Banking crises under a central bank digital currency (CBDC)

Bi e , Lea
A icle
Banking c ises unde a cen al bank digi al cu ency
(CBDC)
Jou nal o Economics and S a is ics
P o ided in Coope a ion wi h:
De G uy e B ill
Sugges ed Ci a ion: Bi e , Lea (2025) : Banking c ises unde a cen al bank digi al cu ency (CBDC),
Jou nal o Economics and S a is ics, ISSN 2366-049X, De G uy e Oldenbou g, Be lin, Vol. 245, Iss.
4/5, pp. 479-526,
h ps://doi.o g/10.1515/jbns -2023-0107
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/333320
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Lea Bi e *
Banking C ises Unde a Cen al Bank Digi al
Cu ency (CBDC)
h ps://doi.o g/10.1515/jbns -2023-0107
Recei ed Decembe 21, 2023; accep ed July 3, 2024
Abs ac : One o he main conce ns associa ed wi h cen al bank digi al cu encies
(CBDC) is he disin e media ing effec on he banking sec o in gene al, and he isk
o bank uns in imes o c isis in pa icula . This pape examines he implica ions o
an in e es -bea ing CBDC on banking c ises in a dynamic bank un model wi h a
financial accele a o . The analysis dis inguishes be ween bank ailu es due o illi-
quidi y and due o insol ency. In a nume ical exe cise, CBDC leads o a educ ion in
he ne wo h o banks in no mal imes bu mi iga es he isk o a bank un in imes o
c isis. The financial s abili y implica ions also depend on how CBDC is accoun ed o
on he asse side o he cen al bank balance shee : i CBDC issuance is complemen ed
by asse pu chases, i delays he onse o bo h ypes o bank ailu es o la ge shocks.
In con as , i CBDC issuance is complemen ed by loans o banks, i subs an ially
impedes ailu es due o illiquidi y, bu only ma ginally affec s bank ailu es due o
insol ency.
Keywo ds: cen al bank digi al cu ency (CBDC); financial in e media ion; financial
s abili y; bank uns
JEL Classifica ion: E42; E58; G01; G21
1 In oduc ion
Digi al ans o ma ion has and will u he change ou paymen and mone a y
sys em. Th oughou his o y, money has adap ed o e ime and i s o m has changed
significan ly – om cow ie shells and commodi y money o commodi y-backed
money and fia money. To adjus o he de elopmen s o an inc easingly digi al
paymen sys em, cen al banks ha e s a ed o explo e he po en ial o a cen al bank
A icle No e: This a icle is pa o he special issue “Cen al Bank Digi al Cu ency”published in he
Jou nal o Economics and S a is ics. Access o u he a icles o his special issue can be ob ained a www.
deg uy e .com/jbns .
*Co esponding au ho : Lea Bi e , Eu opean Cen al Bank, F ank u am Main, Ge many,
E-mail: [email p o ec ed]. h ps://o cid.o g/0000-0003-4148-3718
Jou nal o Economics and S a is ics 2025; 245(4–5): 479–526
Open Access. © 2024 he au ho (s), published by De G uy e . This wo k is licensed unde he
C ea i e Commons A ibu ion 4.0 In e na ional License.
digi al cu ency (CBDC).
1
A e ail CBDC would be inhe en ly isk- ee public money,
simila o cash, bu wi h he addi ional con enience o being digi al and en ailing he
op ion o bea in e es . These ea u es make CBDC a close subs i u e o bank deposi s
which could ad e sely affec he deposi unding o comme cial banks.
One o he main conce ns in he discussion on CBDC e ol es a ound i s impac
on he banking sec o in gene al, bu especially in imes o c isis (e.g. see, Bank o
In e na ional Se lemen s 2018; Ca ney 2018; ECB 2020). The gene al ea is ha an
a ac i e CBDC could s uc u ally disin e media e he banking sec o . These ea s
a e amplified in imes o c isis. Bo h deposi s and CBDC a e digi al, con enien , and
po en ially in e es -bea ing, making hem ela i ely close subs i u es. This ela i ely
close subs i u abili y o deposi s and CBDC, combined wi h he inhe en ly iskless
na u e o public money, aises he conce n ha a CBDC may igge bank uns mo e
easily. When confidence in he financial sys em is low, a CBDC may p o ide an
a ac i e ou side op ion, he eby educing he cos o deposi o s o wi hd awing
hei unds om he bank. The po en ially lowe cos o deposi o s o pa icipa ing in
a bank un aises conce ns ha uns may occu unde a CBDC when deposi o s
o he wise would no ha e un. A u he conce n is ha uns could un old mo e
quickly due o he (po en ially unlimi ed) 24/7 digi al access o CBDC.
This pape shows ha he conce ns ou lined in he p e ious pa ag aph may no
necessa ily subs an ia e. Analysing hese conce ns in a dynamic bank- un ame-
wo k, I find ha a CBDC does indeed make bank uns less dis up i e, bu his does no
impai financial s abili y; on he con a y, i makes i mo e difficul o bank uns o
occu . The esul s highligh he significance o assessing he gene al equilib ium
effec s o he in oduc ion o a CBDC on financial s abili y. Pa icula ly, he asse -side
adjus men s on he cen al bank’s balance shee esul ing om CBDC issuance can
ha e a s abilising impac on he economy ha can domina e he disin e media ing
impac o he ou flow o deposi s in o CBDC.
To s udy he impac o a CBDC on agg ega e banking c ises and he possibili y o
sys emic bank uns, I in oduce an in e es -bea ing CBDC in o he dynamic bank un
amewo k o Ge le and Kiyo aki (2015).
2
The amewo k allows o combine
financial accele a o dynamics wi h he eme gence o bank uns, which a e based on
mac oeconomic undamen als bu ul ima ely igge ed by belie s. Such amewo k
also allows o dis inguish be ween bank ailu es due o illiquidi y and insol ency. In
con as o al e na i e bank un amewo ks in he li e a u e, which a e mos ly
limi ed o wo o h ee pe iods, in he Ge le and Kiyo aki (2015) amewo k
1Acco ding o su eys by he BIS, he sha e o cen al banks s udying CBDC inc eased om below
70 % in 2017 (Ba on ini and Holden 2019) o 93 % in 2022 (Kosse and Ma ei 2023).
2The analysis ocuses only on agg ega e c ises and uns. The impac on indi idual bank ailu es is
no conside ed. The main iew in he li e a u e is ha a CBDC is unlikely o ha e a majo impac on
indi idual bank uns (e.g. see Ba dea and Kumho 2022; Juks 2018; Mancini-G iffoli e al. 2018;
Meaning e al. 2018).
480 L. Bi e
households and banks op imise in an infini e ho izon economy, allowing o iche
and mo e ealis ic dynamics.
The s ylised endowmen economy wi h financial ic ions consis s o households,
banks, and a cen al bank which issues CBDC ollowing an in e es - a e ule. Se ing as
p oxy o he fi m sec o , capi al is modelled as a p oduc i e echnology yielding a e u n
each pe iod ha is subjec o agg ega e shocks. Households can place hei sa ings in
CBDC, bank deposi s, o capi al. The model analysis abs ac s om cash and ocuses only
on deposi subs i u ion in o CBDC which is he mo e ele an case and a mo e conse -
a i e assump ion o financial s abili y conside a ions.
The analysis allows o dis inguish be ween banking ailu es due o illiquidi y
and bank ailu es due o insol ency. While banks a e mos efficien in in e media ing
capi al, hey a e limi ed by a le e age cons ain . In no mal imes, he model ea u es
a unique equilib ium in which banks in e media e he majo i y o asse s. Howe e , i
asufficien ly la ge shock hi s he economy, a second equilib ium eme ges which is
cha ac e ised by a bank un on he en i e banking sys em. Bank ailu es due o
illiquidi y can eme ge i households wi hd aw hei deposi s because hey belie e
ha o he agen s will un, leading o a bank un igge ed by sel - ulfilling belie s. I
he shocks become e en la ge , a a ce ain poin only he bank un equilib ium
emains and banks ail due o insol ency.
The in oduc ion o a CBDC c ea es an addi ional liabili y on he cen al bank
balance shee ha will lead o u he balance shee adjus men s, con ibu ing o he
impac o a CBDC on he economy. This pape accoun s o he po en ially a ying
implica ions o diffe en balance shee adjus men s by analysing CBDC issuance in
he con ex o wo policy scena ios: in he fi s policy scena io, he cen al bank
complemen s CBDC issuance on he liabili y side by lending o banks on he asse side
o he cen al bank’s balance shee (‘c edi policy’); in he second policy scena io, he
cen al bank complemen s CBDC issuance by pu chasing capi al secu i ies (‘asse
policy’). Bo h policy scena ios would lead o an inc ease in he cen al bank balance
shee and a e con as ed o he economy wi hou a CBDC (‘no CBDC’).
3
A simula ion o he model offe s ou main insigh s: fi s , in s eady s a e, a CBDC
does no majo ly affec agg ega e ou pu and p ices bu i does affec he composi ion
o household sa ings, bank unding and capi al in es men , leading o a educ ion in
bank p ofi s. Second, bo h CBDC policy scena ios end o ha e a small bu s abilising
influence in a c isis ha does no igge a bank un. Thi d, hese s abilising effec s
become mo e impo an in a bank un si ua ion: bo h CBDC policy scena ios imp o e
financial s abili y by pos poning he eme gence o bank uns o la ge shocks.
