A Model of Charles Ponzi

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) A Model of Charles Ponzi Gadi Barlevy, Ines Xavier 2025-020 Please cite this paper as: Barlevy,Gadi,andInesXavier(2025). “AModelofCharlesPonzi,”FinanceandEconomics DiscussionSeries2025-020. Washington: BoardofGovernorsoftheFederalReserveSystem, https://doi.org/10.17016/FEDS.2025.020. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

A Model of Charles Ponzi∗ Gadi Barlevy Ineˆs Xavier FederalReserveBankofChicago FederalReserveBoard March 18, 2025 Abstract We develop a model of Ponzi schemes with asymmetric information to study Ponzifrauds. Along-livedagentofferstosaveonbehalfofshort-livedagentsata higherratethantheycanearnthemselves. Thelong-livedagentmaygenuinelyhave asuperiorsavingstechnology,butmaybeanimpostertryingtostealfromshort-lived agents. ThemodelidentifieswhenaPonzifraudcanoccurandwhatinterventions canpreventit. AkeyfeatureofPonzifraudsisthatthelong-livedagentbuildstrust overtimeandimprovestheirreputationbykeepingtheschemegoing. KeyWords: Ponzischeme,asymmetricinformation,reputation,fraud JELCodes: C73,D82,G51,K42,L14 ∗Theviewspresentedherearesolelythoseoftheauthorsanddonotnecessarilyrepresentthoseofthe FederalReserveBankofChicago,theFederalReserveBoard,ortheFederalReserveSystem. Wethank DavidAndolfatto,YairAntler,FernandoArce,MarcoBassetto,DanielBird,RussCooper,IanDew-Becker, BobHetzel,RyoJinnai,GuidoLorenzoni,GeorgeMailath,AlexMonge,KurtMitman,AlessandroPavan, ChrisPhelan,VilleRantala,ManuelSantos,OferSetty,DavidWeiss,NicolasWerquin,NoahWilliams,and ArielZetlin-Jonesforhelpfuldiscussionsandvariousseminarparticipantsatfortheircomments.

1 Introduction EconomistsdefineaPonzischemeasascenarioinwhichanagentborrowsfromothers and then keeps repaying their debt by taking out new debt rather than drawing on their ownresources. Forexample, agovernmentthatrolls over itsdebtwithoutevercollecting taxes, as in Diamond (1965), is said to run a Ponzi scheme. Likewise, a household that keeps taking on new debt to repay its existing debts is said to run a Ponzi scheme, and restrictionsonthetotalindebtednessofhouseholdsareusuallycalledno-Ponziconstraints. ExistingmodelsofPonzischemestypicallyassumesymmetricinformation. Examples includeO’ConnellandZeldes(1988),whostudyPonzischemesindeterministicsettings, andBlanchardandWeil (2001)andAbelandPanageas(2022), whostudyPonzischemes instochasticenvironmentswithequallyinformedagents. Inthesesymmetricinformation models,aPonzischemeistypicallyeithersustainableindefinitely,meaningtheagentcan keeprollingoverdebtasinDiamond(1965),oritcannottakeoffatall.1 In practice, there are many Ponzi schemes that take off even though they cannot be sustained indefinitely. These schemes, including the one hatchedby Charles Ponziafter whomtheseschemesarenamed,tendtoinvolveanelementoffraud: Theborrowerhides thefactthattheyareusingfundsfromnewinvestorstorepaypreviousinvestorsandinstead purports to be using the proceeds from some real investment activity. Indeed, the legal definitionofaPonzischemefocusesonmisrepresentationastowhattheinvestedfundsare usedfor. Frankel(2012)observesthatthesefrauds“appearwithmonotonousregularity” acrossbothtimeand space. Deason etal.(2015)andMarquet(2011) documenthundreds of Ponzi schemeprosecutions by the SECin the past twentyyears, while Springer (2020) documents more than a thousand over the past 60 years. The frequency of such frauds highlights the needfor a framework to analyze this phenomenon. Symmetric information models of Ponzi schemes, in which all agents know what the borrower is doing, are unsuitedforunderstandingPonzifraudsasdistinctfromsustainablePonzischemes. This paper proposes a model of Ponzi frauds based on private information. The operator of the Ponzi scheme claims that they can achieve higher returns for investors than those investorscan achieve on theirown. Investors know that the agent may bean imposterwhoispayingthemfromfundsraisedbynewinvestors,andmustchoosewhether to trust the investor with their wealth or proceed to save on their own. The imposter, in turn,chooseswhethertocontinuetheschemeorabscondwiththefunds. Wederiveconditionsunderwhichanimposterwithnoaccesstoahighreturntechnol- 1OneexceptiontothisdichotomyisBhattacharya(2003). Heconsidersasymmetricinformationsetting inwhichschemeparticipantscanpressurethegovernmenttobailthemoutbytaxingnon-participants. In practice,mostPonzischemesaretoosmalltoconcerngovernments. Evenduringthe1996-7AlbanianPonzi crisisthatinvolvednearlyhalfofGDP,Jarvis(2000)reportsthatthegovernmentresistedcallsforbailouts. 1

ogycanoperateaPonzischemeinequilibrium. ForaPonziequilibriumtooccur,investors must initially be relatively skeptical that the scheme is legitimate. If the scheme were convincingfromthestart,itwouldattractlargeinvestmentimmediatelyandanimposter wouldprefertostealthe fundsrightawayratherthantrytoraise enough fundstopayoff initialinvestorsinthehopeofattractingmoreinvestmentlater. Anotherkeyelementisa lowprobabilityofdetection. Sinceearlyinvestorsareskepticalaboutthelegitimacyofthe scheme,theymustbelieve theimposterwilllikelyrepaythemwithfunds raisedfromnew investors. Butthisrequiresthattheimposterisunlikelytobeexposedasafraudthrough investigations. Finally, we find that Ponzi schemes are harder to sustain if the scheme operatorpromisesinvestorssignificantlymorethan theycanearnontheir own. However, schemeswitharbitrarilyhighpromisedreturnscanbesustainedundersomeconditions. ThePonzifraudinourmodelmustcollapseinfinitetimegiventheamountagentscan investeachperiodisconstantwhiletheobligationsoftheschemeoperatorkeepgrowing. Thereason theschemedoesn’tunravelinthe finalperiodisthat theoperator’sreputation improves the longer the scheme lasts. Early investors are willing to invest despite their skepticismbecausetheyexpecttobepaidfromthefundsofnewinvestorsiftheschemeis indeeda fraud. Bycontrast, investors inthe finalperiod knowthat therewill beno oneto bailthemoutiftheschemeisfraud. Nevertheless,theyarewillingtoinvestbecausethey reasonthatiftheschemesurvivedaslongasitdid,theoperatorislikelytobegenuine. Beyond showing when a Ponzi fraud can occur in equilibrium, the model offers insightsonhowsuchfraudsunfold. Forexample,inourmodelthefraudislikelytolast longer the lower is the promised return, a prediction consistent with what we observe empirically. Our model also suggests which interventions are more likely to deter such frauds. For example, inourmodel educationis lesseffective thanenforcement inruling outPonziequilibria. Thepaperisorganizedasfollows. Theremainderofthissectionsummarizestherelated literature. Section 2reviews Charles Ponzi’soriginalscheme asaway ofmotivatingour modeling approach. Section 3 describes our model. Section 4 solves for the optimal behavior of agents and introduces the notion of a Ponzi equilibrium. Section 5 shows when Ponzi equilibria can be ruled out. Section 6 establishes when Ponzi equilibria exist. Section 7 derives results on the uniqueness of Ponzi equilibria and reports some comparative statics. Section 8 discusses the welfare implications of our model. We concludewithadiscussionofsomeissuesandpotentialgeneralizationsofourmodel. Related Literature OurworkisrelatedtoagrowingliteratureonPonzischemes. Severalpapershavesimilarly studiedPonzischemesfromatheoreticalperspective. EarlyworkbyO’ConnellandZeldes 2

(1988)consideredthepossibilityofPonzischemesindeterministicsettingswithsymmetric information. Theyshowedthatinfinitehorizonsanddynamicinefficiencywerenecessary for Ponzi schemes to arise in such environments. Blanchard and Weil (2001) allowed for uncertainty with symmetric information and showed that Ponzi schemes could also occurindynamicallyefficienteconomies. AbelandPanageas(2022)analyzedadifferent modelofsymmetricuncertaintyandfoundasimilarresult. Inallthreeofthesemodels, Ponzi schemes can either be sustained indefinitely or can be ruled out. Bhattacharya (2003) presented a Ponzi scheme with symmetric information that cannot be sustained indefinitely. He assumed that participants may be able to pressure the government to redistributeresourcesandmakewholethosewhowouldotherwiselosefromthescheme. Artzrouni(2009)looked athowtheamount leftinaPonzi fraudevolvesovertimeas a function of new investment inflows, withdrawals by previous investors, the promised returnoninvestments,andtheactualreturntheoperatorearns. Hisanalysistreatedtherate of investment and withdrawalsas givenrather that derivingthem from optimaldecisions. More generally, he focused on what happens assuming a Ponzi scheme exists without studying whether such a scheme can be an equilibrium. He also abstracted from the outflow of resources stolen by the scheme operator, which figures prominently in our analysis. Ontheempiricalfront,severalresearchershavestudiedPonzifrauds. Frankel(2012) examined why such schemes are so common and identified common characteristics of theirperpetratorsandvictims. Deasonetal.(2015)compileddataon376Ponzischemes prosecutedby theSEC between1988 and2012 andlooked athow theduration, amount invested,andfractionstolenvarywithstate-levelcharacteristics. Marquet(2011)compiled dataon329schemesbetween2002and2011fromvarioussources,andfoundthatthese schemeshavebecomemorefrequentovertime. Springer(2020)constructedadatasetof 1,359Ponzischemesbetween1960and2022andidentifiedsomekeytrendsamongthese schemes. More recent work has focused on Ponzi schemes associated with cryptocurrencies. Bartoletti et al. (2020) identify 184 Ponzi schemes coded as smart contracts on the Ethereum platform. Building on their work, Shuang et al. (2023) identify 512 such contracts. The code for these smart contracts is public and in some cases contained explicit comments explaining that the contract was a Ponzi scheme. Participants who entered these contracts could have figured out that they would only be repaid if others enteredthecontractsafterthem. Thisisincontrasttofraudulentschemesinwhichthefact that repayments come from newcomers remains hidden. Accordingly, the extent of these smartcontractsseemsmorelimitedthantheinvestmentinPonzifraudsthatearlierwork documented. A large share of these smart contracts never attracted any users. Most of 3

