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NBERWORKINGPAPERSERIES

ONDIGITALCURRENCIES

HaraldUhlig

WorkingPaper32159

/papers/w32159

NATIONALBUREAUOFECONOMICRESEARCH

1050MassachusettsAvenue

Cambridge,MA02138

February2024

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OnDigitalCurrencies

HaraldUhlig

NBERWorkingPaperNo.32159

February2024

JELNo.E31,E42,E44,E52,G12,G21

ABSTRACT

Idiscussprivateandcentral-bank-issueddigitalcurrencies,summarizingmypriorresearch.Iarguethatpricesofprivatedigitalcurrenciessuchasbitcoinfollowrandomwalksor,moregenerally,risk-adjustedmartingales.Forcentralbankdigitalcurrencies,Iarguethattheyenhancethe“CBDCtrilemma”facingacentralbank:outofthethreeobjectives,pricestability,efficiency,andmonetarytrust,itcanachieveatmosttwo.

HaraldUhlig

Dept.ofEconomics

UniversityofChicago

1126E59thStreet

Chicago,IL60637

andCEPR

andalsoNBER

huhlig@

1Introduction

Digitalcurrencieshavebeenontherise.Onceanichephenomenonforen-thusiasts,theyhavebecomemainstream,asevidencedbytheintroductionofbitcoinexchange-tradedfunds,thebuild-upofFinTech,risinginterestinthepossibilitiesofblockchaintechnologies,regulatoryframeworkssuchasthemarketsincryptoassets(MICA)regulationintheEU,andthepossibleintroductionofcentralbankdigitalcurrenciesorCBDCs.Therehasbeenanexplosionofliteratureonthetopicfromtheacademicside,andtherestillismuchtounderstand.Theliteratureistoovasttosurveyhere,soIshallpointtosomesurveyswrittenbycolleagues.Allenetal.(2021)providea“SurveyofFintechResearchandPolicyDiscussion”.RegardingCBDCs,Chiuetal.(2023)providealiteraturesurvey,whileKosseandMattei(2023)summarizethedevelopmentsatcentralbanksworldwide.Prassad(2021)isawonderfulbookonthe“FutureofMoney,”whichIhadachancetoreviewinUhlig(2023).Thus,discussingdigitalcurrenciesingreatbreadthherewouldbefutile.

Instead,Iwillsummarizeandreportonmyresearchtoaddresstwoques-tionsinparticular.First,howshouldwethinkaboutpricingcryptocurren-cies?Iwilladdressthisinsection

2,reportingonourfindingsinSchilling

-Uhlig(2019c).Second,whatdoesintroducingCBDCsimplyforthestabilityofcentralbanks?Thiswillbethetopicinsection

3,summarizingthekey

insightsinSchilling,Fern´andez-Villaverde,andUhlig(2024).Thetechnicaldetailsofcryptocurrencies,suchastheinnerworkingsofblockchainsandcryptography,willplaynorole.Myinteresthereisinmonetaryeconomics. Intheinterestofspace,Ishallrefrainfromreportingonmyresearchonthemonetarypolicyconsequencesofcompetingagainstglobalcurrencies,seeBenignoetal.(2019,2022),andofpricinginparalleldigitalcurrencies,analyzedinUhligandXie(2021),fromexpositingmyanalysisoftheLuna

2

stablecoincrash,seeUhlig(2022)andfromexaminingissuesofprivacyinUhligetal(2023).Theseissuesareimportant,though,andIhopeinterestedreaderswillfeelinclinedtofindandreadthesecontributions,complementingtheanalysisreportedhere.

Theaudienceforthispaperisnotjustfellowresearchersbutalsoeducatedlayreadersandpolicymakers.Withthissummary,Iamhopingtohelpinformthepublicdebate.Iconcedethataleveloftechnicalitywillmakeittrickyforthelatteraudiencetoappreciateeverythinghere.Still,hopefully,therest(andtheoccasionalconsultationwithanacademiceconomistathand)willsuffice.

