Creditmarketsplayanincreasinglycentralroleinmoderneconomies.WithintheO - PDF document

Creditmarketsplayanincreasinglycentralroleinmoderneconomies.WithintheOECD,forinstance,domesticcredithasrisenfrom100%ofGDPin1970toapproximately160%ofGDPin2012(Figure1).Thisgrowthmaskslargevariationsacrosscountriesandovertime(Figure2).Andyet,thereisacommonfeaturetoallthesedi¤erentcountryexperiencesthatstandsout:credithasoftenalternatedperiodsofrapidgrowthor“booms”,withperiodsofstagnationorsigni…cantdeclineor“busts”.Moreover,thereissomeevidencethatthesecreditboomsandbustshavebecomemorecommoninrecentyears.1ConsiderthecaseoftheUnitedStates,wherecredithasgrownmoreorlesscontinuouslythroughoutthesampleperiod.Eventhere,itgrewbyapproximately40percentofGDPbetween1990and2010,onlytocontractsharplyafterwards.InGreece,Ireland,SpainandPortugal,thedynamicsofcreditlooksimilarduringthelastdecades:stagnantordecliningcreditbetweenthemid1980sandthemid1990s,aspectacularsurgeincreditbetweenthemid-1990sand2010,andstagnationorasharpdeclinesincethen.Lookingahead,thesedropsincreditneednotbeshort-livedastheJapaneseandSwedishexperiencesshow.InJapan,forinstance,creditgrewrapidlyinthelate1990sandhasfallensteadilysinceits1999peak.InSweden,creditcollapsedduringthe…nancialcrisisoftheearly1990sandittookoveradecadetoreturntoitspreviouspeak.Thesecreditboomsandbuststendtobeaccompaniedbychangesinkeyeconomicvariables.IthasbeenwelldocumentedthatcreditboomsareassociatedwithhighassetpricesandhighgrowthratesofrealGDP,consumptionandinvestment.Accordingtosomeestimatesthegrowthrateofinvestmentdoublesduringbooms.2Inspiteofthis,creditboomsarestillviewedwithconcernbypolicymakersandacademics.Thereasonisthattheyeventuallyend,andtheiraftermathsareoftencharacterizedby…nancialcrisesandloweconomicgrowth.3Thishaspromptedcallsforpoliciesthatrestraincreditduringbooms,inthehopethatsmallerboomswillleadtosmallercrises. 1SeeMendozaandTerrones(2012)andDell’Aricciaetal.(2012)forabriefdiscussionontheformalde…nitionandempiricalidenti…cationofcreditbooms.MendozaandTerrones(2012)analyzetheevolutionofprivatecredittotheprivatesectorin61countriesbetween1960and2010andidentify70creditbooms:eachofthecountriesplottedinFigure2containsatleastoneofthese.Dell’Aricciaetal.(2012)analyzedataonbankcredittothedomesticprivatesectorin170countriesbetween1960and2010.Theyidentify175creditbooms,whichtranslatesintoa14%probabilityofacountryexperiencingacreditboominanygivenyear.Theyalsodocumentthatcreditboomsappeartohavebecomemorecommonovertime,inthesensethatthefractionofcountriesexperiencingacreditboominanygivenyearhasbeenincreasingsincethe1980s.Claessensetal.(2011)useadi¤erentapproachandstudy“creditcycles”,buttheyalso…ndthemtobecommon:inasampleof21advancedeconomiesbetween1960and2007,theyareabletoidentify114suchcycles.2SeeMendozaandTerrones(2012)andDell’Aricciaetal.(2012).3Dell’Aricciaetal.(2012)…ndthatonethirdofcreditboomsendupin…nancialcrises.MendozaandTerrones(2012)…ndslightlylowernumbers.Therecent…nancialcrisisintheUnitedStatesalsoprovidesevidenceinthisregard:itwasthoseregionsthatexperiencedthelargestcreditboomsintherunuptothecrisisthatsu¤eredthegreaterincreaseincreditdelinquencyduringthecrisis(Dell’Aricciaetal.2008,MianandSu…2009).1 Toevaluatethemeritofthesecallsforpolicy,onemusthaveaviewoftheforcesdrivingthesecreditcycles.Creditmay‡uctuateforvarietyofreasons,anddi¤erenttypesof‡uctuationsmaycallfordi¤erentpolicyresponses.Ataverygenerallevel,‡uctuationsincreditmayre‡ectchangesindemandorinsupply.Andthese,inturn,mayre‡ectchangesinavarietyoffactorslikepreferences,technologyorexpectations.Inthispaper,wefollowthelargebodyofrecentmacroeconomicresearchonthe“…nancialaccelerator”andfocusoncredit‡uctuationsthataredrivenby‡uctuationsinborrowingconstraints.Thenarrativeissimple.Whenaborrowerobtainscredit,sheisexchanginggoodstodayforapromisetodelivergoodsinthefuture.Thesepromisesareonlyvaluedbysaversiftheyhavesomeguaranteeofbeingrepaid.Onecanthinkofthisguaranteeasthefutureincomeofentrepreneursthatcanbecrediblypledgedtosavers.Werefertothispledgeableincomeastheeconomy’sstockofcollateral.Ifentrepreneursareconstrained,itisthisstockofcollateralthatdeterminestheamountofpromisesthatcanbeissued.Ifentrepreneursareconstrained,understandingboom-bustepisodeslikethoseshownaboverequiresatheoryofcollateral‡uctuations.Thispaperprovidessuchatheory.Ourkeyinnovationistodistinguishbetweenfundamentalandbubblycollateral.Fundamentalcollateralisthepartofaborrower’spledgeableincomethatcorrespondstofutureoutput,i.e.itconsistsofaborrower’srightstofutureproduction.Bubblycollateralisinsteadthepartofaborrower’spledgeableincomethatcorrespondstofuturecredit,i.e.itconsistsofaborrower’srightstofutureborrowing.Wecallthistypeofcollateralbubblybecauseitconstitutesabubbleorpyramidschemeinwhichpresentcontributions(presentcredit)purchasefuturecontributions(futurecredit):aslongasthereturntothesebubblesorpyramidschemesisnolowerthantheinterestrate,lenderswillbewillingtoacceptthemascollateral.Themacroeconomicliteraturehasexclusivelyfocusedonfundamentalcollateral,studyingitsimplicationsforcredit,investmentandgrowth.Thisviewofcollateralisincomplete,though.Wheneverfundamentalcollateralisinsu¢cient(say,becauseofweakenforcementinstitutions),weshowthatthereisroomforinvestoroptimismtosustainbubblesthatexpandtheeconomy’sstockofcollateralandtotalcredit.Theintuitionbehindthisresultisthatthelackoffundamentalcollateralreducesthedemandforcreditandthustheequilibriuminterestrate,therebyloweringtherateatwhichbubblesmustgrowinorderforthemtobeattractivetosavers.Inthismanner,thelackoffundamentalcollateralrelaxestheconditionsforbubblycollateraltoexist.Whatarethemacroeconomice¤ectsofbubblycollateral?Byde…nition,itenablesentrepreneurstoobtaincreditinexcessoftheirfundamentalcollateral:intuitively,currententrepreneurscan2 obtain“excess”credittodaybecauseitisexpectedthattherewillbe“excess”creditinthefutureaswell.Thisisthecrowding-ine¤ectofbubbles,whichceterisparibusincreasesinvestment.Butthisfuture“excess”creditwilldivertsomeoftheresourcesoffuturegenerationsawayfrominvestment.Thisisthecrowding-oute¤ectofbubbles,whichceterisparibusreducesinvestment.Themacroeconomicconsequencesofbubblesdependontherelativestrengthofthesetwoe¤ects.Inparticular,we…ndthatthecrowding-ine¤ectdominateswhenbubblycollateralislow,andthecrowding-oute¤ectdominateswhenbubblycollateralishigh.Thisgivesrisetoan“optimal”bubblewhichtrades-o¤thesetwoe¤ectsandprovidestheamountofbubblycollateralthatmaximizeslong-runoutputandconsumption.Anessentialfeatureofbubblycollateralisthatitsstockisdrivenbyinvestorsentimentormarketexpectations.Thecreditobtainedbyentrepreneurstodaydependsonmarketexpectationsaboutthecreditthatentrepreneurswillobtaintomorrow,whichinturndependontomorrow’smarketexpectationsaboutthecreditthatentrepreneurswillobtainonthedayafter,andsoon.Becauseofthis,marketsmaysometimesprovidetoomuchbubblycollateralandsometimestoolittleofit,whichcreatesanaturalroleforstabilizationpolicies.Weshowthatacreditmanagementagencywiththeauthoritytotaxandsubsidizecreditcaninfactreplicatetheoptimalbubbleallocation.Todoso,itmustadoptapolicyofleaning-against-the-wind,taxingcreditwhenbubblycollateralisexcessiveandsubsidizingitwhenbubblycollateralisscarce.Weexploretheimplicationsofthispolicyandshowthatitraisesoutputandconsumption,althoughitmaysometimesincreasemacroeconomicvolatility.Wearguethat,tosomeextent,thispolicycanbeinterpretedasaschemeinwhichthecreditmanagementagencypurchasesassetstomanagetheeconomy’sstockofcollateral.Finally,weshowthatthesameconditionsthatmakebubblycollateralpossibleguaranteethatacreditmanagementagencyalwayshasenoughresourcestoimplementthedesiredpolicy.Thepaperisorganizedasfollows.Section1developsourbasicmodelofcollateral,creditandinvestment.Section2exploresbubblyequilibriaandstudiestheimplicationsofbubblycollateralfordynamicsandwelfare.Section3introducesacreditmanagementagencyandshowshowcredit-marketinterventionscanbeusedtomaximizeoutputandconsumption.Section4explorestheissueofa…scalbackstop,i.e.oftheresourcesrequiredtosustainthedesiredpolicy.Section5concludes.Beforemovingon,weo¤erashortreviewofrelatedliterature.Literaturereview:ThemodeldevelopedherebuildsuponpreviousworkbyMartinandVentura(2011,2012).InMartinandVentura(2012),weanalyzedtheexistenceandmacroeconomice¤ectsofbubblesin3 aneconomywithextreme…nancialfrictionsthatimpedecredit.InMartinandVentura(2011),weextendedtheanalysistoaneconomywithcredit,andarguedthatbubblescouldbeasourceofcollateral.Thosepapersusedasimplegrowtheconomytoestablishtwokeyresultsregarding…nancialfrictionsandbubbles.First,…nancialfrictionsthatlimitborrowers’abilitytopledgetheirfutureincome,andthustheeconomy’sstockoffundamentalcollateral,relaxtheconditionsfortheexistenceofbubbles:thereason,aswementionedabove,isthatsuchfrictionslowertheequilibriuminterestrateandthustherateatwhichbubblesmustgrowinorderforthemtobeattractivetosavers.Second,…nancialfrictionsmakeitpossibleforbubblestobeexpansionary:thereasonisthat,byraisingtheeconomy’sstockofcollateral,bubblesmayintheoryraisecreditandinvestment.Relativetothosepapers,wenow:(i)fullycharacterizethee¤ectsofbubblycollateral,identifyingtheconditionsunderwhichitisexpansionaryorcontractionary,and;(ii)analyzethee¤ectsofmacroprudentialpolicies,andoftheresourcesrequiredtoimplementthem.Ourpaperisobviouslyrelatedtoalargebodyofresearchthatdealswith‡uctuationsincredit,bothempiricallyandtheoretically.Theformerhassoughttoidentifyempiricalregularitiesofcreditboomsandbusts:Gourinchasetal.(2001),Claessensetal.(2011),MendozaandTerrones(2012),Dell’Aricciaetal.