BOC-央行数字货币和银行业务选择-Central Bank Digital Currency and Banking Choices 2024

BankofCanadastaffworkingpapersprovideaforumforstafftopublishwork-in-progressresearchindependentlyfromtheBank’sGoverningCouncil.Thisresearchmaysupportorchallengeprevailingpolicyorthodoxy.Therefore,theviewsexpressedinthispaperaresolelythoseoftheauthorsandmaydifferfromofficialBankofCanadaviews.NoresponsibilityforthemshouldbeattributedtotheBank.

DOI:

/10.34989/swp

-2024-4|ISSN1701-9397©2024BankofCanada

StaffWorkingPaper/Documentdetravaildupersonnel—2024-4

Lastupdated:February8,2024

CentralBankDigitalCurrency

andBankingChoices

byJiaqiLi,1AndrewUsher1andYuZhu2

1BankingandPaymentsDepartmentBankofCanada

jiaqili@bankofcanada.ca,

AUsher@bankofcanada.ca

2RenminUniversityofChina

zhuyuzlf57@

i

Acknowledgements

WethankJasonAllen,MatteoBenetton,JonathanChiu,RobertClark,RodGarratt,CharlesKahn,AnnekeKosse,MarianaLaverde,YoumingLiu,ChristineParlour,NunoPaixao,NicolaPavanini,HectorPerez-Saiz,FranciscoRivadeneyra,EnchuanShao,RobertTownsend,StephenWilliamson,andYufengWufortheinsightfulsuggestions.WethankJeffreyWuforexcellentresearchassistanceinverifyingthebranchlocationandgeocodingvariouslocationdatasets.Wethankseminarparticipantsatthe2023CentreforEconomicPolicyResearch(CEPR)—EuropeanCentralBank’sMacroeconomicImplicationsofCentralBankDigitalCurrenciesconference;the2ndConferenceontheEconomicsofCentralBankDigitalCurrencyhostedbytheBankofCanadaandSverigesRiksbank;EconomicsofPaymentsXIIConferencehostedbytheFederalReserveBoard;theEuropeanEconomicAssociation’s2023AnnualConference;theEuropeanAssociationforResearchinIndustrialEconomics2023AnnualConference;the2023SummerWorkshoponMoney,Banking,Payments,andFinance;theSocietyforEconomicDynamics2023Meeting;the1stAnnualConferenceoftheBancoCentraldoBrasil;theSocietyforEconomicMeasurement2023AnnualConference;theInternationalAssociationforAppliedEconometrics2023AnnualConference;the2023InternationalIndustrialOrganizationConference;CanadianEconomicsAssociation’s2022AnnualConference;the2022TokenomicsConference;andvariousseminarsattheInternationalMonetaryFund,CentralBankofBrazil,BankofCanada,andtheUniversityofSaskatchewan.TheviewsexpressedinthispaperarethoseoftheauthorsandnotnecessarilytheviewsoftheBankofCanada.YuZhuacknowledgesthesupportfromtheNaturalScienceFoundationofChina,Grantno.72192801.

ii

Abstract

Towhatextentdoesacentralbankdigitalcurrency(CBDC)competewithbankdeposits?Toanswerthisquestion,wedevelopandestimateastructuralmodelwhereeachhouseholdchooseswhichfinancialinstitutiontodeposittheirdigitalmoneywith.Householdsvaluetheinterestpaidondigitalmoney,thepossibilityofobtainingcomplementaryfinancialproducts,andtheaccesstoin-branchservices.Anon-interest-bearingCBDCthatdoesnotprovidecomplementaryfinancialproductscansubstantiallycrowdoutbankdepositsonlyifitprovidesanextensiveservicenetwork.ImposingalargelimitonCBDCholdingwouldeffectivelymitigatethiscrowdingout.