thesesmartcontractswerecreatedduringathreemonthperiodin2016. Thecontractsthat attractedinvestmentinvolvedstakesofafewhundreddollarsonaverage. Thissuggests such contracts may have been a passing fad written for fun rather than an attempt to commit fraud. That said, cryptographic platforms have also been used to implement Ponzifraudsdisguisedaslegitimateinvestments. Congetal.(2023)discussthe$2billion PlusTokenschemein2019. Springer(2020)citesseveralotherexamplesofcrypto-related Ponzischemesinherdata. OtherresearchershavefocusedonwhatwecanlearnfromparticularPonzischemes. Gurun, Stoffman and Yonker (2018) showed that the collapse of Madoff Investment Securities led to a reduction in assets under management with registered investment advisors in regions that had previously invested more with Madoff. This suggests the performance of schemes affects investment decisions, as is true in our model. Rantala (2019)lookedattheWincapitaPonzischemeinFinlandbetween2003and2008,focusing onwhichinvestorsbroughtin othersgiventhe commissionsofferedforbringing innew investors. Huangetal.(2021)exploredrelatedquestionsonwhichagentsrecruitedothers inlargescalePonzischemeinChinain2016thatdrewinover4800investors. Sinceour modeldoesnotallowagentstorecruit,wecannotrelateourmodeltothesefindings. Finally,ourmodelisrelatedtoadverse-selectionmodelsofreputationinwhichalonglivedagenthasanincentivetopretendtobeatypethatiscommittedtosome particular action. See Mailath and Samuelson (2015) for a comprehensive survey. Our particular modelissimilartoWiseman(2009)andHu(2014)inassumingexogenousinformation that ensures the long-lived agent’s type will be revealed asymptotically almost surely. However, agents in those papers would like to maintain a good reputation indefinitely, while agents in our model have no reason to pretend to be the good type after building enough of a reputation to attract the maximal amount of investment. Our model also featuresastatevariablebeyondthelong-livedagent’sreputation,namelytheobligationto previoussavers. CelentaniandPesendorfer(1996)andBoardandMeyer-terVehn(2013) alsofeaturenon-reputationalstatevariables,althoughtheyarequalitativelydifferentfrom ours. TheclosestpapersinthisliteraturetooursarePhelan(2006)andAmadorandPhelan (2021). Thesepapersconsideredalong-livedagentwhoswitchesexogenouslybetween anopportunistictype(akintoourimposter)andacommitmenttype(akintoourgenuine type). Inourmodel,the long-livedagent’stypeis fixed. However,our Ponziequilibrium hasa similarstructureto theequilibriumin thesemodels: Whenthe long-lived agenthas a bad reputation, theopportunistic type will havean incentive to mixbetween pretending to be the commitment type and revealing it is opportunistic. It keeps doing so until its reputationishighenough,atwhichpointitwillactopportunisticallyandrevealitstype. 4

However,theproblemfacedbytheopportunistictypeisdifferentinourmodel. Phelan (2006)studiedstatic decisions,whileAmador andPhelan(2021)consideredone-period debtwheretheborrowerisalwayssolvent. Inourmodel,bycontrast,theborrowermust eventuallydefault. WhileourPonziequilibriumhasasimilarstructuretotheequilibrium theystudy,theanalysisofwhetherandwhenaPonziequilibriumexistshasnoanalogin theirwork. 2 Historical Context: Ponzi’s Original Scheme To motivate our modeling framework, we turn to Charles Ponzi’s original scheme thatlentthesefraudstheirname. WhilePonzididnotoriginatesuchfrauds,norwashis thelargestoperationofitskind,Ponzi’sschemeiswelldocumentedanddisplaysseveral featurescommontomanyoftheseschemes. OurdescriptionisbasedonZuckoff(2005). In1919,CharlesPonzi–anItalianimmigrantlivinginBostonatthetime–stumbled uponapotentialarbitrageopportunityinvolvinginternationalreplycouponsthatpeople couldbuyoverseasandsendtotheircorrespondentsintheUStotradeforpostage. Ponzi realized that purchasing international reply coupons in Italy and exchanging them for stampsintheUSwascheaperthanbuyingthesameamountofpostageintheUS.Givenhis negativepreviousexperiencewithbanksandconcernedthatbankersmightstealhisidea, Ponzi decided to raise funds from private investors in order to purchase reply coupons. Ponzipromisedinvestorsafixed50%returnontheirinvestmentwithin90days. While Ponzi was quick to raise funds, he was unable to figure outhow to profitably scale his operation. Profiting from discrepancies in postage prices required purchasing couponsinbulkinItaly,bringingthembacktotheUS,exchangingthemforpostage,and thensellingthepostageforcash. Ponzi’sinquirywithpostalofficialsaboutexchanging replycouponsdirectlyforcashwasrejectedoutofhand,andhewaswarnedbytheUS Postmaster’sofficethatitwasillegaltouseinternationalreplycouponsforspeculation. Alarmed by Ponzi’s operation after they learned aboutit, postalofficials moved to block him by pressuring several countries, including Italy, France, and Romania, to suspend salesofreplycouponsinApril1920. InJuly1920,the Postmasterfurther movedtolimit the amount of coupons an individual could redeem in the US at one time. Unable to scaleuphisoperation,Ponzibegantopayearlyinvestorswithfundsheraisedfromnew investors. AsPonzikeptamassinginvestmentandgainednotoriety,skepticsbegantoquestion hisclaims. Earlyon,thestatesupervisorofsmallloans,FrankPope,askedlocalpoliceto investigatePonzi. Thedetectivessenttoinvestigateweresufficiently impressed thatthey investedwithPonziandconvincedotherpoliceofficerstodothesame. Postalinspectors pressedPonzionhowhewasabletogenerateprofitswhenforeigncountriessupposedly 5

stopped issuing coupons, but he managed to evade their questions. Reporters started to investigatePonziafteroneofhisearlyinvestorssuedhimfor1milliondollars. ByJuly 1920, the Boston Post invited financial journalist Clarence Barron to evaluate Ponzi’s operation. Barronpointedouttheimpossibilityofscalingupinawaythatwouldsustain the payoffsPonzi waspromising his investors. ButPonzi was ableto continueto attract investmentbyarguingthatbankersweremerelytryingtoavoidhavingtosharetheirhigh returns with regular depositors. A New York Times article from July 29, 1920 quotes Ponziasfollows: Bankers andbusinessmen can easilyunderstand how Icould make 100 per centformyself,butsimplybecausenoone evermadean added50percent forthegeneralpublictheyreasonthatitcan’tbe. Youremembertheoldrube who saw the giraffe for the first time? He stared at it and remarked ‘There ain’tnosuchanimal.’ Thetruthis,bankersandbusinessmenhavebeendoing plentyforthemselvesunderthepresentbankingsystem,buttheyhavedone littleforanybodyelse. Iwanttochangethisunfaircondition. Thedepositor inthebankstodayisnotgettingasquaredeal... Yes,Iknowitisashockto some ofthese folks whohave beenhogging it all, butit isfair and right, and thedepositorshouldgetafairreturnforhismoney.2 Inearly August1920, Ponzi’spressagent, WilliamMcMasters, contactedtheBoston Post and offered to sell them information that Ponzi was insolvent. The article based on McMasters’ information led to a run on Ponzi’s company. At the same time, the MassachusettsBankCommissionerandAttorneyGeneralbothlaunchedinvestigations intoPonzi’scompany. BymidAugust,thePost reportedthatPonzihadbeenpreviously arrestedforfraudinCanada. Withhisreputationintattersandinvestigatorsclosinginon him,Ponzisurrenderedtoauthorities. ByNovember1920,hepledguiltytomailfraudin federalcourt. ThekeyelementsofPonzi’sschemewewishtoemphasizeare: (1)Ponzipresented himselftoinvestorsashavingalegitimateinvestmentopportunitythatallowedhimtooffer themahigherfixedreturnthantheycouldearnontheirown;(2)thefantasticreturnPonzi promised hisinvestors generated someskepticism and promptedinvestigations; (3) early investigationsthatwereunabletoestablishfraudwerefollowedbyevenmoreinvestment; and(4) acombinationofa tarnishedreputationandthe absenceofnewinvestment forced Ponzitodefault. Wedevelopamodelthataimstocapturethesefeatures. 2Availableathttps://www.nytimes.com/1920/07/29/archives/exchange-wizard-to-fight-bankers-ponzi-ofboston-promises-new.html. ThequotealsoappearsinZuckoff(2005),page209. 6

3 Model Consideraneconomywithoneinfinitely-livedagentandasuccessionofoverlapping generationsofshort-livedagents,eachofwhomlivefortwoperiods. Timeisdiscreteandstartsatt =0. Eachperiod,amass1ofshort-livedagentsisborn. Theseagentsonlycareaboutconsumptionwhenold,i.e.,theutilityofthecohortbornat datet overconsumptionc y whenyoungandconsumptionco whenoldisgivenby t t+1 u(c y ,co )=co (1) t t+1 t+1 Short-livedagentsareendowedwithygoodswhenyoungandnothingwhenold,sotheir concernistosaveforoldage. Theyhaveaccesstoasavingstechnologythatyieldsagross returnof1+R >1. Sincethe returnon investmentexceedsthe (zero)growthrate ofthe L endowment,weknowfromO’ConnellandZeldes(1988)thattheseagentswouldnotbe abletosustainanequilibriumPonzischemeamongthemselves. Thereisalsoonelong-livedagentwhoofferstosaveonbehalfoftheshort-livedagents andpaythemarateR >R . Letx denotetheamountoffundsthat(young)short-lived H L t agents invest with the long-lived agent at date t. They will save the remaining amount y−x ontheirown,earningthemareturnofR perunitsaved. t L Thelong-livedagentcanassumeoneoftwotypes,genuineandimposter. Agenuine long-livedagenthasaccesstoahighreturntechnologyandanincentivetopayshort-lived investors the promised return R . We model a genuine agent as a commitment type H without explicitly modelling their motive to offer a fixed rate of R . One could model H this motive, but we prefer to focus on the behavior of an imposter who wants to mimic thegenuinetypewithouttheadditionaldistractionofwhythegenuinetypebehavesasit does.3 AnimpostertypehasaccesstothesamereturnR thatshort-livedagentscanearnon L theirown. Whilethisimpliesthere isnoscopefor gainsfromtrade between short-lived agentsandanimposter,thelattermightstillwanttoattractinvestmentfromshort-lived agentsinordertostealitanduseittofinancetheirpersonalconsumption. Formally, let θ denote the long-lived agent’s type, where θ ∈{genuine,imposter}. Thelong-livedagentisgenuinewithprobabilityφ ∈(0,1),i.e., Pr(θ =genuine)=φ (2) Since we model the genuine agent as a commitment type, we only need to specify the 3OnewaytoendogenizewhythegenuinetypepaysafixedrateR istoassumeshort-livedagentshold H beliefsthatanyagentwhooffersareturndifferentfromR mustbeanimposter. SeeAmadorandPhelan H (2021)forarelateddiscussiononendogenizingthebehaviorofacommitmenttype. 7