2ATheoryofCryptocurrencyPricing

Bitcoin,thefirstwidelycirculatingandanonymousdigitalcurrencyorcryp-tocurrency,alongsideitsblockchaintechnology,wascreatedbyNakamoto(2008).Thequantityofbitcoinrisesaccordingtoanessentiallydeterminis-ticscheduleandconvergestoanupperlimit;seetheleftpanelofFigure

1.

Thefirsttimebitcoinwasusedforagoodpurchasewason“PizzaDay”onMay22,2010,whenprogrammerLaszloHanyeczpurchasedtwolargepiz-zasfor10,000bitcoin.AsofFebruary7,2024,those10,000bitcoinswouldbewortharound440millionU.S.dollars:quiteaheftypricefortwolargepizzas.Couldonehaveforeseenthatbitcoinwouldriseinprice,seetheleftpanelofFigure

1?Howshouldonethinkaboutpricingintrinsicallyworth

-lesscryptocurrenciessuchasbitcoin?Cancryptocurrenciessuchasbitcoinserveasamediumofexchangedespitetheirpricevolatility?Whataretheconsequencesformonetarypolicyanditsgovernment-issuedcurrency,suchasthedollar?ThesearethequestionsweaddressinSchilling-Uhlig(2019c).Schilling-Uhlig(2019b)summarizesthatpaper,whileSchilling-Uhlig(2019a)providesanextension,allowingfortransactioncosts.

3

BitcoinPrice(1000Dollar)

BitcoinQuantity

BitcoinPrice(1000Dollar)

70

60

50

40

30

20

10

0

1072

1.5

1

0.5

0

20102012201420162018202020222024

Date

70

60

50

40

30

20

10

0

2018201920202021202220232024

Date

Figure1:Leftpanel:bitcoinquantityandpricesince2009.Rightpanel:bitcoinpricesince2018.Source:

/charts/total-

bitcoins

.

Ouranalysisbuildsonandechoestheground-breaking“exchangerateindeterminacy”insightbyKarekenandWallace(1981)andthestochasticextensioninManuelliandPeck(1990).Weshowthatthereisa“fundamen-talpricingequation”forpricingbitcoinandsimilarcryptocurrencies.Asaspecialcase,thebitcoinpriceisamartingaleorrandomwalk:theexpectedfuturepriceisthecurrentprice.Putdifferently,whenLaszloHanyeczpur-chasedthetwopizzasin2010,itwasrationalforhimtoexpectthathisbitcoinwouldnotriseinvalue:spendingthemratherthankeepingthemisthenthewiserchoice.ThismayseematoddswiththepricedevelopmentshownintheleftpanelofFigure

1.However,itwasthatriseinthebitcoin

pricethatgaverisetoacademicslikeustostudybitcoininthefirstplace

.1

Westartedthinkingaboutitanddraftingthefirstversionofthepaperasofearly2018.TherightpanelofFigure

1

showsthepricedevelopmentssincethen,anditappearsrathermartingale-liketous,vindicatingouranal-ysis

.2

Weshowa“nospeculation”theorem:bitcoinarenot“hodled”for

1Thisisakintoananthropicprinciple.

2Obviously,thereisanoverallincreaseinpricesince2018untilthetimeofwriting,

4

speculativepurposes,butrathertheyarespent

.3

Understanding“hodling”must,therefore,stillawaitfutureresearch.Weshowthatthepricevolatilitydoesnotinvalidatethemedium-of-exchangefunctionofbitcoin.Asformon-etarypolicy,weshowthatthebitcoinblockrewardsarenotataxonbitcoin“hodlers”;rather,theyarefinancedbyadollartaxorbyareductionoftheseignorageearnedbythecentralbank.