Topics:Centralbankresearch;Digitalcurrenciesandfintech

JELcodes:E50,E58

Résumé

Dansquellemesureunemonnaienumériquedebanquecentrale(MNBC)concurrence-t-ellelesdépôtsbancaires?Pourrépondreàcettequestion,nousélaboronsetestimonsunmodèlestructureldanslequelchaqueménagechoisitl’institutionfinancièreoùdéposersonargentnumérique.Lesménagesévaluentlesintérêtsservissurl’argentnumérique,lapossibilitéd’obtenirdesproduitsfinancierscomplémentairesetl’accèsauxservicesensuccursale.UneMNBCneportantpasintérêtetnonassortiedeproduitsfinancierscomplémentairesnepeutévincerdansunelargemesurelesdépôtsbancairesquesielleestassociéeàunréseaudeservicesétendu.L’impositiond’unelimiteélevéeàladétentiondeMNBCatténueraitefficacementceteffetd’éviction.

Sujets:Recherchesmenéesparlesbanquescentrales,Monnaiesnumériquesettechnologiesfinancières

CodesJEL:E50,E58

1

1Introduction

Manycentralbanksareconsideringissuingaretailcentralbankdigitalcurrency(CBDC),adigitalformofcentralbankmoneythatisavailabletothegeneralpublicandcanbeusedforretailtransactions

.1

ByofferingthepublictheoptiontoholdthedigitalmoneyinaCBDCinsteadofbankdeposits,theCBDCwoulddirectlycompetewithbankdepositsinthemarketfordigitalmoney.Asaconsequence,thereareconcernsthataCBDCcouldsubstantiallycrowdoutbankdeposits,whichmayunderminefinancialstabilitybyraisingthefundingcostandreducingtheprofitabilityofthebankingsector.However,littleisknownaboutthepotentialmagnitudeoftheCBDCadoptionanditscrowdingouteffectonbankdeposits.

ThekeytoaddressingthisquestionistounderstandhowaCBDCwoulddifferfrombankdeposits.ThispaperquantifiestheimpactofaCBDConbanksinthedepositmarket,takingintoaccounttwokeydifferentiatingfeaturesbetweenaCBDCandbankdeposits:(i)banks

provideproductscomplementarytodeposits,suchasmortgagesandcreditcards,2

whileaCBDCwouldlikelybeastand-alonecentralbankproduct;(ii)mostbankshaveextensivebranchnetworksforin-personservice,whiledependingonthedesign,theservicelocationnetworkofaCBDCcanbeverydifferentfrombanks’branchnetworks.Forexample,aCBDCcanbedesignedtobeentirelydigitalwithnophysicalbranch,oritcanbedesignedtohaveanetworkofbrancheslikebanks.

IncorporatingthesetwofeaturesisimportantforstudyingthecrowdingouteffectsofaCBDC,becausehouseholdshavestrongpreferencesforobtainingcomplementaryfinancialproductsfromtheirdepositbanksandforconvenientlyaccessingin-personservicesatphys-icallocations(bankbranches).AccordingtoCanadianhouseholdsurveydata2010–2017,56%ofmortgageborrowersand45%ofcreditcardholderschoosetheirdepositbanksfortherespectiveproduct

.3

Additionally,bothurbanandruralhouseholdspreferbankswithbranchesclosertotheirresidences.Thesamesurveydatashowthataround60%ofdepositors

1Accordingtothe2021BankforInternationalSettlements(BIS)survey,90%ofthecentralbanksareengaginginCBDCworkand62%areconductingexperimentsorproofsofconcept.Thesurveyrespondentsinclude81centralbanksthatrepresentcloseto76%oftheworld’spopulationand94%ofglobaleconomicoutput(

KosseandMattei,

2022

).TheBahamas,EasternCaribbeanCentralBank,Jamaica,andNigeriahavealreadylaunchedretailCBDCs.Thereare21countriesthathavealreadystartedtheirpilots,includingChina,India,Singapore,SouthKorea,andSweden.

2Wefocusonthecomplementaritybetweendepositsandotherfinancialproductswithinthesamebankthatisenjoyedbyconsumers,whichdiffersfromthecomplementaritybetweencreditlinesanddepositsthatisenjoyedbybanksin

PiazzesiandSchneider

(2020)orthecomplementaritybetweenprintandonline

newspapersin

Gentzkow

(2007)

.Thecomplementarityinourpapercouldreflecttheconvenienceofmanagingthedifferentproductsinthesamebank,theeaseofmakingbillpaymentsfromthedepositaccount,orthebetterdealsfrombundlingdifferentproductsatthesamebank,forinstance.