(2012)andSchularickandTaylor(2012)fallwithinthiscategory.Onthetheoreticalfront,variouspapershavetriedtomodel“creditcycles”asanequilibriumoutcomeofcompetitionin…nancialmarkets.SomeexamplesofthisworkareRuckes(2004),Dell’AricciaandMarquez(2006),Matsuyama(2007),GortonandHe(2008)andMartin(2008).Likeus,thesepapersmodel‡uctuationsincredit.Unlikeus,though,thesepapersemphasizetheroleofinstitutionalfeaturesof…nancialmarkets–likeregulationortheincentivesofcertainmarketparticipants–ingeneratingandmagnifying‡uctuationsincredit.Wetakeinsteadamacroeconomicperspectiveandarguethat,whenfundamentalcollateralisscarce,investorsentimentmaygiverisetocreditbubbles.Fromamethodologicalperspective,thus,thispaperisclosesttotheliteraturethathasstudiedthee¤ectsofbubblesinthepresenceof…nancialfrictions:(i)unlikeus,CaballeroandKrishna-murthy(2006),FarhiandTirole(2011)andMiaoandWang(2011)focusontheroleofbubblesasausefulsourceofliquidity;(ii)likeus,Kocherlakota(2009)focusesontheroleofbubblesascollateralornetworth;and(iii)unlikeus,Ventura(2011)focusesonthee¤ectsofbubblesonthecostofcapital.Thereisalsoalongtraditionofpapersthatview…atmoneyasabubble,likeSamuelson(1958)andKiyotakiandMoore(2008).Ourmodelisalsorelatedtothevastworkonmacroeconomicmodelswith…nancialfrictions,goingbacktoBernankeandGertler(1989)and4 KiyotakiandMoore(1997),inwhichcollateralplaysakeyroleindeterminingthelevelof…nancialintermediationandeconomicactivity.Ourcontributionrelativetothisworkisthatwedistinguishbetweenfundamentalandbubblycollateral.Ourstudyofpolicyisrelatedtotheliteratureonbailouts.Bailoutsarecommonlyrationalizedasawaytostimulateeconomicactivityintheeventofacrisis,bytransferringresourcestowardsdistressedagents.Inamacroeconomicmodelwith…nancialfrictions,forinstance,GertlerandKiyotaki(2012)showhowpoliciesthatprovidecredittodistressedagentscanreducetheseverityofacrisis.Atthesametime,theprevailingviewisthatbailoutsgeneratecostsexantebydis-tortingincentives(e.g.FarhiandTirole2012).Thecreditsubsidiesthatwestudyarenotreallybailoutsbecausetheyare…nancedthroughtaxesonentrepreneursthemselves,i.e.,theyentailnonettransferstoentrepreneursasawhole.Theyareakintobailouts,though,becausetheyaregiventoentrepreneurstorepaytheirdebts.Thee¤ectsofthesesubsidiesaredi¤erentfromthoseemphasizedintheliterature,however.Theyhavepositivee¤ectsoneconomicactivityexantebecause,asinRanciereetal.(2008),theexpectationoffuturebailoutsraisescollateral;buttheyarecostlytoexecuteexpostbecausetheydivertresourcesawayfrominvestment.Finally,thispaperisrelatedtotherecentworkthatstressestheroleofinformationinsensitivecollateralsuchasDangetal.(2011)andGortonandOrdoñez(2012).Thisresearchemphasizestheroleofthe…nancialsystemingeneratingcollateralthatisacceptedbyeveryonewithoutfearofadverseselection,becauseitdoesnotpayforanyagenttoproduceprivateinformationregardingitsunderlyingcharacteristics.Ofcourse,theseincentivesmaychangeduringacrisis,inwhichcaseinformationisgeneratedandthevalueofcollateralmightexperiencelargeswings.Inourmodel,itisalsonaturaltothinkofbubblycollateralasinformationinsensitiverelativetofundamentalcollateral.Thevalueofbubblycollateraldependsoninvestorsentimentormarketexpectationswhicharepublicinformation.Thevalueoffundamentalcollateraldependsinsteadontheregularbusinessactivityoftheunderlying…rmor…nancialinstitution,whichislikelytobein‡uencedbymanagersthroughavarietyofchannelsthataredi¢culttoobserveandgiverisetoprivateinformation.1ThebubblyeconomyThestartingpointofouranalysisistheclassicDiamond(1965)modelofcapitalaccumulationwithoverlappinggenerations.Theyoungsupplylabor,consumepartoftheirlaborincomeandsavethe5 rest.Thesesavingsareusedtopurchasecapitalfromtheoldandtoproducenewcapital.Theoldownthecapital,demandlaborandconsumetheircapitalincome.Tirole(1985)extendedtheDiamondmodelbyintroducinganadditionalmarketforbubblesorpyramidschemes.Participantsinapyramidschememakevoluntarycontributionsthatentitlethemtoreceivethenextcontribution.Thus,wesaythatanindividualpurchasesabubblewhenhe/shemakesacontributiontoapyramidscheme.Similarly,wesaythatanindividualsellsabubblewhenhe/shereceivesacontributionfromthepyramidscheme.Startingabubbleorpyramidschemeyieldsawindfalltothe…rstparticipant,whichconsistsofthe…rstcontributiontothescheme.Weassumethateachgenerationstartsnewbubbleswithrandominitialvalue.Thesenewbubblesaretradedinthemarketforbubblesalongsidetheoldbubblesstartedbypreviousgenerations.4ThemodelwepresenthereaddsacreditmarkettotheTirolemodel,assketchedinthelastsectionofMartinandVentura(2012).IntheDiamondandTirolemodelsthereisonlyintergen-erationaltrade.Herewecreategainsfromintragenerationaltradebyintroducingheterogeneitywithingenerations.Whilesomeindividualshavesavings(the‘savers’),otherindividualshavein-vestmentopportunities(the‘entrepreneurs’).Byinvestmentopportunities,wemeanbothcapitalandbubbles.Youngsaversgivecredittoyoungentrepreneursand,inreturn,oldentrepreneurspaya(possiblycontingent)interestratetooldsavers.Ouremphasisispreciselyontheworkingsofthiscreditmarket.Ifthismarketworkedwell,theequilibriaofourmodelwouldnotdi¤erfromthoseoftheTirolemodel.Thecreditmarketwouldsimplyallowsaverstoe¤ectivelyowncapitalandbubbles,sincecompetitionamongentrepreneurswouldensurethatequilibriumcreditcontractsreplicatethepayo¤sofowningcapitalandbubbles.Butimperfectenforcementinstitutionslimitthecollateralofentrepreneursandimpairtheworkingsofthecreditmarket.5Bycollateral,wemeantheold-ageresourcesthatyoungentrepreneurscanpledgetoyoungsavers.Inthissection,wedescribeourmodelandshowhowthesetofequilibriadependsontheavailabilityofcollateral. 4ThearrivalofnewbubblesisanimportantdeparturefromtheoriginalTirolemodelthatplaysakeyroleinourtheory.5Imperfectenforcementinstitutionsdonotimpairtheworkingsofthemarketsforlabor,capitalorbubblessincetransactionsinthesemarketsdonotinvolveapromisebythesellertodeliverfuturepaymentstothebuyer.Thus,labor,capitalandbubblescanbetradedeveninsituationsinwhichcreditisnotpossibleorseverelyrestricted.6 1.1ThebasicsetupConsideraneconomyinhabitedbyasequenceofequal-sizedandoverlappinggenerationsofyoungandold.Timestartsinperiodt=0andthengoesonforever.Theobjectiveofindividualiofgenerationtistomaximizehis/herutilityUit,givenby:Uit=Cit;t+
EtCit;t+1,(1)where�0;andCit;tandCit;t+1aretheyoung-ageandold-ageconsumptionsofindividualigenerationt.Naturally,consumptionmustbenon-negative,i.e.Cit;t0andCit;t+10.Asusual,Etf
gdenotesthemathematicalexpectationofthecorrespondingvariable.Eachgenerationcontainstwotypesofindividuals.Saversworkduringyouth,savepartoftheirlaborincomeandlenditto…nanceold-ageconsumption.Entrepreneursborrowduringyouthto…nancepurchasesofcapitalandbubbles,andproduceduringoldage.Saverssupplyoneunitoflaborduringyouthanddecidehowmuchoftheirlaborincometosave.Theyusetheirsavingstopurchasecreditcontractsinexchangeforagross,possiblycontingent,interestrateofRt+1unitsofoutputinperiodt+1foreachunitofoutputusedtopurchasecreditcontractsinperiodt.WerefertoRt+1asthecontingentorexpostinterestrate,andwerefertoEtRt+1astheinterestrate.Inoldage,saversreceivethereturntotheirsavingsandconsume.Thus,theirconsumptionsaregivenby:Cit;t=Wt�Lt,(2)Cit;t+1=Rt+1
Lt,(3)whereWtisthewageandLtissavingsorthevalueofcreditcontractspurchased.Equation(2)saysthatyoung-ageconsumptionequalslaborincomeminussavings,whileEquation(3)saysthatold-ageconsumptionequalsthereturntosavings.SaversmaximizeutilityinEquation(1)subjecttothebudgetconstraintsinEquation(2)and(3).Entrepreneursinvestandproduce.Theirproductiontechnologycanbedescribedbythefol-lowingCobb-Douglasfunction:F(Kt;Nt)=At
Kt
� t
Nt
1�,(4)7 whereKtiscapital,2(0;1)and �1.Laborproductivitygrowsattheconstantrate 1.WerefertoAtasaproductivityshockandassumethatit‡uctuatesrandomlyovertime.Theproductionofoneunitofcapitalrequiresoneunitoftheconsumptiongood.Weassumethatcapitalfullydepreciatesinproduction.Entrepreneursalsoinitiateandtradebubbles.WeuseBttodenotethevalueofthebubblesinperiodt.Someofthesebubblesareoldsincetheywereinitiatedbypreviousgenerationsofentrepreneursandwerepurchasedbyentrepreneursofgenerationt�1.Someofthesebubblesarenewsincetheyareinitiatedbyentrepreneursofgenerationt�1.Thus,theaggregatebubbleevolvesasfollows:Bt+1=RBt+1
Bt+BNt+1,(5)whereRBt+1istheincreaseinpriceorreturntothebubblespurchasedfromentrepreneursofgenerationt�1;andBNt+1thevalueofthebubblesinitiatedbytheentrepreneursofgenerationt.WerefertoBNt+1asbubblecreationandweassumethatitisnon-negative,i.e.BNt+10.To…nancetheiractivities,entrepreneurssellcreditcontractstosavers.Thesecontractscanbefullystatecontingentbutmustbecollateralized,thatis,theymustbebackedbycrediblepromisesoffuturepayments.Thisbringsustothekeyfrictionthatunderliesalltheanalysisofthispaper:imperfectenforcementinstitutionslimittheamountofavailablecollateral.Inparticular,entrepreneurscanhideafraction1�oftheircapitalincomefromenforcementinstitutions,sothat:Rt+1
Lt
[F(Kt+1;Nt+1)�Wt+1
Nt+1]+Bt+1.(6)Equation(6)statesthatpromisedpaymentscannotexceedafraction2[0;1]oftheircapitalincomeplustheirbubbles.Wethinkofthe…rsttermasthe“fundamental”collateralofentrepre-neurs,andofthesecondtermastheir“bubbly”collateral.6Throughoutthepaper,werefertoEquation(6)asthecreditorcollateralconstraint.As!1,theentirecapitalincomecanbeusedascollateralforcreditandthecreditconstraintcoincideswiththerestrictionthatold-ageconsumptionbenon-negative,i.e.Cit;t+10.Thislimitingcaseisthefrictionlesseconomy. 6Onewaytothinkaboutthisconstraintisthatcourtscangraballtheconsumptiongoodsthataretradedinthemarketbutonlyafractionofthosethatarenottraded.8 Withtheseassumptions,wecanwritetheconsumptionsofentrepreneursasfollows:Cit;t=Lt�Kt+1�Bt,(7)Cit;t+1=F(Kt+1;Nt+1)�Wt+1
Nt+1+Bt+1�Rt+1
Lt.(8)Equation(7)saysthatyoung-ageconsumptionequalscreditminusinvestmentandthepurchasesbubbles,whileEquation(8)saysthatold-ageconsumptionequalsproductionnetoflaborcostsplusthevalueofbubblesminuscreditpayments.EntrepreneursmaximizeutilityinEquation(1)subjecttothebudgetconstraintsinEquations(7),(8),thecreditconstraintinEquation(6)andthelawofmotionforbubblesinEquation(5).1.2MarketsandpricesWearereadytosolvethemodel.Inthelabormarket,oldentrepreneursdemandlaboruntilthemarginalproductequalsthewage.Sinceyoungsaverssupplyoneunitoflabor,Nt=1,thismeansthat:wt=(1�)
At
kt.(9)Asusual,weuselowercaseletterstorefertovariablesinunitsofe¢cientworkerssothat,forinstance,kt �t
Kt.Equation(9)saysthatlaborincomeisaconstantfraction1�ofproduction.Inthemarketforbubbles,oldentrepreneurssellbubblestoyoungones.Oldentrepreneurssupplyalltheirbubblesprovidedthepriceisnonnegative.Youngentrepreneurs,aswehaveseen,…nancethepurchaseofbubblesbysellingcreditcontracts.Iftheexpectedreturntoholdingbubblesfallsshortoftheinterestrate,youngentrepreneursmakealosswhentheysellcreditcontractsto…nancebubblepurchasesandthedemandforbubblesiszero.Iftheexpectedreturntoholdingbubblesexceedstheinterestrate,youngentrepreneursmakeapro…twhentheyborrowtopurchasebubblesandthedemandforbubblesisunbounded.Thus,equilibriumrequiresthattheexpectedreturntoholdingbubblesequalstheinterestrate:EtRBt+1=EtRt+1.(10)Thisdoesnotmean,ofcourse,thattherealizedreturntoholdingbubblesequalstherealizedreturntocreditcontracts.9 Finally,wemovetothecreditmarketwhereyoungsaverssupplycredittoyoungentrepreneurs.Note…rstthat
EtAt+1
k�1t+1EtRt+1�1inequilibrium.7Usingthisobservation,thesupplyofcreditbyyoungsaverscanbedescribedasfollows:lt8:=wtif
EtRt+1&#x]TJ ;� -1; .63; Td;&#x [00;12[0;wt]if
EtRt+1=1.(11)Iftheinterestrateexceedsthediscountrate,youngsaverssavealltheirincome.Iftheinterestrateequalsthediscountrate,youngsaversareindi¤erentbetweensavingandconsuming.Thedemandforcreditbyyoungentrepreneurscanbewrittenasfollows:EtRt+1=8&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;:
EtAt+1
k�1t+1ifEtbNt+1(1�)

