IMF-投資組合選擇模型中的央行數(shù)字貨幣與銀行去中介（英）

CentralBankDigital

CurrencyandBank

Disintermediationina

PortfolioChoiceModel

HuifengChang,FedericoGrinberg,LucynaGornicka,Marcello

Miccoli,andBrandonJoelTan

WP/23/236

IMFWorkingPapersdescriberesearchin

progressbytheauthor(s)andarepublishedto

elicitcommentsandtoencouragedebate.

TheviewsexpressedinIMFWorkingPapersare

thoseoftheauthor(s)anddonotnecessarily

representtheviewsoftheIMF,itsExecutiveBoard,

orIMFmanagement.

2023

NOV

N

Ar

R

1

CentralBankDigitalCurrencyandBank

*

DisintermediationinaPortfolioChoiceModel

HuifengChangiFedericoGrinbergiiLucynaGornickaiii

MarcelloMiccoliivBrandonJoelTanv

November2023

Abstract

WouldtheintroductionofaCentralBankDigitalCurrency(CBDC)leadtolowerdeposits(disintermediation)andlendinginthebankingsector?Thispaperdevelopsamodelwherehouseholdsheterogeneousinwealthallocatebetweenanilliquidassetandassetsthatcanbeusedforpayments:bankdeposits,cash,andCBDC.CBDCismoree伍cientasameansofpaymentandhasloweraccesscostthandeposits.DepositsareoferedbyanimperfectlycompetitivebankingsectorwhichraisesdepositinterestratesafterCBDCintroductiontopreventsubstitutionawayfromdepositstoCBDC.Weindthattherearetwoopposingmarginsofimpactonthelevelofaggregatedeposits:(1)theintensivemargingainindepositsbyricherhouseholdsincreasingtheirholdingsofdepositsbecauseofhigherinterestrates,and(2)theextensivemarginlossofdepositsamongpoorerhouseholdswhoswitchfromdepositstotheCBDC.Theextensivemarginlossindepositsismorelikelytodominate(yieldingafallinaggregatedeposits)whenthemassofpoorerhouseholdsislargeandwhenitisrelativelycostlytoaccessbankaccounts.Thistendstobethecaseindevelopingandemergingmarketeconomies.However,evenwhentheextensivemarginlossofdepositsdominatesandthereisdisintermediation,theimpactonlendingisquantitativelysmallifbankshaveaccesstootherformsoffunding,suchaswholesaleorcentralbankinancing.

JELClassiication:E50,E58,G21

Keywords:CBDC;bankingdisintermediation;inancialinclusion;monetarypolicy

*TheviewsexpressedinthispaperarethoseoftheauthorsandthereforedonotnecessarilyrelectthoseoftheIMFortheECB.ThisworkwasperformedwhenHuifengChangwasanIMFintern.ThisworkhasbeneitedfrominsightfulcommentsbyDavidAndolfatto,ChrisErceg,TommasoMancini-Grifoli,RobTownsend,Pierre-OlivierWeill,andseminarparticipantsatBancoCentraldoBrasil,BancodeMexico,Bancad’Italia,theEuropeanCentralBank,theFederalReserveBoard,theIMF,andRiksbank.

iFormerIMFintern,nowatFudanUniversity,huifengchangpku@

iiIMF,fgrinberg@

iiiIMF,lgornicka@

ivIMF,mmiccoli@

vIMF,btan2@

WORKINGPAPERS

CentralBankDigitalCurrencyand

BankDisintermediationina

PortfolioChoiceModel

PreparedbyHuifengChang,FedericoGrinberg,LucynaGornicka,MarcelloMiccoli,andBrandonJoelTan

1

1ThisworkwasperformedwhenHuifengChangwasanIMFintern.ThisworkhasbenetedfrominsightfulcommentsbyDavid

Andolfatto,ChrisErceg,TommasoMancini-Griffoli,RobTownsend,Pierre-OlivierWeill,andseminarparticipantsatBancoCentraldoBrasil,BancodeMexico,Bancad'Italia,theEuropeanCentralBank,theFederalReserveBoard,theIMF,andRiksbank.