Fou h, he impac o a CBDC also depends on he accompanying cen al bank
3A CBDC will inc ease he size o he cen al bank balance shee i CBDC issuance is no ully offse by
a educ ion in excess ese es o cash.
Banking C ises Unde a CBDC 481
balance shee adjus men s; CBDC issuance wi h ‘asse policy’delays he onse o bo h
ypes o bank ailu es o la ge shocks. In con as , CBDC issuance wi h ‘c edi policy’
subs an ially impedes ailu es due o illiquidi y, bu has li le impac bank ailu es
due o insol ency.
4
O e all, while CBDC issuance s ains he banking sec o in s eady
s a e by educing deposi s and ne wo h, i imp o es financial s abili y in imes o
c isis by impeding he eme gence o bank uns.
Wha mechanism lies a he oo o he esul ha CBDC issuance delays he
eme gence o bank uns –a esul ha uns con a y o he ini ially s a ed conce ns?
In he model amewo k, CBDC issuance makes bank uns less dis up i e because he
cen al bank channels CBDC inflows back in o he economy, ei he ia c edi o banks
o di ec capi al pu chases. This educes he bu den on households and inc eases he
efficiency and p ice o capi al. These cen al bank in e en ions ha channel he
CBDC inflows back in o he economy slow he financial accele a o dynamics and
make he occu ence o bank uns mo e difficul , ul ima ely pos poning hei
occu ence o la ge shocks.
Na u ally, he cen al bank can ake measu es independen o CBDC issuance o
imp o e financial s abili y and o a e sys emic uns. The main insigh o his model
analysis is ha in a bank un si ua ion inflows in o CBDC, ins ead o inflows in o
o he less efficien asse s, a e aligned wi h financial s abili y measu es and do no
exace ba e bank un isks. In a bank un si ua ion, households do no wi hd aw
deposi s because hey like CBDC bu because hey a e conce ned abou he sa e y o
hei deposi s. By p o iding hem wi h a sa e and mo e efficien al e na i e o un
in o, he issuance o CBDC mi iga es losses in a bank un scena io, s abilising capi al
p ices and making uns less likely om he ou se .
Con ex o he li e a u e: The ela ed li e a u e also suppo s a nuanced pic-
u e o he financial s abili y implica ions o a CBDC. The issuance o CBDC is ound o
ha e ambiguous financial s abili y implica ions, depending no only on adjus men s
on he asse side o he cen al bank’s balance shee bu also on in o ma ion effec s,
he s uc u e o he financial sec o , CBDC emune a ion, and o he no able design
pa ame e s.
5
O e all, he esul s o Keis e and Monne (2022) and Muñoz and Soons (2024) a e
quali a i ely in line wi h he insigh s o his pape , finding a benign impac o CBDC
on financial s abili y. Bo h Keis e and Monne (2022) and Muñoz and Soons (2024)
find ha unde a CBDC banks ha e he incen i e o ebalance hei po olio mo e
4In ac , in he baseline calib a ion unde CBDC wi h c edi policy, sel - ulfilling uns do no occu .
Howe e , his is no a gene al esul and sel - ulfilling uns eme ge in o he calib a ions.
5In con as o his and he ollowing pape s, which ocus on CBDC implica ions and bank uns on
comme cial banks, Fe nández-Villa e de e al. (2021) and Schilling, Fe nández-Villa e de, and Uhlig
(2024) conside bank uns on he cen al bank.
482 L. Bi e

owa ds long- e m liabili ies and lending which educes he ma u i y misma ch o
bank asse s and ends o imp o e financial s abili y. Consis en wi h he mechanism
included in his pape , in he model o Keis e and Monne (2022) he an icipa ion o a
less se e e bank un scena io educes he incen i e o un om he ou se .
Fu he mo e, Keis e and Monne (2022) a gue ha he issuance o CBDC gi es he
cen al bank an in o ma ion ad an age on financial s abili y condi ions h ough
which i can in e ene mo e efficien ly.
6
In he pape s o Kim and Kwon (2023), Ahne e al. (2023), Lucas (2022), and Skeie
(2021), he di ec ional financial s abili y implica ions o a CBDC a e con ex -dependen .
Theimpo anceo heasse sidecoun e pa o CBDC ope a ions is simila ly highligh ed
by Kim and Kwon (2023) in an OLG model wi h a diffe en defini ion o bank uns. In
hei model, bank uns a e no sel - ulfilling bu se up as si ua ions in which deposi o s
who need o wi hd aw money canno be ully epaid. They find ha CBDC issuance ha is
complemen ed by loans o banks inc eases financial s abili y. Howe e , when he
complemen ing asse side ope a ions a e nei he di ec ly (e.g. ia di ec lending) o
indi ec ly (e.g. ia s abilising asse p ices ia asse pu chases) suppo ing banks, hen he
issuance o a CBDC de e io a es financial s abili y.
In Ahne e al. (2023), Lucas (2022), and Skeie (2021), he emune a ion policy o
CBDC and cen al bank efinancing ope a ions is key o de e mine he impac on
financial s abili y. In he global game analysis o he wo-pe iod bank un model o
Ahne e al. (2023), CBDC emune a ion has wo coun e ailing o ces ha can ei he
inc ease o dec ease financial agili y and he p obabili y o a bank un. While a
highe CBDC emune a ion inc eases he incen i e o deposi wi hd awal, i also
induces banks o offe mo e a ac i e deposi condi ions, educing he incen i e o
wi hd aw deposi s. The o e all impac is ul ima ely de e mined by he esponsi e-
ness o deposi condi ions o CBDC emune a ion which depends on he deg ee o
ma ke powe . Lucas (2022) shows ha he cen al bank can p e en bank uns by
allowing cen al bank efinancing ope a ions a sufficien ly a ac i e condi ions o
banks. Simila ly, Skeie (2021) finds ha bank uns in o CBDC can be p e en ed by
dynamically adjus ing emune a ion o CBDC and ese es. The significance o
emune a ion as a design pa ame e is also eflec ed in his pape , in which he
dynamic adjus men o CBDC issuance condi ions in esponse o fluc ua ions in CBDC
demand and ou pu plays an impo an ole in he o e all impac .
The esul ha CBDC issuance makes banking panics less dis up i e is also is in
line wi h he findings o Williamson (2021). Howe e , in con as o Keis e and
Monne (2022) and his pape , he lesse se e i y o uns ends o encou age banking
panics. This opposi e finding in he model o Williamson (2021) can be explained by
CBDC offe ing an imp o ed al e na i e o e ail paymen s and he eby eplacing
6Howe e , simila in o ma ion may al eady be ob ainable h ough RTGS sys ems like T2.
Banking C ises Unde a CBDC 483
paymen s ha would o he wise ha e been made wi h deposi s. The e o e, in hei
model he abili y o CBDC o ensu e paymen efficiency du ing a un, educes he cos
o unning and in his case makes uns mo e likely.
The e a e se e al pape s s udying he s uc u al impac o CBDC on financial
in e media ion, complemen ing he analyses o he impac o CBDC du ing a finan-
cial c isis (e.g. see Andol a o 2021; Chiu e al. 2023; Keis e and Sanches 2023). Ahne
e al. (2022) and Chapman e al. (2023) bo h su ey he g owing CBDC li e a u e wi h a
specific ocus on financial in e media ion and s abili y. Se e al pape s discuss policy
measu es aimed o p e en excessi e CBDC holdings, such as holding limi s
(Assenmache e al. 2021; Bindseil, Pane a, and Te ol 2021; Melle and Soons 2023),
ie ed emune a ion (Bindseil 2019), and limi ed con e ibili y wi h ese es and
deposi s (Kumho and Noone 2021).
This pape con ibu es o he li e a u e on financial s abili y implica ions o a CBDC
in h ee ways. Fi s , i ocuses explici ly on he po en ially a ying implica ions o
diffe en cen al bank balance shee asse side adjus men s in esponse o CBDC issu-
ance. Second, i dis inguishes be ween banking ailu es due o illiquidi y and insol ency
and assessing he implica ions o a CBDC on he ealisa ion o bo h ypes o banking
ailu es. Thi d, while he abo e pape s mos ly analyse financial s abili y implica ion in a
wo o h ee pe iod model, his pape analyses he financial s abili y implica ions in a
mo e dynamic infini e ho izon economy wi h a financial accele a o . The p esen ed
model ocuses on he s o e o alue unc ion o money as i is mos impo an in he
con ex o bank uns.Howe e , heanalysisneglec s he oleo moneyasamediumo
exchange e en hough mos cen al banks aim o design CBDC as a means o exchange
a he han an a ac i e s o e o alue. Fu he mo e, he model poses a e y s ylised
eal se ing which could be ex ended o an economy wi h a iche fi m sec o , nominal
ic ions, con en ional mone a y policy, and a mo e sophis ica ed go e nmen sec o .