Todemonstratetheseinsights,weprovideanovelmodelofanendowmenteconomywithtwointrinsicallyworthlesscurrencies,thedollarandbitcoin,asamediumofexchange.Weseektoallowforthepossibilityofholdingthecurrenciesforspeculativepurposesand,therefore,needtogobeyondthetwo-periodoverlappinggenerationsmodelutilizedbyKarekenandWallace(1981)aswellasManuelliandPeck(1990).ThedetailsarestatedinSchilling-Uhlig(2019c);abirds-eyedescriptionshallsufficehere.Timeisdiscreteandinfinitet=0,1,2,....Somerandomvariableθtisdrawnatthebeginningofeachperiod,generatingthehistoryθt.Allvariablesareassumedtobefunctionsofθt.Thereisoneperishableconsumptiongoodperperiod.Therearetwotypesofagents:“red”and“green”.“Red”agentsconsumeinoddperiods,enjoyingtheconsumptionperu(ct),andhaveendowmentsinevenperiods.Thingsaretheotherwayaroundfor“green”agents.Agentsdiscountthefutureatrate0<β<1.Therearetwomonies:bitcoinsintotalquantityBtanddollarsintotalquantityDt.Acentralbanksteersthequantityofdollarsperlumpsumtransfers,Dt+1=Dt+τt+1,τt+1∈R.ThecentralbankaimstoimplementsomeexogenouslygivenpricepathPt.Anexamplewouldbe

February2024.Itmaybehardertodecidewhetherthatpriceincreaseisduetochanceoranupwarddriftinthebitcoinprice.

3Theterm“hodler”ratherthan“holder”wasintroducedbyatypoon

/index.php?topic=375643.0

andisnowfirmlypartofthevocab-ulary,describingindividualsholdingontoacryptocurrencyratherthanspendingorex-changingit.

5

aconstantinflationratePt/Pt−1=

.4

Weassumethatthebitcoinquantity

isdeterministic,Bt+1=Bt+At,At≥0.TheincreaseAtisexogenouslygiven,reflectingthebehaviorofbitcoinquantityshownintheleftpanelofFigure

1

.Inthebaselineversionofourmodel,weassumethatAtisfreelygivenatthebeginningoftheperiodtotheagentswhoareabouttoconsume.WeshowthatitmakesnodifferenceifweassumethatagentsmustspendminingeffortsinsteadtoreceiveAt.

Thereisalargeamountofliteratureinvestigatingwhyagentswouldusemoneyratherthansomecontractualarrangements.Wedonothaveanythingnewtocontributehereand,therefore,simplyimposethatgoodsaretradedformoney.However,agentsdonotneedtospendalltheirmoney,i.e.,theycanbecome“hodlers”instead.Likewise,theydonotneedtoaccepteithermoneyassellers.Weimposeassumptionssothatalldollarsarealwaysspent.WeareinterestedinwhetherandwhenthisistrueforbitcoinandunderstandingthebitcoinpriceorexchangerateQt=Q(θt)intermsofdollars.

InSchilling-Uhlig(2019c),westateandprovethefollowingresult.

Proposition1Assumeagentsusebothdollarsandbitcoinstobuygoodsattandt+1.Then

Etlu′(ct+1)=Etl（u′(ct+1)(1)

Ifproduction(consumption)isconstantatt+1orifagentsarerisk-neutral,andiffurtherQt+1and1/πt+1areconditionallyuncorrelated,thenthebitcoinpriceQtindollarisamartingale,

Qt=Et[Qt+1](2)

4Thereisalargeliteratureonthedetailsofmonetarypolicyimplementation,andtheyarenotthefocusofourpaper.Wesimplyassumethatthecentralbankcanachievethispricepathexactly.

6

Wecall(1)thefundamentalpricingequation

.Withoutanyrandomness,weobtaintheKareken-Wallace(1981)resultthatQt+1=Qt

.Equation(1)

isakintotheresultinManuelli-Peck(1990),generalizingtheexchangerate

indeterminacyresultinKareken-Wallace(1981).Onecanrewrite(1)inmore

conventionalassetpricingterms.Definethenominalstochasticdiscount

factor

Mt+1=β(3)

ThenominalreturnonholdingbitcoinisRt+1=Qt+1/Qt.Withthat,equa-

tion(1)canbeequivalentlyrestatedas

1=

Et[Mt+1Rt+1]

Et[Mt+1]

(4)

Notethatthisisnottheusualassetpricingformula1=Et[Mt+1Rt+1].Thisdifferencehighlightsthedistinctionbetweenanassetpricingapproachandamonetaryeconomicsapproachtopricingcryptocurrencies.Inessence,thereisnointerestonholdingbitcoin.Itsharesthatpropertywiththeothercurrency,i.e.,thedollar.HoldingadollaryieldsthenominalreturnRt+1=1,

obviouslysatisfying(4).Currenciesdonotearninterestsincetheyareuseful

asamediumofexchange.5

Onaverageandinrisk-adjustedterms,bothhave

thesamereturn.