2

Contents

1Introduction

3

2Baselinemodelwithhomogeneoushouseholds

10

2.1Setup

11

2.2Equilibrium

13

2.3ImpactofCBDCintroduction

16

3Heterogeneoushouseholdsandtheextensivemarginofdepositdis-

intermediation

20

3.1Modelsetup

21

3.2Solutioncharacterization

22

3.3Calibration

23

3.4Results

25

3.5Keymechanismsdrivingresults

29

4Robustnessandextensions

32

4.1Robustness

33

4.2Efectsofdisintermediationinlending

35

4.2.1Quantitativeresults

38

5Conclusions

40

AComparativestaticsforthemodelwithhomogeneoushouseholds

43

BEquilibriuminmodelwithhouseholdheterogeneity

45

3

1Introduction

Recentyearshaveseenasurgeintheuseofnewkindsofprivatelyissueddigitalmoney.1Inresponse,moreandmorecentralbankshaveinitiatedworkonexploringtheissuanceoftheirowndigitalmoney,denominatedcentralbankdigitalcurrency

(CBDC).2

WhilediferentcentralbankspursuediferentobjectiveswithCBDC3,acommonkeyconcernpolicymakershaveisthewhethertheintroductionofCBDCcouldleadtobankdisintermediation(

Bindseil,

2020;

Mancini-Grifolietal.,

2018

).Disinterme-diationwouldbearesultofCBDCcrowdingoutcommercialbankdeposits.Depositsareacheapandstablesourceoffundingforbanks,soifCBDCbecomessuccessfulinsubstitutingbankdepositsasapaymentinstrument,thiscouldhaveanegative

impactonbanks’overallfunding,andthus,theirabilitytolend.

Inthispaperwepresentastandardportfoliochoicemodelwithbanks,inthespiritof

Monti

(1972);

Klein

(1971)and

Drechsleretal.

(

2017

),toanalyzewhetherCBDCcanactuallygeneratebankdisintermediationand,ifso,howbigthisefectmaybe.Inthemodel,householdschoosehowtoallocatetheirwealthbetweenanilliquidassetandthreeimperfectlysubstitutableliquidassets:cash,bankdeposits

andCBDC.Households’utilitydependsontheirinalwealthandonservicesprovided

1Forexample,paymentsystemsprovidedbymobilenetworkoperators,newpaymentsystemproviders,andstablecoins.See

AdrianandMancini-Grifoli

(2019)foradetaileddiscussion

.

2Forexample,China,Canada,Sweden,TheBahamas,andEuropeanUnion.See

Soderberg

etal.

(2022)foradiscussiononthesecentralbanks’policyobjectivesforconsideringaCBDC

.

3CentralbankshaveindicatedthatCBDCmayprovideamoree伍cient,secure,andmoderncentralbankmoneyavailabletoeveryone,andthatitcanalsoincreaseresilience,availability,e伍ciencyandcontestabilityofretailpayments,aswellasbroadeninancialinclusion.

4

bytheliquidassets.Depositsareoferedbybankswhothencaninvestfundsinbondsthatpayaixedreturnorlendtoirms.Bankshavemarketpowerindeposits,whichallowsthemtochargeapositivespreadbetweenthereturnonbondsandthedeposit

rate.

WemeasuretheimpactofCBDConbankintermediationbycomparingthesizeofthebankdepositbaseandthesizeofbanklendingbeforeandaftertheintroductionofCBDC.Inthemodel,CBDCissimplyanewandimperfectsubstitutefortheothertwoliquidassets:depositsandcash.Whenitiscostlesstoaccesstheliquidassets,householdswouldliketouseallthreeofthem,ashouseholdsderiveutilityfromvariety.WeshowanalyticallythatinthiscasetheintroductionofCBDCdoesnotleadtobankdisintermediation.ThereasonisthatCBDCreducesbanks’marketpower,towhichbanksoptimallyrespondbyincreasingtherateofreturnondeposits.Thus,householdschoosetoholdevenmoreofbankdepositsandtheaggregatedepositbaseincreases.WecallthisefecttheintensivemarginofCBDCintroduction,anditisalwayspositive.Atthesametime,duetothelostmarket