He eina e , Sec ion 2 discusses cen al bank balance shee adjus men s o CBDC
issuance in mo e de ail. Sec ion 3 p esen s he gene al model economy and he bank
un scena ios which is subsequen ly simula ed and analysed in Sec ion 4. Finally,
Sec ion 5 concludes.
2 CBDC Issuance and he Cen al Bank Balance
Shee
This sec ion e iews diffe en scena ios o how CBDC issuance can affec he cen al
bank balance shee . The analysis highligh s he ele ance o aking in o accoun
cen al bank balance shee adjus men s in esponse o CBDC issuance o
484 L. Bi e
equilib ium dynamics and p o ides mo i a ion o he wo scena ios employed in
he subsequen model analysis.
2.1 Gene al Cen al Bank Balance Shee Dynamics o CBDC
Issuance
The idea o opening he cen al bank balance shee o deposi s by he public is no
new. Qui e con a y, i was a common cen al bank p ac ice un il mid-20 h cen u y.
As e.g. elabo a ed by Fe nández-Villa e de e al. (2021), his o ically many majo
cen al banks no only allowed deposi s bu also g an ed loans o p i a e fi ms and
indi iduals. The idea o enabling deposi s a he cen al bank o he public was
p ominen ly b ough back on s age and in o he elec onic sphe e by Tobin (1985,
1987) om which he cu en discussion mos ly picks up.
The in oduc ion o a CBDC would c ea e a new ype o liabili y ha can igge
diffe en adjus men s on he cen al bank balance shee , con ibu ing in diffe en
ways o de e mine he equilib ium impac o a CBDC. Fo ins ance, Ba ke , Bhola ,
and Thomas (2017) a gue ha “al hough he e has been a lo o discussion abou how
cen al bank digi al cu ency could adically change paymen sys ems –and e en he
financial sec o as a whole – he implica ion o he asse s on cen al bank balance
shee s could be jus as c i ical”. Howe e , in mos CBDC model analyses, he
modelling choice ha complemen s CBDC issuance on he cen al bank balance shee
emains in he backg ound and i s implica ions a e no explici ly discussed.
7
In
p ac ice, hese balance shee adjus men s will depend on mone a y policy, he eco-
nomic and ins i u ional con ex and will be difficul o link di ec ly o CBDC. None-
heless, o analy ical pu poses, e alua ing he impac o diffe en measu es in
isola ion can p o ide aluable insigh s.
This pape ocuses on he mos ele an CBDC scena ios which would lead o an
inc ease in he cen al bank balance shee . CBDC issuance will ei he lead o ea -
angemen s on he liabili y side o he cen al bank balance shee o o an inc ease in
bo h, liabili ies and asse s. In mos cases, ei he cash o deposi s will be subs i u ed
in o CBDC.
8
The subs i u ion o cash in o CBDC would simply cons i u e a change in
he ype o cen al bank money held by he public and would be neu al om a
financial s abili y pe spec i e.
9
The e o e, he ollowing analysis abs ac s om cash
7No able excep ions a e o ins ance F aschini, Somoza, and Te acciano (2021) o Niepel (2020).
8The a gumen implici ly assumes ha he issuance o CBDC only leads o a edis ibu ion bu does
no affec he o al olume o financial asse s held by households. This ea u e also eme ges om he
model analysis and is consis en wi h wha can be expec ed empi ically.
9In addi ion, ocusing on si ua ions o financial agili y, households may no wan o exchange cash
in o CBDC bu a he inc ease hei holdings o inhe en ly sa e cen al bank money.
Banking C ises Unde a CBDC 485
and ocuses only on deposi subs i u ion in o CBDC. The subs i u ion o deposi s in o
CBDC would ei he lead o a educ ion in excess ese es (no impac on he size o he
cen al bank balance shee ) o equi e a coun e posi ion on he asse side (inc easing
he size o he cen al bank balance shee ).
10
Si ua ions in which deposi ou flows
canno be abso bed by a educ ion in excess ese es pose mo e se ious financial
s abili y conce ns. The e o e, o simplici y he ollowing analysis abs ac s om
modelling ese es and ocuses on scena ios in which CBDC issuance leads o an
inc ease in he cen al bank balance shee .
2.2 Deposi Subs i u ion in o CBDC ia Cen al Bank C edi
(‘C edi Policy’)
As a fi s and mos p ominen ly discussed op ion, he cen al bank could inc ease i s
c edi ope a ions o banks in o de o complemen CBDC issuance. Such adjus men s
would channel los deposi unding back o he banks in he o m o cen al bank
c edi . Lending o financial ins i u ions is al eady a subs an ial componen o cen al
bank asse s in no mal imes bu becomes pa icula ly impo an in imes o c isis.
The ole o he cen al bank as ‘lende o las eso ’, da ing back o Bageho (1873), is
also documen ed by Pa ipeilohy (2016) who shows a clea shi owa ds p i a e
sec o lending (including banks) by he Fede al Rese e, Bank o England and he
Eu osys em in esponse o he financial c isis om 2007 and onwa d.
Gi en he pa icula ele ance o bank lending in imes o c isis, he ollowing
model amewo k will explici ly e iew CBDC issuance wi h ‘c edi policy’as illus-
a ed in Table 1.
11
This scena io also p o ides he basis o he ‘equi alence esul o
Table :CBDC issuance wi h c edi policy in he s ylised model amewo k.
Households Banks Cen al bank
AL ALAL
Capi al Sec. Equi y Capi al Sec. Deposi s Loans CBDC
Deposi s (Endowmen ) Loans
CBDC Equi y
10 See also Adalid e al. (2022) who sys ema ically e iew cen al bank balance shee adjus men o
CBDC issuance and i s impac on he balance shee leng h.
11 The analysis abs ac s om explici ly aking in o accoun colla e al conside a ions o hose c edi
ope a ions.
486 L. Bi e
Rm
+1=δ0+δY
Y−Y
Y
()
−δMM −M
() (9)
Following his ule, he CBDC in e es a e Rm
+1 eac s o fluc ua ions in ou pu Y
om
s eady s a e Yand o fluc ua ions in CBDC demand M
om s eady s a e M. The
cen al bank de e mines he “base issuance”o CBDC in s eady s a e M, while ou side
s eady s a e i s ee s CBDC demand M
ia he CBDC in e es a e. Ou side he s eady
s a e, wo coun e ailing o ces go e n he CBDC in e es a e ule: i eac s s abil-
ising o fluc ua ions in ou pu ( ia δ
Y
) while discou aging excessi e CBDC ake-up ( ia
δ
M
). In esponse o a nega i e shock ha educes ou pu , he CBDC in e es a e
inc eases. While his eac ion may appea coun e in ui i e a fi s glance, i is
impo an o be awa e ha he CBDC a e does no affec c edi condi ions. Ins ead, in
his se ing he CBDC in e es a e is simply a sa e ’s a e, p o iding households wi h
a mo e efficien al e na i e o sa e han di ec ly in es ing in capi al secu i ies which
s abilises capi al p ices and he eby ou pu . I CBDC demand is high, he CBDC
in e es a e ule dec eases and he eby dynamically s abilises he demand o CBDC
by discou aging o hold mo e CBDC when he amoun in ci cula ion is al eady high.
20
While he issuance o CBDC on he liabili y side o he cen al bank balance shee
is he same o bo h analysed policy scena ios, he asse side se up diffe s unde
CBDC issuance wi h ‘c edi policy’and ‘asse policy’.
C edi Policy: The ule alloca ing c edi o he banks unde CBDC wi h ‘c edi
policy’is s aigh o wa d. By assump ion, his policy op ion complemen s CBDC
issuance M
by p o iding c edi o banks L
and hus all CBDC unds a e channeled
back o he banks:
L =M (10)
The cen al bank edi ec s los deposi unding exchanged in CBC back in o he
banking sec o on a 1:1 basis and g an s loans L
o banks equi alen o he amoun o
CBDC. The in e es a e on cen al bank c edi Rl
is endogenously de e mined.
The flow o cen al bank unds is hen gi en by
20 The CBDC in e es a e is de e mined by he join impac o he eac ion o ou pu fluc ua ions and
CBDC demand. To households, holding CBDC, deposi s o capi al secu i ies a e pe ec subs i u es,
equi ing an equal in e es a e ac oss hese op ions in an equilib ium in which all h ee op ions a e
held. Thus, by s ee ing he CBDC in e es a e, he cen al bank also indi ec ly affec s he deposi
in e es a e ia he household’s op imali y condi ions. Allowing o CBDC and deposi s o be
impe ec subs i u es e.g. by in oducing a isk p emium on deposi s o ia p e e ences, would
weaken bu no b eak he impac o he CBDC in e es a e ondeposi condi ions. Indeed, he financial
s abili y conce ns p ima ily a ise om he close subs i u abili y be ween CBDC and deposi s.
The e o e, elaxing his assump ion would no e e se he di ec ion o esul s, bu a he s eng hen
hem.
Banking C ises Unde a CBDC 493

M +Rl
L −1=Rm
M −1+L +T (11)
whe e in each pe iod he cen al bank issues CBDC M
and ecei es he loan e-
paymen s plus in e es om he p e ious pe iod. I uses he unds o epay he claims
and in e es on CBDC om he p e ious pe iod and addi ionally g an s new loans.