EtAt+1
kt+1

EtAt+1
kt+1+Etbt+1 �1
ltifEtbNt+1(1�)

EtAt+1
kt+1.(12)Equation(12)depictsthemaximuminterestratethatentrepreneursarewillingandabletopayperunitofcredit,respectively.Whetherthecreditconstraintisbindingornotdependsonwhetherexpectedbubblecreationexceedsthecapitalincomethatisnotpledgeable.Ifitdoes,thenthecreditconstraintisnotbindingbecauseentrepreneurshaveenoughcollateraltopledgetheentirereturntoinvestment:inthiscase,theinterestrateisequaltotheexpectedmarginalreturntoinvestment.Ifitdoesnot,thenthecreditconstraintisbindingbecauseentrepreneurialcollateralisinsu¢cienttopledgetheentirereturntoinvestment:inthiscase,themaximuminterestratethatentrepreneursareabletopayisdeterminedbytheratioofexpectedcollateral(fundamentalandbubbly)tocredit.8 7Thiscanbeshownbycontradiction.Assume…rstthatEtRt+1�
EtAt+1
k�1t+1.Then,entrepreneurswouldnotwanttosellcreditcontractsand
EtAt+1
k�1t+1=1��1;whilesaverswouldliketospendalltheirincomepurchasingcreditcontracts.Thus,itfollowsthat
EtAt+1
k�1t+1EtRt+1.Assumenextthat�1�EtRt+1.Then,saverswouldnotwanttopurchasecreditcontractsand
EtAt+1
k�1t+1=1��1;whileentrepreneurswouldwanttosellasmanycreditcontractsasallowedbytheirconstraint.Thus,itfollowsthatEtRt+1�1.8ThereisaninterestingwrinklethatariseswhenthecreditconstraintisbindingandEtbNt+1=0.Inthatcase,theonlypossibleequilibriumentailsEtRt+1=

EtAt+1
k�1t+1:atthatinterestrate,though,individualentrepreneurs(whotakethereturntoinvestmentasgiven)areunconstrainedandtheydemandin…nitecredit.Competitioncannotbiduptheinterestrate,moreover,becauseentrepreneursarealreadypledgingthemaximumshareoftheirincomethatisfeasiblegivenenforcementconstraints.Insuchasituation,clearingofthecreditmarketrequiresrationing.Weassumethatthisrationingworksasfollows:given
EtAt+1
k�1t+1entrepreneurso¤ertoborrowanamountLtataninterestrateEtRt+1;intheeventthattheinterestrateo¤eredbyallentrepreneursisthesame,thetotalsupplyofcreditisallocatedequallyacrossentrepreneurs.Therearetwopossibilities:(i)EtRt+1��1,inwhichcasesaversstrictlyprefertolendtheirresourcestoentrepreneursandthesupplyofcreditequalstheeconomy’swagebill;(ii)EtRt+1=�1,inwhichcasesaversareindi¤erentbetweenlendinginthecreditmarketorconsumingduringyouthandequilibriumlendingisexactlyequalto(


EtAt+1)1 1�.ThisissuedoesnotarisewhenEtbt+1�0,howeversmall,anditthereforeplaysaminorroleinouranalysis.10 Tocompletethesolution,weneedtodeterminehowentrepreneursusethecredittheyreceive,andthisisasfollows:
kt+1+bt8:=ltif

EtAt+1
k�1t+1&#x]TJ ;� -1; .63; Td;&#x [00;12[0;lt]if

EtAt+1
k�1t+1=1.(13)Equation(13)saysthatentrepreneursconsumeinthe…rstperiodonlyifthemarginalproductofcapitalequals�1.Otherwise,entrepreneursuseallthecredittheyreceivetoinvestandpurchasebubbles.1.3EquilibriumdynamicsToobtainsomeintuitionsabouttheworkingsofthemodel,itisusefultocollapsethepreviousequationsasfollows:kt+18&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;:=1�
At
kt�bt if
EtRt+1�120;1�
At
kt�bt if
EtRt+1=1,(14)EtRt+1=minf
EtAt+1;Etf(
+nt+1)
At+1gg
k�1t+1,(15)bt+1=EtRt+1+ut+1
bt+nt+1
At+1
kt+1,(16)whereut+1istheunexpectedcomponentofbubblereturnsandnt+1isthevalueofnewbubblesasashareofoutput,i.e.ut+1RBt+1�EtRBt+1andnt+1bNt+1=(At+1
kt+1).Werefertoutandntasbubble-returnandbubble-creationshocks,respectively.Equations(14)and(15)describethesupplyanddemandoffundsforinvestment,respectively.Equation(16)describesthedynamicsofthebubble.Jointly,Equations(14)-(16)provideafulldescriptionofthedynamicsofthestatevariables,i.e.ktandbt;foranyadmissiblesequenceofproductivityandbubbleshocks,i.e.At,utandnt.Thus,werefertothemasthelawofmotionofthesystem.Figure3showshowtheinterestrateandnextperiod’scapitalstockaredeterminedbysolvingEquations(14)and(15).Ifthecapitalstockislow,theinterestrateisabovethediscountrateandyoungsaversusealltheirincometopurchasecreditcontracts.Inthiscase,collateralisabundantandthecapitalstockisdeterminedbythesupplyoffunds.Welabelthisrangeofcapitalstockstheregionoffullintermediation.Ifthecapitalstockishigh,theinterestrateequalsthediscountrateandyoungsaversconsumepartoftheirincome.Inthiscase,collateralisscarceandthecapital11 stockisdeterminedbythedemandforfunds.Welabelthisrangeofcapitalstockstheregionofpartialintermediation.Figure4showshowproductivityandbubblesa¤ectthesupplyanddemandoffundsforin-vestmentand,thus,thelawofmotion.Highrealizedproductivityraisesthesupplyoffunds,whilehighexpectedproductivityraisescollateralandthedemandforfunds.Bubblereturnshocksareembeddedinthesizeofthebubble.Thehigheristhesizeofthebubbletheloweristhesupplyoffunds.Highrealizedbubblecreationraisesthesizeofthebubbleandlowersthesupplyoffunds,buthighexpectedbubblecreationraisescollateralandthedemandforfunds.Theseintuitionsarecrucialtounderstandtheresultsthatfollow.Toconstructequilibriaforthiseconomy,wepropose…rstajointstochasticprocessforbubblereturnandcreationshocks:fut;ntgforallt.Naturally,thisbubbleprocessmustbesuchthatEtut+1=0andnt0forallt.Weadopttheconventionthatb�1=0andsetn0toensurethatb0takesthedesiredstartingvalue.Withthisstochasticprocessathand,wedetermineallpossiblesequencesforthestatevariablesfkt;btgusingEquations(14)-(16).Ifallthesesequencesaresuchthatkt0andbt0,wesaythattheproposedbubbleprocessisanequilibrium.Otherwise,wesaythattheproposedbubbleprocessisnotanequilibrium.Thebubblelessequilibriumisthatoneinwhichthebubbleprocesstakesthisform:fut;ntg=f0;0gforallt.Thebubblelessequilibriumalwaysexistsanditisausefulreference.Butthereareadditionalbubblyequilibriaandthesewe…ndmoreinteresting.1.4DiscussionBeforemovingon,wewriteafewwordsaboutthenatureofbubblesforthosereadersthatremainskepticalaboutthismodelingstrategy.Convincedreaderscanskipthissection.Abubbleisnothingbutapyramidscheme.Participantsinapyramidschememakeavoluntarycontributiontotheschemethatgivesthemrighttothenextvoluntarycontribution.Eventhoughthismightseemquiteabstractorexoticat…rstsight,itiseasyto…ndreal-worldsituationsthatcorrespondfairlywelltothisconcept.Consider,forinstance,creditgiventoa…rminexcessofthenetpresentvalueofthecash-‡owsthatthis…rmwillgenerate.Thiscreditisoftenlabeled“excessive”.Butthis“excessive”creditisrationalifcreditorsexpectthe…rmtoobtain“excessive”creditalsointhefuture.Thus,“excessive”creditcanbeinterpretedasavoluntarycontributiontothe…rmthatgivestherighttothenextvoluntarycontribution.Thisleadsustothinkofreal-world…rmsasportfoliosofcapitalandbubbles.Andthisisexactlywhattheentrepreneursinourmodelstand12 for.Naturally,thesaversinourmodelstandforthosethatrationallyprovidecredittoreal-world…rms.Letustrytopursuethisviewabitfurtherandseehowwecouldre-interpretourmodelintermsof…rmsandthestockmarket.Productionandinvestmentmusttakeplacewithin…rmsthatareownedandmanagedbyentrepreneurs.Youngentrepreneurspurchasepre-existing…rmsinthestockmarketorcreatenewonesatzerocost.LetVtdenotethevalueofall…rms,i.e.thevalueofthestockmarket.To…nancetheiractivities,entrepreneurssellcreditcontractsLt=It+Vt,whereItisinvestmentandthecapitalstockevolvesasfollows:Kt+1=It+(1�)
Kt,whereisthedepreciationrate,whichwehaveassumedtobeoneforsimplicity.Entrepreneurscanonlypledgetotheircreditorsafractionofpro…tsandthere-salevalueoftheir…rms.Asaresult,thefollowingcreditconstraintapplies:Rt+1
Lt
[F(Kt+1;Nt+1)�Wt+1
Nt+1]+Vt+1.Underthisre-interpretation,themodelshowsthattherearemanyequilibriainwhich…rmpricestakethisform:Vt=(1�)
Kt+Bt,where(1�)
KtandBtarethefundamentalandbubblecomponentsof…rmprices.9Inthisre-interpretationofthetheory,thereisnomarketforbubbles,insteadthereisastockmarketinwhichportfoliosofcapitalandbubblesaretraded.Notethat,inthecaseoffulldepreciationthatwefocuson,wecandirectlyre-interpretthemarketforbubblesasthestockmarket.Thus,itisstockmarketbubbles(ormoregenerally,bubblesin…rmvaluesevenifthesearenottradedinorganizedstockexchanges)thatfeedcreditboomsandbusts!Aninterestingissuethatarisesinthisre-interpretationofthemodelreferstothenatureofbubblecreation.DibaandGrossman(1988)arguedthat,ifabubbleexistsonastockor…rmprice,thenitstartedonthe…rstdateoftradingofthe…rm.Toreachthisconclusion,theynotedthatbubblecreationatanydateafterthe…rstdateoftradingwouldinvolveaninnovationinthe…rm 9Thisassumesthatthepriceofcapitalisone.Thisisthecaseifoldandnewunitsofcapitalareindistinguishableandinvestmentisalwaysstrictlypositive.Ifdi¤erentcapitalvintagesweredistinguishable,forinstance,therecouldalsobeabubblecomponentinthepriceofcapital(s).Weignorethiscomplicationhere.13 price.Butthisinnovationmusthavehadazeroexpectedvalueifitwasnotpricedinthe…rstdateoftrading.Uptohere,theirargumentindicatesthatexpectedbubblecreationafterthe…rstdateoftradingofa…rmmustbezero.DibaandGrossmandidnotstophereandnotedalsothatinsomepopularsetupsfreedisposalof…rmsrulesoutnegativebubbles.Inthesesetups,thereisanevenstrongerrestrictiononbubblecreation.Ifnewbubblesmustbenonnegativeandtheirexpectedvaluemustbezeroafterthe…rstdateoftrading,thenbubblecreationcanonlytakeplaceinthe…rstdateoftrading.Sincewestudythroughoutequilibriainwhichexpectedbubblecreationispositive,itmightbeusefultoseehowwegoaroundtheDiba-Grossmanargument.Asthem,weassumefree-disposalandruleoutnegativebubblesastheydo,BNt0.Butonceweknowthatinourmodelthemarketforbubblesisnothingbutthemarketfor…rms,Vt=Bt,itisstraightforwardtore-interpretEquation(5)(whichwereproducehereforconvenience)intermsof…rmprices:Bt=RBt
Bt+BNt,Btisthepriceofallthe…rmsintheeconomyanditincludesoldorpre-existing…rms,RBt
Bt;andnewlycreated…rms,BNt.Theargumentthatbubblecreationmustbezeroafterthe…rstdateoftradingimpliesEtRBt+1=EtRt+1.Andindeed,thisiswhatEquation(10)says.ButtheDiba-Grossmanargumentdoesnotimposeanyrestrictiononthesizeofthebubbleinthe…rsttradingdateof…rms.ThisiswhyEtBNt+1�0ispossible.Positiveexpectedbubblecreationafterthe…rstdateoftradingisalsopossibleoncesomeoftherestrictiveassumptionsusedbyDibaandGrossmanarerelaxed.Butprovingthiswouldtakeustoofara…eldanditisnotneededhere.2BubblyequilibriaEntrepreneurswishtopurchasebubblesifthesegrowatleastasfastastheinterestrate.Entre-preneurscanraiseenoughfundstopurchasebubblesifthesedonotgrowfasterthantheeconomy.Thus,bubblyequilibriaarepossibleifandonlyiftheinterestratedoesnotexceedthegrowthrate.Itiswellknownthatthissituationarisesifthefrictionlesseconomyisdynamicallyine¢cient.Inthiscase,theinterestrateislowbecausethesupplyoffundsishighandtheeconomyoverinvests.Bubblesreduceunproductiveinvestmentandthisallowstheeconomytosustainahigherlevelof14 consumptionandwelfare.Wedonotwanttofocusonthiscasehowever.10Itissomewhatlessknownthat,evenifthefrictionlesseconomyisdynamicallye¢cient,theinterestratemightfallbelowthegrowthrateif…nancialfrictionslimitthestockoffundamentalcollateral.Inthiscase,theinterestrateislowbecausethedemandforfundsislowand,ifanything,theremightbeunderinvestment.Thisisindeedthecasewefocusonhere.Throughout,weassumethat
�1.Thisconditionensuresthatindividualsarepatientenoughtosaveeveniftheinterestrateisbelowthegrowthrate.Withoutthiscondition,theinterestratewouldneverfallshortofthegrowthrateandbubblyequilibriawouldnotbepossibleinourmodel.2.1BubblybusinesscyclesItseemsnaturaltostartwiththecaseinwhichtherearenoproductivityshocks,i.e.At=A.Wealsoadoptthefollowingassumption:1 21 1+.(17)The…rstinequalityensuresthatthefrictionlesseconomyisdynamicallye¢cientandinvestmentisalwaysproductive.Thus,bubblyequilibriaexistonlyifthedemandforfundsisdepressed.Thesecondinequalityensuresthatthisisindeedthecase.Thatis,collateralisscarceandthislowersthedemandforfundsenoughtomakebubblyequilibriapossible.Theexistenceofbubblyequilibriaofthissortdoesnotimplythatthereisunderinvestmentinthebubblelessequilibrium.If1 1+

