea ch model in which a p i a e paymen pla -
o m company issues E-money ha is backed by go e nmen -issued cash, simila o
PayPal and debi ca ds, o add ess he abo e ques ions. The company can es ima e
buye s’ p e e ences using E-money ansac ion da a and sell he p e e ence in o ma-
ion o selle s. A selle can use he p e e ence in o ma ion o p oduce goods ha a e
cus omized o buye s’ p e e ences, which inc eases he o al ade su plus. The p e-
cision o he p e e ence in o ma ion inc eases wi h he amoun o paymen da a, and
he company p o ides ewa ds o using E-money o a ac mo e buye s o use i . Buy-
e s incu disu ili y om p o iding pe sonal in o ma ion, including paymen his o ies, o
he company and, hence, use E-money only i he ewa ds a e highe han he disu ili y;
o he wise, hey use cash.
An inc ease in he ade olume aises he addi ional su plus ha he selle can ob-
ain by selling cus omized goods, so i inc eases he alue o p e e ence in o ma ion and
he company’s p o i . I he in la ion a e is su icien ly high, he ade olume is oo
small o he company o make p o i s, so he E-money business does no exis . Mean-
while, buye s hold mo e eal balances o buy cus omized goods when he E-money busi-
ness exis s. Thus, ein o cing in e ac ions exis be ween he ade olume and he alue
o p e e ence in o ma ion. Because o hese in e ac ions, mul iple equilib ia can exis
wi h di e en ansac ion pa e ns (wi h and wi hou E-money) when buye s and selle s
a e andomly ma ched and ba gain o e he e ms o ade. Howe e , he mul iplici y
disappea s in compe i i e sea ch equilib ium because he pos ed e ms o ade wo k as
a coo dina ing de ice.
Once he company has incu ed he cos o ewa d paymen s o ob ain p e e ence
in o ma ion, he ma ginal cos o selling he in o ma ion o an addi ional selle is ze o.
Thus, a monopoly a ises in he E-money indus y wi h Be and compe i ion. In he
model, he p i a e bene i o he E-money business is lowe han i s social bene i be-
cause o he money holding cos and he ex e nali y p oblem, i.e., he company does no
ully in e nalize he buye s’ bene i o consuming cus omized goods when he E-money
business exis s. Thus, when le o ma ke o ces alone, socially e icien uses o paymen
da a may no occu .1P o iding a subsidy o selle s di ec ly incen i izes in o ma ion
use s, so i is mo e e ec i e o suppo ing e icien uses o paymen da a han subsidiz-
ing he E-money issue .
Al hough he li e a u e on p i acy is ex ensi e, ela i ely li le a en ion has been
gi en o p i acy in mone a y economics.2
Kahn,McAnd ews,andRobe ds(2005) in es-
1The ex e nali y p oblem disappea s unde compe i i e sea ch, bu he socially e icien E-money busi-
ness may no exis when he money holding cos exis s.
2A nonmone a y model ha is closely ela ed o ou model is Be gemann, Bona i, and Gan (2022)in
which a monopolis in e media y buys p e e ence in o ma ion di ec ly om indi idual consume s and e-
sells he in o ma ion in a p oduc ma ke , and hey show he p esence o in o ma ional ex e nali ies. We
di e om hei model as we ocus on he special cha ac e is ics o he digi al cu ency indus y and he
e ec s o go e nmen policies.
Theo e ical Economics 19 (2024) Digi al cu ency and p i acy 133
iga e he ole o p i acy in money ansac ions and Ga a and an Oo d (2021)show
ha indi idual cus ome s do no p ese e hei p i acy in paymen s a he socially op-
imal le el. Guennewig (2023) shows ha inal goods p oduce s issue digi al cu ency
o ob ain hei consume s’ in o ma ion. Mo e ela ed, Ga a and Lee (2021) show ha
paymen da a ha can be used o design u u e goods d i e he o ma ion o a ma ke
monopoly.
Howe e , in hose ea lie s udies, me chan s ob ain cus ome s’ p i a e da a, while
he paymen pla o m company ob ains he paymen da a in ou model, complemen -
ing he p e ious wo ks. Chiu and Koeppl (2022a) in es iga e dynamic eedback loops
be ween he da a and ac i i y sides o he pla o m. In hei model, he pla o m com-
pany ob ains consume s’ in o ma ion, bu he e a e no digi al cu encies and go e n-
men issued money. Fu he mo e, in con as o he p e ious s udies ci ed abo e, ou
model is based on Lagos and W igh (2005), so i admi s he analysis o he e ec s o
mone a y policy on he economic uses o paymen da a. In pa icula , we de i e es able
implica ions ega ding he ela ion be ween he in la ion a e and he p o i abili y o he
E-money business.
We also con ibu e o he g owing li e a u e on digi al cu ency. Chiu and Wong
(2015)andCa li and U as (2023) examine how E-money imp o es he e iciency o he
economy. Chiu and Koeppl (2022b)andKang (2023) in es iga e double spending incen-
i es in he Bi coin sys em, and Choi and Roche eau (2021)andPagno a (2022)s udy
c yp ocu ency p icing. Fe nández-Villa e de and Sanches (2019), Schilling and Uh-
lig (2019), and Kang and Lee ( o hcoming) explo e he mac oeconomic implica ions
o c yp ocu encies ia cu ency compe i ion. While hese pape s ocus on analyzing
he economic implica ions o echnical ea u es o digi al cu ency, such as blockchain
echnology, we ocus on he p i acy issue o digi al cu ency paymen s.3
The es o he pape is o ganized as ollows. Sec ion 2p esen s he model en i-
onmen . In Sec ion 3, we cha ac e ize ba gaining equilib ium, and in Sec ion 4,we
conduc a wel a e analysis. Sec ion 5in es iga es compe i i e sea ch equilib ium. In
Sec ion 6, we ex end he model wi h mul iple E-money issue s. Sec ion 7concludes he
pape . Appendix Acon ains p oo s.
2. En i onmen
The basic amewo k o he model is based on Lagos and W igh (2005)andRoche eau
and W igh (2005). Time is indexed by =0, 1, 2, , and each ime pe iod is di ided
in o h ee subpe iods: mo ning (m), a e noon (a), and e ening (e). A con inuum o
buye s and selle s exis s, each wi h uni mass. Each buye has p e e ences, gi en by
E0
∞
=0
β X −H −1pδ+υ(q )+αu(x ),
3The e also has been ex ensi e esea ch on cen al bank digi al cu ency (see Fe nández-Villa e de,
Sanches, Schilling, and Uhlig (2021), Keis e and Monne (2022), and Williamson (2022a,b), and Chiu,
Da oodalhosseini, Jiang, and Zhu (2023)).
134 Kee-Youn Kang Theo e ical Economics 19 (2024)
and each selle has p e e ences, gi en by
E0
∞
=0
β X −H −c(ha, )−he, .
He e β∈(0, 1)is he discoun a e, and X ,q ,andx a e consump ion in he mo n-
ing, a e noon, and e ening, espec i ely, and H ,ha, ,andhe, a e labo supplies in he
mo ning, a e noon, and e ening, espec i ely. We assume ha υ,u,andca e wice
con inuously di e en iable wi h υ(0)=0, υ <0<υ
,υ(0)=∞,υ(∞)=0, u(0)=0,
u <0<u
,u(0)=∞,u(∞)=0, c(0)=0, c>0, and c >0. Fu he mo e, we assume
ha u(x)u(x)>[u(x)]2 o all x>0. He e α>0 is a pa ame e ha a ec s he buye ’s
u ili y in he e ening, δ>0 is a pa ame e ha cap u es he buye ’s disu ili y by o go-
ing p i acy, he exac de ini ion o which will be p o ided la e , and 1pis an indica o
unc ion ha akes he alue o 1 i a buye o goes p i acy and 0 o he wise.
Agen s can p oduce one uni o he pe ishable consump ion good wi h one uni o
labo supply in each subpe iod. We call goods p oduced in he mo ning, a e noon, and
e ening mo ning goods, a e noon goods, and e ening goods, espec i ely, and we se
mo ning goods as he nume ai e goods.
In he mo ning, he e is a cen alized Wal asian ma ke in which all agen s ade nu-
me ai e goods and asse s. Buye s and selle s mee in la ge g oups ading a e noon
goods in a compe i i e ma ke in he a e noon. Finally, in he e ening, he e a e bila -
e al mee ings be ween buye s and selle s. In pai wise mee ings, a buye and a selle ba -
gain o e he e ms o ade, which a e de e mined acco ding o he ba gaining solu ion
o Kalai (1977), whe e he selle ’s ba gaining powe is θ∈(0, 1).
Ideally, buye s would like o bo ow ou pu om selle s in he a e noon and e ening
ma ke s and o epay loans he nex mo ning. Such c edi a angemen s a e uled ou
he e because agen s a e anonymous and no de ice is a ailable o eco d c edi his o ies.
Consequen ly, any ades be ween buye s and selle s mus occu on a quid p o quo basis
h ough he use o a medium o exchanges.