3Inoursample,30%ofhouseholdshaveamortgageloanand90%ofhouseholdshaveacreditcard.

2

reportedthattheyhadvisitedtheirbranchesatleastonceinthepastmonth.Inparticular,depositorsprefertousebankbranchesfornon-transactionrelatedservices,suchasobtainingcustomerserviceandsupportandmakingcomplaints,comparedtootherbankingmethodssuchasonlineandmobilebanking.

Toincorporatethesetwofeatures,wedevelopastructuralmodelwherehouseholdsobtainutilityfromholdingtheirendowedliquidassetsinphysicalcashanddigitalmoney.IntheabsenceofaCBDC,theonlyavailabledigitalmoneyisdepositsprovidedbyprivatebanks.Toholddeposits,ahouseholdfirstneedstochooseoneoftheprivatebanks,takingintoaccountthatbanksofferdifferentdepositrates,complementaryfinancialproducts,andservicelocations(bankbranches).Eachbankfacesadepositdemandthataggregateseachindividualhousehold’sdepositdemandandcompetesinthedepositrate.Weestimatethemodelprimitives,whichincludehouseholds’preferenceparametersfordifferentattributes(e.g.,depositrate,complementarity,andbranchnetwork)andbanks’marginalcosts.ThesemodelprimitivesareassumedtobeunchangedafterCBDCissuance.

Akeychallengeinestimatingthismodelistoidentifyhouseholds’preferencesforob-tainingcomplementaryfinancialproductsfromtheirdepositbanksandforservicelocations.Toachieveidentification,weuseauniqueCanadianhouseholdsurveydatasetthatcontainsdetailedinformationonhouseholds’bankchoicesforeachfinancialproduct,whichallowsustoestimatehowmuchhouseholdsvalueobtainingagivenfinancialproductfromtheirdepositbank.Inaddition,wecombinethehouseholds’residentiallocationsfromthehouse-holdsurveydatawiththebankbranchlocationdatatomeasurethebranchnetworkthatisspecifictoeachhousehold’slocalareaandestimatehouseholds’preferencestowardsthebranchnetworkwhenchoosingtheirdepositbanks.

Afterobtainingtheestimatesofthemodelprimitives,wethenintroduceaCBDCintothemodelandstudyitsimpactonbanksincounterfactualanalyses.WemodeltheCBDCasanewproductwithattributesthatareexogenouslychosenbythecentralbank.InthepresenceofaCBDC,eachhouseholdchoosestodeposittheirdigitalmoneyineitheranexistingbankortheCBDCbasedontheirutilitiesfromeachproduct,whichinturndependonhowhouseholdsvaluetheattributesofeachproductaswellasthehousehold-specifictastesforeachproduct.IntroducingaCBDCwouldreducethefractionofhouseholdsthatchooseexistingbanksbecausetherewillalwaysbesomehouseholdswithtastesinfavorofaCBDC.Thisleadstoalowerdepositdemandforeachbank.Moreover,theCBDCincreasesthedepositdemandelasticitywithrespecttothedepositratebecausehouseholdsnowhavemoreoptions.

Facingamoreelasticdepositdemand,banksoptimallyraisetheirdepositratetomaintaintheirattractiveness.Withthebetterdepositrate,ahouseholdwouldholdmoredeposits

3

relativetocashiftheyweretochooseanexistingbank.However,thisintensivemarginchangeisdominatedbytheextensivemarginchange(i.e.,fewerhouseholdschooseexistingbanks).Therefore,inequilibrium,households’depositdemandislowerandtheCBDCwouldcrowdoutbankdeposits.