1 1+,thesteadystateofthebubblelessequilibriumliesintheregionoffullintermediation.Theinterestrateislowenoughtomakebubblyequilibriapossible,butnotlowenoughtoleadtounderinvest-ment.If1 21 1+

,thesteadystateofthebubblelessequilibriumliesintheregionofpartialintermediation.Inthiscase,theinterestrateislowenoughnotonlytomakebubblyequilibriapossible,butalsotoleadtounderinvestment. 10Onereasonisthatdynamicine¢ciencyisbelievedtoberareinpractice(seeAbeletal.1988),althoughthishasbeenrecentlychallengedbyGeerolf(2012).Anotherreasonisthat,inthiscasebubblyequilibriaarealwaysassociatedwithdeclinesininvestmentandoutput,whichseemscounterfactual.15 Letusconsider…rsta‘quiet’bubbleprocess:fut;ntg=f0;ngforallt.Thisdeterministicbubbleisanequilibriumifnislowenough.Fromanyinitialcondition,theeconomyconvergestothefollowingsteadystate:k=min1�
A
(k)�b ;[
(
+n)
A]1 1�,(18)b=
n
A
(k) �(
+n)
A
(k)�1.(19)Equation(18)and(19)jointlydeterminethesteadystatevaluesforthecapitalstockandbubble.Onecanthinkofnasanindexofthesizeofthebubble.Ahighervalueofnleadstoahighersteadystatebubble,foratleasttworeasons.The…rstanddirectoneisthatahighervalueofnraisesthevalueofnewbubblesthatarecreatedineachperiod.Thesecondandmoreindirectreasonisthatahighervalueofnraisestheinterestrate,therebyraisingtherateatwhichthevalueofoldbubblesgrows.Figure5showsthee¤ectsofnonthesteadystatevaluesofcapital,thebubble,credit,theinterestrateandutility.Byutilitywemeanthesumofutilitiesofsaversandentrepreneurs,i.e.Ut=PiUit.11TheFigureshowsthecaseinwhichthesteadystateofthebubblelessequilibriumislocatedintheregionofpartialintermediationandthereisunderinvestment.Thee¤ectsofthesizeofthebubbleonthecapitalstockandutilityhaveaninverseU-shape.Smallbubblesraisecreditandinvestment.Thiscrowding-ine¤ectofbubblesraisesthecapitalstockandwelfare.Butifthebubblesaretoolarge,theeconomyenterstheregionoffullintermediationandthee¤ectsofbubbleschange.Largebubblesraisetheinterestrateanddisplaceproductiveinvestment.Thecrowding-oute¤ectreturns.Thelargerthebubblethelargeristhiscrowdingoute¤ect.Indeed,ifthebubbleislargeenough,utilityislowerthanifthereisunderinvestment.Thereisthereforean‘optimal’bubblethatprovidestheamountofcollateralthatmaximizesthecapitalstockandutilityinthesteadystate.Thisbubbleistheonethatplacestheeconomyatthefrontierofthepartialandfullintermediationregions,eliminatingunderinvestmentwithoutraisingtheinterestrate.Asmallerbubblewouldprovidetoolittlecollateralandthefundsavailableforinvestmentwouldbetoolow.Alargerbubblewouldraisetheinterestrateandthefundsavailableforinvestmentwouldalsobetoolow.The‘quiet’bubbleprovidesthebasicintuitionforwhatistocomelater.But,if‡uctuations 11Fornow,thereadershouldfocusonlyonthesolidlinesinFigures5,6,and7.Thedashedlines,whichdepictthebehaviorofthesesamevariablesunderpolicyinterventions,willbeanalyzedinsection3.16 increditaredrivenbybubblesintherealworld,thesebubblesareeverythingbutquiet.Thus,weconsidernextaneconomythatexperiences‘bubblyepisodes’.Inparticular,weassumethattheeconomystartsinthefundamentalstatewherebt=nt=0.Theeconomycantransitiontooneofmanybubblystateswhichweindexbyj=1;:::;J.Let"jbetheprobabilityofreachingbubblystatejfromthefundamentalstate.Whentheeconomyabandonsabubblystate,itreturnstothefundamentalstate.Letjbetheprobabilitythattheeconomyreachesthefundamentalstatefrombubblystatej.Werefertotheintervalthatgoesfromthe…rstperiodtheeconomyreachesbubblystatejtothe…rstperiodinwhichitreturnstothefundamentalstateasa‘bubblyepisodeoftypej’.Letztbethestateoftheeconomy,i.e.zt2fF;B1;:::;BJg.Duringabubblyepisodeoftypej,wehavethatnt=njandut+1=8�&#x]TJ ;� -1; .63; Td;&#x [00;:[
+(1�j)
nj]
A
k�1t+1
j 1�jifzt+1=Bj�[
+(1�j)
nj]
A
k�1t+1ifzt+1=F.(20)Theassumptionthatbubblecreationisconstantisjustforconvenience.Theassumptionaboutbubblereturnssaysthat,duringabubblyepisode,theexpectedreturntoholdingthebubbleequalsthatofcreditcontracts.Conditionalonthebubblyepisodenotending,thereturnonthebubblewillexceedthatofcreditcontracts.Butthisisonlycompensationfortheriskofthebubblyepisodeendingandthebubblebursting.12Withtheseassumptions,thedynamicsoftheeconomycanbewrittenasfollows:kt+18��&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;:min1�
A
kt;[
(
+"
n)
A]1 1�ifzt=Fmin1�
A
kt�bt ;[
(
+(1�j)
nj)
A]1 1�ifzt=Bj,(21)bt=8�&#x]TJ ;� -1; .63; Td;&#x [00;:0ifzt=F(
+(1�j)
nj)
A
k�1t
(1�j)
bt�1+nj
A
ktifzt=Bj,(22)where"
n=Pj"j
nj.Equation(21)describesthebehaviorofthecapitalstock.Itsevolutionineachperioddependsonwhetherinvestmentisdeterminedbythesupplyordemandforfunds.Itisworthmentioningthattheprobabilityofabubblyepisodestartingraisesthedemandfor 12Naturally,theeconomywiththe‘quiet’bubbleanalyzedinFigure3isthelimitingcaseoftheeconomywith‘bubblyepisodes’inwhichthereisabubblystatewith"j!1,j!0andnj=n.Theeconomyentersthisstateatthebeginningoftimewithprobabilityoneandneverleavesit.17 fundsinthefundamentalstate.Thus,thebehavioroftheeconomyinthefundamentalstateisnotequivalenttothatofthebubblelessequilibrium.13Equation(22)describestheevolutionofthebubbleduringanepisodeoftypej.Notethatthegrowthofthebubbledependsonhowlargeandriskyitis,i.e.njandj.Thelargerandriskier,thefasteritgrows.Weshownextsomesimulations.Toproducethem,wechooseparametervaluesthatensurethatthesteadystateofthebubblelessequilibriumislocatedintheregionofpartialintermediation.Then,wecreatefourbubblystatesthatdi¤erintermsofthesizeofthebubbleanditsrisk: SizenRisk safe risky small (nS;S) (nS;R) large (nL;S) (nL;R) wherenSnLandSR.Wealsoassumethatallthesebubblesareequallylikely,i.e."j=".Finally,wegenerateasequenceof10,000valuesforzt,andstarttheeconomyinthesteadystatesothatwerecord10,000periodsofsteadystatebehavior.Figure6plotsthesimulatedtimeseriesofkt,EtRt+1,lt,Utandbtduringawindowof200periods.TheFigurealsoshowsthemeanandstandarddeviationofthesevariablescomputedwiththeentireseriesof10,000periods.Giventhelengthofthesimulation,thesesamplestatisticsareexcellentapproximationstotheircorrespondingtheoreticalcounterparts.The…gureillustratesthat,dependingontheircharacteristics,somebubblescrowd-incapitalwhileotherscrowd-outcapital.Takethe…rstbubbleinthesample:inthiscase,thebubbleissmallandsafeanditraisestheeconomy’scapitalstock.Theinterestrateremainsconstantat�1throughouttheepisode,whichcon…rmsthatthereisnotenoughbubblycollateraltofullyintermediatewages.Contrastthesee¤ectswiththoseofthenexttwobubbles,whicharelarge.Initially,bothbubblesraisetheeconomy’scapitalstock:afterawhile,though,theybothbecometoolargeandreducecapitalaccumulation.Theinterestrisesabove�1duringbothepisodes,con…rmingthatthereistoomuchbubblycollateral.The…gurealsoillustratesthatriskierbubblesgrowfasterbecausetheymustcompensatepotentialbuyersfortheirriskofbursting.Toseethis,comparethetwoepisodesaroundperiods8030and8040.Bothbubblesareofthesamesizebutthe…rstoneisriskier:thus, 13Wehaveassumedthattheprobabilityandtypeofbubblyepisodesareconstantovertimeandthereforeexpectedbubblecreationisconstantwhentheeconomyisinthefundamentalstate,i.e."
n.Thisassumptionisnotnecessary.Wecouldhaveassumedinsteadthatthereare‡uctuationsinexpectedbubblecreationinthefundamentalstate.Inthiscase,eveninthefundamentalstatetheeconomywouldbesubjecttochangesininvestorsentiment.18 eventhoughthe…rstepisodelastslessthanthesecondone,themaximumsizeofthebubbleislarger.Oneimplicationofthisisthat,conditionalonnotbursting,riskierbubblesbecomelargerandhavemoreseverecrowding-oute¤ects.Theseexamplesillustratetwokeyaspectsofcreditbubbles.The…rstoneisthattheyhavemixede¤ectsontheeconomy.Bubblecreationprovidescollateraltothecurrentgenerationandthisallowsittoraisecreditandexpandinvestment.Butbubblecreationalsorequiresfuturegenerationstocancelthisadditionalcreditandthisdivertssomeoftheirresourcesawayfrominvestment.Inthisregard,wecanthinkofthecrowding-oute¤ectofbubblesasan“overhang”e¤ect,inthesensethatgrowthishurtbybubblesthatwerecreatedinthepast.Thesecondpointisthatcreditbubblescanfollowverydi¤erentpaths,potentiallymuchmorecomplicatedthantheonesexploredhere,andtheymayhaveverydi¤erente¤ectsonoutputandwelfare.Thisnaturallysuggestsaroleforpolicyinselectingorreplicatingadesirableequilibrium.Beforegoingthere,though,weneedtodealwithanotherlayerofcomplexity:real-worldeconomiesarenotonlysubjecttobubbleshocksbutalsotofundamentalones.2.2BubblyandrealbusinesscyclesLetusbringbackproductivityshocks.Inparticular,assumethatAtcantaketwopositivevalues:At2fAL;AHgwithALAH.Thetransitionprobabilityis0:5,i.e.Prob(At+16=At)=.Wede…neEALEtfAt+1=At=ALgandEAHEtfAt+1=At=AHg,andnotethatALEALEAHAH.Then,weadoptthefollowingassumption:AH AH+EAHAL AL+