Definerisk-adjustedexpectationst[Xt+1]foranyrandomvariableper

Et[Xt+1]=Et[Mt+1Xt+1]/Et[Mt+1].Equation(4)showsthatQtisarisk-

adjustedmartingale,

Qt=t[Qt+1].(5)

Thedifferenceto(2)isonlythatexpectationsandprobabilitiesareadjusted

forriskin(5).WeusethisformulationinBenignoetal.(2022)

.

5Fromanassetpricingperspective,onecanthinkofliquidityservicesasthedividendofmoney.Onestillneedstobecarefulwhenapplyinganassetpricingapproachsincedoublingthepriceofbitcoinalsodoublesitsliquidityservices.

7

Undermildconditionsonpreferences,our“nospeculationtheorem”arises:allbitcoinsarespenteveryperiod.Thisisaconsequenceofthefeaturethatcurrenciesdonotpayinterestbutthatagentsdiscountthefuture.Itisthenbettertospendallcurrencyineveryperiodthantoholdontoit.IntheonlineappendixtoSchilling-Uhlig(2019c),availableat

https://ars.els-

/content/image/1-s2.0-S0304393219301199-mmc1.pdf,

weshowthatanypriceprocessQt

,whichsatisfies(1)or,equivalently,(4)andsomemild

boundednesscondition,canariseasthebitcoinpriceinequilibrium.6

Inotherwords,thefundamentalpricingequationisallonecansayaboutthebitcoinprice.Inparticular,therearenorestrictionsonthebitcoinpricevolatilitybeyondthispricingequationandsomemildboundednesscondition.

Asadirectconsequenceoftheseresults,weobtainthatvolatilitydoesnotinvalidatethemedium-of-exchangefunctionofbitcoin.Allbitcoinsarealwaysusedintransactions,regardlessofthevolatilityoftheunderlyingpriceprocess.Theeconomicreasonforthisresultisthefundamentalpricing

equation(1):buyersandsellersarecompensatedforthebitcoinrisk.Tosee

this,examineequation(4)

.Considerabuyerandasellerfacingeachotheratdatet,knowingthestochasticnatureofthepriceQt+1inthefuture.The

expressionontheright-handsideof(5)ishowsellersvaluebitcoinoncethe

appropriateriskcompensationistakenintoaccount.7

IfQtwashigherthan

theright-handsideof(5),thensellerswouldinsistononlyreceivingdollars

sincebitcoinistooexpensive.Thus,buyersseekingtopaywithbitcoinmustacceptalowervalue:thepricewouldbebiddownuntilitreachesthevalue

expressedin(5)

.Conversely,ifQtwerelowerthantheright-handsideof

6Theboundednessconditionarisesfromthefeaturethatthetotalvalueofbitcoin

cannotexceedthetotalvalueofallgoodswhenallbitcoinsarealwaysspent.

7Notethatthesellerbecomesabuyerint+1,i.e.,itistheutilityfunctionofthe

currentsellerontheright-handsideof(5).Theriskcompensationcanalsotaketheform

ofan“insurancepremium”,i.e.,therearegenerallypriceprocessesQtsothattheexpectedreturnonbitcoinisbelowunity.

8

(5),sellerswouldinsistononlyreceivingbitcoin,andthepricewouldbe

bidup.Inequilibrium,equation(5)holds,andbitcoinsareusedfreelyasa

mediumofexchange,despitetheirpricevolatility,sinceagentsareexactlycompensatedfortheriskthattheyentail.