The e exis s ia money ha is aded a p ice φ in e ms o nume ai e goods in
he mo ning in pe iod . Money is supplied by he go e nmen a he beginning o he
mo ning wi h a lump-sum ans e τ =(γ−1)φ M −1 o each buye , whe e M −1is he
money s ock in pe iod −1. Thus, he money s ock g ows a a cons an g oss a e γ.
We ocus on policies whe e γ≥βbecause equilib ium does no exis o he wise. When
γ=β, we conside equilib ium ob ained by aking he limi γ→β. Fu he mo e, we
assume ha γ<β/θbecause money is no used in he e ening o he wise.
A he beginning o he mo ning, buye s a e subjec o an idiosync a ic shock, which
de e mines whe he hey consume ea ly (in he a e noon) o la e (in he e ening). Le
ρ∈(0, 1)deno e he p obabili y ha a buye goes o he a e noon ma ke and a buye
goes o he e ening ma ke wi h p obabili y 1 −ρ. No e ha his shock is ealized a
he beginning o he mo ning. Thus, buye s know which ma ke hey will go o in he
subsequen pe iod when hey make decisions in he mo ning. We call buye s who go
o he a e noon ma ke ea ly buye s and hose who go o he e ening ma ke la e buy-
e s.
Theo e ical Economics 19 (2024) Digi al cu ency and p i acy 135
Indi idual p e e ence in he e ening In he model economy, N∈N+, di e en as es
exis o e ening goods, and a buye has one o hose as es wi h p obabili y 1/N.The
e ening as e is ealized in he a e noon, and buye s’ e ening as es a e hei p i a e
in o ma ion. I a la e buye consumes cus omized goods ailo ed o his/he as e in he
e ening, hen α=αH;o he wise,andα=αL,whe e0<α
L<α
H. Gi en he alue o α,
we de ine he h eshold alues o ade olume in he e ening ma ke and eal balances
as
x∗
i=u−11
αiand m∗
i=θαiux∗
i+(1−θ)x∗
i
β(1)
o each i∈{H,L}.
We assume ha a selle can p oduce a cus omized p oduc ailo ed o a pa icula
as e only i he selle p epa ed he p oduc ion o ha p oduc by incu ing ς>0 uni s
o labo a he beginning o he e ening. Fu he mo e, he e exis s a collec ion o in ini e
numbe s o di e en e ening as es, and in each pe iod, Ne ening as es a e andomly
chosen om ha collec ion. Thus, he se o e ening as es changes o e ime, simila
o a passing ad, and he p obabili y ha a pa icula e ening as e is ealized in a gi en
pe iod is ze o. Consequen ly, selle s canno p epa e he p oduc ion o a cus omized
p oduc unless hey know wha e ening as es a e ealized in he cu en pe iod.
A selle may a emp o con ac indi idual buye s o lea n hei ealized e ening
as es. Howe e , he e is no way o e i y whe he an indi idual buye p o ides he co -
ec in o ma ion. In pa icula , we assume ha buye s incu some disu ili y om p o id-
ing pe sonal in o ma ion o o he s due o p i acy conce ns, which will be discussed u -
he la e . Fu he mo e, a la e buye has a ze o p obabili y o mee ing he selle o whom
he/she p o ided he p e e ence in o ma ion because o he andom ma ching p ocess
in he e ening. Thus, buye s always ha e an incen i e o p o ide inco ec in o ma ion
o keep hei p i acy, so selle s canno ob ain in o ma ion abou ealized e ening as es
by asking indi idual buye s.
Digi al cu ency and p i acy In his economy, he e is a p i a e company ha can issue
elec onic money (hence o h, E-money) and we assume ha E-money mus be backed
by go e nmen issued money.4Fo example, he go e nmen can p ohibi p i a e sec-
o s om issuing pu e ia money o main ain he e icacy o mone a y policy o agen s
a e eluc an o ecei e money issued by a p i a e company unless i is backed by go -
e nmen issued money.5Speci ically, i an agen deposi s money o he company by
4We can cons uc a model such ha he company suppo s online ansac ions by issuing c edi ca ds
ha ha e wo dis inc i e ea u es: (i) C edi ca d use s pu chase goods wi h c edi and make mon hly pay-
men s on c edi ca d bills, and (ii) he c edi ca d company ans e s money o me chan s quickly, in con-
as o s anda d loans, on behal o i s use s. Howe e , he main implica ions om he model wi h he
c edi ca d company do no di e om hose in he model wi h he E-money company.
5The Chinese go e nmen , o example, o bids using any p i a ely issued money in China, while i al-
lows people o use Alipay and WeCha Pay ha a e basically backed by Renminbi. Fu he mo e, al hough
i is no illegal o use c yp ocu encies o ades in coun ies, such as Japan, Sou h Ko ea, and he Uni ed
S a es, c yp ocu encies a e no widely used in e ail ansac ions in hose coun ies in con as o debi
ca ds and PayPal ha a e backed by go e nmen issued money.
136 Kee-Youn Kang Theo e ical Economics 19 (2024)
opening an accoun , hen he company pu s he same amoun o E-money in o he
agen ’s accoun . In his sense, E-money is equi alen o debi ca ds and Alipay in eali y.
We assume ha he company is a monopoly, bu in Sec ion 6we show ha a
monopoly a ises al hough mul iple companies can issue hei own E-money. Agen s
mus open a new accoun in he mo ning o use E-money in he cu en pe iod and
he accoun is closed he nex mo ning. The company canno es ic he numbe o
use s and dis ibu es i s p o i s o buye s in he mo ning. Finally, we assume ha he
company can collec each indi idual’s E-money ansac ion da a, which could p o ide
use ul in o ma ion as desc ibed below.
In eali y, an indi idual who canno ind p oduc s ha mee his/he p e e ence pe -
ec ly because hose p oduc s a e no ye a ailable in he ma ke can ca e o his/he
p e e ence o some deg ee by consuming a ailable goods app op ia ely. Fo example,
suppose ha John wan s o enjoy high-quali y co ee a home in he mo ning, bu does
no ha e a sophis ica ed esp esso machine and good skills in making co ee. In his case,
he can sa is y his needs in pa by going o S a bucks in he mo ning, al hough wha
he eally wan s is o enjoy quali y co ee on his balcony seeing he sun ise. As shown
in his example, p e e ences could a ec consump ion beha io s, so consump ion pa -
e ns could p o ide some in o ma ion abou indi iduals’ p e e ences ha can be used
o de elop mo e cus omized p oduc s, such as ad anced Nesp esso capsule co ee ma-
chines in ou co ee example.
Simila o he example desc ibed abo e, ea ly buye s’ e ening as es could a ec
hei consump ion beha io s in he a e noon al hough hey do no go o he e ening
ma ke because hey wan o consume goods ea ly in he a e noon. This implies ha
he company can es ima e e ening as es ealized in he cu en pe iod by analyzing
E-money ansac ions in he a e noon. Ob iously, he accu acy o es ima ion would in-
c ease wi h he sample size by he undamen al ule in s a is ics. Based on his a ionale,
we assume ha he company ob ains he co ec p e e ence in o ma ion wi h p obabil-
i y κ(B)∈[0, 1]by analyzing E-money ansac ions in he a e noon, whe e B∈[0, ρ]
is he mass o ea ly buye s who use E-money in he a e noon and κis an inc easing
unc ion wi h κ(0)=0andκ(ρ)=1.6,7
Selle s would buy p e e ence in o ma ion because hey can p epa e he p oduc ion
o cus omized goods ailo ed o ealized e ening as es wi h p e e ence in o ma ion,
which inc eases he o al ade su plus in he e ening. Le ϕ deno e he p ice o each
as e in o ma ion in pe iod in e ms o mo ning goods in he nex pe iod +1. We
assume ha selle s can commi o making paymen s o in o ma ion pu chases he nex
mo ning.
6In he model, p e e ence in o ma ion o an indi idual buye e eals p e e ence in o ma ion o o he
buye s who sha e he same e ening as e. In his sense, ou model esona es da a ex e nali y, whose eco-
nomic implica ions ha e been in es iga ed in Be gemann, Bona i, and Gan (2022) and Ichihashi (2021).
Howe e , in con as o he models o da a ex e nali y, he company canno lea n indi idual buye ’s e ening
as e because a buye always has incen i es o p o ide inaccu a e in o ma ion abou himsel /he sel . Fu -
he mo e, analyzing a e noon ansac ion da a does no e eal each buye ’s p e e ence and i only p o ides
in o ma ion abou ealized e ening as es in he cu en pe iod.
7We assume ha κ(ρ)=1 o ob ain a closed o m solu ion o equilib ium alloca ions. I κ(ρ)<1, we
canno ob ain he closed o m solu ion, bu main implica ions do no change based on he nume ical anal-
ysis.