WhenaCBDCisdesignedtobemoreattractive,itwouldleadtomorecrowdingout.ApartfromtheCBDCinterestrate,weconsiderdesignfeaturessuchascomplementarity,servicelocationnetwork,andapotentiallimitonCBDCholdings.WeassumethattheCBDCdoesnotoffercomplementaryfinancialproductssuchasmortgageloansandcreditcards

.4

Dependingonthedesign,theCBDCmayhavenoservicelocationormayhaveanetworkofservicelocationsthatisidenticaltothenetworkofallbankbranchesorthenetworkofCanadaPostoffices

.5

Additionally,weuseourmodeltostudytheeffectofintroducingalimitonCBDCholdings,whichisadesignchoicefrequentlydiscussedbypolicymakersasapotentialtooltoreducethecrowdingouteffectoftheCBDC.

Themainfindingsareasfollows.First,thenetworkofservicelocationsandtheabsenceofcomplementaryfinancialproductsaretwokeyfactorsthatdeterminethetake-upofaCBDCanditscrowdingouteffectonbankdeposits.Anon-interest-bearingCBDCthatdoesnotcomewithcomplementaryfinancialproductswouldrequirebetterservicelocationstobeattractive.Ifithasnoservicelocation,itwouldhaveanegligibleimpact,crowdingoutbankdepositsbyonly1%.Incontrast,ifitusesbothCanadaPostofficesandbankbranchesasservicelocations,itwouldcrowdout12%ofbankdeposits.Underthisextensivenetworkdesign,ifwemisspecifythemodelbyneglectingthecomplementaritybetweendepositsandotherfinancialproducts,wewouldsignificantlyoverestimatetheimpactofaCBDCandconcludeincorrectlythataCBDCcancrowdout39%ofbankdeposits.Therefore,itiscrucialtoaccountforthesetwodifferentiatingfactorswhenquantifyingtheeffectsoftheCBDConbankdeposits.

Second,alargelimitonCBDCholdingscansubstantiallyreducethecrowdingout,whileitsimpactonreducingthedepositors’welfaregainsismuchsmaller.Bysolvingthemodelforarangeofdifferentlimits,wefindthatevenalargeholdinglimitof25,000Canadian

4Asaconsequence,ifahouseholdchoosestheCBDCfortheirdigitalmoney,theystillneedtogotoaprivatebankforamortgageloanoracreditcardandwouldnotenjoythesynergybetweentheCBDCandthesefinancialproducts.Wethinkthisisarealisticdesignfeaturesincethecentralbankisunlikelytomakeloanstothepublic.

5ThesecounterfactuallocationchoicesaremotivatedbythepolicydiscussionsonwhetheracentralbankshoulddirectlydelivertheCBDCtothepublic(i.e.,single-tiersystem)orpartnerwiththeprivatesector(i.e.,two-tiersystem).The2022BISsurveyshowsthat87%ofcentralbanksengagedinretailCBDCworkareconsideringatwo-tiersystemthatinvolvesarolefortheprivatesector(

KosseandMattei,

2023)

.Wehighlighttheimplicationsofthedeliveryarrangementonservicelocations.Whileasingle-tiersystemislikelytoimplynoservicelocationfortheCBDC,atwo-tiersystemwouldimplyanetworkofCBDCservicelocationssimilartothoseoftheprivatesectorproviders.

4

dollars,whichismuchhigherthanthelimitof3,000eurosconsideredbytheECB(

Panetta,

2022

),wouldreducetheshareofliquidassetsheldintheCBDCbyhalfregardlessoftheservicelocationnetwork.Thisisbecausetheconstrainedhouseholds,whilerepresentingonlyaround10%ofhouseholds,arefromthetopendoftheliquidassetdistributionandthustendtoholdalargeamountofliquidassets.Sincethislargelimitonlyaffectsasmallproportionofhouseholds,thereductioninhouseholds’welfaregainsfromtheCBDCismuchsmaller.ThissuggestspolicymakerscanreducethecrowdingoutwhilemakingCBDCattractivetomosthouseholds.