Finally,considertwoeconomiesthatdifferinthegrowthpathsforthebitcoinquantitybutwherethecentralbankachievesthesamepathforprices.Tomakethingsmorespecific,imaginethatthefirsteconomyhasaconstantsupplyofbitcoinwhilethequantityisgrowingforthesecondeconomy,asshownintheleftpanelofFigure

1.

Duetotheno-speculationtheorem,aversionofthequantitytheoryequationhastoholdinbotheconomies,

Ptyt=Dt+QtBt(6)

Asarguedabove,thereisnorestrictiononthepricepathQt.Therefore,thesamepricepathmayariseinbotheconomies,providedsomemildbound-ednessconditionissatisfied.TheequationimmediatelyimpliesthatmorebitcoinsBtmeanslessdollarsDt,keepingeverythingelsethesame.Thedol-larquantityinthesecondeconomygrowsmoreslowly(orfallsfaster)thanthedollarquantityinthefirsteconomypreciselybythevalueofthebitcoinincreasesinthesecondeconomy.Thecentralbankpaysforthebitcoinblockrewards.TheeconomicreasonisthatthecentralbankkeepsthepricepathPtincheck.It,therefore,hastolivewiththeconsequencesofthatpolicychoice.

3CentralBankDigitalCurrencyandtheSta-

bilityofCentralBanks

CentralbanksworldwideareconsideringtheintroductionofcentralbankdigitalcurrenciesorCBDCs,seeKosseandMattei(2023).Whataretheconsequences?Inparticular,couldtheyleadtoinstabilityofthecentral

9

banksthemselves?Diamond-Dybvig(1983)providestheclassicframeworktoexaminethestabilityofbanks.Thatpaperhasspawnedahugeliterature,butithasrarely,ifever,beenappliedtotheissueofthestabilityofthemon-etarysystemandthecentralbankitself.WedosoinSchilling,Fern´andez-Villaverde,andUhlig(2024),considerablyextendingFern´andez-Villaverdeetal.(2021),addressingthekeyquestion.Itturnsoutthatouranalysisappliestocentralbanksmorebroadly,evenabsentCBDCs.However,ouranalysisbecomesparticularlyrelevantforawidelyusedCBDC,giventhespeedatwhichprivatecitizenscanthenexecuteelectronictransactionsincentralbankmoney.

Theessenceofourmodelisthis:whileacentralbankcanobviouslyalwaysmakegoodonitsnominalobligations,agentsmaynonethelessbeconcernedthatthesenominalobligations,i.e.,thatmoney,willbeworthlessinthefuture.Agentswillthenseektospendtheirmoneyasquicklyaspossible,eventhoughtheydonotneedtoconsumeurgently.Usually,a“bankrun”isarunawayfromdepositsintocash.Here,therunonthecentralbankisarunawayfrommoneyintogoods.Theybothsharethatitisarunawayfromwhatthebankcanissuetosomethingitcannot.Arunonthecentralbankisaspendingrun.Inaspendingrun,money,whetherintraditionalformorintheformofaCBDC,losesits“storeofvalue”function.ThetrustinthemonetarysystemandCBDCevaporates,andmonetaryinstabilityresults.Thisphenomenonshouldbefamiliarfromhyperinflationsorcurrencycrises.Morerecently,wehaveseenversionsofthisintheformoftemporarypandemicstockouts.

Thebaselineversionofourmodelishighlystylizedand“stripped-down”tothebarebonesessence:thereareonlyhouseholdsandacentralbank,andthecentralbankisrunningthecountry.Obviously,thisisnotthecaseinpractice,andwedonotmeantoimplythatthisisso.Anextensioninthepaperdescribesadecentralizationwithfirmsandprivatebanks.Nothingof

10

essencechanges.