powerthenetefectofCBDCintroductiononbankproitsisnegative.4

However,therecanbebarriersandcoststoaccessinancialassets.Whileget-

tingcashusuallyhasnocostsforretailusers5,gainingaccesstodepositscanbe

Althoughthisismostcertainlythecaseforretailusersandsmallamountsofcash,storage

4Inarelatedmodelof

Andolfatto

(2021)theintroductionofCBDCalsohasanon-negative

impactonaggregatebankdeposits.Similarlytooursetup,thishappensbecausecompetitionfromCBDCmakesbanksoferhigherratesofreturnondeposits.However,in

Andolfatto

(2021)agents

cannotholdpositiveamountsofcash,CBDCanddepositsatthesametime,butstrategicallychooseonlyonemeansofpayment.Asaresult,alsotheextensivemargin(seebelow)hasalwaysapositiveimpactonthesizeofthebankdepositbase,whichisnotthecaseinourmodel.Allowingdeposits,cashandCBDCtobeimperfectsubstitutesallowsustostudytheconsequencesofCBDCintoductionforamoregeneralsetofhouseholdpreferences.

5

cumbersomeandcostly.6.

WeintroducehouseholdheterogeneityinwealthinthemodeltogetherwithixedcostsofholdingbankdepositsandCBDC.WhenaccesscoststoCBDCarelowerthanthoseofbankdeposits,theintroductionofCBDCcangeneratebankdisinter-mediation.Thishappenswhen(i)CBDCismoreliquidthandepositsandcashand(ii)thehighcostofaccesstobankdepositsleadspoorerhouseholdstoabandonde-positsandtouseCBDCandcashonly,evenwithhigherdepositrates.WecallthistheextensivemarginofCBDC.Ifthisefectislargeenough,theextensivemargin

canmorethanofsettheintensivemargin.

WecalibratethemodelwithheterogeneoushouseholdstoUSdataandsolveitnumerically.WeindthataggregatebankdepositsfallfollowingCBDCintroductionwhenitiseasytoaccessCBDCandwhenthemassofpoorerhouseholdsislarge.Inthiscase,morehouseholdsinddepositsixedcoststoohighandswitchcompletelytoCBDC.Banksdonotaggressivelyincreasedepositratestopreventtheoutlowofcustomersduetotherelativelysmallwealthheldbythepoorhouseholds.This

leadstoanaggregatedecreaseinbankdepositsof4percent.

ToanalyzetheimpactofCBDConbanklending,weextendthemodeltoirmsthatdemandcreditfrombanks.Weindthat,evenwhenCBDCgeneratesdepositdisintermediation,thenegativeefectonlendingisquantitativesmallandbelow0.2

percent.Banks’accesstootherformsoffunding,suchaswholesaleorcentralbank

costscanbenon-negligiblewhentheyinvolvelargeramounts.Cashalsopaysalowerrealreturncomparedtobankdeposits(aslongasinterestratesondepositarepositive).

6Forexample,banksmayrequirecustomerstoprovideaproofofresidenceandemploymentinordertoopenanaccount.Somebanksmayalsochargeixedfeesforsettingupanaccount

6

inancing,allowsbankstocompensatethedeclineindepositswithouthavingtoreducelendingtoomuch.Ontheonehand,whenthesealternativefundingsourcesarerelativelycheap,itiseasyforbankstosubstituteawayfromdeposits.7Ontheotherhand,whenalternativeformsoffundingareexpensive,banksightfordepositsmoreaggressively,furtherincreasingdepositrates,andthusreducingtheirlossof

deposits.8

Overall,ourresultsshowtheimportanceoftakingintoaccounthouseholdschar-acteristics,marketstructure,andbanks’strategicresponseswhenassessingtheim-pactofCBDConthebankingsystem.Policymakersaimingtoexamineresiliencyofbanklendingtotheintroductionofpotentiallyveryattractivemeansofpaymentshouldtakeintoaccountthemechanismsthatweunveil.Ourindingspointalsototheneedforqualitydataonhouseholds’preferencesovermeansofpaymentto

estimatethedemandforliquidassets.