preferencesandchoicesofthelong-livedagentasanimposter. Denotetheimposter’sconsumptionatdatet byc . Theimposterhasutility t ∞ U({c }∞ )= ∑β tc (3) t t=0 t t=0 whereweassumetheimposterisrelativelyimpatient,specifically, 1 β ≤ (4) 1+R H Assumption(4)impliesthattheimposterwouldnotwanttokeeppostponingconsumption unlessitgrewfasterthanR . Assumption(4)alsoimpliesβ < 1 ,meaningtheimposter H 1+R L willnotwanttodelayconsumptionforachancetosaveatrateR . L In terms of resources, let w denote the imposter’s wealth at the start of date t. We t assumetheimposterisendowedwithnoinitialwealth,i.e.,w =0. Wealsoassumethe 0 imposterearnsnoincomeandcanonlyobtainresourcesfromshort-livedagents. Finally,weassumethatonceapositivemeasureofshort-livedagentsinvestwiththe long-livedagent,thelong-livedagentwillbeinvestigatedforaslongastheirtyperemains uncertain. Thisismeanttocapturehow schemespromisinghighreturnsattractattention andscrutinyoncetheytakeoff. Formally,startingrightafterdatet =inf{t :x >0},there 0 t isa constantprobability µ ∈(0,1)per periodthat ifthe long-lived agentis animposter, theirtypewillberevealed. Theinvestigationprocessisasymmetric: Itcanconfirmthatthe long-livedagentisanimposterbutitcannotvalidatethatthelong-livedagentisgenuine.4 Inwhatfollows,weabstractfromthepossibilitythatanagentwillbepunishedifthey are revealed to be a fraud. Our results should carry over if we introduce an arbitrarily smallpenalty (ora penaltythat isapplied with arbitrarilysmall probability). Inpractice, a non-negligiblefractionofschemesthatwereprosecutedhavenorecordedsentence. Timing The timing of the model is as follows. At the start of date 0, young short-lived agents choosetheamountx toinvestwiththelong-livedagent. Thelong-livedagentmovesnext. 0 Agenuinetypewouldinvestx inthehigh-returntechnology. Animpostermustdecide 0 howmuchofx toconsumeandhowmuchtosaveatareturnR . 0 L The timing for any datet >0 is the same if there was no investment prior to datet. 4Ponziequilibriacanexistevenwhenµ=0.Theassumptionthatµ>0impliesthatwhilethelong-lived agent’styperemainsuncertain,theirreputationwillimproveovertimeregardlessoftheirdefaultstrategy. When µ =0, their reputation will still improve if the imposter defaults with positive probability every period. 8

That is, if x =···=x =0, young short-lived agents choose an amount x to invest 0 t−1 t withthelong-livedagent,afterwhichthelong-livedagentmoves. Afterthe firstperiodin whichinvestmentis positive, thetiming within eachperiod is abitmoreelaborate. Atthestartoftheperiod,thelong-livedagentisinvestigated. Ifthe long-livedagentisanimposter,theirtypewillberevealedwithprobability µ. Otherwise, nosignalwillbeproduced. Short-livedagentsobservewhetherthelong-livedagentwas exposedasanimposteratthestartoftheperiod(orinanypreviousperiod). Theycanalso observe whetherthe long-lived agentdefaulted in the past. However, they cannotobserve anything else about the long-lived agent, including their investments, consumption, or earnings. Atthispoint,short-livedagentsdecidehowmuchtoinvestwiththelong-lived agent. Aftershort-livedagentsinvestanamountx ,thelong-livedagentmoves. Agenuine t type would pay previous investors the amount (1+R )x and invest the new amount H t−1 x . Animpostertypemustdecidewhethertorepaytheirobligation,knowingthatdefault t wouldrevealthattheyareanimposter. Sincethelong-livedagentchoosestodefaultafter short-livedagentsinvestx ,defaultatdatet canonlyaffectinvestmentx ,x ,...after t t+1 t+2 datet. Animpostermust alsochoose how muchof the resourcestheyhave left atthe end oftheperiodtoallocatetoconsumptionc andhowmuchtosaveatrate1+R . t L 4 Optimality and Equilibrium Now that we have described the model, we can discuss what agents should do and defineanequilibriuminwhichallagentschooseoptimallygivenwhatotherschoose. We startwiththedecisionsofthelong-livedagentandthenmoveontoshort-livedagents. Optimal Decisions for the Imposter Since we model the genuine long-lived agent as a commitment type, we only need to solve for the decisions of the imposter. These decisions depend on what short-lived agentsknowaboutthelong-livedagent’stype. LetI denoteanindicatorthatequals1if t short-livedagentsknowthatthelong-livedagentisanimposterwhentheyinvestatdatet and 0 otherwise. Since thereare nogainsfrom tradebetween short-lived agentsand an imposterandtheimposterwouldstealanyfundstheyreceive,thenx =0fors≥0if t+s I =1. Given β(1+R )≤1 from (4), the fact that the imposter will receive no further t L investments regardless of what they do means they have no reason to delay consuming their wealth if I =1. The imposter’s utility in this case equals the wealth w they start t t withinperiodt. If I =0 and the imposter has yet to be revealed by date t, they can keep their type t hidden by repaying their obligation in the same way a genuine type would. However, they can only do this if their resources w +x are at least as large as their obligation t t 9

(1+R )x . Let S be an indicator that is equal to 1 if the long-lived agent is solvent, H t−1 t meaningtheycanrepaytheirobligationafterraisingtheinvestmentx : t (cid:40) 1 ifw +x ≥(1+R )x t t H t−1 S = (5) t 0 ifw +x <(1+R )x t t H t−1 IfS =0andthelong-livedagentisinsolvent,theyhavenochoicebuttorevealtheir t type by the end of date t. The imposter will thus attract no investment beyond date t regardlessofwhattheydo,andsoshoulddefaultontheirdebtandconsumeanyresources theyhaveattheendofdatet withoutdelay. Theirutilityinthiscasewillequalw +x . t t Thisleavesthecasewhere I =0andS =1,meaningthelong-livedagent’stypehas t t yet tobe revealedand theimposter is solvent. If theimposter repays theirobligation to the previous cohort (1+R )x in full, they would leave short-lived agents unsure if H t−1 the long-lived agent is an imposter or not. Since they would reveal their type by doing otherwise,theirchoiceamountstoeitherpayingtheirobligationinfullordefaultingand consumingimmediately,inwhichcasetheirutilitywouldbew +x . t t Defineanindicatord whered =1ifthesolventimposterdefaultsatdatet andd =0 t t t if they repay their obligation in full. We will use c to denote the solvent imposter’s t consumptionatdatet ifd =0,i.e.,ifthesolventimposterchoosesnottodefault. t LetV denote the imposter’s maximal utility as of datet if I =0 after receiving the t t investment x from short-lived agents. This utility depends on the imposter’s wealth w t t as well as whether they are solvent. As noted above, if S = 0, then V = w +x . If t t t t S =1,thepayoffV willdependontheinvestmentofshort-terminvestors. Thismeans t t (cid:0) (cid:1) V =V w ,S ;{x }∞ . To simplify the notation, we suppress these arguments and use t t t s s=0 V. t Before any investment takes place, i.e., at dates t <t ≡ min{t : x > 0}, there is 0 t nothing for the imposter to do given they have yet to raise any wealth. The non-trivial decisionsoccuronlyfort ≥t . Theimposter’sdecisionproblematdatest ≥t depends 0 0 ontheirobligationx toshort-livedagentsfromthepreviousperiod,wherewedefine t−1 x ≡0tocapturethefactthatthereisnooutstandingdebtatdate0. −1 Ifx =0,theimposterhasnoobligationtheycandefaulton. Theironlydecisionis t−1 howmuchtoconsumeandhowmuchtosave. Sincet ≥t ,theimposterknowsthatthey 0 willbeinvestigatednextperiod. Theirmaximizationproblematdatet isgivenby V =maxc +βµw +β(1−µ)V (6) t t t+1 t+1 ct 10

subjectto 1. w =(1+R )(w +x −c ) t+1 L t t t 2. w ≥0andc ≥0 t+1 t If x > 0, the imposter will also have to decide whether to default and reveal t−1 themselvesormaintainambiguityabouttheirtype. Thismaximizationproblemisgiven by V =maxd (w +x )+(1−d )[c +β(µw +(1−µ)V )] (7) t t t t t t t+1 t+1 ct,dt subjectto 1. w =(1+R )[w +x −(1+R )x −c ]whend =0 t+1 L t t H t−1 t t 2. w ≥0andc ≥0 t+1 t Inspecting (7), a necessary condition for the imposter to not default is that the continuationvalueV besufficientlyhigh. Theonlyreasonforan imposter tonotdefault t+1 immediatelygiventheirhighdegreeofimpatienceistheprospectofhighinvestmentat datet+1orlaterthatwouldallowthemtostealmore. In principle, more than one path for {c }∞ and {d }∞ may allow the imposter to t t=0 t t=1 attain the maximal possible utility that solves (6) and (7). This will indeed be true in equilibrium. If multiple paths are optimal, the impostershould be willing to play a mixed strategythatrandomizesoverpaths. Theimposter’sstrategyisthusaprobabilitydistributionπ overpathsfor{c }∞ and{d }∞ . Sinceamixedstrategyassignsprobabilitiesto t t=0 t t=1 bothpaths,theimpostermaycoordinatebetweenthetwodecisionswhentheyrandomize. For example, the imposter might randomize between a low value for c when d =0 t t+1 andahighvalueforc whend =1. Thatis,theimpostermaychoose tosaveatdatet t t+1 toavoiddefaultinperiodt+1butwouldprefernottodelayconsumptioniftheyintendto defaultnextperiod. Givenastrategyprofileπ,wecancomputetheconditionalprobabilitythattheimposter willdefaultinperiodt iftheirtyperemainsuncertainandtheyaresolvent,i.e., σ =Pr(d =1|I =0,S =1) (8) t t t t As we shall now see, the probability σ will be a key object that short-lived agents care t about. In short, the optimal decision for an imposter involves when to default and how much toconsumeorsavebeforehand. Impatiencewoulddictatethatanyfundsleftafterpaying previousinvestorsshouldbeconsumedratherthansaved. However,theagentmaychoose to save to keep the scheme going for longer. The imposter will consume any resources 11

theyhaveattheirdisposaloncetheydefault. Optimal Decisions for Short-Lived Investors We now turn to the decisions of short-lived agents. Each cohort must allocate its endowment y when young between saving on their own and investing with the long-lived agent. They make this decision after observing all previous investments, defaults, and investigationsofthelong-livedagent. Wecanexpressthispublichistoryatdatet as ht ={x ,...,x ,d ,...,d ,I ,...,I }≡{x ,d ,I } (9) 0 t−1 1 t−1 1 t t−1 t−1 t Short-lived agents willchoose whichever option offers themthe highest expected return. Savingyieldsareturnof1+R . Theexpectedreturnfrominvestingwiththelong-lived L agent depends on the probability that the long-lived agent is an imposter as well as the strategy π that the imposter chooses. Let Φ denote the probability that short-lived t investorsatdatet assigntothelong-livedagentbeinggenuinegivenpublichistoryand what they believe the imposter’s strategy to be, i.e., Φ = Pr(θ = genuine | ht,π). In t equilibrium,theirbeliefsabouttheimposter’sstrategycoincideswiththeimposter’sactual strategy. Short-lived investors at datet expect to earn 1+R if either the long-lived agent is H genuineorifthelong-livedagentisanimposterandtheimposter(i)issolventatdatet anddoesnotdefaultoninvestorsafterraisingx sothattheycangoontoraisefundsat t datet+1; (ii) is not exposed at the start of periodt+1; and (iii) is solvent at datet+1 anddoesnotdefaultafterraisingx . Iftheseconditionsarenotmet,thosewhoinvested t+1 atdatet receive0. Theexpectedreturnfrominvestingwiththelong-livedagentinperiod t isthus 1+R =[Φ +(1−Φ )S (1−σ )(1−µ)S (1−σ )](1+R ) (10) t t t t t t+1 t+1 H Short-livedagentsinperiodt arewillingtoinvestwiththelong-livedagentif 1+R ≥1+R t L Ifwedefinez≡ 1+R L,wecanrewritethisconditionas 1+R H Φ +(1−Φ )S (1−σ )(1−µ)S (1−σ )≥z t t t t t+1 t+1 12