Weimagineacentralbankthatinteractswithhouseholdsbyprovidingthemwithmoney,perhapsintheformofaCBDC,topurchasegoods.WebuildonDiamond-Dybvig(1983)andthenominalversionsinSkeie(2008)andAllenetal.(2014).Indeed,therealportionisjustaminorvariationofDiamond-Dybvid(1983).Therearethreeperiods,t=0,1,2andacontinuum[0,1]ofagents.Theseagentsareendowedwithoneunitofarealgoodint=0.Int=1,theirtypeisrevealed.Agentsmaybe“impatient”withprobabilityλand“patient”withprobability1−λ.Probabilitiesarefractions,usingthelawoflargenumbersforacontinuum;seeUhlig(1996).Impatientagentsmustconsumeint=1,whilepatientagentsmayconsumeineitherperiod.Theutilityfunctionu(·)isstrictlyincreasing,concave,andhasarelativeriskaversionexceedingunity,−x·u′′(x)/u′(x)>1.Aninvestmenttechnologyturns1unitofthegoodint=0into1unitofthegoodint=1orRunitsofthegoodint=2.Theproblemofasocialplannerwhomaximizesex-antewelfareis

maxλu(x1)+(1−λ)u(x2)s.t.λx1+(1−λ)=1(7)

Theuniquesolutionsatisfiesu′(x)=Ru′(x)andx>1.Consequently,y∗=λxarethetotalresourcesliquidatedinperiodt=1.

Wenowaddanominallayer.Inperiodt=0,theagentsselltheirgoodstothecentralbankforMunitsofmoneyorCBDC.Thecentralbankinveststherealgoods,usingtheinvestmenttechnology.Int=1,theimpatientagentsspendtheirbalancesM,whilepatientagentsmay.Letnbethefractionofspendingagents,withλ≤n≤1.Thecentralbankobservesλ.ThisisparticularlyappropriateintheCBDCcontextwhenthecentralbankcanobserveprivatemonetarytransactionsinrealtime.Thecentralbankthenliquidatesafractiony=y(n)∈[0,1]oftheinvestedgoodsandsellsthem.ThemarketclearingpriceisP1.Int=2,theremainingagents

11

receiveaninterestpaymenti(n)ontheirbalances:again,thisisparticularlyeasytodowithaCBDC.Theythenspend(1+i(n))M.ThecentralbanksellstheremaininginvestedgoodsR(1−y).ThemarketclearingpriceisP2.NotethatwehaveassumedthatthecentralbankonlyissuesCBDCs.Nothinghingesonthatformally.Onecaninterpretthemodelequallywellasacentralbankissuingcashorextendingtheenvironment,allowingforbothcashandaCBDC:detailsareinthepaper

.8

Thereareseveralchoicesforthecentralbanktomake.Weassumethatitannouncesthemint=0andthatagentsexpectthecentralbanktosticktoit.InSchilling,Fern´andez-Villaverde,andUhlig(2024),wedefineacentralbankpolicyasfollows.

Definition1Acentralbankpolicyisatriple(M,y(·),i(·)),wherey:[0,1]→[0,1]isthecentralbank’sliquidationpolicyforeveryobservedfrac-tionnofspendingagents,andi:[0,1]→[−1,∞)isthenominalinterestratepolicy.

AboringexampleofacentralpolicyisshowninFigure

2.

Howshouldthecentralbankchoose?Weassumethattherearethreecompetingobjectives:

1.PriceStability:thetraditionalobjectiveofacommitmenttostableprices.Here,wemeanthatthepriceP1(n)remainsthesame,regardlessofcircumstancesn.

2.Efficiency:here,wemeanthatthecentralbankimplementstheso-ciallyoptimalrisksharing.Itisachieved,ify(λ)=y*=λx.

3.MonetaryTrust:avoidspendingrunsandmonetaryinstability.

InSchilling,Fern´andez-Villaverde,andUhlig(2024),andasIshalloutlinebelow,weshowthatthecentralbankcanonlyachieveatmosttwoofthese

8WithcashandaCBDC,thereisa“boring”runwhereagentsexchangeCBDCforcashorviceversa.Thisisaratherbenignscenarioandnotourfocus.