RelatedLiteratureOurpapercontributestotheemergingliteratureonCBDCs.OurmainfocusisontheefectsthatCBDCintroductioncanhaveonthedepositbaseofcommercialbanksandonbanklending.Otherpapersthathavealsoconsidered

thisquestioninclude

Andolfatto

(2021),

Chiuetal.

(2023)and

Aguretal.

(2022)

.

In

Andolfatto

(2021),bankingsectorisalsomonopolistic,butCBDCisaperfect

substituteforcurrencyandbankdeposits.Thus,agentschoosetoholdonlyone

7Themodelignorestheefectthatthechangesinthefundingstructuremayhaveonregulatoryratiosor,moregenerally,oninancialstability.

8Animportantcaveatisthatourmodelisstatic,sothecompressionofbankproitsandcapitalerosiondoesnotafectlending.Thisisclearlyachannelthatcanhaveanimpactonlendinginadynamicsetting.See,forinstance,(

VandenHeuveletal.,

2002

).

7

meansofpaymentandbanksalwaysmatchtheratepaidondepositswiththereturnonCBDC.Additionally,costsofaccessingbankdepositsandCBDCarethesame.Asaresult,whilethereisanextensivemarginofCBDCintroductionsimilartooursetting,

Andolfatto

(2021)alwayshasapositiveimpactonbankdepositsleadingto

intermediationofbankdeposits.

Chiuetal.

(2023)consideramodelcalibratedtotheUSeconomywherecash

anddepositsareusedfordiferenttransactionsandwherearemuneratedCBDCisaperfectsubstituteforbankdepositsonly.Theyshowthatifbankshavemarketpowerinthedepositmarket,CBDCcanenhancecompetition,raisingthedeposit

rateandexpandingintermediation.

InourmodelCBDCcanincreasebankdepositbasebyreducingthemarketpowerofbanks—asin

Andolfatto

(2021)and

Chiuetal.

(

2023

).OurmodelalsonestskeymechanismsofthesetwopaperswhileourrichersetupunveilsthatCBDC’sefectondepositsdependsonparametrizationofhouseholdspreferences,wealthdistribution,andthecostsofaccessingCBDCrelativetobankdeposits.Incontrasttothesetwopapers,wemodelCBDCasanimperfectsubstituteforbothcashandbankdepositsinasimpleportfoliochoicemodel.Thus,CBDCisheldregardlessofwhetheritpaysaninterestornot.9;10Diferentfrom

Chiuetal.

(2023)andasin

Andolfatto

(

2021

),inourmodelhouseholdsareheterogeneousinwealthwhichleadstoanintensiveand

extensivemarginofdepositholding.Diferentfrom

Andolfatto

(

2021

),weshowthat

9ThisisrelevantasmostcentralbanksarenotconsideringpayinganinterestrateonaretailCBDC.

10Inboth

Chiuetal.

(2023)andin

Andolfatto

(2021)CBDCincreasescompetitioninthedeposit

marketbyservingasanoutsideoptiontodepositorsandsettingtheinterestrateondepositsandCBDCisnotheldinequilibrium.

8

theextensivemargincanbestrongerandleadtodepositdisintermediationandlowerbanklendingifboththecostofsettingaCBDCaccountissmallerthansettingup

abankdepositaccountandifthereislargermassofpoorerhouseholds.

Aguretal.

(2022)considerasetupwherehouseholdschoosethemeansofpay

-mentsdependingontheirpreferencesoverthelevelofanonymityandsecurityoftransactions.Whilecashofersmostanonymity,bankdepositsprovidemostsecu-rity.Similarlytoourmodel,varietyinpaymentinstrumentsincreaseswelfare,butthisisbecauseoftheheterogeneityinhouseholdpreferences.Contrarytooursetup,

Aguretal.