The emainde is dis ibu ed as ans e s T
o he households (o as ax in he case o
a nega i e esidue). In he absence o a bank un, he in e es a e on loans is highe
han he in e es a e on CBDC; he e o e, he ans e s a e posi i e despi e being
small.
Asse Policy: In con as , unde ‘asse policy’, he cen al bank does no
accommoda e CBDC wi h c edi bu buys capi al secu i ies i sel . When in es ing in o
capi al secu i ies Kcb
, i is easonable o assume ha he cen al bank is less efficien
han he bank bu mo e efficien han p i a e households. The e o e, when he
cen al bank pu chases capi al secu i ies i aces –simila o households bu a a
lowe le el –an inc easing and con ex cos o K
cb
()
=αcb
2Kcb
()
2wi h α
cb
<α. The
cen al bank ully complemen s he issuance o CBDC by in es men s in capi al and
he e o e buys as much capi al secu i ies, such ha he g oss in es men equals he
amoun o CBDC supplied o households, leading o:
Q Kcb
+αcb
2Kcb
()
2=M (12)
The flow o unds o he cen al bank e ol es simila ly o Equa ion (11): he unds
om CBDC and las pe iod’s capi al secu i y e u ns a e used o epay he claims o
he CBDC om he p e ious pe iod and o in es men s in o new capi al, plus
managemen cos s:
M +(Z +Q )Kcb
−1=RM
M −1+Q Kcb
+αcb
2Kcb
()
2+T (13)
3.1.4 Agg ega ion, Timing and Equilib ium
Summa ising he indi idual bank measu es (deno ed in lowe case le e s) leads o
he banking sec o agg ega e (deno ed in capi al le e s). As he e a e a cons an
numbe o symme ic banks no malised o one and an equal amoun o banks exi
en e each pe iod, agg ega e ne wo h N
is a combina ion o he ne wo h o banks
in ope a ion and he endowmen W bo newly en e ing banks:
N =σ(Z +Q )Kb
−1−Rd
D −1−Rl
L −1
[]
+Wbwi h (1−σ)wb=Wb(14)
Capi al is o fixed supply and no malised o uni y, he e o e capi al holding o
households Kh
, banks Kb
and he cen al bank Kcb
add up o:
494 L. Bi e
Kh
+Kb
+Kcb
=1 (15)
Unde CBDC wi h ‘asse policy’Kcb
≥0 and unde CBDC wi h ‘c edi policy’Kcb
=0as
he cen al bank does no in es in o he p oduc i e echnology i sel .
G oss ou pu , amoun ing o he sum o capi al ou pu Z
, household endowmen
Z
Whand bank endowmen Wb, is ei he consumed by households Ch
, banke s Cb
o
managemen cos s o households α
2Kh
()
2o he cen al bank αcb
2Kcb
()
2:
Ch
+Cb
+α
2Kh
()
2+αcb
2Kcb
()
2=Z +Z Wh+Wb(16)
Howe e , ou pu is defined wi hou managemen cos s i.e.
Y =Ch
+Cb
(17)
The sequence o e en s is as ollows: a he beginning o pe iod , he p oduc i i y o
capi al Z
is ealised. The new alloca ion o deposi s, CBDC, loans, and capi al in es men s
a e hen de e mined. Deposi s and loans a e issued in a way ha he mo al haza d
cons ain is sa isfied and he banke does no ha e he incen i e o di e asse s. The nex
pe iod begins wi h he ealisa ion o e u n on he capi al in es ed he p e ious pe iod.
Unde CBDC wi h ‘c edi policy’, he equilib ium is gi en by he ec o o eal p ices
Q ,Rd
,Rm
,Rl
and a ec o o quan i ies Z ,Kb
,Kh
,D ,M ,L ,N ,
V
N ,Ch
,T .Unde CBDC
wi h ‘asse policy’ he equilib ium is de e mined by he sligh ly diffe en ec o o eal
p ices Q ,R ,RM
and a ec o o quan i ies Z ,Kb
,Kh
,Kcb
D ,M ,N ,
V
N ,Ch
,T .A
compila ion o he equilib ium equa ion unde bo h policies can be ound in Appendix C.
3.2 Bank Failu es in he Model En i onmen
The equilib ium ou lined abo e is based on he mu ual belie s o households abou
he deposi decisions o o he households, in pa icula ha hey will no pa icipa e
in a bank un. Unde ce ain condi ions, a second equilib ium among he abo e
ou lined ‘banking equilib ium’exis s, in which he belie o a bank un igge s a
bank un in equilib ium (‘bank un equilib ium’). The ollowing sec ion examines
he condi ions unde which a bank un equilib ium exis s and how i un olds unde
he diffe en CBDC policy scena ios. The analysis conside s only sys emic bank uns,
whe e banks a e iden ical and subjec only o agg ega e shocks. In addi ion, he
p oduc i i y shock o he economy ha opens up he possibili y o a bank un is
unan icipa ed and he e o e bank uns a e also unan icipa ed.
21
21 Ge le and Kiyo aki (2015) and Ge le , Kiyo aki, and P es ipino (2016a) u he s udy an icipa ed
banking panics, allowing o a isk p emium on deposi s ha is based on bank un p obabili ies.
Banking C ises Unde a CBDC 495
3.2.1 Types o Bank Failu es
3.2.1.1 Bank Failu es Due o Illiquidi y
In each pe iod, households decide whe he o oll o e hei deposi s o he nex
pe iod. The indi idual decision o oll o e deposi s c i ically depends on he ex-
pec a ions o o he households. I a household expec s ha o he households will no
oll o e deposi s and i his would lea e he bank wi h ze o asse s, hen i is indi-
idually a ional no o oll o e deposi s as well. In such cases o a sys emic bank, all
households wi hd aw hei deposi s and banks a e o ced o liquida e hei asse s a
he fi e-sale p ice Q*
. I he unds om he sold asse s a e less han he claims on he
deposi s, deposi o s mus accep an equal hai cu x
wi h
x =Q*
+Z
()
Kb
−1
Rd
D −1
(18)
on hei claims Rd
D −1.
22
A bank un equilib ium exis s i he hai cu alue becomes less han 1 (x
< 1).
Con e sely, i x
≥1, all unds can be paid ou in ull and he e is no eason o un,
e en i all o he households un. Thus, he exis ence o a bank un equilib ium is
de e mined by economic undamen als, wi h he fi e-sale alue o he asse s de e -
mining he hai cu a e x
. Whe he a bank un is igge ed ul ima ely depends on he
belie s o households, wi h he bank un being induced in he model ia a sunspo
shock. I a bank un esul s in a de aul , he s a egic complemen a i y in beha iou
makes he belie o a bank un a sel - ulfilling p edic ion. Sel - ulfilling bank uns can
be conside ed as bank ailu es due o illiquidi y and a e possible i :
Q*
+Z
()
Kb
−1<Rd
D −1(19)
3.2.1.2 Bank Failu es Due o Insol ency
I a sufficien ly la ge shock hi s he economy and he bank’s liabili ies exceed he
alue o i s capi al e en a he non-fi e-sale p ice Q
, i is a ional o households o
wi hd aw hei deposi s – ega dless o he belie s o o he households. These bank
uns d i en by economic undamen als can be conside ed as bank ailu es due o
insol ency and a e possible i :
(Q +Z )Kb
−1<Rd
D −1(in he CBDC asse policy & no CBDC case)
(Q +Z )Kb
−1<Rd
D −1+Rl
L −1(in he CBDC c edi policy case)(20)
22 In con as o Diamond and Dyb ig (1983), he e is no sequen ial se ice cons ain assump ion
and deposi o s sha e he emaining unds equally ac oss all deposi o s.
496 L. Bi e
The sole diffe ence be ween he illiquidi y and insol ency condi ions in he ‘no CBDC’
scena io and he ‘asse policy’CBDC scena io is he p icing o capi al. The e is a u he
diffe ence in he ‘c edi policy’CBDC scena io: In he case o bank ailu es due o
illiquidi y, i is assumed ha only households bu no he cen al bank pa icipa e in he
bank un. Ins ead, he cen al bank con inues o lend o banks (see a mo e de ailed
desc ip ion below). In con as , in he case o bank ailu es due o insol ency, he cen al
bank also wi hd aws i s unding suppo and banks ail when he asse alua ion canno
co e he ull liabili ies (i.e. deposi s and cen al bank loans). Hence, he condi ion o a
bank ailu e due o insol ency is me when he undamen al alua ion o asse s is less
han he o al liabili ies (deposi s and c edi ). In con as , he condi ion o a bank ailu e
due o illiquidi y is me when he fi e-sale alua ion o asse s is less han he alue o
deposi s, i.e. only a subse o he o al liabili ies.
3.2.1.3 Equilib ium Ranges
The abo e scena ios lead o h ee possible equilib ium egions: in he fi s egion,
he e is a unique equilib ium wi h unc ioning financial in e media ion (‘banking
equilib ium’). In he second egion, he banking equilib ium and he bank un
equilib ium coexis . Las ly, in he hi d egion, only he bank un equilib ium e-
mains. Based on he abo e condi ions o he wo ypes o bank ailu es, Table 3
shows he h esholds o he diffe en equilib ium anges, condi ional on he e o-
lu ion o capi al p oduc i i y Z
.