Thetotalamountinvestedbyshort-livedinvestorswillthusbe  y ifΦ +(1−Φ )S (1−σ )(1−µ)S (1−σ )>z  t t t t t+1 t+1  x = anyx∈[0,y] ifΦ +(1−Φ )S (1−σ )(1−µ)S (1−σ )=z (11) t t t t t t+1 t+1   0 ifΦ +(1−Φ )S (1−σ )(1−µ)S (1−σ )<z t t t t t+1 t+1 Next, wespecifyhowshort-lived investorsupdateΦ . Ift <t =min{t :x >0},no t 0 t investigationwillbelaunchedatdatet+1. Thereisalsonoobligationforthelong-lived agenttodefaultonthatcanrevealitisanimposter. Short-livedagentswillthennotrevise theirbeliefsbetweendatest andt+1ift <t . Ifinsteadt ≥t ,thelong-livedagentwill 0 0 beinvestigatedinperiodt+1andcouldhavedefaultedinperiodt. Ifthelong-livedagent isrevealedasanimposterbeforeshort-livedagentsmaketheirinvestmentdecisionsatdate t+1,thenΦ =0. Otherwise,short-livedagentsshouldupdatetheirbeliefsaccording t+1 to Bayes rule. Since an imposter who is insolvent at date t would default at date t and revealtheirtype,I =0impliesS =1. Thelawofmotionforbeliefsisthusgivenby t+1 t    Φ t ift <t 0        Φ Φ = t ift ≥t andI =0 (12) t+1 0 t+1 Φ +(1−Φ )(1−σ )(1−µ)  t t t         0 ift ≥t andI =1 0 t+1 whereweadopttheconventionthatΦ =φ andσ =0. 0 0 Since µ >0,condition(12)impliesthatΦ >Φ ift ≥t andI =0. Eachperiod t+1 t 0 t+1 inwhichthe long-livedagentavoidsbeing exposedasanimposterconvinces short-lived agentstofavorablyrevisetheirlikelihoodoffacingagenuinetype. Animpostercanthus improvetheirreputationbykeepingtheschemegoing. Definition of Equilibrium Anequilibriumconsistsofadistributionπ overpathsoffunctions {c }∞ ,{d }∞ anda t t=0 t t=1 path of functions {x ,Φ }∞ that map public history ht into the relevant strategy space t t t=0 such that all agents choose their actions optimally given the strategy others play and short-livedagentsupdatetheirbeliefsinlinewithBayesrule. Formally,foralldatest ≥0, 1. Ifx =0,anyc thatisassignedpositiveprobabilityatdatet underπ solves(6) t−1 t 2. Ifx >0,any{c ,d }thatisassignedpositiveprobabilityatdatet underπ solves t−1 t t (7) 3. Investmentx satisfies(11) t 13

4. BeliefsΦ satisfy(12)giventheimposter’sstrategyπ t We will refer to an equilibrium as a Ponzi equilibrium if after the first datet for which 0 x >0,theshort-livedagentswhoinvestatdatet willberepaidwithpositiveprobability t 0 atdatet +1. Thatrequires(i)S =1sotheimposterissolventatdatet +1,and(ii) 0 t +1 0 0 theprobabilityofdefaultσ impliedbyπ isstrictlylessthan1. t +1 0 WewillsaythataPonziequilibriumcanlastT ≥2periodsifstartingfromthefirst datein whichshort-livedagents firstinvest,t =min{t :x >0},the agentswhoinvestin 0 t periodst ,...,t +T −2 willberepaidwithpositive probability, but theagentswhoinvest 0 0 inperiodt +T −1willforsurenotberepaidinperiodt +T. Thatis, 0 0 1. S =1for j=1,...,T −1 t +j 0 2. Theprobability∏ T j= − 1 1(1−σ t 0 +j )ofnodefaultbeforeT periodsispositive 3. EitherS =0orσ =1,sodefaultafterT periodsiscertain t +T t +T 0 0 AlthoughtheschemecanpotentiallylastT periods,itmightendearlieriftheimposteris exogenouslyexposedoriftheydefaultbeforeT periodsareup. 5 Non-Existence Results WebeginwithresultsonwhenPonziequilibriacanberuledout. Wefirstestablishan intermediateresultthatanimposterwillnotwaitmorethanoneperiodtodefaultifthey expecttheamountinvestedwiththemtoequalyinallperiods. Lemma1. Supposex =yforallt ≥0. Thenanimposterwoulddefaultinperiod1. t Intuitively,ifagentsalwaysinvesty,thereisnopointfortheimpostertowait: They willnotbeabletoraisemoreinvestmentbywaiting,andwaitingiscostlygivenimpatience, theriskofbeingexposedasanimposter,andthecostofkeepingtheschemegoing. AnimplicationofLemma1isthatPonziequilibriacanberuledoutwheneverφ >z, i.e., when the initial belief that the long-lived agent is genuine is sufficiently high. In period 0, short-lived agents expect to earn at least φ(1+R ) from investing with the H long-lived agent. When φ >z, this will exceed the 1+R they can earn on their own. L Youngagentsatdate0willthusinvestalloftheirendowmentywiththelong-livedagent. Since Φ is increasing int as long as the long-lived agent’s type is not revealed, young t agentswillcontinuetoinvestalloftheirendowmentwiththelong-livedagentuntilthey observeadefault. Lemma1thenimpliestheimposterwilldefaultinperiod1. Inshort, when φ >z, there will not be a Ponzi equilibrium in which the funds raised from new investors are used to pay the original investors. However, the equilibrium will feature fraud. Short-livedagentswhoinvestinperiods0and1willlosealloftheirinvestmentsin period1iftheyfaceanimposter. 14

Proposition 1. Suppose φ > z. Then a Ponzi equilibrium is not possible. The only equilibriumfeaturesx =yandx =yifI =0,i.e.,short-livedagentsinvestalloftheir 0 1 1 wealthwith thelong-lived agentunless thelong-lived agentis exposed. An imposterwill defaultinperiod1. Next, suppose φ = z. In this case, short-lived agents in period 0 might no longer strictlyprefertoinvestastheywouldwhenφ >z. GivenLemma1,aPonziequilibrium canonlyoccurifthefirstpositiveinvestmentx islessthany,i.e.,ifshort-livedagents t 0 are exactly indifferent between investing with the long-lived agent and saving on their ownwhen theyfirstinvestapositive amount. Theexpectedreturntoinvestinginperiod t isgivenby[φ+(1−φ)(1−µ)S (1−σ )](1+R ). Forshort-livedagentstobe 0 t +1 t +1 H 0 0 indifferent between investing with the infinitely-lived agent and saving on their own in thisperiod,thisexpressionmustequal1+R . Equatingthesetwoanddividingby1+R L H impliesthatwhenφ =z,indifferencewillonlyholdifeitherS =0or1−σ =0. t +1 t +1 0 0 There is thuszero probabilitythat short-lived agentswho investin period0willbe repaid. This again rules out the possibility of Ponzi equilibria while admitting equilibria with fraud. Proposition 2. Suppose φ = z. Then a Ponzi equilibrium is not possible. There exist equilibriainwhichthefirstpositiveinvestmentx <yfort =min{t :x >0}. However, t 0 t 0 intheseequilibria,theimposterwilldefaultinperiodt +1. 0 Finally, suppose φ <z. If µ is large, a long-lived agent who is an imposter is likely to be exposed as an imposter one period after the first positive investment at datet , in 0 whichcasethoseinvestorswillreceivenothing. Theexpectedreturnfrominvestingwith thelong-runagentforagentsatdatet willthenbelow,andshort-livedagentswillstrictly 0 prefertosaveontheirown. Inthiscase,therewillbenoPonziequilibrianorfraud. Proposition3. Supposeφ <z. ThenaPonziequilibriumisnotpossibleif µ ≥ 1−z. 1−φ To recap, when φ ≥ z, a Ponzi scheme cannot occur but outright fraud can: The likelihoodthattheinfinitely-livedagentisgenuine ishighenoughtoattractlargeamounts ofinvestment,anditwillnotbepossiblefortheinfinitely-livedagenttorepaythem. When φ <z,fraudmaybeavoidedifshort-livedagentsbelievethatanimposterislikelytobe revealedandwillnotbeabletoattractnewfunds. ForaPonziequilibriumtobepossible requiresamixofinitialskepticismaboutthelong-livedinvestorisgenuine(φ <z)and limitedinvestigativecapacity(µ < 1−z). Thefirstensuresagentsdonotinvestenmasse 1−φ early on, while the latter implies early investors expect that an imposter can still repay them. 15