(2022)donotconsidertheroleofbanksmarketpowerinthedeposits

.There,banksaremodeledasprice-takersinbothdepositandloanmarkets.TheimplicationsofCBDCintroductioncruciallydependonhowcloseitresemblescashordeposits:acash-likeCBDCcanreducethedemandforcashbeyondthepointwherenetworkefectscausethedisappearanceofcash,whileadeposit-likeCBDCcancauseanincreaseindepositandloanrates,andacontractioninbanklendingtoirms.TheoptimaldesignofCBDCinvolvesatrade-ofbetweenlossofutilityfromvarietywhenCBDCcrowdsoutcashandlossofbankintermediationinthepresence

ofseverelendingfrictions.

OtherrelatedpapersonmacroeconomicimplicationsofCBDC,include

Barrdear

andKumhof

(2022),

Burlonetal.

(2022),

KeisterandSanches

(

2023

),

Brunnermeier

andNiepelt

(2019),

Williamson

(2019),

PiazzesiandSchneider

(2022),

Garrattetal.

(2021),

WangandHu

(2022),

GrossandLetizia

(2023),

Whitedetal.

(2023),and

Lietal.

(2023).Inparticular,

KeisterandSanches

(

2023

)showthatbychoosinga

properinterestonCBDC,policymakerscanensurethatCBDCintroductionnever

9

decreaseswelfare.

BarrdearandKumhof

(

2022

)introduceCBDCinaDSGEmodelwithcompetitivebutregulatedbankingsector.TheyindthatCBDCalwaysspurseconomicactivity,lowersthepolicyanddepositratesandincreasesbanklending.

Burlonetal.

(2022)alsoutilizeaDSGEmodelwhereCBDC,cash,anddepositsare,

asinourmodel,imperfectsubstitutesfortherepresentativehousehold.TheyindthatCBDCexertsasmoothingefectaslendingandrealGDPbystabilizingdepositholdswhenthereareshockstoliquidityservices.

Garrattetal.

(

2021

)consideramodelwithbanksthathaveheterogenousmarketshares,andanalyzehowaninterest-bearingCBDCcanafectconcentrationinthebankingsystem.TheyindthattheimpactcruciallydependsonthedesignofCBDC.

WangandHu

(2022)studythe

linkbetweenCBDCandinancialdevelopment.Theyarguethatinlessinanciallydevelopedeconomies,retailCBDCscanbeusefulforpromotinginancialinclusion,whileincountrieswithhighlevelsofinancialdevelopment,CBDCcanenhanceinancialstabilitybysubstitutingoutmoreriskynon-banke-money.

Grossand

Letizia

(2023)developanagent-basedmodelwherehouseholdshaverandomutility

overCBDC,cash,anddepositsandcalibratedtotheUSeconomythatgeneratesmoreorlessdisintermediationdependingonwhetherCBDCisdesignedtobemoretodepositsortocash.

Whitedetal.

(2023)and

Lietal.

(

2023

)estimatepotentialdemandofCBDCusinghouseholddataandstructuralchoicemodels.Whiletheapproachofthesepapersisverydiferentthanours,theirbaselineresultsfordeposits

andlendingareinlinewithours.11

Ourpaperalsobelongstothevastliteraturestudyingimplicationsofimperfect

11Forexample,

Whitedetal.

(2023)indthattheimpactofCBDConlendingismuchsmaller

thanondepositsasbankshavetheoptiontousewholesalefunding.

10

competitioninbankingsystem(Drechsleretal.,

2017,

Repulloetal.,

2020)

.Inparticular,webuildonatmodeldeveloped

Drechsleretal.

(

2017

)tostudythedepositchannelofmonetarypolicy.Themodelcontainstwofeaturesthatmakeitsuitableforourpurposes:i)imperfectsubstitutionbetweenliquidassetsasameansofpayment,andii)imperfectlycompetitivebankingsystem.Finally,ourworkbuildsonmodelsthatdistinguishbetweentheextensiveandintensivemargins

ofadjustmentasin

Hopenhayn

(1992)and

Melitz

(2003)

.