3.2.2 Bank Runs in he Diffe en Policy Scena ios
How does a bank un un old?
23
As a e e ence poin , i is use ul o fi s e iew he
baseline scena io wi hou a CBDC: a la ge shock is ealised, igge ing a bank un and
Table :Equilib ium anges based on capi al p oduc i i y Z
.
Equilib ia Type o bank
ailu es
CBDC asse policy &
no CBDC
CBDC c edi policy
Banking None Z ≥Rd
D 
Kb

Q*
Banking +bank
un
Illiquidi y Rd
D 
Kb

Q*
>Z ≥Rd
D 
Kb

Q
Rd
D 
Kb

Q*
>Z ≥Rd
D þRl
L 
Kb

Q
Bank un Insol ency Z >Rd
D 
Kb

Q Z >Rd
D þRl
L 
Kb

Q
23 The se o he condensed equilib ium condi ions a he ime o he bank un can be ound in he
Appendix C.
Banking C ises Unde a CBDC 497
households ha e no o he op ion bu o in es in capi al hemsel es. Howe e ,
households a e only willing o pu chase capi al a a lowe p ice due o hei
inc easing ma ginal cos s o managing capi al: he highe cos s o managing capi al
lead a fi e-sale p ice o capi al ha is much lowe han he non-bank- un p ice o
capi al. Al hough socially de imen al, indi idually i is s ill op imal o un and
ecei e a hai cu on deposi claims a he han losing he ull deposi alue.
In he ‘asse policy’CBDC scena io, bank uns un old in a simila way: house-
holds wi hd aw all deposi s om he bank, he bank ails and closes down. The main
diffe ence he e is ha households can hold CBDC as a second asse . The ela i e
e u n o households be ween holding capi al and CBDC, de e mines he sha e ha
flows in o CBDC and in o capi al. In his case, households bu also he cen al bank
in es in o capi al in he pe iod o he bank un.
In he ‘c edi policy’CBDC scena io, bank uns emain simila o households bu
un old diffe en ly o banks. As long as he bank is me ely illiquid bu no insol en
(see he condi ions ou lined abo e), only households pa icipa e in he un and
wi hd aw hei deposi s while he cen al bank con inues o g an c edi . Households
wi hd aw hei deposi s wi h a hai cu , lea ing he bank wi h no asse s bu
ou s anding deb o he cen al bank. In he pe iod o he un, he bank de aul s on i s
claims on he cen al bank. Wi hou an in e en ion, he bank would go bank up .
Ins ead, he cen al bank, in i s spi i as ‘lende o las eso ’, con inues o lend o he
bank in he amoun o i s CBDC inflows which enables he bank o con inue o
ope a e.
24
In he pe iod o he bank un, he bank in es s in o capi al only wi h
cen al bank c edi as i has ze o ne wo h and ze o deposi s. Due o he ou s anding
claims o banks o he cen al bank, he capi al e u n in he pe iod o he un is ully
paid ou o he cen al bank. Ul ima ely, hese measu es p e en bank ailu es bu
also lea e banks wi h ze o ne wo h in he pe iod a e he un.
The cen al bank usually makes p ofi as he in e es a e on c edi is la ge han
he a e on CBDC. Howe e , in he pe iod o he bank un, he cen al bank abso bs
he loss on he de aul ed loans and he losses a e dis ibu ed o households ia lump-
sum axes ( he ans e s become nega i e). Ne e heless, in he pe iod a e he bank
un, he cen al bank eco e s some losses as i collec s he ull p ofi on in es ed
capi al by he bank and edis ibu es i o households.
25
In he pe iod a e he bank
un, he banking sec o s a s o eco e . Thus, in he ‘c edi policy’scena io, banks
do no ail and hei abili y o in e media e capi al efficien ly can be main ained in
he pe iod o he bank un. A he same ime, banks s ill s a he pe iod a e he
bank un wi h ze o ne wo h and he e o e a e in he same si ua ion as in he
24 No e ha his app oach may aise mo al haza d conce ns, which could be mi iga ed by close
supe ision.
25 The se o he equilib ium condi ions a he ime o he bank un can be ound in Appendix C.
498 L. Bi e

scena ios wi h ‘no CBDC‘and CBDC wi h ‘asse policy’: wi h ze o ne wo h i is as i
hese banks had ailed and he sec o mus eco e ia newly en e ing banks.
4 Model Analysis
This sec ion examines he dynamics o he economy unde h ee diffe en scena ios:
CBDC wi h ‘c edi policy’, CBDC wi h ‘asse policy’, and ‘no CBDC’. A e explaining he
calib a ion o he nume ical exe cise in Sec ion 4.1, Sec ion 4.2 analyses he impli-
ca ions o hese h ee model scena ios in s eady s a e. Subsequen ly, Sec ion 4.3
examines he esponse o he economy o capi al p oduc i i y shocks ha do no lead
o bank ailu es, whe eas Sec ion 4.4 ocuses on bank ailu es and analyses he
impac o he diffe en CBDC policy scena ios. Specifically, he la e sec ion in-
es iga es he equi ed shock sizes a e which sel - ulfilling bank uns become
possible (Sec ion 4.4.1) and he shock sizes ha lead o insol ency (Sec ion 4.4.2). The
aim o he analysis in his s ylised endowmen economy se up is no o aim o
p ecisely ma ching empi ical p ope ies bu a he o in es iga e he mechanisms
influencing financial s abili y h ough he issuance o CBDC in he diffe ing policy
scena ios ou lined abo e.
4.1 Calib a ion and Simula ion
Table 4 lis s all he pa ame e choices. Mos o he pa ame e s p esen in he ‘no
CBDC’scena io a e aken om GK15, wi h he excep ion o he sha e o di e able
asse s θ= 0.22, which is upda ed o he alue o Ge le , Kiyo aki, and P es ipino
Table :Calib a ion o pa ame e alues.
Pa ame e Value Sou ce Desc ip ion
α. GK Ma ginal managemen cos s households
α
cb
. New Ma ginal managemen cos s cen al bank
β. GK Discoun a e
σ. GK Banke s su i al p obabili y
θ. GK Sha e o di e able asse s
ω.GK Ad an age in seizu e a e o cen al bank c edi
ρ. GK Se ial Co ela ion o p oduc i i y shock
Wh. GK Household endowmen
Wb. GK Banke s endowmen
δ

. New CBDC base pa ame e
δ
Y
. New CBDC ule ou pu coefficien
δ
M
.New CBDC ule demand coefficien
GK = Ge le and Kiyo aki (), GK = Ge le and Ka adi ().
Banking C ises Unde a CBDC 499
(2020). The in oduc ion o a CBDC in oduces addi ional pa ame e s. Unde CBDC
issuance wi h ‘c edi policy’, cen al bank c edi o banks is, simila o deposi
unding, also subjec o a mo al haza d cons ain bu only o hal he ex en and hus
ωis se o 0.5 (e.g. as in Ge le and Ka adi 2013). Simila ly, unde CBDC issuance wi h
‘asse policy’, he managemen cos o capi al pu chases α
cb
ha he cen al bank
aces is se o hal he efficiency cos s ha households ace.
The e is no empi ical coun e pa o guide he choice o alues o he CBDC
in e es a e ule. The cen al bank supply o CBDC in s eady s a e M(‘base issuance’)
in se o equal 10 % o o al household sa ings.
26
As mos cen al banks aim o design
CBDC as a means o exchange a he han an a ac i e s o e o alue (G oup o
Cen al Banks 2020), his is a ela i ely high and conse a i e alue and in line wi h
he cash- o-GDP a io o 2021 o he US (9.2 %) and eu o a ea (12.8 %) (Bank o
In e na ional Se lemen s 2020).
27
The CBDC in e es a e is de e mined by he join impac o he CBDC demand
and ou pu coefficien . The ou pu coefficien o he CBDC ule δ
Y
se o 0.26, adop ing
he alue o Minesso, Mehl, and S acca (2022) who analyse he in e na ional im-
plica ions o CBDC wi h a Taylo - ype ule. I is impo an o diffe en ia e ha , in
con as o con en ional mone a y policy ia a Taylo ule, in his model con ex an
inc ease CBDC in e es a e has a s abilising effec . By inc easing he oppo uni y
cos s, i makes i less a ac i e o households o in es in less efficien capi al
secu i ies which s abilises capi al p ices and hence ou pu . The CBDC ule demand
coefficien δ
M
is se o 0.1 such ha an inc ease in he demand o CBDC has a dis inc
bu mode a e effec on he CBDC in e es a e. Finally, pe iods a e calib a ed o be
equal o one qua e .