6 Equilibria when φ ≤ z In theprevioussection, we establishedconditions thatrule out thepossibility ofPonzi equilibria. Wenowturntothequestionofwhatequilibriaarepossibleintheremaining cases. Proposition1establishesthatwhenφ >z,theuniqueequilibriumhasshort-lived agents invest all of their endowment with the long-lived agent unless they learn the long-livedagentisanimposter. Sothequestioniswhatequilibriaarepossiblewhenφ ≤z. Ourfirstresultconcernsthepossibilityofno-tradeequilibriawhenφ ≤z. Supposethat short-livedagentsdidnotinvestwiththelong-livedagentbeforeperiodt,andshort-lived agents at date t expect x =0. In this case, short-lived agents at date t know that the t+1 imposterwouldnothaveenoughresourcestopaytheminfullinperiodt+1. Theexpected returnfrominvestingwiththelong-livedagentisthusφ(1+R ). Ifφ ≤z,thiswillnot H exceed1+R ,sox =0willbeoptimal. Wesummarizethisinthefollowingproposition: L t Proposition 4. Suppose φ ≤ z. Then there exists an equilibrium in which there is no investmentatanydate. Inthisequilibrium,x =0andx (ht)=0forht ={x ,d ,I }= 0 t t−1 t−1 t {0,0,0}ift >0. Theequilibriumabovereliesonthefactthatagentsareinfinitesimalandcannotaffect x by deviating and investing. That is, if any single agent chose to deviate and invest at t datet,x wouldremainequalto0andnoinvestigationswouldbelaunchedinperiodt+1. t Next, we turn to the possibility of Ponzi equilibria. From the previous section, we already know that such equilibria can only arise when φ < z and µ < 1−z. Lemma 1 1−φ implies that Ponzi equilibria are not possible if x = y for all t, since in that case the t imposterwoulddefaultatdate1. TosustainaPonziequilibriumwheretheimposterdoes notdefaultimmediatelythusrequiresthatx <yinitially. t Our first observation is that when φ < z and µ < 1−z, any equilibrium in which 1−φ investmentx >0atsomedatet >0whenthelong-livedagent’stypeisuncertainmust t featurepositiveinvestmentstartingatdate0. Thatis,inanequilibriumwithinvestment, thedateofthefirstinvestmentt =min{t :x >0}ist =0. Formally: 0 t 0 Lemma2. Supposeφ <zandµ < 1−z. Thenifx (ht)>0inequilibriumforsomehistory 1−φ t ht ={x(cid:48) ,0,0} for somet >0, then investment must be positive all along that history, t−1 i.e.,eachentryinx(cid:48) mustbepositive. t−1 Inwords,ifagentsexpectsomeinvestmentnextperiod,itcannotbeanequilibrium fornobodytoinvestthisperiod. Thisisbecausethelong-livedagentwillbesolventand haveanincentivetonotdefaultwheninvestmenttodayispositivebutsmall. SincePonziequilibriaareonlypossiblewhenφ <zand µ < 1−z,Lemma2implies 1−φ thatPonziequilibriamustbeassociatedwithpositiveinvestmentfromdate0. Welookfor 16

Ponziequilibriainwhichx isstrictlybetween0andyuntiljustbeforethedateT atwhich t thePonziequilibriummustendindefault. Investmentx mustbepositiveatdateT −1 T−1 fordefaulttobepossibleatdateT. Lemma2thenimpliesthatx mustbestrictlypositive t for t =0,...,T −1. Imposing that x must also be below y makes it easier to sustain a t Ponziequilibrium, since thatcondition isnecessary forx >x to helpkeepthe scheme t+1 t going. NotethatthisconditionisnotneededatdateT −1,sincetheimposterwilldefault anywayatdateT. Wewillthereforeonlyrequirethatx <yfort =0,...,T −2. Wenow t deriveconditionsforaPonziequilibriumtosatisfythisconstraint. If0<x <yatsomet,short-livedagentsmustbeindifferentbetweensavingontheir t ownandinvestingwiththelong-livedagent. Fromequation(11),weknowthatshort-lived agentsatdatet areindifferentbetweenthetwoifandonlyif Φ +(1−Φ )S (1−σ )(1−µ)S (1−σ )=z (13) t t t t t+1 t+1 If a Ponzi equilibrium can last until date T, the imposter must be solvent through date T −1anddefaultwithprobabilitylessthan1beforedateT. Thatis,S =1andσ <1for t t allt =1,...,T −1. ThismeanswecanreplaceS andS in(13)with1fort ≤T −2. t t+1 AtdateT,theimpostermusteitherbeinsolvent(S =0)ordefaultwithprobability1 T despite being solvent (σ =1) to ensure the scheme cannot last beyond date T. Either T way,thecohortthatinvestswiththelong-livedagentatdateT −1expectstoearnareturn ofΦ (1+R ). Thiscohortwillbewillingtoinvestwiththelong-livedagentiff T−1 H Φ ≥z (14) T−1 TurningtothebeliefsofagentsinaPonziequilibrium,weknowthatshort-livedagentsat date0willrelyontheirpriorfortheprobabilitythatthelong-livedagentisanimposter, i.e., Φ =φ (15) 0 As long as the long-lived agent isn’t exposed as a fraud, short-lived agents will update theirbeliefsaccordingto(12),i.e., Φ t Φ +(1−Φ )(1−σ )(1−µ)= fort =0,...,T −1 (16) t t t Φ t+1 APonzi equilibrium thatlastsuntilperiod T mustsatisfycondition (13)ateachdatet in which0<x <y,togetherwithconditions(14),(15),and(16). Webeginwithwhatthese t equilibriumconditionsimplyaboutbeliefsΦ anddefaultprobabilitiesσ . t t 17

Equilibrium Beliefs Φ and Default Probabilities σ t t Considertheinfinitesystemofequationsin{Φ ,σ }∞ definedby(13)and(16)without t t t=0 imposinganyenddateT,i.e., Φ +(1−Φ )(1−σ )(1−µ)(1−σ )=z (17) t t t t+1 Φ t Φ +(1−Φ )(1−σ )(1−µ)= (18) t t t Φ t+1 fort =0,1,2,...,togetherwiththeinitialconditionsΦ =φ andσ =0. Weestablishthe 0 0 followinglemmaregardingthesolutiontothissystem: Lemma3. Supposeφ <zand µ < 1−z. ThenthereexistsaT∗ where2≤T∗ <∞such 1−φ thattheuniquesolution{Φ ,σ }∞ to(17)and(18)satisfies t t t=0 (i) Φ ∈[φ,z)fort =0,...,T∗−2 t (ii) σ ∈[0,1)fort =0,...,T∗−1 t (iii) Φ T∗−1 ≥zandσ T∗ ≥1 Also,Φ isincreasingint fort <T∗,i.e.,Φ >Φ fort =0,...,T∗−1. t t+1 t Animplicationof Lemma 3isthataPonzi equilibriuminwhich0<x <yuntiljust t beforetheschemecollapsesmustlastuntilthespecificdateT∗ definedinthelemma. Proposition5. Supposeφ <zand µ < 1−z. AnyPonziequilibriumthatsolves(13),(14), 1−φ (15),and(16)forsomeT musthaveT =T∗ whereT∗ isdefinedinLemma3. Essentially, if ascheme wereto lastuntil somedate T >T∗, thevalue ofσ att =T∗ t wouldhavetobeatleast1att =T∗. ButthentheschemewouldendbeforeT. Conversely, if a scheme were to end by some date T < T∗, then short-lived agents would assign probabilityΦ <zatt =T −1giventhatT −1≤T∗−2,inviolationof(14). t Lemma3alsoimpliesthatthepathforbeliefsΦ andtheprobabilityofdefaultσ are t t uniquely determined in any Ponzi equilibrium inwhich short-lived agents are indifferent about investing with the long-lived agent until just before when the scheme must end. Since this path has Φ T∗ >Φ T∗−1 ≥z, we know that short-lived agents will invest their entire endowment y with the long-lived agent at date T∗ (and thereafter) as long as the long-livedagentwasnotrevealedtobeanimposter. Lemma3furtherimpliesthatinanequilibriumwhereshort-livedagentsareindifferent, σ isstrictlybetween0and1indatest =1,...,T∗−1. Hence,forthistypeofequilibrium, t theimpostermustbeindifferentbetweendefaultingandrepayingthepreviousperiod’s investors at each of these dates. Whether the imposter should default or maintain the schemedependsonthepathofinvestmentx overtime,sinceitgovernswhattheimposter t 18

willreceiveiftheydefaultandwhattheywillreceiveiftheywait. Thisleadsustoexamine whatpathsforinvestmentwillkeeptheimposterindifferentaboutwhentheydefault. Equilibrium Paths for Investment {x }T∗ t t=0 InlookingforaPonziequilibriumwhereshort-livedagentsareindifferentuntiljustbefore the scheme must collapse, i.e., an equilibrium that solves (13), (14), (15), and (16) for some T, we showed that suchan equilibrium willbe associated witha unique value T∗ forT andauniquepathforbeliefsΦ anddefaultprobabilitiesσ fort =0,...,T∗. t t Per Lemma3, the unique pathfor beliefs willnecessarily haveΦ T∗ >z. Thismeans investment x T∗ mustequal y, sinceshort-lived agentswould expect toearn ahigher return from investing with the long-lived agent even if they expected the imposter to default with certainty. Lemma 3 further implies that Φ T∗−1 one period earlier can either equal z or strictly exceed z. If Φ T∗−1 >z, investment x T∗−1 at date T∗−1 must also equal y. If Φ T∗−1 =z, the value of investment x T∗−1 is indeterminate and can assume any value between 0 and y. In the latter case where x can be flexibly assigned, it is easy to T∗−1 constructapathfor{x }T∗−1 thatwillleavetheimposterindifferentaboutwhentodefault t t=0 aswellassolventuntildateT∗. ThisissummarizedinthenextProposition. Proposition6. Supposeφ <zand µ < 1−z. If 1−φ (cid:20) (cid:21)T∗−t β(1−µ) 2 x = y fort =0,...,T∗ (19) t 1+R H thentheimposterisalwayssolventbefore dateT∗ andwouldbeindifferentaboutwhich datet between 1 and T∗ to default on. Hence, if the solution to the system of equations (17)and(18)impliesΦ T∗−1 =z,aPonziequilibriumexists. To construct a Ponzi equilibrium, we set x (ht) equal to the value for x in (19) for t t allhistoriesht inwhichthelong-livedagent’stypeisuncertain,i.e.,whered =0and t−1 I =0. t The path for {x }T∗ in (19) implies investment grows at a constant rate x t+1 = t t=0 xt (cid:104) (cid:105)1/2 1+R H until x reaches y in period T∗. Since β(1+R ) ≤ 1, investment grows β(1−µ) t H atrate thatexceeds1+R . Newinvestmentx thusexceeds theobligation(1+R )x H t H t−1 tooldinvestorsateacht. Theimposterwillremainsolventevenwithoutsaving. We next confirm that the path in (19) leaves the imposter indifferent about when to default. If the imposter intends to default in periodt, they should not carry any wealth intodatet,i.e.,theyshouldsetw =0. Theirutilityasofdatet wouldthusbex . Ifthey t t wait to default in periodt+1, it will be optimal for them to carry over zero wealth into 19