Third,impactsofaCBDConbothbanksandhouseholdsareheterogeneous.WefindthatbankswithhighermarketsharestendtorespondmoretotheintroductionofaCBDC,

raisingdepositratesbymoreandlosingfewerdeposits

.6

HouseholdsinruralareaswouldbenefitmorefromaCBDCthanurbanhouseholdseveniftheCBDCdoesnothaveservicelocations,becauseofthelackofaccesstobanksinruralareas.ThedifferenceislargerwhentheservicelocationsoftheCBDCincludetheCanadaPostoffices,whicharemoreevenlydistributedacrossruralandurbanareascomparedtobankbranches.Therefore,fromafinancialinclusionperspective,itmightbedesirabletoconsiderlocationsasbroadasCanadaPostofficesasservicelocationsfortheCBDC.

ThispapercomplementsthegrowingtheoreticalliteraturethatstudiestheeffectsofaCBDConbanking(e.g.,

Changetal.,

2023;

Chiuetal.,

2023;

KeisterandSanches,

2023;

Garratt,YuandZhu,

2022;

Andolfatto,

2021)

.7

ThisliteraturetypicallyassumesthattheCBDCisaperfectsubstituteforbankdeposits

.8

MostpapersinthisstreamalsohavelimiteddiscussionsonthedesignsoftheCBDC,focusingmainlyontheinterestrate.Bycontrast,weuserichmicro-leveldatatoestimatehouseholds’preferencesoverproductcharacteristicssuchasbranchnetworksandcomplementaritywithotherfinancialproducts,andweuse

6Similarly,

Whited,WuandXiao

(2023)findthatwhenthebankingsectorismoreconcentrated,banks

losefewerdepositstoaonedollarincreaseinCBDC.

Garratt,YuandZhu

(2022)findthatthelargebank

raisesitsdepositratebymoreinresponsetoahigherCBDCinterestrate.Incontrast,withthehousehold-leveldata,wecanalsostudytheheterogeneousimpactofaCBDConhouseholds.

7ExistingtheoreticalliteraturealsostudiestheimpactofCBDConfinancialstability(e.g.,

Ahnert

etal.,

2023;

Fern´andez-Villaverdeetal.,

2021;

Williamson,

2021;

Schilling,Fern´andez-VillaverdeandUhlig,

2020;

BrunnermeierandNiepelt,

2019;

Skeie,

2019),monetarypolicy(e.g

.,

Abad,Nu˜noandThomas,

2023;

Davoodalhosseini,

2022;

JiangandZhu,

2021;

BordoandLevin,

2017),macroeconomicvolatility(e.g

.,

As-

senmacher,BitterandRistiniemi,

2023;

BarrdearandKumhof,

2022;

George,XieandAlba,

2022;

Minesso,

MehlandStracca,

2022),paymentplatformcompetition(

Liu,ReshidiandRivadeneyra,

2023),andwelfare

(e.g.,

Williamson,

2022;

Assenmacheretal.,

2021;

PiazzesiandSchneider,

2020

).ForpolicydiscussionsonthemacroimplicationsofCBDCissuance,see

GrossandLetizia

(2023),

Davoodalhosseini,Rivadeneyraand

Zhu

(2020),

Garc´ıaetal.

(2020),

BerentsenandSchar

(2018),

Mancini-Griffolietal.

(2018),

Meaningetal.

(2018),

EngertandFung

(2017),etc

.

8

Garratt,YuandZhu

(2022)consideralargebankandasmallbankthatdifferinbothdepositratesand

conveniencevaluesintheirtheoreticalanalysis,buttheystillmaintaintheperfectsubstitutionassumptioninthesensethatCBDCdemandwouldbezeroifitsinterestrateandconveniencevaluecombinedislower.

5

thesepreferencestopredictthesubstitutionpatternsbetweendepositsandtheCBDCwithdifferentdesigns

.9

ThispaperismostcloselyrelatedtotheempiricalliteratureonCBDC,whichisscarceatthispointduetothelackofdataonCBDC.Toaddressthisissue,oneapproachistodevelopastructuralmodelandestimatethehouseholds’preferencesforproductattributesusingthedataonexistingproductsthatareclosesubstitutesfortheCBDC.TheestimatedmodelisthenusedtoconductcounterfactualanalysisontheCBDC.Usingthisapproach,

Li

(2023)studiesthehouseholds’potentialholdingsoftheCBDCrelativetocashanddemand

deposits,and

Huynhetal.