Theo e ical Economics 19 (2024) Digi al cu ency and p i acy 137
In eali y, many people a e inhe en ly eluc an o e eal p i a e in o ma ion o o h-
e s. We cap u e his ea u e in he model wi h disu ili y δsimila o Choi, Doh-Shin, and
Byung-Cheol (2019): When a buye opens an accoun a he company, he buye incu s
δ>0 uni s o ixed disu ili y in he mo ning o ag ee ha he company ob ains his/he
pe sonal in o ma ion, including paymen his o ies, and can use he ob ained in o ma-
ion o comme cial pu poses.8This disu ili y is associa ed wi h, o ins ance, p i acy
conce ns such as da a hacking o p i acy cos s o igina ing om he agen ’s own as e
o keeping p i acy. Consequen ly, he company mus compensa e buye s o using E-
money, such as he Boun y Paymen s p og am o PayPal and asso ed bene i s p o ided
by debi ca d issue s.
The ewa d can ha e wo o ms: ixed and p opo ional ewa ds. In Appendix B,we
show ha he p o i maximizing company does no p o ide a p opo ional ewa d be-
cause i dis o s buye s’ decisions abou money holdings, which gene a es an addi ional
cos o he company. Thus, in his pape , we assume ha he company p o ides only
ixed ewa ds. This implies ha he company does no compensa e la e buye s o using
E-money in he e ening because e ening ansac ion da a ha e no alue o he company
and p o iding a ixed ewa d does no a ec he quan i y o la e buye s’ money holdings
in he e ening. On he o he hand, he company p o ides R +1≥0 uni s o nume ai e
goods o a buye in he mo ning in pe iod +1 i he buye used E-money o a e noon
ansac ions in pe iod .9
Figu e 1summa izes he sequence o e en s in a ep esen a i e pe iod. Th oughou
he pape , he E-money business means he business o ob aining and selling e ening
as e in o ma ion by issuing E-money. In wha ollows, we call money supplied by he
go e nmen P-money o emphasize ha i is pape money a he han elec onic money.
3. Equilib ium
In his sec ion, we cha ac e ize equilib ium o he model economy as ollows. Fi s , we
s udy agen s’ alue unc ions in each subpe iod. Second, we s udy he op imal decisions
o buye s, selle s, and he company. Thi d, we s udy ma ke clea ing condi ions. Then
we cha ac e ize equilib ium.
3.1 Value unc ions
Mo ning ma ke In he mo ning, agen s consume nume ai e goods, supply labo ,
and eadjus hei po olios. We de ine an indica o a iable ι ∈{0, 1}such ha
8In he model, we assume ha buye s incu p i acy cos δwhen hey open an accoun a he company o
cap u e wo ies ha an indi idual expe iences when he/she p o ides pe sonal in o ma ion, such as name,
phone numbe , and social secu i y numbe , o open a new accoun , o example, a a bank o on social
media. Al e na i ely, we can assume ha buye s incu disu ili y δwhen hey use E-money. Howe e , he
main esul s do no change because, in equilib ium, buye s will ne e accumula e E-money in he mo ning
unless hey use i in he a e noon o e ening ma ke o ansac ions.
9Al e na i ely, we can assume ha he company ans e s R +1uni s o money (o E-money) in e ms o
mo ning goods in he nex pe iod o an ea ly buye a he end o he a e noon in pe iod i he/she used
E-money o a e noon ansac ions in pe iod . This al e na i e assump ion aises he company’s cos in
equilib ium, bu he main implica ions do no change.
138 Kee-Youn Kang Theo e ical Economics 19 (2024)
Figu e 1. Timeline o a ep esen a i e pe iod.
ι =1 i a buye used E-money in he a e noon in pe iod and ι =0 o he wise. Le
Vb,ea ly
m, (mp,me,ι −1)deno e he ea ly buye ’s alue unc ion a he beginning o he
mo ning in pe iod wi h mpuni s o eal P-money, meuni s o eal E-money, and p e i-
ous ac ion ι −1.
Then, by i ue o he quasi-linea i y o p e e ences, Vb,ea ly
m, (mp,me,ι −1)is gi en as
Vb,ea ly
m, (mp,me,ι −1)=mp+me+τ +π +R ι −1
+max
m
p,m
e−φ
φ +1m
p+m
e−1{m
e>0}δ+Vb
a, m
p,m
e.(2)
He e, π is di idend paymen s om he company, m
pand m
ea e he eal balances o P-
money and E-money, espec i ely, bo h in e ms o nume ai e goods in he nex pe iod,
1{m
e>0}is an indica o unc ion ha akes he alue o 1 i m
e>0 and 0 o he wise, and
Vb
a, (m
p,m
e)is he alue o he ea ly buye wi h po olio (m
p,m
e)in he a e noon in
pe iod . Simila ly, he alue unc ion Vb,la e
m, (mp,me,ι −1)o a la e buye wi h po olio
(mp,me)and p e ious ac ion ι −1a he beginning o he mo ning in pe iod is gi en as
Vb,la e
m, (mp,me,ι −1)=mp+me+τ +π +R ι −1
+max
m
p,m
e−φ
φ +1m
p+m
e−1{m
e>0}δ+Vb
e, m
p,m
e,(3)
whe e Vb
e, (m
p,m
e)is he alue o he la e buye in he e ening wi h po olio (m
p,m
e).
Nex , he selle ’s alue unc ion Vs
m, (mp,me,n)in he mo ning in pe iod wi h po -
olio (mp,me)and he numbe o unpaid in o ma ion pu chases n∈{0, 1, ,N}in he
p e ious e ening is gi en as
Vs
m, (mp,me,n)=mp+me−nϕ −1+max
m
p,m
e−φ
φ +1m
p+m
e+Vs
a, m
p,m
e,(4)
Theo e ical Economics 19 (2024) Digi al cu ency and p i acy 145
and he in o ma ion p ice as (23). Thus, he company’s p o i is gi en as
π=(1−ρ)D(m)−Nς−ρδ
β, (31)
whe e mis he la e buye ’s eal P-money balances in he e ening ma ke in equilib ium.
Then i mus be ha D(m)≥(Nς+ρδ)/(1−ρ) o E-equilib ium o exis because he
company has no incen i e o un i s business o he wise. On he o he hand, i mus
be ha D(m)<(Nς+ρδ)/(1−ρ) o P-equilib ium o exis because, o he wise, he
company can make nonnega i e p o i s om unning i s business. These a gumen s
lead o he nex p oposi ion.
P oposi ion 2. S a iona y mone a y equilib ium exis s as ollows:
(I) Suppose ha (Nς+ρδ)/(1−ρ)≤D(m∗
L). Then he e exis γ1>γ
2≥βsuch ha
(a) o all γ∈[β,γ1], E-equilib ium exis s, and (b) o all γ>γ
2, P-equilib ium
exis s.
(II) Suppose ha D(m∗
L)<(Nς+ρδ)/(1−ρ)≤D. Then, he e exis s γ3≥βsuch ha
(a) o all γ∈[β,γ3], E-equilib ium exis s, and (b) o all γ≥β, P-equilib ium
exis s.
(III) Suppose ha D<(Nς+ρδ)/(1−ρ). Then, o all γ≥β, P-equilib ium exis s.
P oposi ion 2shows how he alue o (Nς+ρδ)/(1−ρ)and he in la ion a e γ o-
ge he de e mine he exis ence o each ype o equilib ium. Figu e 2depic s how he
pa ame e space is subdi ided wi h (Nς+ρδ)/(1−ρ)on he e ical axis and γon he
ho izon al axis, illus a ing P oposi ion 2g aphically.
The e ec s o pa ame e s δ,ς,N,andρon he ype o equilib ium a e s aigh o -
wa d. He e ρδ is he ea ly buye s’ disu ili y om o going p i acy o using E-money, Nς
Figu e 2. Typology o equilib ia in (γ,Nς+ρδ
1−ρ)space.
146 Kee-Youn Kang Theo e ical Economics 19 (2024)
is he o al in es men cos ha selle s incu o p epa e he p oduc ion o cus omized
goods in he e ening, and 1 −ρis he p obabili y ha a selle mee s a buye in he
e ening ma ke . Thus, i δ,ς,N,andρa e su icien ly high, as illus a ed in he hi d
case o P oposi ion 2, he cos s o ob aining and ha nessing p e e ence in o ma ion a e
highe han he expec ed payo ; hence, he company canno make nonnega i e p o i s
om i s business.
Nex , P oposi ion 2shows ha he company is mo e likely o un he E-money busi-
ness when γis low, while E-money is no ci cula ed when γis su icien ly high.1112 The
economic mechanism o his esul is in line wi h ou ea lie obse a ion. As γin-
c eases, la e buye s ca y less P-money in o he e ening ma ke , which educes he alue
o p e e ence in o ma ion o selle s by he esul s o Lemma 3. Consequen ly, he p ice
o p e e ence in o ma ion and he company’s p o i dec ease. Thus, i is mo e likely ha
he company uns he E-money business making nonnega i e p o i s when γis low and
ice e sa.13
One no iceable esul in P oposi ion 2is ha he model can gene a e mul iple s a-
iona y mone a y equilib ia: Fo all γ∈(γ2,γ1]when (Nς+ρδ)/(1−ρ)≤D(m∗
L)o
o all γ∈[β,γ3]when D(m∗
L)<(Nς+ρδ)/(1−ρ)≤D, bo h E-equilib ium and P-
equilib ium can exis . The in ui ion o his inding is as ollows. Fo in e media e in-
la ion, i la e buye s expec ha hey can buy cus omized goods in he e ening, hey
accumula e a su icien amoun o money, which mo i a es selle s o buy p e e ence
in o ma ion om he company. On he o he hand, i la e buye s expec ha hey can-
no buy cus omized goods in he e ening, hey hold oo li le money in he e ening o
incen i ize selle s o buy p e e ence in o ma ion, which jus i ies la e buye s’ ini ial ex-
pec a ion. This complemen a i y leads o mul iple equilib ia.