EAL.(23)ThisassumptionconvergestothatinEquation(17)asproductivityshocksbecomesmall.The…rstinequalityensuresthatthefrictionlesseconomyisdynamicallye¢cientand,asaresult,investmentisalwaysproductive.Thesecondinequalityensuresthat…nancialfrictionsarestrongenoughtomakebubblyequilibriapossible.Consider…rstthebubblelessequilibriumwithproductivityshocks.Theeconomygoesthroughperiodsofhighandlowproductivity.Periodsofhighproductivityarecharacterizedalargesupplyoffundsandbyahighexpectedproductivity,whichraisesthedemandforfunds.Theformerlowerstheinterestratewhilethelaterincreasesit(seeFigure4).Therelativestrengthofthesee¤ectsdependsonthepersistenceofproductivityasmeasuredby,andtheimportanceofdiminishing19 returnstoinvestmentasmeasuredby.If�0:5
,thee¤ectofproductivityonthesupplyoffundsdominates.Inthiscase,whenproductivityishightheinterestrateislowandthegapbetweentheactualandtheoptimallevelsofinvestmentishigh.Ifinstead0:5
,thee¤ectofproductivityonthedemandforfundsdominatesandtheoppositeistrue.Toillustratehowproductivitya¤ectstheoptimalbubble,weextendtheeconomywith‘bubblyepisodes’tointroduceproductivityshocks.Tosimplify,weassumethattransitionprobabilitiesforproductivityandbubbleshocksareindependent.Withtheseassumptions,wehavethatthedynamicsoftheeconomyarenowgivenby:kt+1=8&#x]TJ/;༕ ;.9;‘ ;&#xTf 2;�.35; 0 ;&#xTd [;&#x]TJ/;༕ ;.9;‘ ;&#xTf 2;�.35; 0 ;&#xTd [;&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;:min1�
At
kt;[
(
+"
n)
EtAt+1]1 1�ifzt=Fmin1�
At
kt�bt ;[
(
+(1�j)
nj)
EtAt+1]1 1�ifzt=Bj,(24)bt=8�&#x]TJ ;� -1; .63; Td;&#x [00;:0ifzt=F(
+(1�j)
nj)
Et�1At
k�1t
(1�j)
bt�1+nj
At
ktifzt=Bj.(25)Equations(24)and(25)aresimplegeneralizationsofEquations(21)and(22)toallowforproduc-tivity‡uctuations.Figure7showssimulationsforthiseconomy.Asbefore,thesimulationsstartinthesteadystateandrunfor10,000periods.Toproducethem,weusethesamemethodologyasintheprevioussection,exceptthatwenowhaveaprocessforproductivity.Inchoosingtheparametersforthisprocess,wehaveassumedthat�0:5
.Asmentioned,thisparameterconditionimpliesthatthebubblethatfullyintermediateswagesgrowswithproductivity.ThisiswhyinFigure7bubblesraisethecapitalstockwhenproductivityishighbutreduceitwhenproductivityislow.Thisisbestexempli…edbythelongepisodethatstartsafterperiod6960andlastsuntilafterperiod7000,inwhichthebubbleislargeandsafe.Thecapitalstockincreasesduringtheepisodebutit‡uctuateswithproductivity.Whenproductivityishigh,thecapitalstockgrowsbecausewagesarehighandthelargebubblechannelspartofthesewagestoinvestment.Whenproductivityfalls,however,thecapitalstockscontractsfortworeasons:notonlyarewageslower,butthelargebubbledivertspartofthemawayfrominvestment.Theevolutionoftheinterestratecon…rmsthisnarrativesince,byrisingabove�1wheneverproductivityislow,itindicatesthatthereis“toomuch”bubblycollateralandinvestmentisbeingcrowdedout.Inthisexample,theequilibriumbubbleistoosmallwhenproductivityishighbutitistoolargewhenproductivityissmall:an‘optimal’bubble20 wouldthusbeprocyclical,amplifyingtheresponseofoutputtoproductivityshocks.14Thegeneralinsightofthisexampleisthattherelativestrengthofcrowding-inandcrowding-oute¤ectschangeswiththestrengthoffundamentals.Thesameamountofbubblecreationtoday,forinstance,mayhavesmallorlargecrowding-oute¤ectsinthefuturedependingontheevolutionofproductivity.Iffutureproductivityishighandtheeconomyisintheregionofpartialintermedia-tion,thecrowding-oute¤ectofcurrentbubblecreationwillbesmall.Ifinsteadfutureproductivityislowandtheeconomyisintheregionoffullintermediation,thecrowding-oute¤ectofcurrentbubblecreationwillbehigh.Intheterminologyoftheprevioussection,bubblyepisodesmayhavelarge“overhang”e¤ectsshouldtheeconomysu¤erfromafallinproductivity.Takentogether,theexamplesofthelasttwosectionsshowthatbubblesmayexpandtheecon-omybyprovidingcollateral,buttheymayalsolowerinvestmentandoutputiftheygrowtoolarge.Canpublicpolicybeusedtoprovidetheeconomywiththedesiredamountofbubblycollateral?Weshowinthissectionthattheanswertothisquestionisa¢rmative.3AcreditmanagementagencyWeconsidernextanagencythatmanagesbubblycollateralthroughcredit-marketinterventions.Subsidizingoldentrepreneurs,thisagencyexpandstheircollateral.Taxingoldentrepreneurs,thisagencyreducestheircollateral.The…scalimplicationsoftheseinterventionsarebornebyyoungentrepreneurs.Althoughtheseinterventionsdonotdirectlya¤ecttheequilibriumvalueofthebubble,theyenabletheagencyto“select”anequilibriumallocation.Werefertothisagencyasacreditmanagementagency(CMA),sinceitguaranteesthecollateralofentrepreneursatthetargetedlevel.Ifex-postthebubbleturnsouttobetoosmallgiventhetarget,theCMAprovidesadditionalresourcestoentrepreneurs.Ifex-postthebubbleturnsouttobetoolargegiventhetarget,theCMAtakesawayresourcesfromthem.Beforeturningtotheanalysis,though,itisimportanttoponderbrie‡yonwhatitmeanstodesignpolicyinourenvironment.Thecompetitiveequilibriumisdrivenbyinvestorsentimentorexpectations,ascapturedbytheprocessfnt;utg.Thisposesaproblemfortheevaluationofpolicy,sincetheimplementationofagivenpolicymayin‡uenceexpectationsthemselvesandchangethe 14Moregenerally,thecyclicalityofthebubblethatattainsfullintermediationdependsontheshockthatiscon-sidered.Wehavealsoexplored,forinstance,aspeci…cationofthemodelwith…nancialshocks,i.e.withrandom‡uctuationsin.Theseshocksa¤ectthedemandforfundsbutnottheirsupplyand,asaresult,collateralispro-cyclical.Thismeansthat,tointermediatewagesfully,abubblemustbecounter-cyclicalwithrespectto,sothatitreducestheresponseofoutputto…nancialshocks.21 underlyingprocessfnt;utg.Weavoidthisproblembyfocusingonpoliciesthatareexpectationallyrobust,inthesensethattheyimplementthesameallocationsregardlessofagentexpectationsorinvestorsentiment.3.1ThemodelwithacreditmanagementagencyWeintroducenowaCMAthatpromisesentrepreneursinperiodtatransferofStunitsoftheeconomy’sconsumptiongood.ThetransferStcanbenegativeandcontingentonthestateoftheeconomy.Ifthetransferispositive,itscostis…nancedbyyoungentrepreneurs.Ifthetransferisnegative,itsbene…tisdistributedtoyoungentrepreneurs.LettingXtdenotetheagency’staxrevenues,werequirethattheCMArunabalancedbudgetsothatSt=Xtineveryperiodt.De…neSNt+1asthenetresourcesthatthispolicyprovidestotheentrepreneursofgenerationtintermsofgoodsinperiodt+1:SNt+1St+1�EtRt+1
Xt.(26)Thatis,SNt+1isthedi¤erencebetweenthesubsidiesthatwillbeobtainedinoldageandthetaxespaidinyoungage,bothexpressedintermsofgoodsinperiodt+1.Insofarasexpectedsubsidiescanbepledgedbyentrepreneurstocancelcreditcontracts,thisisawealthtransferthata¤ectscollateral.IfSNt+1�0,thepolicycreatescollateralinperiodt.IfSNt+10,thepolicydestroyscollateralinperiodt.Thus,werefertoSNt+1ascollateralcreationbytheCMA.UsingthisnotationandthebudgetconstraintoftheCMA,wecanexpressthedynamicsofthepolicyasfollows:St+1=EtRt+1
St+SNt+1.(27)TheformalsimilaritybetweenEquations(27)and(5),respectivelydescribingthedynamicsofpolicyandthebubble,hintsatsomeoftheresultsthatfollow.TheintroductionofaCMAdoesnota¤ecttheproblemofsavers,butitdoesa¤ecttheproblemofentrepreneurs.Sinceentrepreneurscanusesubsidiestocancelcreditcontracts,thecollateralconstraintinEquation(6)nowbecomesRt+1
Lt
[F(Kt+1;Nt+1)�Wt+1
Nt+1]+Bt+1+St+1,(28)22 whiletheirconsumptionsarenowgivenbyCit;t=Lt�Kt+1�Bt�Xt,(29)Cit;t+1=F(Kt+1;Nt+1)�Wt+1
Nt+1+Bt+1+St+1�Rt+1
Lt.(30)Theonlydi¤erencesbetweenEquations(29)-(30)andEquations(7)-(8)areduetothetransfersthataremadebytheCMA.BorrowersmaximizeutilityinEquation(1)subjecttothebudgetconstraintsinEquations(29),(30),thecreditconstraintinEquation(28)andthelawsofmotionofbubblesandpolicyinEquations(5)and(27).TheintroductionofaCMAdoesnota¤ecttheequilibriuminthemarketsforlabororbubbles,whicharestillrespectivelydescribedbyEquations(9)and(10).Itdoesa¤ecttheequilibriuminthecreditmarket,though.WhilethesupplyofcreditisstilldescribedbyEquation(11),thedemandoffundsforinvestmentbyyoungentrepreneurscanbewrittenasfollows:EtRt+1=8�&#x]TJ ;� -1; .63; Td;&#x [00;:
At+1
k�1t+1ifEt�bNt+1+sNt+1
(1�)