The simula ion la gely ollows he compu a ional p ocedu e o Ge le and
Kiyo aki (2015), ou lined in mo e de ail in he Online Appendix. The esponse o a
shock wi hou a bank un is compu ed as a shock o he nonlinea pe ec o esigh
model. To simula e a shock ollowed by a bank un, he equilib ium pa h is spli in o
h ee componen s and calcula ed s a ing backwa ds. Fi s , he pa h o he economy
a e he bank un is calcula ed, s a ing om one pe iod a e he bank un. This is
possible because he e minal alues (i.e. s eady s a e alues) and he ini ial alues o
he backwa d-looking a iables a e known. Based on he pos -bank un pa hs o he
a iables, he o wa d-looking bank un equilib ium can be calcula ed. Finally, un il
26 The pa ame e de e mining he CBDC in s eady s a e δ
0
mus be se o align wi h he household’s
op imali y condi ions.
27 Se e al cen al banks a e conside ing capping he maximum supply o CBDC bu he anges unde
discussion a y conside ably ac oss ju isdic ions. Fo example, he ECB has men ioned p elimina y
cap o up 1.5 illion eu os, e e encing he simila size o bankno es in ci cula ion, which would
ansla e o 3,000–4,000 eu os pe capi a (Pane a 2022). In con as , he Bank o England has e e ed
o holding limi s o 10,000–20,000 pounds pe capi a (Bank o England and HM T easu y 2023).
500 L. Bi e
he pe iod in which he bank un occu s, he equilib ium pa h is he same as unde
he shock wi hou a bank un. While in p inciple a bank un can occu in any pe iod,
he pe iod o he shock is he pe iod mos suscep ible. The e o e, he ollowing
analysis ocuses on bank uns eme ging in he pe iod o he shock and only las one
pe iod. Fu he mo e, he analysis es ic s deposi s, bank ne wo h, and bank
consump ion o be non-nega i e.
4.2 S eady S a e Implica ions o a CBDC
In s eady s a e, he issuance o a CBDC does no affec agg ega e quan i ies, only
alloca ions. Table 5 compa es he s eady s a e alues o he model unde ‘c edi
policy’,‘asse policy’, and he baseline scena io wi h ‘no CBDC’.
28
S eady s a e ou pu
and mos p ices a e he same ac oss all h ee scena ios. Acco dingly, s eady s a e
wel a e, defined as agg ega e consump ion, is he same ac oss all scena ios.
29
While o al wel a e is unchanged by CBDC, he e is edis ibu ion om banks o
households. Household consump ion sligh ly benefi s om CBDC issuance and he
banking sec o –especially banks’ne wo h and consump ion –is s ained by CBDC
in s eady s a e. Howe e , he inc ease in household consump ion by less han 1 % is
ma ginal and o igina es om he lump-sum ans e s o he cen al bank o
households. While he po olio alloca ion o household sa ings changes mo e sub-
s an ially (Figu e 1), his does no affec household weal h as equilib ium e u ns a e
equal ac oss asse s and p ices emain unchanged. Pe calib a ion a ge , CBDC
holdings accoun o 10 % o o al household sa ings. The sha e o capi al in es men
emains ela i ely cons an . The e o e, flows in o CBDC o igina e en i ely om de-
posi s and educe he pe cen age o deposi holdings by abou 10 %-poin s.
The consump ion o banke s is lowe due o lesse ne wo h and p ofi abili y o
banks. Unde CBDC wi h ‘asse policy’, he issuance o CBDC leads o a educ ion in
he size o he balance shee , while unde CBDC wi h ‘c edi policy’ he los deposi
unding is eplaced by cen al bank c edi , leading o a balance shee size ha
emains simila (Figu e 2). Bo h policy scena ios s ain p ofi abili y: while he
issuance o CBDC unde ‘c edi policy’ educes p ofi abili y h ough highe unding
cos s o cen al bank c edi , he issuance o CBDC unde ‘asse policy’ educes ne
wo h h ough a smalle balance shee size. This smalle balance shee , and hence
he capi al holdings o banks unde CBDC wi h ‘asse policy’is compensa ed by
28 Values a e ounded up o he ou h decimal. Diffe ences beyond he ou h decimal a e so small
ha alues will be conside ed as he same.
29 Wel a e alls sho o he fi s bes scena io wi hou financial ic ions in which only banks in es
in o capi al.
Banking C ises Unde a CBDC 501
Table :Compa ison o s eady s a e alues.
Va iable CBDC c edi policy CBDC asse policy No CBDC
Ou pu Y
. . .
Capi al
Bank capi al Kb
. . .
Household capi al Kh
. . .
Cen al bank capi al Kcb
/. /
Consump ion
Household consump ion Ch
. . .
Banke consump ion Cb
. . .
Financial asse s
Deposi s D
. . .
CBDC M
. . /
Loans o banks L
. //
Banking sec o
Bank ne wo h N
. . .
Bank le e age ϕ
. . .
In e es a es & p ices
P ice o capi al Q
. . 
Capi al e u n Rb
. . .
Deposi a e Rd
. . .
CBDC a e RM
. . /
Loan a e RL
. //
No e: e u ns a e exp essed in annualised alues.
Figu e 1: Households sa ings in s eady s a e.
502 L. Bi e
The e a e wo o ces ha join ly impede he occu ence o bank uns unde
CBDC wi h ‘c edi policy’. Fi s , in he e en o a bank un, only a ac ion o he
bank’s liabili ies a e wi hd awn, as bo h deposi s and cen al bank c edi a e on he
bank’s balance shee . The e o e, he fi e-sale p ice o asse s needs o co e only he
sha e o liabili ies ha a e deposi s as he cen al bank does no engage in he bank
un. In his way, cen al bank c edi p o ides he bank wi h a s able unding sou ce
which c ea es an addi ional buffe on he bank’s balance shee . This addi ional
cen al bank c edi buffe esul s in a lowe sha e o asse s ha would be wi hd awn
in a bank un scena io and makes he unding ou flow easie o abso b. The la ge
he p e- un cen al bank c edi holdings a e, he highe he hu dle o he eme gence
o a sel - ulfilling bank un. Second, ins ead o wi hd awing unds, he cen al bank
simul aneously p o ides he bank wi h addi ional liquidi y in he amoun o inflows
in o CBDC. The con inued c edi p o ision enables he bank o in es in capi al e en
in case o a bank un. Wi h he bank being mo e efficien in managing capi al, his
g ea ly s abilises he capi al p ice. Ac ing as lende o las eso , he cen al bank
p o ision o c edi u he impedes he eme gence o bank uns. Those wo o ces
b eak he ‘doom-loop’o sel - ulfilling uns in he baseline calib a ion.
How can banks unde CBDC wi h ‘c edi policy’become insol en wi hou
opening he possibili y o sel - ulfilling uns? As desc ibed abo e, he p ice
h eshold
¯
Q*
ha igge s sel - ulfilling banks is only condi ional on he ac ion o
deposi s
¯
Q*
=Rd
D −1
Kb
−1
−Z . In con as , he p ice h eshold
¯
Q ha igge s insol ency is
easie o be eached because he cen al bank also s ops unding in he e en o
insol ency:
¯
Q =Rd
D −1
Kb
−1
+Rl
L −1
Kb
−1
−Z . While Q
and Q*
a e he wo equilib ium p ices in
he banking and bank un equilib ium, Rd
D −1
Kb
−1
−Z a e he same in bo h scena ios. Fo
his eason, unde he baseline calib a ion, e en on he b ink o insol ency, he fi e-
sale alue o asse s would be sufficien o co e deposi claims and would p e en a
sel - ulfilling un.
32
A he same ime, he non-fi e-sale alue o asse s will no be
32 Why a e bank uns no possible unde he baseline calib a ion, e en on he e ge o insol ency?
A e all, swi ching om deposi unding o c edi unding comes a a highe in e es a e, which
should igge insol ency? The answe is no, in his case he non-fi e-sale asse p ice Q
wound
emain ma ginally abo e he insol ency h eshold and he e o e he cen al bank will con inue o
p o ide c edi as he bank is no ye insol en . Bea ing in mind, he insol ency h eshold is condi-
ional on households no expec ing a bank un. The e o e, i he belie s o households’would change
and hey would un on he bank, his will only affec he fi e-sale p ice bu no he non-fi e sale p ice.
Because unde no- un belie s he bank would emain sol en , he cen al bank will con inue o
p o ide c edi in he e en o a bank un, he eby p e en ing he bank un e en a he b ink o
insol ency. In sho , a sel - ulfilling un canno igge insol ency because hey a e dis inc equi-
lib ium concep s ha a e condi ional on household belie s.
Banking C ises Unde a CBDC 509

sufficien o co e all liabili ies o he bank (i.e. deposi s and cen al bank c edi ).
The e o e, unde he baseline calib a ion, he bank can ail due o insol ency wi hou
opening up he possibili y o a sel - ulfilling un.
O e all, by in e ceding as lende o las eso wi h he unds om CBDC, he
cen al bank al oge he elimina es he possibili y o a un in he con ex o he
nume ical example. Howe e , his does no p eclude sel - ulfilling bank uns unde
c edi policy in he gene al se ing. Sel - ulfilling bank uns unde CBDC wi h c edi
policy a e possible unde al e na i e calib a ions.