(2020)studytheconsumeradoptionandusageoftheCBDCas

apaymentinstrumentrelativetocash,debitcards,andcreditcards,usingtheCanadianconsumersurveydata.Morerecently,

Whited,WuandXiao

(2023)buildastructuralmodel

toquantifytheimpactofaCBDConbanklendingusingUSbank-leveldata.Ourpapercontributestothisstreambyhighlightingtheimportanceofthecomplementaritybetweendepositsandotherfinancialproductsfromtheconsumers’pointofviewandtheservicelocationswhenstudyingtheimpactofaCBDConbankdeposits

.10

AnotherapproachistoconductsurveysorexperimentstodirectlyaskpeopleabouttheirintentiontoadoptahypotheticalCBDC

.11

Forexample,surveyshavebeenconductedintheNetherlands(

Bijlsmaetal.,

2021),inAustria(Abramovaetal.,

2022),andbythe

EuropeanCentralBank(ECB)(

KantarPublic,

2022

).IntheabsenceofaCBDCoraconcretedesignforCBDC,thissurveyapproachischallengingbecausetheresultswouldrelyheavilyonconsumers’understandingoftheCBDCbasedonabroaddescriptionfortheCBDC.

Choietal.

(2022)useadiscretechoiceexperimentapproach,whereeachrespondent

isaskedtochoosebetweenapairofhypotheticalpaymentmethods(constructedbyvaryingthevaluesofnineproductattributes).Again,theaccuracyoftheresultsreliesonconsumers’

understandingofeachhypotheticalpaymentinstrument.Differentfromthesurveyorthe

9Whilethereareafewpapersstudyingthenon-pricedesignfeaturesofaCBDC,suchasanonymity

(ChengandIzumi,

2023;

ChiuandMonnet,

2023;

Agur,AriandDell’Ariccia,

2022;

Ahnert,Hoffmann

andMonnet,

2022)andexpirydateonofflineCBDCbalances(

Kahn,vanOordtandZhu,

2021),weare

thefirsttoquantifytheimpactoftheCBDCdesignsintermsofthenetworkofservicelocationsandthecomplementarity.

10Thispaperalsoaddstotheliteratureondepositmarketcompetitionusingstructuralestimation(e.g.,

Albertazzietal.,

2022;

Wangetal.,

2020;

Xiao,

2020;

Abrams,

2019;

Aguirregabiria,ClarkandWang,

2019;

Egan,Horta¸csuandMatvos,

2017;

HoandIshii,

2011;

Dick,

2008),whichoftenuseaggregatebank-level

dataanddonotconsidertheeffectsoffinancialproductbundlingwithinthesamebank.Weshowhowhousehold-leveldatacanbeusefulinthislineofresearch.

Carbo-Valverde,Perez-SaizandXiao

(2023)use

household-leveldatatostudytheimpactofgeographicalandculturalproximityonhouseholds’choicesofcreditproductsandthepricingreceived,whilewefocusonthedepositmarketcompetitioninthispaper.

11

Grodecka-MessiandZhang

(2023)useahistoricalevent,theopeningoftheBankofCanadain1935,

tostudyhowtheissuanceofcentralbank(physical)moneycouldaffectthecharteredbanks,whichoffersinsightsforthepotentialeffectofthedigitalmoneyissuedbythecentralbank.

6

experimentalapproach,weusehouseholds’preferencesthatarerevealedfromtheirchoicesofdepositproductsfromdifferentbanks.

Therestofthepaperisorganizedasfollows.Section

2

presentsthemodelwithoutaCBDC.Section

3

discussesidentificationandestimationofthemodel.Section

4

presentsourdatasourcesandestimationresults.Section

5

showsthecounterfactualanalysesfordifferentCBDCdesigns,includingthenetworkofservicelocationsaswellasahardlimitonCBDCholdings.Section

6

concludes.