EtAt+1
kt+1

EtAt+1
kt+1+Etbt+1+Etst+1 kt+1+ �1
(bt+st)ifEt�bNt+1+sNt+1
(1�)

EtAt+1
kt+1.(31)Asbefore,Equation(31)saysthattheinterestrateiseithertheexpectedmarginalproductofcapitalortheratioofentrepreneur’sexpectedcollateraltototalcredit.Theonlynoveltyisthattheconstraintisa¤ectedbytheCMA’spolicies:expectedsubsidiesEtst+1providecollateraltoentrepreneursandtheyraisethedemandforcreditandthusinvestment,butcurrenttaxesstdiverttheresourcesofentrepreneursandtheyreducetheshareofcreditthatcanbeusedforinvestment.Tocompletethesolution,weneedtodeterminehowentrepreneursusethecredittheyreceive,andthisisasfollows:
kt+1+bt+st8:=ltif

EtAt+1
k�1t+1&#x]TJ ;� -1; .63; Td;&#x [00;12[0;lt]if

EtAt+1
k�1t+1=1.(32)Equation(32)saysthatentrepreneursconsumeinthe…rstperiodonlyifthemarginalproductofcapitalequals�1.Otherwise,entrepreneursuseallthecredittheyreceivetoinvest,purchasebubbles,andpaytaxes.23 3.2EquilibriumdynamicsWecannowcollapsethemodelwithaCMAintothefollowingfourequations:kt+18��&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;:=(1�)
At
kt�bt�st if
EtRt+1�120;(1�)
At
kt�bt�st if
EtRt+1=1,(33)EtRt+1=minf
EtAt+1;Etf(
+nt+1+mt+1)
At+1gg
k�1t+1,(34)bt+1=EtRt+1+ut+1
bt+nt+1
At+1
kt+1,(35)st+1=EtRt+1
st+mt+1
At+1
kt+1.(36)wheremt+1measurescollateralcreationbytheCMAasashareofoutput,i.e.mt+1sNt+1=(At+1
kt+1).Weshallthinkofmtasthekeypolicyinstrumentinthemodel,whichisdeterminedbytheCMA.Equations(33)-(36)provideafulldescriptionofthedynamicsofthestatevariables,i.e.kt,btandst;foranyadmissiblesequenceofproductivityandbubbleshocks,i.e.At,utandntandforanysequenceofpolicyinterventionsmt.Thus,werefertothesefourequationsasthelawofmotionofthesystemwithpolicy.Equations(33)-(36)showhowpolicyshocksa¤ectthesupplyanddemandoffundsforinvest-mentandthusthelawofmotion.Likebubbleshocks,pastandpresentpolicychoicesascapturedbym0;m1;:::;mtareembeddedinsubsidiesst.Throughtaxation,thesepolicychoicesreducethepresentsupplyoffundsavailableforinvestmentasdepictedinEquation(33).Simplyput,youngentrepreneursmustdevotepartoftheircredittopayingforthesubsidiespromisedtothepreviousgeneration.Futurepolicychoicesascapturedbymt+1insteadraisethecollateralofentrepreneurs,asshownbyEquation(34),enablingthemtoexpandtheirdemandforfundsandtheirinvestment.OnceaCMAisintroducedintothemodel,anequilibriumcannotbede…nedwithoutspecifyingpolicy.Throughout,wetakepolicyasgivenanddescribeitbyastochasticprocessfmtg.Weadopttheconventionthats�1=0sothats0isdeterminedbym0.Wethenproposeajointstochasticprocessforbubblereturnandbubblecreationshocks:fut;ntg,withEtut+1=0,bt0andnt0forallt.Withthisprocessathand,wesearchforasequenceforthestatevariables(kt;bt;st)thatsatis…eskt0aswellasEquations(33)-(36).Ifsuchasequenceexists,wesaythattheproposedstochasticprocessisanequilibrium.Ifsuchasequencedoesnotexist,wesaythattheproposedstochasticprocessisnotanequilibrium.24 3.3ManagingcollateralduringbusinesscyclesWhatcanaCMAachievebytaxingandsubsidizingcredit?A…rstimportantresult,whichfollowsdirectlyfromcomparingEquations(33)-(36)withEquations(14)-(16),isthataCMAcanusepolicytoreplicateanyoftheequilibriaoftheoriginaleconomy.Takeanyprocessfut;ntgforthebubblereturnandcreationshocksandacorrespondingbubblebt.Then,theCMAcanreplicatetheequilibriumallocationthatwouldariseunderanalternativeprocessf^ut;^ntg;andthecorrespondingbubble^bt,bychoosingapolicyfmtgthatsatis…esmt
At
kt=^ut
^bt�1�ut
bt�1 +(^nt�nt)
At
kt(37)forallt�0:Thispolicyguaranteesthatst+bt=^btandthatEtf(mt+1+nt+1)
At+1g=Etf^nt+1
At+1ginallperiods.Thus,thetotalcrowding-outandcrowding-ine¤ectsofthepol-icysttogetherwithbubblebtareidenticaltothoseofbubble^btintheabsenceofpolicy.Notethatthispolicyisexpectationallyrobustinthesensethatitworksforanyprocessfut;ntg.Thisresultisimportantbecausewehaveseenthatmarketsmayprovidetoolittlebubblycollateral,whichisinsu¢cienttointermediateallwages,ortoomuchofit,whichcrowdsoutinvestmentandreducesoutputandconsumption.Itthereforeseemsnaturaltoconsiderpoliciesthatmanagetheeconomy’scollateral.Inparticular,weconsiderpoliciesfmtgthatsimultaneouslysatisfyEtfmt+1
At+1g=�1
k1�t+1�Etf(
+nt+1)
At+1g,(38)kt+1=(1�)
At
kt�bt�st ,(39)inallperiodst,andstudytheire¤ectsonsteady-stateconsumptionandwelfare.Equation(38)guaranteesthattheproposedpolicystabilizesEtRt+1andsetsitequalto�1atalltimes.Equa-tion(39)saysthatthisstabilizationmustbeachievedwhileguaranteeingthatwagesarefullyintermediatedinequilibrium.Throughthispolicy,theCMAe¤ectively“leansagainstthewind”,taxingcreditwhencollateralisexcessiveandEtRt+1��1andsubsidizingitwhencollateralisscarceandthereispartialintermediation.Notethatthepolicydependsonlyonobservables,sinceitneedonlybecontingentonaggregateproductivityAtandonthebubbleshocksfut;ntg.Wecanillustratethee¤ectsofsuchapolicybyrevisitingtheexampleofa‘quiet’bubblyprocess,inwhichAt=Aandfut;ntg=f0;ngforallt.Wehavealreadyarguedthatthereisanoptimallevelofbubblecreationinthisexample,whichmaximizesoutputandwelfareinthesteady25 state.Thisbubble,whichplacestheeconomyatthefrontierofthepartialandfullintermediationregions,correspondsexactlytotheoneprescribedbyourpolicyrule.Thus,ifweusentodenotesuchanoptimallevelofbubblecreationandfollowEquation(37),apolicythatsetsm=n�ninallperiodsattainsthemaximumlevelsofoutputandwelfareinsteadystate.ThedashedlinesinFigure5depictthesteadystatevaluesofcapital,credit,theinterestrate,utility,thebubbleandthesubsidysasafunctionofnundertheproposedpolicy.Inthiscase,allvariablesareindependentofn.Onceagain,thereasonisthattheCMAintervenesinthecreditmarkettomaximizeintermediationwhileminimizingcrowdingoute¤ects.Intheregionofpartialintermediationitdoessobysubsidizingcreditpaymentsbyoldentrepreneurs,i.e.bysettings�0,whereasintheregionoffullintermediationitdoessobytaxingcreditpaymentsbyoldentrepreneurs,i.e.bysettings0untilEtRt+1=�1.Theseinsightsareeasilyextendedtotheeconomywith“bubblyepisodes”thatwasanalyzedinSection2.1.Thisisthesameeconomyasbefore,withtheonlydi¤erencethatittransitionsbetweenfundamentalandbubblystatesaccordingtoasunspotvariablezt2fF;Bg.Thus,theoptimalbubbleisalsothesameasbefore,andthecorrespondingallocationcanbeimplementedbytheCMAbysettingmt
At
kt=(n�nt)
At
kt�ut
bt�1 ,inallperiods.ThedashedlinesinFigure6showthee¤ectsofthispolicyonthemainvariablesoftheeconomy.TheCMA’sintervention,whichisnowstatecontingent,raisesthecapitalstockinallperiodsandfullystabilizesit.Wheneverbubblycollateralisscarceandtheeconomyisinthepartialintermediationregion,creditissubsidizeduntilitequalstotalwages.Whenevercollateralisabundantandtheinterestrateexceeds�1,creditistaxeduntiltheEtRt+1=�1.ThisexampleillustratesthattheCMA’sinterventiononlya¤ectsthemarketforbubblesthroughitse¤ectontheinterestrate.Theeconomystillexperiencesbubblyepisodesandthepolicyrulehasnodirecte¤ectoverthem.Instead,theCMAmonitorsthese‡uctuationsandintervenesinthemarketforcreditaccordingly.Whenbubblesburst,theCMAstepsinandhelpsentrepreneurscanceltheircreditcontracts.Whenbubblesbecometoolarge,asinthesecondandthirdepisodesoftheexample,theCMAstepsinandtaxesentrepreneurs.Bydoingso,theCMAreducesthedemandforcredit,eliminatesthecrowdingoute¤ectandboostscapitalaccumulation.Ofcourse,noteveryoneishappywiththispolicy:generationswithalotofbubblecreationlosebecausepart26 oftheirwealthistaxedaway.Yet,asthetableinFigure6con…rms,averagewelfareincreasesasaresultoftheintervention.Figures5and6depictthee¤ectofourpolicyruleineconomieswithnofundamentaluncertainty.Inthiscase,theruleisoptimalinthesenseofmaximizingaveragewelfareandoutputinthesteadystate.Theanalysisbecomesmorecomplicatedineconomieswithfundamentaluncertainty,though.Aswehavediscussed,alevelofbubblecreationthatboostsgrowthwhenproductivityishighmightleadtolargecrowding-oute¤ectswhenproductivityfalls.Thiscomplicatesthecharacterizationofanoptimalbubble.Nonetheless,weapplynextoursamepolicyruletothiseconomyandshowthatitalsoraisestheaveragelevelsofoutputandwelfareinthesteadystate.Figure7buildsontheexampleofSection2.3anddepictstheimpactofthepolicyruleinaneconomywithboth,productivityandbubbleshocks.Asbefore,thesolidlinesrepresentthevaluesofthedi¤erentvariablesintheabsenceofpolicywhilethedashedlinesrepresenttheirvaluesunderthepolicyrule.The…gureshowsthat,oncemore,thepolicyrulestabilizestheinterestrateanditalsoraisestheaveragecapitalstockandwelfare.Thenewinsightisthatthepolicyhasanambiguouse¤ectoneconomicvolatility.Fluctuationsthataredrivenbybubbleshocksarestabilizedbythepolicy,while‡uctuationsthataredrivenbyproductivityshocksareampli…edbyit.Thereasonforthislastresultisthat,underourassumptions,thepolicyruleisprocyclicalrelativetoproductivityshocks.Whenproductivityishigh,investmentisconstrainedbythedemandoffunds.Wagesarehighintheseperiodsandcollateralisinsu¢cienttointermediatethemall.Byfollowingthepolicy,theCMAincreasesexpectedsubsidies,raisescreditandexacerbatestheboom.Whenproductivityislow,however,investmentisconstrainedbythesupplyoffunds.Inthesetimes,thereisnotmuchthatthepolicycandotoexpandoutputbesidestaxingborrowingtoreducecrowding-oute¤ects.Hence,thepolicyruleincreasesthestrengthofproductivity-drivenboomsrelativetoproductivity-drivenrecessions.Itsoveralle¤ectoneconomicvolatility,however,dependsontherelativevarianceofbubblevis-à-visproductivityshocks.Inourparticularexample,asthetableinFigure7shows,itincreasesthevolatilityofboththecapitalstockandofwelfare.3.4DiscussionTheexamplesofthissectionshowthatthebubblyeconomyprovidesanewrationaleformacropru-dentialpolicy.Whenbubblycollateralislowerthandesired,theCMAsupplementsitbyprovidingfundstoentrepreneurs.Whenbubblycollateralisinsteadhigherthandesired,theCMAreducesit27 bysubstractingfundsfromentrepreneurs.Thisleaning-against-the-windpolicycanreplicatethe“optimal”bubbleandmaximizesteady-stateoutputandconsumption.Naturally,thispolicygeneratesresourceswhenbubblycollateralishighanditrequiresresourceswhenbubblycollateralislow.Underourassumptions,theselossesandgainsarebornefullybyyoungentrepreneurs.Thepolicycanthereforebeinterpretedasanentrepreneurialinsuranceor“counter-cyclical”fund,towhichentrepreneurscontributeduringyouthintheexpectationofreceivingatransferduringoldageiftheircollateralturnsouttobelow.Notehoweverthatthisinsuranceisnotactuariallyfair.IfEtSNt+1�0,itprovidesnetresourcestotheentrepreneursofgenerationt.IfEtSNt+10,itsubstractsnetresourcestotheentrepreneursofgenerationt.Couldthispolicyalsobeinterpretedasconventionalmonetarypolicy?Afterall,itfullysta-bilizestheinterestrateandsetsitequalto�1.Butthisisonlyonepartofthepolicy,though.Therearemanypoliciesthatstabilizetheinterestrateatthatlevel(i.e.,theysatisfyEquation(38))withoutfullyintermediatingwages(i.e.,theydonotsatisfyEquation(39)).Thecrucialpartofthepolicyisthatitmanagescollateral,providingfundstoentrepreneurswhencollateralislowandtakingfundsfromthemwhencollateralishigh.Butperhapsourleaning-against-the-windpolicycouldbeinterpretedasunconventionalmon-etarypolicy.Inparticular,itcouldberecastasanassetpurchasescheme,notunliketheonesadoptedbyvariousgovernmentssincetheonsetoftherecent…nancialcrisis.Underthisinterpre-tation,theCMApurchasescreditcontractsfromsaversatapriceof