12

y(n)

i(n)

1

0.8

0.6

0.4

0.2

0

0.2

0.1

0

-0.1

-0.20

Aliquidationpolicy

y

0

0.20.40.60.81

n

Aninterestratepolicy

0.20.40.60.81

n

Figure2:Aboringexampleforacentralbankpolicy.SetMsothatP1=1clearsthemarket,ifn=λagentsspendint=1.Setliquidationy(n)alwaysequaltoy*=λx;seetheleftpanel.Seti(n)=0foralln;seetherightpanel.

threeobjectives.Thisresultappliesregardlessofwhetherthecentralbankusestraditionalformsofcash,CBDC,orboth.Still,theremarksaboveshowitisparticularlyrelevantforawidelyusedCBDC.We,therefore,callthisresulttheCBDCtrilemma;seeFigure

3.

Tothatend,letustakeacloserlook.

MarketClearingyields

nM=P1y(n)(

8)

(1−n)(1+i(n))M=P2R(1−y(n))(

9)

Thecentralbankpolicyandn,therefore,pinsdownthepricelevels

P1(n)=andP2(n)=(10)NotethatP2(n)canbe“anything”peri(n),buttheinterestpolicyi(n)doesnotaffectP1(n).Whileperhapsobviousintermsoftheequations,thereisadeepereconomicmeaninghere.IntypicalKeynesian-stylemodels,economic

13

Price

Stability

Efficiency

Mone-

tary

Trust

Figure3:TheCBDCTrilemma:outofthreecompetingobjectives,acentralbankcanachieveatmosttwo.Whilethattradeoffappliestoallcentralbanks,thetrilemmaisparticularlyrelevantwhenissuingawidelyusedCBDC.

activitycanbesubduedbyraisinginterestrates.Appliedtothemodelhere,thiswouldleadonetothinkthatexcessivespendingn>λinperiodt=1canbecombatedwithraisinginterestratesi(n).However,thishasnoeffect.InterestrateshereareFisherian,notKeynesian:ariseinthenominalinterestrateraisestheinflationfromt=1tot=2anddoesnotaffectpricesint=1.Therefore,therealallocationandthedecisionofpatientagentstospendornotinperiodt=1dependsonlyontheliquidationpolicyandthefractionnofspendingagents.Agentswillconsumex1(n),iftheyspendint=1andx2(n),iftheyspendint=2,where

x1(n)=

=andx2(n)==R

(11)

Givennandthecentralbankpolicy,patientagentsspendint=1or“run”,iffx1(n)≥x2(n).Arunonthecentralbankisaspendingrun,visualizedinFigure

4

:

14

t=0

t=0

Norun:

t=1

Run:

t=1

t=2

t=2

Figure4:Visualizationofarun.Inperiodt=0,agentsreceivemoneyfortheirgoods.Ifthereisnorun,someagentswillspendint=1,andsomewillspendint=2.Inarun,agentsfearthattoofewgoodsareavailableforpurchaseint=2,thereforepurchasinggoodsingreaterquantitythanusualint=1.

Definition2Arunoccursifn>λ:patientagentsalsospend.

Toachievethethirdobjective,thecentralbankmustchooseapolicysuchthataruncannotoccurinequilibrium.Apolicyis“run-proof”,ifnλis“offequilibrium”,i.e.ifx1(n)<x2(n)foralln.Fortheaggregate

liquidation,thismeansthatthecentralbankmustchoosey(n)<(n)=

nR/(1+n(R−1)).TheleftpanelofFigure

5

showsthislimittoliquidationandcomparesittotheboringpolicyofFigure

2:theboringpolicysatisfies

it.

15

y(n)

P1(n)

4

3

2

1

0

Run-deterringliquidationlimit

1

nR/(1+n(R-1))

0.8

0.6

y

0.4

0.2

0

00.20.40.60.81

n

PriceP1(n)

00.20.40.60.81

n

Figure5:Run-proofpoliciesmusthaveliquidationstrategiesnotexceedingthelimitshownintheleftpanel.Forexample,theboringpolicyshowninFigure

2

isrun-proof.However,run-proofpoliciesinducerisingpricesof