2Model

Themodelconsistsoftwotypesofagents:householdsandbanks.Eachhouseholdfirstchooseswhichbankjtogotofordepositingtheirdigitalmoney.Conditionalonthede-positbankchoice,theythenallocatetheirendowedliquidassetsbetweendigitalmoneyandphysicalcashbasedontherelativeattributesofthetwoassets.Households’utilitiesfromholdingtheliquidassetsdependontheirdepositbankchoicebecausebanksofferdifferentdepositrates,forinstance.Afterthedepositbankchoice,ahouseholdmayneedtogetafinancialproductk.Inthiscase,theychooseabankn,whichcanbedifferentfromthedepositbankj.However,theyenjoysomeextrautilityiftheygetthisproductfromtheirdepositbankduetothecomplementaritybetweendepositsandthefinancialproductwithinthesamebank.

Wesolvethehouseholds’problemusingbackwardinduction.Wefirstsolvetheportfo-lioallocationproblemandthebankchoiceproblemsforotherfinancialproductstoobtainhouseholds’utilitiesfromliquidityholdingandfromcomplementaryfinancialproducts,re-spectively.Thenwesolvehouseholds’depositbankchoiceproblem,takingintoaccountthesetwoutilitiesandtheutilityfrombranchnetworksthatdifferentdepositbanksprovide.

Bysolvingthehouseholds’problem,weobtaineachhousehold’sdepositdemand.Wethenaggregatetoobtainthedepositdemandforeachbankatthenationallevel.BankstakethedepositdemandfunctionsasgivenandengageinBertrandcompetitionwithdifferenti-atedproductsinthedepositmarket.Eachbankchoosestheirdepositratetomaximizetheirprofit.TofocusonthedepositmarketwhereaCBDCwillhavethemostdirectimpact,wesimplifythebank’sproblembyabstractingfrombanks’lendingdecisionsandhowthecom-plementarityaffectstheloandemand.Wediscussthebank’sproblemandtheimplicationsofthesesimplifyingassumptionsinSection

2.2.

Wenowdiscussthehousehold’sprobleminSection

2.1

.

7

2.1Household’sProblem

Eachhouseholdiisendowedwithwiliquidassetbalance,whichtheywouldliketoholdinboththephysicalform(cash)andthedigitalform(deposits).BankdepositsaretheonlydigitalliquidassetavailableintheabsenceofaCBDC.Toholddeposits,thehouseholdfirstchoosesabankjthatgivesthemthehighestutilityfromtheirchoicesetJi.WedenotethesetofallbanksbyJ.

Conditionalonchoosingadepositbankj,thehouseholdoptimallyallocateswibetween

cashanddepositsandobtainsutilityVifromholdingtheliquidassets.GiventhatVi

dependsonbankj’sattributes,suchasthedepositrateandthebranchnetwork,itaffectsahousehold’sdepositbankchoice.

Afterchoosingthedepositbank,thehouseholdmayneedtoobtainafinancialproductk∈K,suchasamortgageloanandacreditcard,withanexogenousprobabilityωk

.12

Foreachproductk∈K,thehouseholdwouldneedtochooseabanknfromtheirchoicesetJik.Tocapturethecomplementaritybetweendepositsandotherfinancialproductswithinthesamebank,weallowthehouseholdtohaveahomebankpreferenceforeachfinancialproduct.Thatis,theyenjoyextrautilityiftheyobtainthefinancialproductfromtheirdepositbank.

Becauseofthishomebankpreference,theexpectedvalueE[Vi]fromobtainingthefinancial

productkdependsonthedepositbankchoicej.

Wesolvethehousehold’sproblemusingbackwardinduction.Wefirstsolveforeach

household’sutilityfromholdingtheportfolioofliquidassetsViandtheirexpectedvalue

fromobtainingthecomplementaryproductE[Vi].Thesefactorsthenenterintothehouse-

hold’sutilityVifromopeningadepositaccountatbankj,whichalsodependsonthebranch

networkXi,j,unobservedqualitycapturedbythebankfixedeffectη,andtheidiosyncratic

tasteε,jforbankj:

ThetermωkE[Vi]istheexpectedutilityfromobtainingthefinancialproductk,whichisthe

productoftheprobabilityofrequiringthefinancialproduct,ωk,andt