Therestofthepaperisorganizedasfollows.Section

2

introducesthebaselinemodelwithhomogeneoushouseholdsandnoixedcostsforholdingbankdepositsandCBDC.Section

3

discussestheenrichedmodelwithheterogeneoushouseholdsandixedcostsofholdingdepositsandCBDC.Section

4

addslendingandwholesale

fundingforbankstothemodel.Section

5

concludes.

2Baselinemodelwithhomogeneoushouseholds

Thissectionintroducesthebaselinemodelwithhomogeneoushouseholds.Thepur-poseistointroducesomeofthemechanismsthatareatplayinthelargermodelwithheterogeneousagents.Weshowthatwhenhouseholdsarehomogeneousinwealth,

theintroductionofCBDCwillalwaysleadtoanincreaseintotalbankdeposits.

11

2.1Setup

Weconsideraportfoliochoicemodelwithanimperfectlycompetitivebankingsector.

Therearethreetypesofagentsinthemodel:households,banks,andacentralbank.

HouseholdsHouseholdsarehomogenousandhaveinitialwealthofW0,whichtheyallocateamongfourtypesofassets:(i)notes(cash),denotedbyN,earnsnoreturn;(ii)CBDC,denotedbyC,earnsreturnrC0;(iii)deposits,denotedbyD,earnrD;and(iv)bonds,earnanon-negativeratef.Thebondsarerisk-free,andfistherisk-freeratesetbythecentralbank.Bondsarealso“illiquid”astheyarenotusefulasmeansofpayment.Cash,CBDC,anddepositscaninsteadbeusedfor

payments,creatingliquidityservicesvalueinhouseholds’utilityfunction.

Households’utilityisafunctionofinalwealthWandliquidityservicesL:

U(W0)=max（W-1+λL-1）-1,

(1)

wherewealthandliquidityarecomplements,withtheelasticityofsubstitutionρ<

1.12Liquidityservicesarisingfromholdingcash,CBDCanddepositsaredeinedby:

E-1E-1E-1E

L(N,C,D)=(NE+δCCE+δDDE)E-1.(2)

Thethreeliquidassetsareimperfectsubstitutesforhouseholds,hencetheelasticity

ofsubstitutionisgreaterthanone,E>1.δCandδDrepresenttherelativeusefulness

12Weabstractfromadjustmentcostinnonliquidwealth.Ifthesecostsweretakenintoaccount,impactswouldbemoremuted.See

Kaplanetal.

(2018)foranexampleofanilliquidassetsportfolio

choicemodelingeneralequilibrium.

12

ofCBDCanddepositsasmeansofpaymentscomparedtocash.

Householdsfacethefollowingbudgetconstraint:

W=W0(1+f)-Nf-C(f-rC)-D(f-rD),(3)

rearrangedtohighlighttheopportunitycostsofholdingtheliquidassetswithrespecttobonds.Ascashearnsnoreturn,householdsfaceanopportunitycostoff,thereturnonbonds,whenholdingcash.TheopportunitycostsofholdingCBDCanddepositsarelowerthanforcash,astheyguaranteenon-negativereturnsrCandrD,

respectively.

BanksAggregatedeposits(D)areacompositegoodproducedbyasetofJbanks

(Dj),indexedbyj2f1,2...,Jg:

η

D=Djη-1,(4)

whereη>1istheelasticityofsubstitutionbetweendepositsofdiferentbanks.It

isgreaterthanone,relectingimperfectsubstitutabilitybetweenbankdeposits.

TofocusontheefectthatCBDChasonthedepositmarket,fornowweas-sumethatbanksarefullyfundedbydepositsandcanonlyinvestinbonds.TheseassumptionsarerelaxedinSection

4.

Asdepositsareimperfectsubstitutes,bankshavemarketpowerandsetthereturnondepositsrD,jwiththeobjectiveofmaxi-

mizingtheirproits,(f-rD,j)Dj,subjecttodepositsdemand.Thereturnonag-

13

gregatedepositsisdeinedbytheweightedaverageofeachbank’srateofreturn,i.e.

rD=1D,j.