datet onceagaingiventheycanrepaytheirobligation(1+R )x outofx . Theirutility H t−1 t as of datet would thus be x −(1+R )x +β(1−µ)x . Equating the payoff from t H t−1 t+1 defaultingatt anddefaultingatt+1implies β(1−µ) x = x (20) t−1 t+1 1+R H Thisconditionensuresthattheadditional(1+R )x animpostercanconsumeifthey H t−1 defaultatt exactlyoffsetstheutilityfromconsumingx withprobability1−µ ifthey t+1 (cid:104) (cid:105)1/2 defaultatt+1. Since(19)implies x t+1 = 1+R H then x t+1 = x t+1 xt = 1+R H . xt β(1−µ) x t−1 xt x t−1 β(1−µ) Alongthispath,theimposterwillconsumex indate0,thenx −(1+R )x atdates 0 t H t−1 t =1,...,T −1,andfinallyx =yatdateT iftheydonotdefault. Iftheimposterdefaults T atdatet <T,theywouldconsumex atthatdate. Iftheyareexogenouslyexposedatdate t t <T,theywouldreceivenonewinvestmentthatperiodandconsumenothing. Theinvestmentpathdefinedby(19) isnottheonlypaththatensurestheimposterwill bebothsolventandindifferentaboutwhentheydefault. Condition(20)definesasecond orderdifferenceequationinx ,whichrequirestwoboundaryconditionstodefineaunique t path. Whenx T∗−1 isindeterminate,theonlyboundaryconditionisx T∗ =y. Thatmeans thereisacontinuumofpathsindexedbyx thatallsatisfy(20). Supposeweset T∗−1 (cid:20) (cid:21)1/2 β(1−µ) x T∗−1 = y+ε 1+R H and then solve backwards x at t = 0,...,T∗−2 using (20). For ε = 0, the path for x t t satisfies the strictly inequality x >(1+R )x . By continuity, as long as |ε| is small, t H t−1 it will still be the case that x ≥(1+R )x . Thus, we can find additional equilibrium t H t−1 paths {x }T∗ beyond (19) which ensure the imposter does not need to save to remain t t=0 solventandwhichleavetheimposterindifferentaboutwhentodefault. Can we also find equilibria in which |ε| is large? Suppose we set x T∗−1 = y, the maximalvalueforx . Ifwesolvebackwardsusing(20),theresultingpathwillbe T∗−1 β(1−µ) x = y=x T∗−3 T∗−2 1+R H The imposter’s obligation at date T∗−2 equals (1+R )x , which exceeds new H T∗−3 investment x . To avoid default at date T∗−2, the imposter would have to save in T∗−2 period T∗−3. Condition (20) ensures the imposter is indifferent between defaulting at T∗−3andT∗−2assumingtheydonotsaveinperiodT∗−3. Ifthereisanequilibrium inwhichx =y,itwillsatisfyadifferentconditionthan(20). T∗−1 The condition for indifference when agents can save, which they must to remain 20

solvent when x = y, will be particularly important if the solution to (17) and (18) T∗−1 implies Φ T∗−1 >z rather than Φ T∗−1 =z as we have considered so far. In that case, we musthavex =y,andsothequestionbecomeswhetheraPonziequilibriumispossible T∗−1 atall,notjustinadditiontotheequilibriumin(19). Wenowturntothisscenario. Equilibrium Paths for Investment with x = y T∗−1 Condition (20) dictates when the imposter is indifferent between defaulting in periodt anddefaultinginperiodt+1ifx ≥(1+R )x . Using(20)tosubstituteinforx ,we t H t−1 t−1 canbesurethattheimposterisbothsolventatdatet andindifferentbetweendefaultingin periodt andperiodt+1whenever β(1−µ) x = x ifx ≥β(1−µ)x (21) t−1 t+1 t t+1 1+R H If x < (1+R )x , the imposter will need positive wealth w > 0 at date t to t H t−1 t delay default until t+1. If the imposter intends to default at date t, they should carry no wealth into period t (i.e., they should set w = 0) and then consume x . Let w t t t−1 denote the imposter’s wealth at the end of period t−1 before consuming. Since the imposterwillconsumew atdatet−1,theirutilityasoftheendofperiodt−1willbe t−1 w +β(1−µ)x . t−1 t Iftheimposterintendstodefaultatdatet+1insteadoft,theyshouldsavejustenough atdatet−1toremainsolventindatet. Thatis,theyshouldsave (1+R )x −x H t−1 t s = t−1 1+R L andconsumetherest atdatet−1. Atdatet,theywouldconsume(1+R )s iftheyare L t−1 exposed. Otherwise,theywouldrepaytheirdebtandwaittodefaultinperiodt+1. Their utilityasoftheendofperiodt−1wouldbe (w −s )+βµ(1+R )s +β(1−µ)·0+β2(1−µ)2x t−1 t−1 L t−1 t+1 Usingtheexpressions = (1+R H )x t−1 −xt andequatingthetwoutilitiesimplies t−1 1+R L 1−β(1+R ) x β(1−µ)(1+R )β(1−µ) L t L x = + x t−1 t+1 1−βµ(1+R )1+R 1−βµ(1+R ) 1+R L H L H (cid:18) (cid:19) x β(1−µ) t ≡ α +(1−α) x (22) t+1 1+R 1+R H H whereα = 1−β(1+R L ) isbetween0and1andindependentofR . Savingss mustbe 1−βµ(1+R L ) H t−1 21

positiveifx <(1+R )x . Substitutinginforx from(22),savingswillbepositiveif t H t−1 t−1 (1−α)(β(1−µ)x −x )>0 t+1 t Combining with (21), the condition that ensures the imposter will be indifferent about defaultingatdatet andwaitingtodefaultatt+1reducesto  β(1−µ)    1+R H x t+1 ifx t ≥β(1−µ)x t+1 x = (23) t−1    α (cid:16) 1+ x R t H (cid:17) +(1−α) β 1 ( + 1− R H µ) x t+1 ifx t <β(1−µ)x t+1 Thepaththatensurestheimposterisindifferentisthusstillcharacterizedbyasecond-order difference equation. Given x T∗ =y and a value for x T∗−1 , including but not limited to x =y,condition(23)definesauniquepaththatensurestheimposterwillbeindifferent T∗−1 aboutdefaulteveniftheyhavetosaveatdatet−1todelaydefault. The path in (23) ensures that the imposter is indifferent about default for all t. An equilibriumrequiresthattheimposterissolvent,i.e.,thattheimposter’swealthw at t−1 theendoft−1isenoughforthemtosavetheamounts = (1+R H )x t−1 −xt theyneedat t−1 1+R L datet. Ournextresultprovidesconditionsunderwhichthepath(23)withx =yisan T∗−1 equilibrium. Proposition7. Supposeφ <zand µ < 1 1 − − φ z. APonziequilibriumwithx T∗−1 =yexistsif T∗ =2 or if z≥z∗ for some cutoff z∗ <1. The path of investment in this equilibrium is givenby(23). Recallthatwhenthesolutiontothesystemofequations(17)and(18)impliesΦ T∗−1 > z, investment x will have to equal y at date T∗−1. In that case, the only candidate T∗−1 Ponziequilibriuminwhichshort-livedagentsareindifferentuntiljustbeforedateT isthe pathdefinedby(23)withterminalconditionsx T∗ =x T∗−1 =y. Proposition7statesthata PonziequilibriumwillindeedexistifeitherT∗ =2orifzissufficientlyhighandcloseto 1,i.e.,ifthepromisedreturn1+R isnottoolargerelativeto1+R . H L To see that a Ponzi equilibrium might not exist for low values of z when T∗ > 2, considerthecasewhereT∗ =3andx =x =y. From(23),theinvestmentsthatensure 2 3 theimposterwillbeindifferentaboutwhentheydefaultaregivenby β(1−µ) x = y 1 1+R H (cid:104) (cid:105) x = α + 1−α β(1−µ)y 0 (1+R H )2 1+R H AsweletR →∞,theinvestmentx thatwillkeeptheimposterindifferentwillconverge H 0 22

to0. Theamounttheimpostermustsaveatdate0isgivenby s = (1+R H )x 0 −x 1 = 1−α R H β(1−µ)y 0 1+R L 1+R L1+R H (1−α)β(1−µ) This amount converges to y>0 as R →∞. This will inevitably exceed x . 1+R L H 0 Hence,forlargevaluesofR ,theimpostercannotbebothsolventandindifferent. H Intuitively, the tension forthe existenceof Ponziequilibria isthat x mustsatisfy two 0 conditions. Fortheimpostertobeindifferentbetweendefaultingatdate1anddefaulting at date 2, thevalue ofx can’t betoo small: Waitingto defaultat date2is verytempting 0 given how large x is, so getting to consume x instead of saving it must be sufficiently 2 0 attractive. Atthesametime, x can’tbetoolargefortheimpostertobeabletocoveran 0 obligationof(1+R )x bysavingandusingthenewinflowsx . Fortheseforcesnotto H 0 1 beintensionrequiresthat1+R becloseto1+R ,implyingzwillbecloseto1. H L ThisexampledoesnotmeanPonzischemesareneverpossibleforlowvaluesofz. Our nextpropositionreinforcesthispointbyshowingthatforanyz∈(0,1),therearealways parametersthatensureΦ T∗−1 =zandhenceaPonziequilibriumexistsperProposition6. Proposition 8. For any z∈(0,1), there exist values of φ and µ that ensure Φ T∗−1 =z andthataPonziequilibriumexists. To summarize, when the long-lived agent has a low initial reputation (φ < z) and thereislimitedenforced(µ < 1−z),Ponziequilibriainwhichanimposterkeepsrepaying 1−φ earlier investors with the funds raised from new investors may be possible. Such Ponzi equilibriaare hardertosustainwhen thepromisedrepayment isgenerous(i.e., whenzis close to 0), but Ponzi equilibria can occur even when the promised return is arbitrarily large. Strongerenforcementthatraises µ caneliminatePonziequilibria,whileeducating peopletobemoreskepticalofsuchschemescannoteliminatesuchequilibriasolongasφ remainspositive. ThereasonPonzischemesendinfinitetimewithoutunravelingisthatthereputation ofthe long-lived agentrisesovertimetime ifthescheme keepsgoing. Earlyon, agents are willingto invest despite thebelief thatthe long-livedagent islikely an impostersince theystillexpecttoberepaidfromthefundsofnewinvestors. Thelastcohortthatinvests withanimposterknowstherewillnofundstorepaythemincaseoffraud. However,they reckonthelong-livedagentislikelytobegenuinegivenhowlongtheschemehaslasted. 7 Uniqueness and Comparative Statics Inthissection,weexaminewhetherPonziequilibria,whentheyexist,canbeuniquely characterized. Proposition 4 tells us that when a Ponzi equilibrium exists, a no-trade equilibriummustexistaswell. APonziequilibriumthuscannotbetheuniqueequilibrium 23