A mul iplici y o equilib ia due o s a egic complemen a i ies has been discussed in
p e ious s udies. Fo ins ance, in he pla o m model wi h wo-sided ma ke s, such as
Roche and Ti ole (2003)andHagiu and Spulbe (2013), one g oup’s bene i o using he
pla o m inc eases wi h he size o he o he g oup ha uses he pla o m, which gen-
e a es s a egic complemen a i ies and mul iple equilib ia. The E-money business in
ou model is simila o a pla o m in he sense ha i connec s supplie s o in o ma ion
(ea ly buye s) and use s o in o ma ion (selle s). Howe e , ea ly buye s’ choices do no
depend on selle s’ decision on in o ma ion pu chases: Ea ly buye s only ca e abou he
p i acy cos s and he ewa ds o using E-money. S a egic complemen a i ies in ou
model exis be ween he selle ’s in es men decision and he eal balances o la e buye s
who do no use E-money.
11Be en sen, Came a, and Walle (2007) show ha he exis ence o a mone a y equilib ium wi h c edi
equi es some posi i e in la ion, while E-money and P-money coexis when in la ion is su icien ly low in
ou model.
12This esul implies ha an inc ease in he in la ion a e can dec ease he a ie y o goods ha selle s can
p oduce because selle s can p oduce cus omized goods only i he E-money business exis s. In his sense,
ou model is ela ed o She chenko (2004) and Dong (2010) who in es iga e how economic en i onmen s,
such as in la ion and sea ch ic ions, a ec he a ie y o goods.
13No e ha ou model deli e s po en ially es able p edic ions abou he ela ion be ween he in la ion
a e γand he p o i abili y o he E-money business: An inc ease in γ educes he company’s p o i and
he eby d i es ou he E-money business om he economy.
Theo e ical Economics 19 (2024) Digi al cu ency and p i acy 147
Mo e ela ed, in he money sea ch li e a u e, Les e , Pos lewai e, and W igh (2012)
show ha i agen s can e i y he asse ’s quali y a a cos , hen s a egic complemen-
a i ies exis be ween buye s’ asse demand and selle s’ in o ma ion in es men . This
s a egic complemen a i y can gene a e mul iple equilib ia simila o ou model.14
Howe e , ou model di e s om Les e , Pos lewai e, and W igh (2012) in he way ha
selle s ob ain in o ma ion. In hei model, selle s pay an exogenous ixed cos o ob ain
in o ma ion abou he asse ’s quali y, while he p ice o p e e ence in o ma ion in ou
model is de e mined by he p o i maximizing company in equilib ium. Fu he mo e,
he company in ou model is a big playe , so i may be able o lead o E-equilib ium
when mul iple equilib ia a e easible as ollows.
Suppose ha when he company sells p e e ence in o ma ion, i p omises o com-
pensa e a selle i he/she does no make a su icien su plus om selling cus omized
goods because he ma ched buye in a bila e al mee ing holds less han
dH(γ)uni s o
eal P-money.15 No e ha he company can co ec ly in e he selle s’ ade su plus in
he e ening in equilib ium by obse ing agg ega e a iables, such as agg ega e ade ol-
umes in he e ening, al hough he company canno di ec ly obse e he ade olume
o indi idual selle s in he e ening ma ke . Thus, he company can make compensa ion
con ingen on hose agg ega e a iables.
Then selle s will buy p e e ence in o ma ion wi hou he conce n o making insu -
icien su plus in he e ening. An icipa ing ha selle s will p epa e o p oduce cus-
omized goods in he e ening ma ke , la e buye s will ca y
dH(γ)uni s o eal P-money
in o he e ening. Thus, he company does no need o compensa e selle s in equilib-
ium, and gua an eeing he selle ’s su plus can elimina e P-equilib ium when mul iple
equilib ia a e easible, simila o inding ha a bank—a big playe —can p e en bank
un equilib ium by announcing he suspension o con e ibili y in Diamond and Dyb ig
(1983).
Howe e , he company canno use he su plus gua an ee i some la e buye s do no
ha e an indi idual as e o e ening goods. Suppose ha he μ∈(0, 1) ac ion o la e
buye s does no ha e e ening as es and ha hei u ili y in he e ening is αLu(x).Then
he μ ac ion o la e buye s will b ing
dL(γ)uni s o eal balances in o he e ening ma -
ke e en hough selle s can p oduce cus omized goods, and he o he 1 −μ ac ion o
la e buye s will hold
dH(γ)uni s o eal balances. Thus, he e a e selle s who me a
la e buye wi h
dH(γ)uni s o eal balances and selle s who me a la e buye wi h
dL(γ)
uni s o eal balances in he e ening.16 In his case, he company needs o p o ide com-
pensa ion based on in o ma ion abou ade su pluses epo ed by indi idual selle s.
14The e a e o he asse exchange models, such as T ejos and W igh (2016), Bu de , T ejos, and W igh
(2017), and He and W igh (2019), ha show he exis ence o mul iple equilib ia. Howe e , in hose models,
mul iple equilib ia exis because o a sel - ul illing p ophecy o he asse ’s liquidi y—i an agen hinks o h-
e s alue he asse high, hen he agen would gi e high paymen o ge he asse —no because o a s a egic
complemen a i y. Fu he mo e, in hose models, he asse is indi isible and mul iple equilib ia do no exis
i he asse is ia .
15Speci ically, wha he company does is o gua an ee θ[αHu(ˆ
xH(
dH(γ))) −ˆ
xH(
dH(γ))] uni s o ade
su plus o a selle in he e ening i he selle buys p e e ence in o ma ion.
16O he wise, in oducing he assump ion ha he μ ac ion o la e buye s does no ha e e ening as e
does no change equilib ium ou comes excep ha an inc ease in he selle ’s ade su plus in he e ening by
p epa ing he p oduc ion o cus omized goods wi hou he su plus gua an ee is now gi en as (1−μ)D(mp).
148 Kee-Youn Kang Theo e ical Economics 19 (2024)
Howe e , he company canno e i y each selle ’s ade olume in he e ening, so selle s
will always misin o m he company o hei su plus o ob ain compensa ion. Thus, he
company canno use he su plus gua an ee as a ool o suppo ing E-equilib ium when
mul iple equilib ia a e easible.
4. Wel a e analysis
In his sec ion, we examine he model’s no ma i e p ope ies in e ms o social wel a e,
in es iga e he op imal mone a y policy, and explo e he e ec s o go e nmen in e -
en ions, such as p o iding subsidies, on eal alloca ions and wel a e. We de ine he
sum o expec ed u ili ies in a s eady s a e equilib ium ac oss agen s wi h equal weigh
as ou wel a e measu e,
W=ρυ(q)−c(ρq)+(1−ρ)αiu(x)−x−1{e=E}(Nς+ρδ),
whe e i=Hin E-equilib ium, i=Lin P-equilib ium, and 1{e=E}is an indica o unc ion
ha akes he alue o 1 i he economy is in E-equilib ium and 0 o he wise. Speci ically,
gi en he esul s o P oposi ion 1,wel a eisgi enas
W=WP(γ)≡ρυ
q(γ)−cρ
q(γ)+(1−ρ)αLu
xL(γ)−
xL(γ)(32)
in P-equilib ium and
W=WE(γ)≡ρυ
q(γ)−cρ
q(γ)+(1−ρ)αHu
xH(γ)−
xH(γ)−(Nς+ρδ)(33)
in E-equilib ium. Because he company uns he E-money business in E-equilib ium
and does no in P-equilib ium, WE(γ)−WP(γ)measu es he social ne bene i o he E-
money business. The nex lemma shows how he in la ion a e γa ec s his social ne
bene i .
Lemma 4. The social ne bene i WE(γ)−WP(γ)dec eases wi h γ.
To ob ain he in ui ion o he esul s o Lemma 4, no e ha he ade olume xin he
e ening ma ke alls as γinc eases. A dec ease in x, in u n, educes he con ibu ion o
he E-money business o wel a e—an inc ease in he ade su plus αu(x)−xby aising
α om αL o αH—while social cos Nς+ρδ s ays a a cons an le el. Thus, he social ne
bene i o he E-money business WE(γ)−WP(γ)dec eases wi h γ.