At
kt+bt+st.Itthencollects

At
kt+btinpaymentsfromoldentrepreneurs.Whenst&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;
.51; 0 ;&#xTd [;0,thisinterpretationisverynatural:itmeansthattheCMApays“toomuch”forassetsandsu¤ersalossequaltost.Bydoingso,though,itrelaxesentrepreneurialconstraintsandraisesborrowingexante.Whenst0,instead,thisinterpretationofthepolicyisabitlessnatural:itmeansthattheCMApays“toolittle”forassetsandmakesapro…tofst.Butwhywouldasaversellhercontractsatsuchalowprice?Indeed,thisinterpretationonlymakessenseiftheCMAhasthelegalabilitytoforcesuchsales.Thereisanotherinterestingaspectofinterpretingourpolicyasanassetpurchasescheme.Uptonow,wehaveassumedthatalllossesgeneratedbythepolicy(i.e.,st&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;
.51; 0 ;&#xTd [;0)are…nancedthroughtaxation.Inreality,though,theseschemesaremostly…nancedbyissuinggovernmentsecurities.Aswenowargue,though,nothingchangesifweallowtheCMAto…nanceitslossesbyissuingdebt.28 4The…scalbackstopWehaveassumedsofarthattheCMAcanguaranteeanyamountofcollateralprovidedthatitdoesnotexceedsavings,i.e.theupperboundonXtisgivenbytheeconomy’sentiresavings.Thisisobviouslyanunrealisticassumption.WhathappensiftheCMA’sabilitytotaxyoungentrepreneursislimited?Weshownextthatthesameconditionsthatmakeapolicydesirablealsoguaranteethatacreditmanagementagencyhasenoughresourcestoimplementit.4.1ThemodelwithdebtWeaddressthisquestionbyassumingthattheCMAhasataxation“capacity”�0,i.e.amaximumamountthatitcantaxpere¤ectiveworker.Clearly,thiscapacityconstraintdoesnotbindforpoliciesthatsatisfystinallperiods.Tosustainpoliciesthatrequiretransfersst&#x-285;insomeperiods,though,theCMAneedstoissuecreditcontracts.Todistinguishthesecontractsfromthoseissuedbyentrepreneurswerefertothemaspubliccreditcontractsorpublicdebt.Wethenrefertocreditcontractsissuedbyentrepreneursasprivatecreditcontractsorprivatedebt.Publiccreditcontractsissuedinperiodtpayagross,possiblycontingent,amountDt+1unitsofoutputinperiodt+1.Weuseqttodenotetheperiod-tpriceofapubliccontractthatpromisestodeliver,inexpectation,oneunitofoutputinperiodt+1.Withthisnotation,wecanwritethebudgetconstraintoftheCMAasfollows:St+Dt=Xt+qt
EtDt+1.(40)whereqt
EtDt+1isthevalueofpubliccreditcontractsissuedinperiodt.Equation(40)saysthattotalspendingbytheCMAintransfersanddebtpaymentscannotexceedtotalrevenuesfromtaxationandthesalesofpubliccreditcontracts.Themainchangerelativetoourpreviousanalysisisthatnowwehaveanadditionalmarketforpubliccreditcontracts.Inthismarket,theCMAsuppliesdebtinelastically.Lendermaximizationimpliesthat,inequilibrium,thereturntopubliccreditcontractsmustequalthereturntoprivatelending,sothatqt=(EtRt+1)�1.15Thisdoesnotimply,however,thattheex-postinterestrateonpublicandprivatecreditcontractscoincide.De…neDNt+1measuresthenetresourcesthattheevolutionofpubliccreditprovidestothe 15ToseethisnotethattheconsumptionsoflendersarenowgivenbyCSt;t=Wt�Lt�qt
EtDt+1andCSt;t+1=Rt+1
Lt+Dt+1.Clearly,thedemandforpubliccreditcontractswillbeeitherzeroorin…nityifqt6=(EtRt+1)�1:29 entrepreneursofgenerationtintermsofgoodsinperiodt+1:DNt+1Dt+1�EtRt+1
Dt.(41)Thatis,DNt+1isthedi¤erencebetweenthepublicdebtthatgenerationtleavesinoldageandthepublicdebtthatthisgenerationreceivesinyoungage,bothexpressedintermsofgoodsinperiodt+1.Itrepresentsawealthtransfertogenerationtanda¤ectsitscollateral.IfDNt+1�0,thepolicycreatescollateralinperiodt.IfDNt+10,thepolicydestroyscollateralinperiodt.Thus,werefertoDNt+1asadditionalcollateralcreationbytheCMA.WhileSNt+1isthecollateralcreationthatisdeterminedbytheevolutionofsubsidies,DNt+1isthecollateralcreationthatisdeterminedbytheevolutionofdebtandultimatelyoftaxes.4.2EquilibriumdynamicsFollowingstepsthatarenowfamiliar,wecanrewritethemodelwithaCMAasfollows:kt+18&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;
.98; 0 ;&#xTd [;&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;
.98; 0 ;&#xTd [;&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;:=(1�)
At
kt�bt�st�dt if
EtRt+1�120;(1�)
At
kt�bt�st�dt if
EtRt+1=1(42)EtRt+1=min
EtAt+1;Et�
+nt+1+mSt+1+mGt+1

At+1
k�1t+1(43)bt+1=EtRt+1+ut+1
bt+nt+1
At+1
kt+1(44)st+1=EtRt+1
xt+mSt+1
At+1
kt+1andxt(45)dt+1=EtRt+1
dt+mGt+1
At+1
kt+1(46)wheremSt+1andmGt+1re‡ectnowcollateralcreationthroughbothsubsidies/taxesandpublicdebt,i.e.mSt+1sNt+1=�At+1
kt+1
andmGt+1dNt+1=�At+1
kt+1
.AcomparisonbetweenEquations(42)-(43)and(33)-(34)showsthattheintroductionofdebtdoesnotsubstantiallya¤ectthesupplyanddemandoffundsforinvestment.Aslongass0t=st+dtinallperiods,anypolicywithsubsidiess0tthatisfully…nancedthroughtaxationcanbeperfectlyreplicatedwithanalternativepolicywithsubsidiessts0tthatis…nancedpartlythroughtaxationandpartlythroughdebt.Equations(42)and(43)alsoshowthat,likebubblesandsubsidies,theintroductionofdebt…nancinghascon‡ictinge¤ectsonintermediationandcapitalaccumulation.30 Allelseequals,debtissuedinthepastrequirestaxationanditreducesthesupplyoffundsavailableforinvestment.ThisiswhydtenterswithanegativesigninEquation(42).Debtissuedinthepresent,though,makesitpossibletoreducetaxesonthecurrentgenerationofentrepreneurstherebyenablingthemtoexpandtheirdemandforcredit.ThisiswhymGt+1enterswithapositivesigninEquation(43).Nowpolicyconsistsofasequenceforthetripletxt;mSt;mGt .Weadopttheconventionthats�1=d�1=0sothats0andd0isdeterminedbyx0;mS0andmD0.Wethenproposeajointstochasticprocessforbubblereturnandbubblecreationshocks:fut;ntg,withEtut+1=0,bt0andnt0forallt.Withthisprocessathand,wesearchforasequenceforthestatevariables(kt;bt;st;dt)thatsatis…eskt0aswellasEquations(42)-(46).Ifsuchasequenceexists,wesaythattheproposedstochasticprocessisanequilibrium.Ifsuchasequencedoesnotexist,wesaythattheproposedstochasticprocessisnotanequilibrium.4.3Aretherelimitstodebt…nancing?AquickanalysisofEquations(42)-(46)suggeststhatdebt…nancingcanhelpachievepoliciesthatwouldnotbefeasibleotherwiseduetothe…scalconstraintthatxt.Whatensures,though,thatthegovernmentcanissuepublicdebtcontracts?Thatis,whatarethelimitstodebt…nancing?Thisquestionisrelevantbecauseanimportantdi¤erencebetweentaxesanddebtisthatpayingtaxesismandatorywhilepurchasingdebtisnot.Inanyperiodt,thedebtissuedbytheCMAisbackedbytaxesbutalsobytheexpectationofnewdebtthatwillbeissuedinperiodt+1.Ifthisexpectationchanges,somightthewillingnessofsaverstoholddebt.Thus,debt…nancingseemssubjecttochangesininvestorsentimentthatmightlimittheCMA’sabilitytomanagetheeconomy’scollateral.Inshort,publiccreditmightbejustasbubblyasprivatecredit.Toseewhetherthisconcernisjusti…ed,wefocusonthecaseinwhichtheCMAusesthepolicyrulederivedinSection3.3tointermediateallwageswhilesettingEtRt+1=�1forallt.Weassumethatissmall,sothatdebt…nancingisisrequiredtofundthedesiredsubsidies.Inequilibrium,themostthattheCMAcanpromisetorepayattimet+1isgivenbyEtdt+1+