CentralbankThecentralbankchoosestherisk-freeratef,i.e.remunerationonbonds,andtheinterestrateonCBDC,rC.ItalsosuppliesbondsandCBDCwith

anininiteelasticity.

2.2Equilibrium

Thebehaviourofhouseholdsischaracterizedbyfourirst-orderconditions.First,

householdschoosebetweenliquidassetsandbondsaccordingto:

L

W

=λρsρ,(5)

wheresL=(f1-E+δ(s*)1-E+δ(f-rC)1-E

liquidassets.Ahigherforgoneinterestdecreasestheshareofwealthkeptinliquid

assets.13

Second,householdschoosebetweenliquidassetsaccordingtothetwoirst-order

conditions

CN

CD

=δCE-E,

=()E(--)-E.

(6)

(7)

13Notethatthehouseholds’budgetconstraintcanbealsorewrittenasW=W0(1+f)-LsL.

14

ItfollowsthathouseholdswillwanttoholdmoreCBDCifitismoreusefulasmeans

ofpaymentsrelativetootherliquidassets,andifitearnsahigherreturn.

Third,householdschoosebetweendepositsofdiferentbanksaccordingto:

=-η.(8)

Banks’havemarketpower,hencetheycanremuneratedepositsbelowthecentralbank’srisk-freerate:thespreadwithrespecttotheratef,f-rD;j,ispositive.The

irst-orderconditionforbanksisgivenby:

?Dj(f-rD;j)

?(f-rD;j)Dj

=-1.

(9)

Wecanusetheinterbankmargin(8)tocalculatetheelasticityin(9).Following

Drechsleretal.

(2017),wefocusonthesymmetricequilibriumwithD

j=Dwhere

itisgivenby:

=-η（1-.(10)

Substituting(10)into(9),wegettheequilibriumcondition

.

-η（1-=-1.(11)

Itfollowsfrom(11)thatwhenbanksareataninterioroptimum,theelasticityof

15

aggregatedepositdemandwithrespecttothespread(f-rD)isequalto:

?D(f-rD)

?(f-rD)D

=1-(η-1)(J-1)=w.

(12)

Theelasticityofdemandwithrespecttothespreaddecreasesinthelevelofcompetitioninthedepositmarket.Inturn,thecompetitivenessofthedepositmarketincreaseswiththenumberbanksJ,andwithhighersubstitutabilityofdepositsacross

banks,η.

Aclosed-formsolutiontothemodelcanbeobtainedforthelimitcaseinwhichλ!0.Inthiscase,followingproposition1in

Drechsleretal.

(

2017

),ifE>w>ρ,

thedepositremunerationandaggregatedepositsaregivenby:

f-rD*=δ

and

1

ε-1

lf1-E+δ(f-rC)1-E]

1

1-ε>0,(13)

ρ-ε

D*=δ(f-rD*)-ρl1+δEE-1+-EE-1]ε-1.

(14)

Ifw<ρ,thenrD*=f.

Throughouttheanalysiswefocusonthecasewhenthereturnondepositsisstrictlylessthanthepolicyratef,henceweimposethatE>w>ρ.ThisholdsasEisgreaterthanone(asliquidassetsaresubstitutes)andwhasanupperboundof

one(whentherepresentativebankactsasamonopolist).Wefocusontheempirically

16

relevantcaseinwhichbankshavemarketpower,sorislowerthanf,soMislarger

thanρ.

Asin

Drechsleretal.

(2017)theequilibriumspreads

*=f-rD*isnon-decreasing

andtheamountofdepositsD*isnon-increasinginthepolicyratef,givingarise

toa”bankdepositchannel”:

≥0(15)

?D*

?f

0.

(16)

Inequilibrium,ahigherratefincreasestheopportunitycostofusingcashorCBDCinsteadofdepositstoserviceliquidityneeds,allowingbankstoincreasetheratepaidonbankdeposits,butbynotasmuchasf,hencethespreads*increases.Inresponsetothehigheropportunitycostofho