In he model economy, wha he company conside s when i decides whe he o un
he E-money business is he p o i om he business, no i s social bene i . Thus, he
company may un (may no un) he E-money business al hough i is socially unde-
si able (desi able). Howe e , he nex p oposi ion shows ha whene e he economic
en i onmen desc ibed by (N,ς,ρ,δ,γ)suppo s nonnega i e p o i om he E-money
business in E-equilib ium, wel a e is highe wi h he ac i e E-money business han wi h-
ou i .
Theo e ical Economics 19 (2024) Digi al cu ency and p i acy 149
P oposi ion 3. Suppose ha he company can make nonnega i e p o i s om i s busi-
ness when la e buye s hold
dH(γ)uni s o eal balances, i.e., E-equilib ium can exis . Then
WE(γ)>W
P(γ), so wel a e is highe wi h he E-money business han wi hou i .
The in ui ion o he esul s o P oposi ion 3is as ollows. As shown in (31)and(33),
he p i a e cos o unning he E-money business equals i s social cos s Nς+ρδ,so
he company ully in e nalizes he social cos o he E-money business. On he o he
hand, he main sou ce o he E-money business’ e enue is he inc ease in he selle ’s
ade su plus by selling cus omized goods in he e ening ma ke (p i a e bene i ), while
he E-money business’ posi i e e ec s on wel a e a e he inc ease in he o al ade su -
plus om ading cus omized goods in he e ening (social bene i ), and he e is a wedge
be ween he p i a e and social bene i s o he E-money business o he ollowing wo
easons.
Fi s , expec ing o consume cus omized goods, la e buye s hold mo e eal balances
in E-equilib ium han in P-equilib ium. Thus, he ade olume o noncus omized
goods, and he eby he ade su plus, is highe in E-equilib ium han in P-equilib ium
when he money holding cos exis s, i.e., ˆ
xL(
dH(γ)) >
xL(γ) o all γ>β. This implies
a highe social bene i han he p i a e bene i o he E-money business ce e is pa ibus,
because he main sou ce o he p i a e and social bene i s o he E-money business is
ob ained by sub ac ing he su plus om ading noncus omized goods om he su -
plus om ading cus omized goods.17 Second, in he model, he selle ’s ade su plus is
he θ∈(0, 1) ac ion o he o al ade su plus in he e ening ma ke gi en he ba gain-
ing ule. Thus, he p i a e bene i is scaled down by θin con as o he social bene i .
Fo hese wo easons, he social bene i is highe han he p i a e bene i . Thus, when-
e e he company makes a nonnega i e p o i om unning he E-money business, i
mus be ha WE(γ)>W
P(γ).
No e om (1), (12), (29), and (30) ha ˆ
xL(
dH(β)) =
xL(β)=x∗
L.Thus,whenγ=β,
he su plus om ading noncus omized goods unde each ype o equilib ium is he
same, so he i s channel, desc ibed abo e, ha gene a es he wedge be ween he so-
cial and p i a e bene i s o he E-money business disappea s. Nex , he second channel
o igina es om he ba gaining assump ion in bila e al mee ings, and his channel may
disappea unde a di e en ma ke s uc u e in he e ening. Fo ins ance, suppose ha
la e buye s and selle s ade in he e ening h ough a ma ching pla o m ope a ed by he
company. Then he company can ex ac he addi ional su plus ha la e buye s ob ain
om ading cus omized goods by aising he la e buye s’ ees o using he pla o m.
Fu he mo e, in Sec ion 5, we show ha he company can also ex ac he la e buye ’s
addi ional su plus wi hou unning he ma ching pla o m i he sea ch is di ec ed wi h
he p ice pos ing in he e ening.
17Speci ically, no e om (19), (31), (32), and (33) ha he p i a e and social bene i s a e ob ained by
sub ac ing αLu(ˆ
xL(
dH(γ)))−ˆ
xL(
dH(γ))and αLu(
xL(γ))−
xL(γ), espec i ely, om αHu(
xH(γ))−
xH(γ).
Because la e buye s ca y mo e eal balances in E-equilib ium han in P-equilib ium expec ing o consume
cus omized goods, i mus be ha ˆ
xL(
dH(γ)) ≥
xL(γ)wi h he s ic inequali y when γ>β.Thus,we
sub ac a highe alue om αHu(
xH(γ)) −
xH(γ)when we calcula e he p i a e bene i han when we
calcula e he social bene i .
150 Kee-Youn Kang Theo e ical Economics 19 (2024)
The al e na i e in e p e a ion o he esul s o P oposi ion 3is ha he company does
no ully in e nalize he posi i e e ec s o using paymen da a on wel a e, so he socially
e icien E-money business may no exis in equilib ium. Speci ically, no e om (30),
(31), P oposi ion 1, and Lemmas 3and 4 ha WE(γ)−WP(γ)and he company’s p o i π
dec ease wi h γ.Thus,i γ∗and γ∗∗ exis such ha WE(γ∗)=WP(γ∗)and he company
ea ns ze o p o i when γ=γ∗∗,i mus be ha γ∗>γ
∗∗ by he esul s o P oposi ion 3.
Then, o all γ∈(γ∗∗,γ∗), he company does no un he E-money business al hough
he E-money business imp o es social wel a e. This esul is o mally s a ed in he nex
p oposi ion.
P oposi ion 4. Assume ha (Nς+ρδ)/(1−ρ)<D/θ. Then he e exis γ∗>βand
γ∗∗ ∈[β,γ∗)such ha o all γ∈(γ∗∗,γ∗),WE(γ)>W
P(γ), bu he company does no un
he E-money business and he economy is in P-equilib ium.
As explained ea lie , socially e icien uses o paymen da a may no exis because
he p i a e bene i o he E-money business is lowe han i s social bene i . Thus, he
go e nmen may ix his p oblem by subsidizing he E-money indus y. Speci ically, we
conside wo schemes o subsidy policy: The go e nmen p o ides uni s o nume ai e
goods o he company he nex mo ning i he company issues E-money (subsidy o he
company) and he go e nmen p o ides a selle /Nuni s o nume ai e goods he nex
mo ning o buying each e ening as e in o ma ion (subsidy o selle s).18,19 We assume
ha he go e nmen aises unds o he subsidy wi h lump-sum axes on buye s in he
mo ning.
No e ha he company can ex ac he go e nmen ’s subsidy o selle s by selling
p e e ence in o ma ion a a highe p ice. Howe e , he iden i y o he subsidy ecipi-
en could a ec selle s’ incen i es o p epa e he p oduc ion o cus omized goods and,
he eby, wel a e in a di e en way, as desc ibed in he nex p oposi ion.
P oposi ion 5. Suppose ha WE(γ)>W
P(γ)and D(
dH(γ)) <(Nς+ρδ)/(1−ρ),so he
company does no un he socially e icien E-money business. I he go e nmen suppo s
he E-money indus y wi h ≥[Nς+ρδ −(1−ρ)D(
dH(γ))]/β uni s o subsidy, hen
wel a e Wunde each subsidy scheme is gi en as ollows:
(I) Unde he subsidy o he company, W=WE(γ)i D(
dH(γ)) ≥Nς/(1−ρ)and W=
WP(γ)−ρδ i D(
dH(γ)) <Nς/(1−ρ).
(II) Unde he subsidy o selle s, W=WE(γ).
When D(
dH(γ)) ≥Nς/(1−ρ), selle s p epa e he p oduc ion o cus omized goods
unless p ice ϕis oo high as shown in (20), and he company can make nonnega i e
18The go e nmen can also subsidize ea ly buye s o using E-money in he a e noon. Howe e , i s
economic e ec s a e he same as hose o subsidizing he company, because he company can make ea ly
buye s use i s E-money wi h less (e en nega i e) ewa ds o using i in he a e noon.
19No e ha he o al amoun o subsidies unde each scheme is , because selle s buy Nnumbe o
e ening as e in o ma ion.
Theo e ical Economics 19 (2024) Digi al cu ency and p i acy 151
p o i unde each subsidy scheme. Howe e , when D(
dH(γ)) <Nς/(1−ρ), he inc ease
in he selle ’s expec ed su plus om selling cus omized goods is lowe han he ixed in-
es men cos Nς. Thus, selle s would no p epa e he p oduc ion o cus omized goods
wi hou any suppo . In his case, i he go e nmen gi es subsidies o he company, he
company issues E-money o ob ain subsidies wi hou selling p e e ence in o ma ion.
Thus, eal alloca ions a e he same as in P-equilib ium excep ha ea ly buye s incu
p i acy cos s, so W=WP(γ)−ρδ. On he o he hand, i he go e nmen subsidizes sell-
e s o buying p e e ence in o ma ion, hey will p epa e he p oduc ion o cus omized
goods. Thus, he economy is in E-equilib ium and W=WE(γ). As shown om he abo e
analysis, he subsidy o selle s is mo e e ec i e han he subsidy o he company because
he o me di ec ly incen i izes selle s, who a e he inal use s o p e e ence in o ma ion,
o ha ness p e e ence in o ma ion.