Which o he wo channels p e en ing he bank un unde he baseline cali-
b a ion is s onge and wha a e he implica ions o educing bo h o ces, indi id-
ually and join ly? The fi s o ce wo ks ia he sha e o c edi on he bank’s liabili ies
in he pe iod be o e he bank un. Reducing δ
0
in he cen al bank’s CBDC ule o
δ
0
= 1.01015 diminishes cen al bank c edi in s eady s a e o almos ze o, wi h
L
s .s .
= 0.0005 (ins ead o L
s .s .
= 0.0941 unde he baseline calib a ion wi h
δ
0
= 1.0194). Mu ing he fi s channel is no sufficien o enable sel - ulfilling bank
uns be o e he insol ency h eshold. The second channel o p o iding cen al bank
c edi ia CBDC inflows is sufficien ly s ong o s ill p e en he bank un om he
ou se . The second channel can be elaxed by inc easing he eac ion o CBDC de-
mand in he CBDC ule o δ
M
o disincen i ise CBDC up ake. Se ing δ
M
= 0.5 (ins ead
o δ
M
= 0.1 unde he baseline calib a ion), sel - ulfilling bank uns become possible
om shock sizes abo e 6.3 %. Relaxing bo h o ces join ly, wi h δ
0
= 1.01015 and
δ
M
= 0.5, enables sel - ulfilling bank uns al eady om shock sizes abo e 2.9 %.
4.4.1.4 The Role o F ic ion o he Eme gence o Sel -Fulfilling Runs
Bo h ypes o CBDC issuance –wi h ‘asse policy’and ‘c edi policy’–delay he
eme gence o he bank un equilib ium o la ge shocks by educing ic ions du ing
a bank un. This s abilises he fi e-sale p ice o asse s, mi iga es wel a e losses and
makes banks uns mo e difficul o eme ge. The con ibu ions o he diffe en model
ic ions o he eme gence o bank uns a e examined in mo e de ail below.
The main ic ion ha makes bank uns cos ly is he capi al managemen cos o
households α. Reducing he household capi al managemen cos s o almos ze o wi h
α= 0.001 (α= 0.008 in he baseline calib a ion) delays he eme gence o sel - ulfilling
uns in he ‘no CBDC’scena io o he b ink o insol ency a shocks abo e 21.9 % ( s.
2.5 % unde he baseline calib a ion). Con e sely, inc easing capi al managemen
cos s o α= 0.015 in he scena io wi hou CBDC enables a bank un equilib ium
al eady in s eady s a e in he absence o any shocks.
While addi ional ic ions in he CBDC scena ios also play a ole o he eme -
gence o bank uns, hey a e no as cen al as household capi al managemen cos s α
and he pa ame e s o he CBDC eac ion ule. In he scena io o CBDC issuance wi h
‘asse policy’, capi al managemen cos s o he cen al bank a e in oduced as
510 L. Bi e
addi ional ic ion. Reducing he cen al bank’s capi al managemen cos s o
α
cb
= 0.001 (ins ead o α
cb
= 0.004 in he baseline calib a ion) delays he eme gence o
sel - ulfilling uns om shocks abo e 8.6 % unde he baseline calib a ion o shocks
abo e 9.7 %. Con e sely, se ing cen al bank capi al managemen cos s o he capi al
managemen cos s o he households wi h α
cb
= 0.008 educes he shock h eshold o
he eme gence o bank uns o shocks abo e 7.7 %. In he CBDC scena io wi h ‘c edi
policy’, he le e age cons ain o cen al bank c edi ωis in oduced as an addi-
ional ic ion. Se ing ω= 1 emo es he le e age cons ain bu inc eases he cos s o
cen al bank c edi which leads o he possibili y o sel - ulfilling bank uns a shocks
abo e 2.1 %. Con e sely, se ing ω= 0 leads o he same le e age cons ain bu also o
he same unding cos s as on deposi s. Simila ly o he baseline calib a ion, his does
no open up he possibili y o sel - ulfilling uns be o e he insol ency h eshold.
Mo eo e , bank uns a e no an icipa ed. How would he esul s change i bank
uns we e an icipa ed, o example as in Ge le and Kiyo aki (2015) o Ge le ,
Kiyo aki, and P es ipino (2016a)? Accoun ing o he possibili y o bank ailu es
would lead o a isk p emium on deposi s and inc ease bank unding cos s. As he
in oduc ion o CBDC in he model amewo k makes bank uns less likely by
delaying hei occu ence o la ge shocks, he isk p emium on deposi s would
dec ease in he p esence o CBDC. The e o e, in oducing an icipa ion o bank uns
in o he analysis would no ma e ially change he insigh s o his analysis and esul s
would be s eng hened u he . In addi ion, allowing o an icipa ion o bank ail-
u es, CBDC would be mo e a ou able o he p ofi abili y o banks as he in o-
duc ion o CBDC would educe he isk p emium and hus he deposi unding cos s
and he e o e u he mi iga e financial agili y.
4.4.2 Bank Failu es Due o Insol ency
Figu e 8 displays he e olu ion o deposi s o inc easing shock sizes unde he
diffe en policy scena ios. Banks become insol en in esponse o a shock when he
non-fi e-sale alue o asse s becomes smalle han he liabili ies. Thus, he condi ions
o a de aul due o illiquidi y and de aul due o insol ency only diffe in he p ice o
capi al o he ‘no CBDC’and CBDC wi h ‘asse policy’scena ios. Fo CBDC wi h ‘c edi
policy’, he condi ions o bank ailu e due o illiquidi y and insol ency also diffe in
he asse s ha a e aken in o accoun . While he cen al bank does no engage in sel -
ulfilling uns and he fi e-sale alue o asse s only needs o co e deposi s, he
cen al bank c edi claims a e aken in o accoun in he calcula ion o he insol ency
condi ions.
The size o shocks ha h us banks in o insol ency unde CBDC wi h ‘c edi
policy’is simila bu a bi la e han he scena io wi hou CBDC. While in bo h
scena ios he o e all balance shee size is simila , he s abilisa ion o capi al p ices
Banking C ises Unde a CBDC 511
unde CBDC wi h ‘c edi policy’leads o a de e al o insol ency o sligh ly la ge
shock. In con as , he shock size ha igge s insol ency unde CBDC wi h ‘asse
policy’is subs an ially la ge . The highe insol ency h eshold can be explained by
he smalle bank balance shee unde CBDC wi h ‘asse policy’and also by he
s abilisa ion o he capi al p ice.
To summa ise, unde bo h policy scena ios CBDC affec s he equilib ium egions
in a way ha suppo s financial s abili y. Unde CBDC wi h ‘asse policy’, he
eme gence o bo h bank un ypes is pos poned o subs an ially la ge shock. Unde
CBDC wi h ‘c edi policy’, he possibili y o sel - ulfilling uns does no a ise in he
baseline calib a ion, while insol ency is igge ed a a simila shock size as in he ‘no
CBDC’scena io.
5 Conclusion
One o he main conce ns abou a CBDC is i s disin e media ing effec on he banking
sec o and especially he inc eased isk o a bank un in imes o c isis. This pape
Figu e 8: E olu ion o deposi s in he pe iod o he shock wi h inc easing capi al shock sizes unde he
diffe en policy scena ios and wi hou eme gence o sel - ulfilling uns.
512 L. Bi e
analyses he impac o CBDC on financial s abili y as an ex ension o he dynamic
bank un model wi h a financial accele a o by Ge le and Kiyo aki (2015).
CBDC issuance c ea es an addi ional ype o liabili y on he cen al bank balance
shee which will lead o u he balance shee adjus men s. In he model analysis, I
accoun o he po en ially diffe en impac o hese balance shee adjus men s by
analysing CBDC issuance in he con ex o wo diffe en asse side policies: by
g an ing loans o banks (CBDC wi h ‘c edi policy’) and by pu chasing capi al (CBDC
wi h ‘asse policy’).
The s ylised model analysis offe s se e al insigh s: in he s eady s a e, a CBDC
does no affec ou pu and wel a e. Ins ead, i affec s he composi ion o household
sa ings, bank unding and capi al in es men , ul ima ely educing bank p ofi s. In
esponse o shocks ha do no igge a bank un, he issuance o CBDC does no
exace ba e, bu a he ends o mi iga e, ou pu and wel a e losses by s abilising
asse p ices. A he same ime, he p esence o CBDC also leads o la ge deposi
ou flows. Mos impo an ly, he s abilisa ion o asse p ices imp o es financial s a-
bili y by de e ing he eme gence o bank ailu es due o illiquidi y (caused by sel -
ulfilling uns) and bank ailu es due o insol ency o la ge shocks. O e all, I find
ha a CBDC s ains he banking sec o in no mal imes by educing deposi s and ne
wo h. Ye , con a y o p e ailing conce ns, CBDC imp o es financial s abili y in
imes o c isis by s abilising capi al p ices h ough asse -side adjus men s ha ollow
CBDC issuance.
This analysis is ca ied ou in a s ylised se ing ha abs ac s om se e al
ea u es ha could be conside ed in u u e esea ch. Fo ins ance by: (i) b inging he
analysis om a eal o a nominal se ing ha allows o infla ion dynamics and
con en ional mone a y policy, (ii) in oducing a iche financial sec o and
embedding a mo e complex s uc u e o he cen al bank balance shee ha includes
ese es, cash, go e nmen bonds and a colla e al amewo k, (iii) inco po a ing he
unc ion o money as a medium o exchange in o he analysis and (i ) allowing o
iche dynamics and ic ions a fi m le el.