outcome. However,theremaystillbeauniquePonziequilibriumoutcome. Inthatcase, we can carry out comparative statics conditional on being in a Ponzi equilibrium. That cantell uswhat variation toexpectin dataon avarietyof Ponzischemes suchas inthe datasetscompiledinsomeoftheempiricalworkthatwediscussedintheIntroduction. Proposition 5establishes thatany Ponzi equilibriumthat solves (13),(14), (15), and (16)isassociatedwithauniqueterminaldateT∗ atwhichanimpostermustdefault. Such aPonziequilibriumisalsoassociatedwithauniquepathfortheprobabilityofdefaultσ at t eachdatet andauniquepathforthebeliefsΦ ofshort-livedagentsoverthelikelihoodof t facingagenuine investor isalsouniquelydetermined. Thepathof investments{x }T∗ is t t=0 uniquelydeterminedifthelatterpathimpliesΦ T∗−1 >z,sinceinthiscasex T∗ =x T∗−1 =y and {x t } t T = ∗ 0 can be solved backwards using (23). However, if Φ T∗−1 = z, the path of investments will notbe uniquelydetermined even thoughT∗,σ , andΦ are. In thatcase, t t there will bea continuum of differentequilibrium paths for how investment evolves over time. ThatleavesthequestionofwhetherthereexistadditionalPonziequilibriathatdonot solve(13),(14),(15),and(16). Ournextresultshowsthatiftheimposterissufficiently impatient,therewillbenosuchPonziequilibria. (cid:16) (cid:105) Proposition 9. There exists a β ∈ 0, 1 such that if 0 < β < β and there exists 1+R H a Ponzi equilibrium that lasts until date T, then 0 < x < y for t = 0,...,T −2 in that t equilibrium. In essence, a highly impatient agent will want to consume if investment attains its maximal possible value. However, even a highly impatient agent will not default when x <yiftheyexpectmuchlargerinvestmentinthefuture. Highimpatiencedoesnotrule t outPonziequilibria,butitwilllimitwhichPonziequilibriaarepossible. Imposing β < β, we can describe how the maximal date T in the unique Ponzi equilibriumvarieswithφ andz,atleastforvaluesof µ thataresufficientlycloseto0. Proposition10. Supposeφ <z. Ifµ issufficientlysmall,thenT∗inLemma3isdecreasing inφ andincreasinginz. Thatis,withminimaloversight,aPonzischemecanlastlonger ifthe long-livedagenthasa worse initial reputation (lowerφ) oroffers alowerrelative return(higherz). Intuitively,themaximaldurationofthePonzischemecorrespondstothetimeittakes beliefstorisefromtheirinitialvalueofφ tothevaluezthatdrawsinallshort-livedagents to invest and induces the long-lived agent to default. Reducing the starting point φ or increasingtheendpointzrequiresbeliefstogrowmorebeforetheschememustcollapse. However, the time it takes to travel from φ to z depends on the endogenous variable σ t 24

which governs the rate at which short-lived agents revised their beliefs. In principle, a lowerreputationoralessgenerousreturnshouldmakeinvestingwiththelong-livedagent less attractive, requiring the long-lived agent to default with lower probability to keep short-lived agents indifferent. In that case, making it another period would not be as informative,andsoshort-livedagentswouldnotrevisetheirbeliefsupwardsasmuch. It iseasytoshowthattheprobabilityofdefaultσ inthefirstperiodfallswhenwelowerφ 1 orincreasez. Butbecausetheexpectedpayofffrominvestingwiththelong-livedagentat eacht dependsonbothσ andσ ,wecannotprovethisfortheentirepathofσ exceptin t t+1 t thespecialcasewhere µ =0. Althoughouranalyticalresultonlyholdsinaneighborhood of µ =0,numericallyweconfirmedtheresultforallvalues µ < 1−z weconsidered. 1−φ SincetheproofofProposition10usesthefactthatalowerφ orhigherzisassociated with (weakly) lower σ , it follows that the Ponzi scheme not only can last longer but is t alsolesslikelytoendeachperiod. Ahigherzorlowerφ willthereforebeassociatedwith alongerlastingschemeonaverage. Thispredictionaccordswithdatafromprosecution recordsreportedinMarquet(2011)thatconfirmsschemespromisinghigherreturnsare shorteronaverage. ThispatternremainsevenafterexcludingtheMadoffscheme,which did not involve a particularly high return and lasted an exceptionally long period of multiple decades, in contrast to most schemes that last a few years at most.5 Marquet arguesthereasonforthispatternisthathigherpromisedpaymentsaremoredifficultto sustain. Our model offers a more nuanced take. In principle, we can sustain a scheme foraslongaswewantbystartingatasmallerinitialinvestmentthatallowsthescheme roomtogrow. However,imposterswhopromisehighreturnshaveagreaterincentiveto steal, which means they build more reputation when they survive. Such schemes draw large-scaleinvestmentearlier. 8 Welfare OurresultsimplythatwhenaPonziequilibriumexists,notradeisalsoanequilibrium. AnaturalquestiontoaskwhentherearemultipleequilibriaiswhethertheycanbePareto ranked. IssocietyworseoffunderaPonziequilibrium? Answering this question requires taking a stand on the welfare of the genuine type. Weassumethatthistypeisweaklybetteroffiftheyreceivefundingthaniftheydonot. Presumably,thereasonagenuinetypeiswillingtoofferareturnof1+R isthatithas H access to a technology that yields a return of at least 1+R and which they can scale H beyondtheirownendowment. Ifthatwerethecase,thelong-livedagentwouldweakly benefitfromtradewithshort-livedagents.6 Underthisassumption,wegetthefollowing 5Madoffpromisedclientsastablereturnratherthanahighreturn. Sinceagentsinourmodelarerisk neutral,itishardtouseourmodeltoanalyzethatparticularPonzioperation. 6Aseparatequestioniswhythecommitmenttypehastopaysaversmorethanwhattheseagentscanearn 25

result. Proposition11. IfthereexistsbothaPonziequilibriumandano-tradeequilibriumfor the same parameter values, then the Ponzi equilibrium Pareto dominates the no-trade equilibriumexante. Theargumentforthisresultisasfollows. Byassumption,thegenuinetypeisweakly better off with trade. The imposter type benefits, since they would consume nothing without trade but can consume at least x > 0 in the Ponzi equilibrium. As for short- 0 lived agents, they earn a return of 1+R in the no-trade equilibrium. Given that they L alwayshavetheoptiontosaveontheirown,theymustbeweaklybetterunderthePonzi equilibrium. Noagentisex-anteworseoffinthePonziequilibrium,whiletheimposteris strictlybetteroff. The reason that the Ponzi equilibrium is superior to the no-trade equilibrium is not because Ponzi schemes are welfare improving. Rather, it is because trade with the commitmenttypecreatessurplus. Along-livedimpostercancutintosomeofthissurplus bypassingthemselvesoffasgenuine. However,aPonzischemeisnotnecessarytoachieve gainsfromtrade. Aseverepenaltyforimposterswilldriveoutimpostersandallowagents togainfromtradingwithacommitmenttypewithoutrequiringaPonzischeme. NotethatourmodelalsoabstractsfromvariousfeaturesthatwouldimplythataPonzi fraud can destroy the surplus. With those features, it would no longer be the case that Ponzi equilibria are better than no trade. For example, investigations in our model do not consume resources. The fact that a Ponzi equilibrium triggers investigations would therefore not cut into the gains from trade with genuine long-lived agents. We also do notmodelentryforcommitmenttypes. Assuch,impostersinourmodeldonotdriveout productiveagentsthatcancreatesurplusbytradingwithshort-livedagents. ThekeytakeawayfromProposition11isnotthatPonzischemesshouldbeencouraged. Rather,itillustratesthatPonzischemesrequirenotonlyalowinitialreputationandlow enforcement, but alsoan inefficient initial allocation thatlets agents gain from trade. This iswhatallowsanimpostertotakeadvantageandprofitattheexpenseofsavers. 9 Discussion Weconcludewithseveralobservationsabouthowourmodelrelatesbothtoreal-world Ponzischemesandtothetheoreticalliteratureonbubblesandpyramidscams. ontheirown. Onepossibilitywealreadyalludedtoaboveisthatsaverscannotdistinguishthecommitment typeandhastoworryaboutthepossibilityofanimposter. Outsidersmightbeskepticalthatthelong-lived agentisgenuineiftheyofferlessthan1+R . Thismicrofoundation,alsoraisedbyAmadorandPhelan H (2021),doesnotexplainwhytherelevantrateis1+R asopposedtosomeothervalue. H 26

Theft versus Excessive Risk-Taking WhileCharlesPonzineveractuallyinvestedinthepostagearbitragehetoldhisinvestors he was undertaking, other Ponzi schemes began as a a genuine attempt to undertake an investmentthatfailedtopayoff,andtheschemeoperatorborrowedtopaypreviousdebt holders while trying to get the investment to pan out. Our model does not capture this scenario,althoughamodifiedversionofitcan.7 To explore this possibility, we consider an extension of our model in Appendix B thatallowsanimpostertoeitherstealfundsorinvesttheminariskytechnology. Inthis case, borrowing until an investment pays off can still be understood as a Ponzi fraud: An imposterpoolswithacommitmenttypethatissafetoinvestwithanddoesnotdiscloseto its lenders that they are in fact undertaking a risky investment. If lenders knew that the investmentwasrisky,theywouldnotfundit. The detailed analysis of this case is in Appendix B. Here, we sketch out the main ideas. Suppose that each period, the imposter can consume, save at the risk-free rate 1+R , or undertake a risky investment that yields a return of 1+R with probability λ L and0withprobability1−λ. Short-livedagentscannotobservewhattheimposterdoes with their funds. We impose parameter restrictions such that if agents knew they were facing the imposter, they would refuse to trade. The interest rate they would require to let the long-lived agent invest in the risky technology would be high given the risk of getting nothing. At such a high interest rate, the imposter would prefer to steal rather thaninvest inthe riskytechnology. Thus, short-livedagents wouldrefuse totradewith a long-livedagentiftheyrevealthattheyareanimposter. Iftheimposterinsteadpoolswith the commitment type and offers to pay 1+R , our parametric assumptions ensure that H they will have an incentive to undertake the risky investment even if they pay investors 1+R iftheysucceed. H Wepresentanumericalexampleofanequilibriuminwhichanimposterusesthefunds theyreceive tooperateariskytechnology. Ifthe technology paysoff,short-livedagents observe the success and can force repayment of 1+R . The scheme then ends. If the H riskyinvestmentpaysnothing,theimposterborrowstopayoffpreviousinvestorsanduses theresttooperatetheriskytechnology. Incontrasttoourbenchmarkmodel,wherethe imposter intends to steal and will default on some agent, here there is a possibility that all agents will be repaid in full. Nevertheless, the imposter hides the fact that they are undertakingriskyinvestments,sinceshort-livedagentswouldrefusetoinvestwiththemif 7Springer(2020)distinguishesbetweenintentionalPonzischemesthatarefraudsfromthestartand unintentionalschemeswheregenuineinvestorshitbyashockborrowtocovertheirshortfallsuntilthey recover. Shefindsthatamongthe1,359schemesinherdatasetbasedonprosecutedcases, 1,225were intentionalanddidnotinvolveactualinvestment. Thismayreflectthatunintentionalschemesarelesslikely tobeprosecuted. However,outrightfraudwithnoactualinvestmentasinourmodelappearsquitecommon. 27