We now analyze he e ec s o mone a y policy, i.e., changes in γ,onwel a e. As
shown in P oposi ion 1,qand xdec ease wi h γin each ype o equilib ium. Because
he ade olumes in he a e noon and e ening ma ke s a e ine icien ly low o all
γ>β,wel a eWdec eases wi h γin each ype o equilib ium. Fu he mo e, as shown in
P oposi ion 2, a dec ease in γ ends o change he equilib ium ype om P-equilib ium
o E-equilib ium, he eby discon inuously inc easing wel a e by suppo ing socially e i-
cien uses o paymen da a. Thus, wel a e mono onically dec eases wi h γ. This implies
ha he op imal mone a y policy is he F iedman ule as s a ed in he nex p oposi ion,
whose p oo is omi ed.
P oposi ion 6. Op imal mone a y policy is he F iedman ule, i.e., γ=β.
The esul s o P oposi ion 6, howe e , do no necessa ily imply ha he F iedman
ule always achie es he highes wel a e. Al hough he F iedman ule elimina es he
money holding cos , he socially e icien E-money business may no exis due o he
ex e nali y p oblem. Speci ically, i D<(Nς+ρδ)/(1−ρ)<D/θ, he economy is in P-
equilib ium unde he F iedman ule as shown by P oposi ion 2, while wel a e is highe
wi h he ac i e E-money business han wi hou i as shown by (32)and(33). Howe e ,
wi h app op ia e use o he subsidy o selle s, he F iedman ule achie es he highes
wel a e in he model economy.
5. Compe i i e sea ch in he e ening
In his sec ion, we adop he concep o compe i i e sea ch, whe e selle s pos hei
e ms o ade o e ening ansac ions and la e buye s di ec hei sea ch in he
e ening, o unde s and how equilib ium ou comes depend on he ma ke s uc u e in
he e ening. We assume ha he economy is composed o di e en subma ke s in he
e ening, whe e a subma ke is iden i ied by i s e ms o ade pos ed by selle s.20 We u -
he assume ha wi hin any subma ke , la e buye s and selle s a e andomly ma ched.
20In his sec ion, we emo e he uppe bound β/θ o γ, because he selle ’s ba gaining powe θdoes no
ma e in he model wi h compe i i e sea ch.
152 Kee-Youn Kang Theo e ical Economics 19 (2024)
No e ha he equilib ium ou comes in he a e noon ma ke s a e he same as in ba -
gaining equilib ium, so we ocus on he equilib ium ou comes in he e ening ma ke .
The sequence o e en s is as ollows. A he beginning o he mo ning, he com-
pany announces i s ewa d policy Rand p ice ϕ o each e ening as e in o ma ion.
A e obse ing (R,ϕ), each selle decides whe he o p epa e he p oduc ion o cus-
omized goods in he e ening. Gi en he linea p e e ence o selle s in he mo ning and
e ening, a selle will ei he buy all e ening as e in o ma ion o buy no in o ma ion.21
I he selle chooses o sell cus omized goods in he e ening unde he assump ion ha
he/she can buy he co ec p e e ence in o ma ion, he selle pos s he e ms o ade
(xH,dH) o cus omized goods in he mo ning.22 O he wise, he selle pos s he e ms
o ade (xL,dL) o noncus omized goods in he mo ning. Finally, based on he ob-
se ed e ms o ade in he mo ning, la e buye s decide which pa icula subma ke
iden i ied by (xi,di)i∈{H,L} hey will isi in he e ening and he quan i y o eal balances
ha hey ca y in o he e ening ma ke .
Le Ubdeno e he expec ed su plus o a la e buye in he e ening gi en (R,ϕ),ne
o he money holding cos . Then, in any ac i e subma ke (xi,di) o each i∈{H,L},we
ha e
−(γ−β)di+min1, 1
niαiu(xi)−βdi=Ub, (34)
whe e nideno es he a io o buye s pe selle in a subma ke (xi,di).
When a selle pos s his/he e ms o ade in he mo ning, he selle akes Ubas
gi en, and (xi,di)de e mines he leng h o he queue, ni, in his/he subma ke . Speci i-
cally, i he selle chooses o sell noncus omized goods, his/he expec ed su plus is gi en
as
Vs
L=max
xL,dL,nLmin{1, nL}(−xL+βdL)(35)
subjec o (34)wi hi=L. On he o he hand, i he selle decides o sell cus omized
goods unde he assump ion ha he selle can buy he co ec p e e ence in o ma ion
om he company, hen his/he expec ed su plus is gi en as
Vs
H=max
xH,dH,nHmin{1, nH}(−xH+βdH)−N(ς+βϕ)(36)
subjec o (34)wi hi=H.
The ype o goods ha selle s can sell depends on whe he he company ac i ely
uns he E-money business. Fi s , i R≥δ/β, hen hecompanywillob ain heco ec
p e e ence in o ma ion, and selle s will op imally decide he ype o e ening goods ha
hey will sell. Thus, he selle ’s expec ed su plus, deno ed by Vs, om ac i ely pa ici-
pa ing in he e ening is gi en as Vs=max{Vs
H,Vs
L}. On he o he hand, i R<δ/β, hen
21We assume ha selle s choose o p epa e he p oduc ion o cus omized goods i hey a e indi e en .
22I a selle chooses o p epa e he p oduc ion o cus omized goods, he/she can pos he e ms o ade
o cus omized goods and noncus omized goods. Howe e , he selle p epa es he p oduc ion o cus-
omized goods only i i is be e o sell cus omized goods han noncus omized goods. Thus, we assume
ha he selle only pos s he e ms o ade o cus omized goods in he mo ning wi hou loss o gene ali y.
Theo e ical Economics 19 (2024) Digi al cu ency and p i acy 153
he company does no ob ain p e e ence in o ma ion, so selle s ha e no choice bu o
sell noncus omized goods in he e ening and Vs=Vs
L.
The equilib ium alue o he la e buye ’s expec ed su plus Ubgi en (R,ϕ)is de e -
mined such ha he a io o buye s pe selle in he di e en subma ke s is consis en
wi h he mass o la e buye s and selle s in he e ening. To make he analysis clea , we
index each selle by i∈[0, 1], de ine ⊆[0, 1]as he se o ac i e selle s pa icipa ing in
he e ening, and le n(i)deno e he measu e o la e buye s pe selle in he subma ke
o selle i∈. Then, in compe i i e sea ch equilib ium, he condi ion
Nd≡
n(i)di=Ns∈[0, 1 −ρ]
mus hold, whe e Ndis he agg ega e demand o ac i e la e buye s by selle s and Nsis
he agg ega e supply o ac i e la e buye s in he e ening.
To se he s age o equilib ium cha ac e iza ion, we i s de i e he uppe bound o
he la e buye ’s su plus in any equilib ium gi en (R,ϕ).Le
Ub
Land Ub
Hdeno e he
uppe bound o he la e buye ’s su plus when he/she chooses o buy noncus omized
goods and cus omized goods, espec i ely. Then, om (34)–(36), we ob ain
Ub
L=−γ
β¯
xL(γ)+αLu¯
xL(γ)(37)
Ub
H=−γ
β¯
xH(γ)+αHu¯
xH(γ)−γ
βN(ς+βϕ), (38)
whe e
¯
xi(γ)=u−1γ
βαi(39)
o each i∈{H,L}. Using hese de ini ions, we cha ac e ize he compe i i e sea ch equi-
lib ium gi en (R,ϕ)in he nex p oposi ion.
P oposi ion 7. De ine he cu o le el o he p ice o e ening as e in o ma ion as
ϕ∗=1
γN−γ
β¯
xH(γ)+αHu¯
xH(γ)−−γ
β¯
xL(γ)+αLu¯
xL(γ)−ς
β. (40)
Then he equilib ium ou comes wi h compe i i e sea ch gi en (R,ϕ)a e as ollows:
(I) I R≥δ/β and ϕ≤ϕ∗, hen all ac i e selle s sell cus omized goods, (xH,dH)=
(¯
xH(γ),[¯
xH(γ)+N(ς+βϕ)]/β),nH=1,Ub=Ub
H,Vs=0,andNd=Ns=1−ρ.
(II) I R<δ/βo ϕ>ϕ
∗, hen all ac i e selle s sell noncus omized goods, (xL,dL)=
(¯
xL(γ),¯
xL(γ)/β),nL≤1,Ub=Ub
L,Vs=0,andNd=Ns=1−ρ.
154 Kee-Youn Kang Theo e ical Economics 19 (2024)
In he e ening, la e buye s a e on he sho side o he ma ke because a uni mass
o selle s go o he e ening ma ke while a 1 −ρmass o la e buye s go he e. As a e-
sul , la e buye s ha e all he ma ke powe and ex ac he en i e ma ch su plus in com-
pe i i e sea ch equilib ium. Thus, selle s ecei e ze o su plus.23 Nex , he company
canno ob ain p e e ence in o ma ion i R<δ/βbecause no ea ly buye s use E-money.