Acknowledgmen s: Fo help ul commen s, I would like o hank wo anonymous
e e ees, Lukas Al e ma , Ul ich Bindseil, Michael Bu da, Lukas Geh ing, Rou en
Geisma , Pe a Ge lach, Jonas G oss, F ank Heinemann, Jona han Schille , as well as
pa icipan s a semina s and con e ences o ganised by he ECB, Ve ein ü
Socialpoli ik, TU Be lin, HU Be lin, Uni e si y o Basel and Ruh G adua e School
in Economics. Suppo by Deu sche Fo schungsgemeinscha h ough CRC TRR 190
(p ojec numbe 280092119) is g a e ully acknowledged. The iews exp essed a e
hose o he au ho which do no necessa ily eflec hose o he ECB, he au ho is no
pa o he ECB’s digi al eu o p ojec eam and mos o he p ojec was done a he TU
Be lin be o e joining he ECB.
Banking C ises Unde a CBDC 513
Appendix A: Compa ison o Al e na i e Model
Specifica ions
Mos pa ame e alues o he simula ion a e adop ed om Ge le and Kiyo aki
(2015). Howe e , he e a e a ew pa ame e s ha ha e been newly in oduced and
a e less es ablished in he li e a u e. This sec ion p esen s insigh s om: (i) a ying
he dynamic coefficien s δ
Y
and δ
M
in he CBDC in e es a e ule; (ii) a ying he
capi al managemen cos s o he cen al bank α
cb
; and (iii) a ying he le e age
cons ain o cen al bank c edi o banks ω.
O e all, i he CBDC in e es a e ule is mo e esponsi e o ou pu fluc ua ions
om s eady s a e, his leads o a s onge s abilisa ion o ou pu , consump ion,
capi al p ices, and bank ne wo h. Such s abilisa ion comes a he cos o highe
deposi disin e media ion and a la ge inc ease in CBDC, also esul ing in a la ge
cen al bank balance shee . In con as , a s onge esponse o CBDC demand makes
he CBDC in e es a e less a ac i e and leads o s onge au oma ic s abilisa ion o
CBDC issuance, which in u n comes a he cos o lowe s abilisa ion in ou pu and
capi al p ices.
The same mechanisms explain he op imal alues o δ
Y
and δ
M
: he op imal δ
Y
ha would maximise wel a e (agg ega e discoun ed o al consump ion) in esponse
o a capi al quali y shock would be as high as possible, whe eas he op imal δ
M
would
be as low as possible. Howe e , hese wel a e conside a ions do no ake in o accoun
he nega i e implica ions o excessi e cen al bank in e en ion in esponse o a
shock, such as po en ial mo al haza d side effec s.
Relaxing he assump ions abou he cen al bank’s capi al managemen cos s in
case o CBDC wi h ‘asse policy’, and he le e age cons ain o cen al bank c edi o
banks in he case o CBDC wi h ‘c edi policy’, ends o lead o a highe effec i eness o
cen al bank measu es (i.e. a highe s abilisa ion and smalle inc eases in CBDC and
he cen al bank balance shee ). Ye , one may ques ion how easonable i would be o
ully elax hose assump ions. The opposi e pic u e eme ges when hese assump ions
a e u he es ic ed. Al hough, o e all he impac o a ia ions is ela i ely small,
especially o a ia ions in capi al managemen cos s unde CBDC wi h ‘asse policy’.
Appendix A.1: Va ying CBDC Rule Pa ame e s
The ollowing sec ion in es iga es how he esponse o a capi al p oduc i i y shock
changes, a ying he wo coefficien s δ
Y
and δ
M
in he CBDC in e es a e ule Rm
+1=
δ0+δYY−Y
Y
()
−δMM unde bo h ‘c edi policy’and ‘asse policy’.
514 L. Bi e

The coefficien δ
Y
cap u es he eac ion s eng h o he CBDC in e es a e ule o
fluc ua ions in ou pu om i s s eady s a e alue. Figu es 9 and 10 display a ia ions
in δ
Y
in he ‘c edi policy’and ‘asse policy’scena io. Compa ed o he baseline
calib a ion (solid lines, ollowing he calib a ion o Minesso, Mehl, and S acca 2022),
δ
Y
is doubled o 0.52 (highe ou pu s abilisa ion, dashed lines) and hal ed o 0.13
Figu e 9: Response o a 5 % shock o p oduc i i y unde CBDCwi h ‘c edi policy’ o highand low CBDC
ule ou pu coefficien δ
Y
.
Figu e 10: Response o a 5 % shock o p oduc i i y unde CBDC wi h ‘asse policy’ o high and low
CBDC ule ou pu coefficien δ
Y
.
Banking C ises Unde a CBDC 515
(lowe ou pu s abilisa ion, do ed lines). Va ia ions in δ
Y
ha e he same quali a i e
implica ions o CBDC issuance wi h ‘c edi policy’and ‘asse policy’: a highe
esponse o ou pu fluc ua ions inc eases he s abilisa ion o ou pu , consump ion,
he p ice o capi al, and bank ne wo h. Ye , his comes a he cos o a highe deposi
disin e media ion and a g ea e inc ease in CBDC issuance. The effec s a e e e sed
in he case o a lowe esponsi eness o ou pu fluc ua ions i.e. lowe s abilisa ion bu
lowe disin e media ion.
The coefficien δ
M
cap u es he esponsi eness o he CBDC in e es a e ule o
CBDC demand and he eby s abilising fluc ua ions in CBDC. Figu es 11 and 12 show
a ia ions in δ
M
in he ‘c edi policy’and ‘asse policy’scena io. Compa ed o he
baseline calib a ion (solid lines) o δ
M
=0.1,δ
M
is inc eased o 0.15 (highe CBDC
demand esponse, dashed lines) and educed o 0.075 (lowe CBDC demand
esponse, do ed lines). Va ia ions in δ
M
lead o simila obse a ions as he a i-
a ions in δ
Y
and ha e again ha e he same quali a i e implica ions o CBDC
issuance wi h ‘c edi policy’and ‘asse policy’. Making he CBDC in e es a e ule
less a ac i e in esponse o inc eased CBDC demand (i.e. a highe CBDC demand
esponse) leads o a less s abilisa ion, bu also o a smalle inc ease in CBDC supply
(and ice e sa). Howe e , he diffe ence in he a ia ions is ela i ely small,
excep o he implica ions on he deposi esponse whe e he diffe ences a e mos
p onounced.
Figu e 11: Response o a 5 % shock o p oduc i i y unde CBDC wi h ‘c edi policy’ o high and low
CBDC ule CBDC demand coefficien δ
M
516 L. Bi e
Appendix A.2: Va ying CBDC C edi and Asse Policy
Assump ions
The ollowing sec ion in es iga es how he esponse o a capi al p oduc i i y shock
changes, a ying he wo addi ional assump ions unde lying CBDC issuance wi h
‘c edi policy’and ‘asse policy’.
Fo CBDC issuance wi h ‘c edi policy’, i is assumed ha he le e age cons ain
o cen al bank c edi o banks is less binding han o deposi s, due o supe io
supe iso y powe s and colla e al equi emen s. Figu e 13 displays a ia ions in he
ela i e s eng h o he le e age cons ain V ≥θQ kb
−ωl
()
o cen al bank c edi ,
as cap u ed by ωwhich is se o 0.5 in he baseline calib a ion (blue solid lines).
Assuming no le e age cons ain on cen al bank c edi (dashed u quoise lines)
leads o lowe deposi disin e media ion and sligh ly highe s abilisa ion o ou pu
and consump ion, while equi ing he smalles inc ease in CBDC. Con a y, applying
he same le e age cons ain s eng h as deposi s (blue do ed lines) leads o he same
ou pu and consump ion esponse as in he case wi hou CBDC, he highes deposi
disin e media ion and he highes inc ease in CBDC.
Simila ly, o CBDC wi h ‘asse policy’, he cen al bank is assumed o ace lowe
capi al managemen cos s han households, bu which a e o he same unc ional
o m K
cb
()
=αcb
2Kcb
()
2. Figu e 14 displays a ia ions in he capi al managemen
Figu e 12: Response o a 5 % shock o p oduc i i y unde CBDC wi h ‘asse policy’ o high and low
CBDC ule CBDC demand coefficien δ
M
.
Banking C ises Unde a CBDC 517
cos s o he cen al bank α
cb
which is se o hal o he capi al managemen cos s o
households in he baseline calib a ion ( ed solid lines). Assuming no capi al man-
agemen cos s o cen al bank capi al pu chases (dashed o ange lines) and he same
capi al managemen cos s as households (pu ple do ed lines) leads o e y simila
Figu e 14: Response o a 5 % shock o p oduc i i y unde CBDC wi h ‘asse policy’and a ying cen al
bank capi al managemen cos s.
Figu e 13: Response o a 5 % shock o p oduc i i y unde CBDC wi h c edi policy and a ying le e age
cons ain o cen al bank c edi .
518 L. Bi e
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