EtfEt+1dt+2g,i.e.expecteddebtpaymentscannotexceedtheexpectedresourcesobtainedthroughtaxationandthroughtheissueofnewdebt.ButtheexpecteddebtpaymentsthattheCMAcanmakeattime31 t+2arethemselveslimitedbyexpectedtaxationandthesaleofnewdebtatt+3,andsoon.Iteratingthisprocessforwardweobtainanupperboundfordebtpayments:Etdt+1
1Xj=0(
)j+limT!1h(
)T
Etdt+Ti.(47)Equation(47)providesanexpressionfortheCMA’s…scalbackstop.Itsaysthatexpecteddebtpaymentsarebackedbyfundamentalresources,whichcorrespondtothenetpresentvalueoftaxes,andbybubblyresources,whichcorrespondtothepublicdebtthatisexpectedtoberolledoverforever.Thekeyaspectofthisexpressionisthat,since
�1,thefundamentalresourcesoftheCMAarein…niteforanypositivevalueof,regardlessofhowsmallitis.Thus,eveniftheCMAmustissuedebtto…nanceitspolicies,itcanalwaysdosowithoutbeingexposedtoroll-overcrises:thereasonisthat,nomatterhowlargedebtpaymentsare,theyarealwaysfullybackedbyfuturetaxes.Inthebubblyeconomy,the…scalbackstopoftheCMAisunlimited!Theintuitionbehindthisresultisquitenatural.TheCMA’s…scalbackstopincludesallfuturetaxes,includingthosethatwillbepaidbygenerationsthatareyettobeborn.Thesetaxrevenuesgrowwiththeeconomy.Whencollateralisscarceandoutputisbelowpotential,though,thisgrowthrateishigherthanthemarketinterestrateandthenetpresentvalueoftaxrevenuesisunbounded.Thus,thesameconditionthatmakesbubblyepisodespossibleguaranteethat,eveniftheCMAmustissuedebtto…nanceitspolicies,thisdebtisnotpronetoarollovercrisisbecauseitisfullybackedbytaxrevenues.5Whathavewelearned?Sowhatdowetakehomefromthispaper?Creditboomsandbustsareafactoflifeinmoderneconomies.Thereisawidespreadviewamongmacroeconomiststhat‡uctuationsincollateralareanimportantdriverofthese‡uctuationsincredit.Thispaperbuildsonthisviewandmakesthefollowingobservations:1.Economieswithbindingborrowingconstraintshavetwotypesofcollateral,fundamentalandbubbly.Fluctuationsinbothtypesofcollateralgenerateboom-bustcyclesincredit,investmentandgrowth.These‡uctuationsmightbedrivenbytraditionalorfundamentalshocks,butalsobychangesininvestorsentimentormarketexpectations.Bothtypesofshocka¤ecttheamountsoffundamentalandbubblycollateral.32 2.Bubblycollateralraisesequilibriumcredit(“crowding-in”)butdivertspartofthiscreditawayfrominvestment(“crowding-out”).Whenbubblesaresmall,thecrowding-ine¤ectdominatesandinvestmentandoutputincrease.Whenbubblesarelarge,thecrowdingoute¤ectdominatesandinvestmentandoutputarelow.Thereisthereforean“optimal”bubblesizethattradeso¤thesetwoe¤ectstomaximizelong-termoutputandconsumption.3.Marketsaregenericallyunabletoprovidetheoptimalamountofbubblycollateral,butacreditmanagementagencycanreplicatethe“optimal”bubbleallocationbytaxingcreditwhenbubblycollateralisexcessiveandsubsidizingitwhenitisinsu¢cient.Suchapolicycanbecharacterizedasleaning-against-the-wind,butnotasapolicyofpreventingbubbles.Itcannotbecharacterizedasabailoutpolicyeither,sincethecredit-marketinterventionsthatarerequiredtoimplementitpayforthemselves.Tosomeextent,wearguethatthispolicycanbeinterpretedasaschemeinwhichthecreditmanagementagencypurchasesassetstomanagetheeconomy’sstockofcollateral.Theseresultsprovideacoherentandrichviewofcreditboomsandbusts,inwhichbothfunda-mentalandbubblycollateralplayakeyrole.Theyalsoprovideausefulblueprinttoguidepolicyindealingwithcreditbubbles.Butthetheoryhaslimitationsandthereismuchworktobedone.A…rstlimitationofthemodelisthatitdepictssituationsinwhichonlythetotalamountofbubblycollateralmatters.Buttherearesituationsinwhichitisnotjusttheamount,butalsothetypeofbubblycollateralthatisimporttant.Aneconomywith…nancialintermediariesthatcollectdepositsandlendtoentrepreneurs,forinstance,willrequiredi¤erenttypesofbubblycollateraltobackdepositandloancontracts.Aneconomyinwhichsaversdi¤eraccordingtotheirriskaversion,inturn,mightrequiredi¤erenttypesofcollateraltobacksafeandriskycreditcontracts.Inbothcases,theremaybetoomuchcollateralofonetypebuttoolittlecollateraloftheothertype.Thus,toimproveonmarketoutcomes,theCMAmustsimultaneouslysubsidizesometypesofcreditwhiletaxingothers.16Asecondlimitationofthemodelisthatitconsidersaclosedeconomy.Butoneofthecharac-teristicsofrecentcreditboomsandbustsisthattheyhavebeenassociatedwithsurgesandsuddenstopsincapital‡owstogetherwithlarge‡uctuationsinrealexchangerates.Itwouldbeinter-estingtodevelopanopen-economyversionofthemodelthatallowsustoinvestigatetheroleofcapital‡owsandexchangeratesinshapingthesecreditboomsandbusts.Moreover,thisextension 16Wereferthereadertoanonlineappendixforamoreformaltreatmentoftheseissues.33 wouldallowsusalsotostudytheinternationaltransmissionofcreditbubblesandtheinternationalcoordinationofcreditmanagementpoliciesliketheoneswehaveanalyzedhere.17Athirdlimitationofthemodelisthateveryoneobservesthegapbetweentheoptimalbubbleandtheexistingone.Thus,theroleofpolicyissimplytobridgethisgap.Realityismorecomplicatedbecausemarketparticipantsandpolicymakersmaybeuncertainastowhether‡uctuationsaredrivenbyfundamentalorbubblycollateral.Introducingthistypeofuncertaintyisanimportantnextstepinthisresearchagenda.Afourthlimitationofthemodelisthatitessentiallytakesfundamentalcollateralasexogenous.Onemaythink,though,thatfundamentalcollateralis“produced”bythe…nancialsystemwhenitinvestsinthescreeningandmonitoringofentrepreneurs.Bubblycollateralprovidesaninexpensivealternativetofundamentalcollateralbecauseitissustainedbymarketexpectations.Thismayentailcosts,however:bydiscouraginginvestmentinfundamentalcollateral,bubblycollateralmaybeexpansionaryintheshortrunattheexpenseoflargerdownturnswhenbubblyepisodesend.Takingintoaccountthesenegativeincentivee¤ectsofbubbles,andanalyzinghowtheyimpactthedesignofpolicy,isanotherexcitingnextstepinthisresearchagenda.References[1]Aoki,K.andK.Nikolov(2011),“Bubbles,BanksandFinancialStability”,workingpaperCARF-F-253,TheUniversityofTokio.[2]Bernanke,B.andM.Gertler(1989),“AgencyCosts,NetWorthandBusinessFluctuations”,AmericanEconomicReview79,14-31.[3]Caballero,R.andA.Krishnamurthy(2006),“BubblesandCapitalFlowVolatility:CausesandRiskManagement”,JournalofMonetaryEconomics53(1),33-53.[4]Claessens,S.,A.KoseandM.Terrones(2011),“FinancialCycles:What?How?When?”,NBERInternationalSeminaronMacroeconomics,UniversityofChicagoPress,7(1),303-344.[5]Dang,T.V.,G.GortonandB.Holmstrom(2012),“Ignorance,DebtandFinancialCrises”,mimeo,Yale. 17Forsuchanextension,seeMartinandVentura(2015).34 [6]Dell’Ariccia,G.,D.Igan,L.LaevenandH.Tong,withB.BakkerandJ.Vandenbussche(2012),“PoliciesforMacro…nancialStability:HowtoDealwithCreditBooms”,IMFSta¤DiscussionNote,SDN12/06.[7]Dell’Ariccia,G.,D.IganandL.Laeven(2012),“CreditBoomsandLendingStandards:Evi-dencefromtheSubprimeMortgageMarket”,JournalofMoney,Credit,andBanking44(2-3),367-384..[8]Dell’Ariccia,G.andR.Marquez(2006),“LendongBoomsandLendingStandards”,TheJour-nalofFinance51(5),2511–2546.[9]Diba,B.andH.Grossman(1988),"TheTheoryofRationalBubblesinStockPrices,"Eco-nomicJournal,98(3),746-754.[10]Farhi,E.andJ.Tirole(2011),“BubblyLiquidity”,ReviewofEconomicStudies79,678-706.[11]Farhi,E.andJ.Tirole(2012),“CollectiveMoralHazard,MaturityMismatchandSystemicBailouts,”AmericanEconomicReview102,60-93.[12]Geerolf,F.(2013),“ReassessingDynamicE¢ciency”,mimeo,ToulouseSchoolofEconomics.[13]Gertler,M.,andN.Kiyotaki(2010),“FinancialIntermediationandCreditPolicyinBusinessCycleAnalysis”,inFriedmanandWoodford,HandbookofMonetaryEconomics,Vol.3A,Elsevier.[14]Gorton,G.andP.He(2008),“BankCreditCycles”,ReviewofEconomicStudies75(4),1181-1214.[15]Gorton,G.andG.Ordoñez(2014),“CollateralCrises”,AmericanEconomicReview104(2),343-378.[16]Gourinchas,P.O.,R.ValdesandO.Landerretche(2001),“LendingBooms:LatinAmericaandtheWorld”,Economia1(2),47-99.[17]Kiyotaki,N.,andJ.Moore(1997),“CreditCycles”,JournalofPoliticalEconomy105,211-248.[18]Kiyotaki,N.andJ.Moore(2008),“Liquidity,BusinessCyclesandMonetaryPolicy”,mimeo,PrincetonUniversity.35 [19]Kocherlakota,N.(2009),“BurstingBubbles:ConsequencesandCures”,mimeo,FederalRe-serveBankofMinneapolis.[20]Martin,A.(2008),“EndogenousCreditCycles”,mimeo,CREI.[21]Martin,A.andJ.Ventura(2011),“TheoreticalNotesonBubblesandtheCurrentCrisis”,IMFEconomicReview59,6-40.[22]Martin,A.andJ.Ventura(2012),“EconomicGrowthwithBubbles”,AmericanEconomicReview102,3033-3058.[23]Martin,A.andJ.Ventura(2015),“InternationalTransmissionofCreditBubbles:TheoryandPolicy”,mimeo,CREI.[24]Matsuyama,K.(2007),“CreditTrapsandCreditCycles”,AmericanEconomicReview97(1),503-516.[25]Mendoza,E.andM.Terrones(2012),“AnAnatomyofCreditBoomsandtheirDemise”,WP18379,NBER.[26]Mian,A.andA.Su…(2009),“TheConsequencesofMortgageCreditExpansion:EvidencefromtheU.S.MortgageDefaultCrisis”,TheQuarterlyJournalofEconomics124(4),1449-1496.[27]Miao,J.andP.Wang(2011),“BubblesandCreditConstraints”,mimeo,BostonUniversity.[28]Rancière,R.,A.TornellandF.Westermann(2008),“SystemicCrisesandGrowth”,TheQuarterlyJournalofEconomics123(1),359-406.[29]Ruckes,M.(2004),“BankCompetitionandCreditStandards”,TheReviewofFinancialStud-ies17(4),1073-1102.[30]Samuelson,P.(1958),“AnExactConsumption-loanModelofInterestwithorwithouttheSocialContrivanceofMoney”,JournalofPoliticalEconomy66,467-482.[31]Schularick,M.andA.Taylor(2012),“CreditBoomsGoneBust:MonetaryPolicy,LeverageCyclesandFinancialCrises,1870-2008”,AmericanEconomicReview102(2),1029–1061.[32]Tirole,J.(1985),“AssetBubblesandOverlappingGenerations”,Econometrica53(6),1499-1528.36 [33]Ventura,J.(2011),“BubblesandCapitalFlows”,JournalofEconomicTheory147(2),738-758.37 79207940796079808000802080408060808081000.70.80.91InterestRate,EtRt+17900792079407960798080008020804080608080810000.020.04BubbleCreation,nt79007920794079607980800080208040806080808100!0.100.10.2Bubble,bt79007920794079607980800080208040806080808100!0.100.10.2PolicyIntervention,st7900792079407960798080008020804080608080810011.21.4Welfare,EtUt+1790079207940796079808000802080408060808081000.350.40.450.5Credit,lt7900792079407960798080008020804080608080810000.20.4BurstingProbability,!t790079207940796079808000802080408060808081001.522.5Productivity,AtWithoutPolicyWithPolicyWithoutPolicyWithPolicykt+1µ0.2970.337s.d.0.0270EtUt+1µ1.1361.202 InterestRate,EtRt+168606880690069206940696069807000702000.020.040.06BubbleCreation,nt68606880690069206940696069807000702000.51Bubble,bt686068806900692069406960698070007020456Productivity,At68606880690069206940696069807000702051015Welfare,