Fu he mo e, selle s mus buy p e e ence in o ma ion o p epa e o he p oduc ion o
cus omized goods. Thus, selle s sell cus omized goods only i R≥δ/β and he p ice o
p e e ence in o ma ion is su icien ly low as ϕ≤ϕ∗. Since all ac i e selle s sell ei he
cus omized goods o noncus omized goods, we call equilib ium whe e selle s sell non-
cus omized (cus omized) goods P-equilib ium (E-equilib ium) simila o he baseline
model wi h he ba gaining.24
We now in es iga e he company’s op imal decision. Gi en he esul s o P oposi-
ion 7, he company will se (R,ϕ)=(δ/β,ϕ∗)i i chooses o un he E-money busi-
ness. No e ha nH=1andNd=1−ρin E-equilib ium. Thus, he company sells
p e e ence in o ma ion o he 1 −ρmeasu e o selle s and he p o i is gi en as πcs =
[(1−ρ)βNϕ∗−ρδ]/β. Using (40), we can exp ess he company’s p o i as a unc ion o
γsuch ha πcs =(γ)/β,whe e
(γ)=(1−ρ)−¯
xH(γ)+β
γαHu¯
xH(γ)+¯
xL(γ)−β
γαLu¯
xL(γ)−Nς−ρδ. (41)
Then he company will un he E-money business and he economy is in E-equilib ium
i and only i (γ)≥0. This leads o he nex p oposi ion.
P oposi ion 8. S a iona y mone a y equilib ium wi h compe i i e sea ch exis s as ol-
lows:
(I) I αHu(x∗
H)−x∗
H−[αLu(x∗
L)−x∗
L]≥[(1−ρ)Nς+ρδ]/(1−ρ), he eisγcs ≥β
such ha (a) o all γ∈[β,γcs], E-equilib ium exis s, and (b) o all γ>γ
cs,P-
equilib ium exis s.
(II) I αHu(x∗
H)−x∗
H−[αLu(x∗
L)−x∗
L]<[(1−ρ)Nς+ρδ]/(1−ρ), hen o all γ≥β,
P-equilib ium exis s.
The e ec s o δ,N,ς,ρ,andγon he equilib ium ype a e simila o hose in he
ba gaining model by he same a ionale. Howe e , in con as o he ba gaining model,
he e is a unique equilib ium unde compe i i e sea ch. This is because he e ms o
ade pos ed by selle s wo k as a coo dina ing de ice and in e nalize any s a egic com-
plemen a i y be ween la e buye s’ money demand and selle s’ decisions abou he ype
o goods.
23One hing o no e is ha when ac i e selle s sell cus omized goods, hey can o e la e buye s hei ma -
ke expec ed u ili y Ub
Hwi hou gene a ing a nega i e payo o hemsel es only i he e is no conges ion
on he selle ’s side, i.e., nH=1, in con as o he case when hey sell noncus omized goods. This is because
selle s mus incu cos N(ς+βϕ)be o e being ma ched o sell cus omized goods as shown in (36).
24All open subma ke s ha e he same e ms o ade because he solu ions o (35) and (36) a e unique.
Theo e ical Economics 19 (2024) Digi al cu ency and p i acy 161
Gi en ha Ub=Ub
H≥Ub
L,i mus be ha nL=0, so an ac i e subma ke o noncus-
omized goods does no exis .29 Because Vs=Vs
H=0, some selle s may be inac i e, so
Nd∈[0, 1]. On he o he hand, Ns=1−ρbecause Ub>0.
(ii) Suppose ha Ub∈(0, Ub
H). Then, om (34)wi hi=Hand (36), we ob ain (46),
and i mus be ha nH=1 because Ub< Ub
H. Nex , om (34)wi hi=Land (35),
we ob ain (47). Gi en ha Ub
H≥Ub
L,i mus be ha Vs
H≥Vs
Lwi h s ic inequali y
i nL<1o Ub
H> Ub
L. Thus, all ac i e selle s sell cus omized goods, (xH,dH,nH)=
(¯
xH(γ),[αHu(¯
xH(γ)) −Ub]/γ,1
)by (34)and(46), and Vs=Vs
H>0. Since Vs>0and
Ub>0, all selle s and la e buye s pa icipa e in he e ening, i.e., Nd=1andNs=1−ρ.
(iii) Suppose ha Ub=0. Then la e buye s a e indi e en be ween ac i ely pa -
icipa ing o no in he e ening. I some la e buye s pa icipa e, all ac i e selle s sell
cus omized goods, (xH,dH,nH)=(¯
xH(γ),[αHu(¯
xH(γ)) −Ub]/γ,1
),Vs=Vs
H>0, and
Nd=1 by he same easoning as he case whe e Ub∈(0, Ub
H). Since Ub=0, we ha e
Ns≤1−ρ.
F om he analysis o h ee cases abo e, one can show ha Nd=Nsonly i Ub=
Ub
H.Thus,i ϕ≤ϕ∗, hen all ac i e selle s sell cus omized goods, (xH,dH,nH)=
(¯
xH(γ),[¯
xH(γ)+N(ς+βϕ)]/β,1
),Ub=Ub
H,Vs=0, and Nd=Ns=1−ρin compe i i e
sea ch equilib ium.
Case 2. Assume ha ϕ>ϕ
∗,soUb
H< Ub
L. In wha ollows, we ocus on he case whe e
Ub≤Ub
Lbecause o he wise, no selle s will pa icipa e in he e ening.
(i) Suppose ha Ub=Ub
L.F om(34)wi hi=Hand (36), we ob ain (46). Since
Ub
H< Ub
L,i mus be ha nH=0 and he e exis s no ac i e subma ke in which cus-
omized goods a e aded. Nex , he solu ion o (35)isxL=¯
xL(γ),dL=¯
xL(γ)/β,and
nL≤1. Thus, any ac i e selle s pos (¯
xL(γ),¯
xL(γ)/β) o sell noncus omized goods in
he e ening and Vs=Vs
L=0. Since Vs=0andUb>0, Nd∈[0, 1]and Ns=1−ρ.
(ii) Suppose ha Ub∈(0, Ub
L). Then he solu ion o (35)is(xL,dL,nL)=(¯
xL(γ),
¯
xL(γ)/β,1
)and Vs
L>0. Fu he mo e, gi en ha Ub
H< Ub
L, i can be e i ied om (34)–
(36) ha Vs
L>Vs
H. Thus, any ac i e selle s sell noncus omized goods and Vs=Vs
L.Be-
cause Vs
L>0andUb>0, we ha e Nd=1andNs=1−ρ.
(iii) Suppose ha Ub=0. Then la e buye s a e indi e en be ween ac i ely pa ici-
pa ing o no in he e ening. I some la e buye s pa icipa e, all ac i e selle s sell non-
cus omized goods, (xL,dL,nL)=(¯
xL(γ),¯
xL(γ)/β,1
),Vs=Vs
L>0, and Nd=1 o he
same easons as he case whe e Ub∈(0, Ub
L). Because Ub=0, we ha e Ns≤1−ρ.
The analysis o he h ee cases abo e shows ha Nd=Nsonly i Ub=Ub
L.Thus,i
ϕ>ϕ
∗, henUb=Ub
L,Vs=0, all ac i e selle s pos (xL,dL)=(¯
xL(γ),¯
xL(γ)/β) o sell
noncus omized goods, nL≤1, and Nd=Ns=1−ρin compe i i e sea ch equilib ium.
29In he kni e edge case wi h Ub
H=Ub
L,nL≥0 is easible and Vs
L=0. Howe e , we assume ha ac i e sell-
e s sell cus omized goods when hey a e indi e en be ween selling cus omized goods and noncus omized
goods. Thus, no selle s sell noncus omized goods when Ub=Ub
H≥Ub
L.
162 Kee-Youn Kang Theo e ical Economics 19 (2024)
Now suppose ha R<δ/β. Then he company canno ob ain e ening as e in o ma-
ion in he e ening, so any ac i e selle s sell noncus omized goods. Thus, equilib ium
ou comes a e equi alen o he case wi h R≥δ/β and ϕ>ϕ
∗. By combining he equi-
lib ium ou comes in Cases 1 and 2, and he case wi h R<δ/β, we ob ain he esul s o
P oposi ion 7.
P oo o P oposi ion 8.No e om(1), (39), and (41) ha limγ→∞ (γ)=−
(1−
ρ)Nς−ρδ,(β)=(1−ρ){αHu(x∗
H)−x∗
H−[αLu(x∗
L)−x∗
L]}−(1−ρ)Nς−ρδ and
(γ)<0. Thus, i αHu(x∗
H)−x∗
H−[αLu(x∗
L)−x∗
L]≥[(1−ρ)Nς+ρδ]/(1−ρ), he e
exis s γcs ≥βsuch ha (γcs )=0. Then, o all γ∈[β,γcs],(γ)≥0 and E-equilib ium
exis s, and o all γ>γ
cs,(γ)<0 and P-equilib ium exis s. On he o he hand, i
αHu(x∗
H)−x∗
H−[αLu(x∗
L)−x∗
L]<[(1−ρ)Nς+ρδ]/(1−ρ), hen o allγ≥β,(γ)<0,
so P-equilib ium exis s.
P oo o P oposi ion 9. A beha io s a egy o company j∈{1, ,J}in he mo ning
is he ixed ewa d Rj o using i s E-money in he a e noon and a beha io s a egy
in he e ening is he p ice o e ening as e in o ma ion ϕj. No e ha se ing Rj<δ/β
is equi alen o no unning he E-money business because no ea ly buye s use j’s E-
money, so company jdoes no ob ain e ening as e in o ma ion. Thus, ϕjis i ele an .
Wi hou loss o gene ali y, we se Rj=0andϕj=ε>0i companyjchooses no o un
he E-money business. Le L⊆{1, ,J}deno e he se o companies ha issue hei E-
money wi h he ixed ewa d R≥δ/β. In wha ollows, we sol e subgame pe ec Nash
equilib ium by using backwa d induc ion s a ing om a game in he e ening.
In he subgame in he e ening, suppose ha |L|=1, whe e |L|is he ca dinali y o L.
Then he op imal beha io s a egy o company j∈Lin he e ening is monopoly p ic-
ing gi en by (23) and i is Nash equilib ium o his subgame. Nex , suppose ha |L|>1.
Selle s will buy p e e ence in o ma ion om one o he companies ha sell he in o ma-
ion a he lowes p ice, because all companies in he se Lha e he co ec in o ma ion.
Suppose ha ϕmin ≡minj∈Lϕj>0. Then any company j∈Lcan aise i s e enue by
se ing he p ice o p e e ence in o ma ion as ϕ
j=ϕmin −ε o a su icien ly small ε>0
o a ac all selle s. Thus, he only Nash equilib ium in he subgame in he e ening wi h
|L|>1 consis s o ϕj=0 o allj∈L.
We now analyze companies’ op imal decisions in he mo ning. The abo e analy-
sis shows ha whene e |L|>1, companies in he se Lend up making nega i e p o -
i s as π=−ρδ/β. Suppose ha he e is one company j∈{1, ,J}wi h he s a egy
{Rj,ϕj}={δ/β,ϕ},whe eϕ=[(1−ρ)D(mp)−Nς]/βNi |L|=1
0i |L|>1. Then he bes e-
sponses o o he companies j= ja e no o un he E-money business. Gi en ha
all o he companies j= jdo no un he E-money business, company j’s s a egy
{Rj,ϕj}={δ/β,ϕ}is he bes esponse. Thus, o any j∈{1, ,J}, a p o ile o s a e-
gies {{Rj,ϕj},{Ri,ϕi}i=j}={{δ/β,ϕ},{0, ε}} cons i u es subgame pe ec Nash equilib-
ium.
Theo e ical Economics 19 (2024) Digi al cu ency and p i acy 163
Appendix B: Rewa d policy o using E-money
In he main body, we assumed ha he company p o ides only he ixed ewa d o ea ly
buye s o using E-money. Howe e , he company can also p o ide p opo ional e-
wa ds: The company can subsidize he κa∈[0, 1]and κe∈[0, 1] ac ions o he E-
money paymen s in he a e noon and e ening, espec i ely.30 In his appendix, we
show ha he assump ion ha he company only p o ides he ixed ewa d is wi hou
loss o gene ali y by illus a ing ha he p o i maximizing company does no p o ide
p opo ional ewa ds.
Gi en he ixed ewa d R, p i acy cos δ, and p opo ional ewa d κa≥0, ea ly buy-
e s will ei he use E-money o P-money. I he ea ly buye chooses o use E-money, hen
his/he su plus is
Sea ly
e=max
qeβR −δ−γp(1−κa)qe+υ(qe), (48)
and i he ea ly buye chooses o use P-money, hen his/he su plus is
Sea ly
p=max
qp−γpqp+υ(qp). (49)
Gi en he monopoly powe , he company will se Rand κasuch ha Sea ly
e=Sea ly
p.
By subs i u ing he op imali y condi ions, γp(1−κa)=υ(qe)and γp =υ(qp)in o (48)
and (49), espec i ely, he condi ion ha Sea ly
e=Sea ly
pgi es
βR =δ−−γp(1−κa)υ−1γp(1−κa)+υυ−1γp(1−κa)
+−γpυ−1(γp)+υυ−1(γp).
Nex , he company pays pκaqeuni s o E-money, which is backed by P-money, o selle s
in he a e noon as subsidies o buying goods wi h E-money in he a e noon ma ke .
Combined wi h he ixed ewa d, he o al cos o a ac ing each ea ly buye is gi en as
δ+γpυ−1γp(1−κa)−υυ−1γp(1−κa)−γpυ−1(γp)+υυ−1(γp).
No e ha he e m in he squa e b acke is s ic ly posi i e o all κa>0andze owi h
κa=0. Thus, i is op imal o he company o se κa=0.
We now show ha a company’s p o i dec eases wi h κe, so i is op imal o he com-
pany o se κe=0. No e ha he e is no eason o he company o p o ide any ewa ds
o la e buye s o using E-money in he e ening i ea ly buye s do no use E-money in he
a e noon. Thus, we assume ha ea ly buye s use E-money. Simila o ea ly buye s, la e
buye s will ei he use P-money o E-money o e ening ansac ions gi en he p opo -
ional ewa d κe≥0 and he ixed p i acy cos δ. To analyze how κea ec s a company’s
p o i , we assume ha la e buye s choose o use E-money o e ening ansac ions.
30Because e ening ansac ion da a ha e no alue o he company, i he company p o ides any ewa d
o la e buye s, i mus be a p opo ional ewa d ha could a ec la e buye s’ ade olume in he e ening
ma ke .
164 Kee-Youn Kang Theo e ical Economics 19 (2024)
Gi en ha la e buye s hold E-money, we can ew i e he ba gaining p oblem in he
e ening as
max
x,dαiu(x)−x+βκed(50)
subjec o
−x+βd =θαiu(x)−x+βκed(51)
(1−κe)d≤me(52)
o each i∈{H,L}.Le ˆ
xe
i(me)and ˆ
de
i(me)deno e he solu ion o he abo e maximiza-
ion p oblem gi en i∈{H,L}and he la e buye ’s E-money holdings me.No e ha
ˆ
xe
i(me)inc eases wi h κewhene e cons ain (52) binds.
Gi en he p opo ional ewa d κeand la e buye ’s E-money holdings me, selle s buy
all p e e ence in o ma ion i De(me)≥N(ς+βϕ)/[(1−ρ)κ(B)],whe e
De(me)=θαHuˆ
xe
H(me)−ˆ
xe
H(me)+βκeˆ
de
H(me)
−αLuˆ
xe
L(me)−ˆ
xe
L(me)+βκeˆ
de
L(me)(53)
is an inc ease in he selle ’s ade su plus in he e ening by p epa ing he p oduc ion o
cus omized goods when he company p o ides p opo ional ewa ds o using E-money
in he e ening.
Using he monopoly powe , he company will se he p ice o p e e ence in o ma ion
as ϕ=[(1−ρ)κ(B)De(me)−Nς]/βNand sell all p e e ence in o ma ion o all selle s.
The cos o unning he E-money business consis s o he ixed ewa ds o ea ly buy-
e s o using E-money in he a e noon and κedeuni s o E-money ans e s in each
bila e al mee ing in he e ening as p opo ional ewa ds o la e buye s. Then we ob ain
discoun ed ne p o i as
βπ =(1−ρ)De(me)−Nς−ρδ +(1−ρ)γκede, (54)
whe eweimpose hecondi ion ha κ(B)=1, because all ea ly buye s use E-money in
he a e noon ma ke , i.e., B=ρ, gi en ha R=δ/β.
No e ha la e buye s will minimize idle E-money ha is no used in he e ening
ma ke . This implies ha (1−κe)ˆ
de
H(me)=meby (52). Then, om (51)–(53), we ob ain
De(me)=−ˆ
xe
H(me)+βme
1−κe
−θαLuˆ
xe
L(me)−ˆ
xe
L(me)+βκeˆ
de
L(me). (55)
No e ha he e m αLu(ˆ
xe
L(me)) −ˆ
xe
L(me)+βκeˆ
de
L(me)inc eases wi h κeas shown in
(50)–(52). Then, om (51), he binding (52), (54), and (55), we ob ain
∂(βπ)
∂κe
≈−∂ˆ
xe
H(me)
∂κe
−me(γ−β)
(1−κe)2−θ∂αLuˆ
xe
L(me)−ˆ
xe
L(me)+βκeˆ
de
L(me)
∂κe
<0.
Theo e ical Economics 19 (2024) Digi al cu ency and p i acy 165
This implies ha i is op imal o he company o se κe=0. Thus, he p o i maximizing
company does no p o ide any p opo ional ewa ds: κa=κe=0.
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Co-edi o Flo ian Scheue handled his manusc ip .
Manusc ip ecei ed 18 Oc obe , 2021; inal e sion accep ed 23 Feb ua y, 2023; a ailable on-
line 3 Ma ch, 2023.