theyknew. Thefactthat repaymentscomefrom new investment isconcealedfrom agents inequilibrium. How Ponzi Schemes End CharlesPonzi’sschemecollapsedwhentoomanyofhisinvestorsdemandedrepayment. No such run occurs in our model. In the Ponzi equilibrium of our model, the investor is willing to wait to default only because they expect more investment in the future. Equilibrium investment must thus increase over time. Default occurs when investment peaks,notwhenitfalls. Whatforcestheschemeinourmodeltocollapseisthatinvestment stopsgrowingratherthanthatinvestmentstartstofall. The notion that a Ponzi scheme must collapse when investment stops growing as opposed to when it starts to decline is consistent with some historical episodes. For example,Madoff’sschemecollapsedwheninsufficientgrowthinnewinvestmentforced Madoff to admit his transgressions to his sons, who then turned him in to authorities. BeforeMadoff’stwosonsturnedhimin,Madofftriedtogiveoutsomeofthefundsthat remainedtohisfriendsandfamily,consistentwiththedynamicsinourmodel. The reason our model cannot generate a run is that it lacks two elements that seem to be essential for a run to occur. First, the imposter must have a reason not to default immediately when investment declines. The scenario in Appendix B, where the agent is waiting for investment to pan out, may create such an incentive. Alternatively, the endowment y might be random, and the imposter might prefer to wait to see if it rises again. Second,therehastobeareasonforinvestmenttodeclinegraduallyratherstopall atonce. Onepossibilityisthesametemporaryfallintheendowmentythatmayleadto lower investment. Alternatively, short-lived agents may receive a less stark signal than weassume,causingthemtolower Φ butnottostopinvestingaltogether. Weleavesuch t modellingforfuturework. Asymmetric Information versus Naivety Another feature of Ponzi frauds that is arguably missing in our model is the gullibility and naivety of some investors. The agents who invest with the long-lived agent in our model are fully aware of the risks they face. Indeed, short-lived agents are essentially co-conspirators,willingtoinvestwithanagenttheybelieveisquitelikelytobeanimposter becausetheyknowthatinthiscasetheymightstillbepaidbytheinvestmentofsubsequent cohorts. Theevidencesuggeststhatbothtypesofforcesplayarole. Frankel(2012)surveysthe evidenceonvictimsofPonzischemesandconcludes 28

This research on the attitudes of Ponzi scheme victims suggests that they are driven by twostrong tendencies, which render them morevulnerable to the lure of the schemes. One powerful tendency is the drive to trust; it is a tendency that borders on gullibility. The other tendency is ‘heightened risk tolerance.’ ” (p137) She cites studies which find that victims of these schemes tend to more educated than non-victimsonaverage,althoughthismaybeduetoselectionifwealthiertargetsaremore attractive to scammersandalso tendtobe moreeducated. Shealsoarguesthatinvestors insuchschemesoftendemonstrategreed,alackofempathy,andawillingnesstoinvestin schemestheyfindsuspicious. AniceillustrationofthiscomesfromworkbyBakerand Faulkner (2003) who surveyed participants in a scheme involving the California-based Fountain Oil and Gas Company in the late 1980s. This scheme involved separate jointventuresinspecificwells. Anassistantdistrictattorneyinvolvedinthecaseexplainsthe Ponzinatureoftheschemeinthearticle: “[M]oneyhadcomeinforinvestorstobeused for a specific well. In some cases it was diverted to other wells, or other expenses of Fountain, or to specific personal purchases.” While Fountain pressured its investors to refertheirventuretoothers, only24%ofrespondentsdidso, andeachonlymadeasingle referral. 31%ofrespondentsexplainedthisreluctanceasduetotheinvestmentbeingtoo risky. 23%ofrespondentssaidtheydidn’ttrustFountain. Onerespondentexplainedthat “Iwas always alittlesuspicious alltheway along. Theseguys seemedalittle slimy... My greedandtheirsmoothsuperficialsuccessfulexteriorovercamemysuspicions.” (p1194) Leuzetal.(2023)foundsimilarresultsforpump-and-dumpschemesinGermanythat involvedtoutingastockinordertoselltoeagerinvestorsatahighprice. Specifically,they found thatthere appears tobe an investortype thatinvests in theseschemes early onand tendstoearnhigherreturnsthanothertypes. Theyinterpretthispatternasevidencethat someagentsunderstandtheriskoftheseschemesandinvestinthemdespitethepotential lossesinherenttosuchascheme. Animportantfeatureofourmodelinwhichagentsareskepticalratherthangullibleis itsimplicationthatagentswouldadjusttheirinvestmentinresponsetoinformation. Here, thereisbroadconsensusintheliteraturethatpositiveactionsbyPonzischemeoperators arekeytoattractingadditionalinvestors. Frankel(2012)writes Timelypaymentsatshortintervalshelpestablishareputationfortrustworthiness. Eachpaymentbringsaddedproofoftheconartist’scredibility. (p39) Victimsofparticular Ponzi schemesreportinsurveysthat thefactanoperationhad been runningforawhilehelpedthemovercometheirconcensandinvest. 29

Comparisons with Bubbles and Pyramid Scams Finally, we turn to how the Ponzi schemes we study are related to phenomena such as bubbles and pyramid schemes. We define a bubble as an asset whose price exceeds the expectedpresentdiscountedvalueofthedividendsitpaysoutoveritslifetime. Sinceitis notoptimaltobuyanassetatthispriceandholditindefinitely,theexistenceofabubble typically requires that agents who hold the asset can find someone else to sell it to, just as a lender in a Ponzi scheme must rely on subsequent lenders to be repaid. A pyramid scheme is a setup in which agents pay for the right to recruit new participants into the scheme. Anagentwhobuysintotheschemecanonlyprofitiftheycanfindotherstoalso buyintothescheme. While economists sometimes use the terms Ponzi schemes, bubbles, and pyramid scams to describe thesame phenomena,our modelreveals somedistinctions betweenthe three. Insymmetricfullinformationsettings,theconceptsarecloselyrelated. Specifically, theexistenceofaPonziequilibriumimpliesthatabubbleandapyramidschemecanalso exist. For suppose there exists an equilibrium in which a borrower keeps paying their olddebtwithnewdebt. Insymmetricfullinformationsettings,theschemeistransparent to all agents. We can therefore eliminate the borrower and let lenders interact directly withoneanother. Inparticular,lenderscanspendtheamounttheywouldhavelentoutto buy anintrinsicallyworthlessassetand thensell tothelenders whosefunds theywould havereceivedunderthePonzischeme. Thiscorrespondstoabubble. Likewise,wecan get agents to spend the amount they would have lent out to join a pyramid scheme and thenrecruitthosewhosefundstheywouldhavereceivedunderthePonzischemeintothe scheme. Withprivateinformation,thisequivalencecanbreakdown. Inparticular,lendersinour modelareunsurewhethertheirreturnsarecomingfromnewinvestorsorfromanactual technology that the scheme operator has access to. The fact that the proceeds investors earnmay comefrom aproductivetechnology ratherthan fromother investors meansthat we cannot just eliminate the scheme operator and construct equivalent equilibria with bubblesorpyramidschemes. Whilethereareprivateinformationmodelsofbubblesand pyramidschemes,thosemodelsarequalitativelydifferentfromourmodel. Examples of private information models of bubbles include Allen, Morris and Postlewaite(1993),Conlon(2004),Doblas-Madrid(2012),andAwaya,IwasakiandWatanabe (2022). Inthesemodels,ifabubbleoccurs,itwillcollapsebyafinitedate,similarlyto howaPonzischemecollapsesbyafinitedateinourmodelifthelong-livedagentisan imposter. Inthese models,all agents know that the asset isovervalued. However, thefact thattheassetisovervaluedisnotcommonknowledge. Agentsarewillingtobuyanasset theyknow isovervaluedin thehope ofselling itto anagent who doesnot know theasset 30

is overvalued, or who does not know that all other agents know the asset is overvalued, andsoon. Agentsinthesemodelsknowthattheywillprofitattheexpenseofothers. By contrast,theagentsinourmodeldonotknowuntilthelong-livedagent’stypeisrevealed whethertheirprofitscomefromothersorfromsomerealunderlyinginvestment. Our model is arguably more similar to dynamic private informationmodels of asset trade than to models of bubbles. For example, Awaya and Krishna (2022) develop a dynamicmodelinwhichaninformedsellersellsanassettheyknowisworthlesstoagents who are unsure whether the asset is valuable. The price of the asset rises over time as agents become more convinced that the asset is valuable given the absence of negative news. Thepricecrashesoncetheassetispubliclyrevealedtobeworthless. However,the price of the asset in their model is always equal to what uninformed buyers expect the assetwillpayoutandsoisnotatruebubble. Theirmodelalsodoesnotallowagentsto resellanassetafterbuyingit,andsodoesnotinvolvePonzi-liketransfers. Turningtopyramidschemes,Stivers,SmithandJin(2019)andAntler(2023)develop models of multilevel marketing in which there is a scheme operator similar to the longlived agent in our model. The operator offers to sell distribution rights for a good they produce,aswellasbonusestodistributorsbothforsellingthegoodandforrecruitingnew distributors. Apurepyramidschemecorrespondstothecasewherethegoodisintrinsically worthlessandagentsbuythedistributionrightsonlytoearnrecruitingbonuses. Thekey issueinthesemodelsishowtosustainapurepyramidschemeifitiscommonknowledge thattherearefinitelymanyagents. Stivers,SmithandJin(2019)assumeagentsincorrectly estimatetheprobabilitytheycanrecruitnewbuyers,whileAntler(2023)assumesagents areboundedlyrational andholdbeliefsthat arecorrecton averageacrossagents butnot necessarilytrueforanygivenagent. Similartoprivateinformationmodelsofbubbles,all agentsunderstandthattheirprofitcomesfromthefundsofothers,incontrasttoourmodel. Agents gamble that they cansuccessfully recruitenough new agents tomakeparticipation profitable. Aversion ofthese modelsthat wouldbe closerin spirit toours isone inwhich agents areuncertainwhetherthegoodproducedbytheschemeoperatorisvaluable. Theycould then beunsure whether their profits comefrom selling thegood or whetherthey would needtorecruitnewagentstoprofit. Whilethisissimilartoourmodel,someaspectswould likelybedifferent. In particular,akeyquestioninourPonzischememodeliswhetherthe long-lived agentis solvent and can keep thescheme going. Uncertainty about thequality ofthegooddoesnotinvolvetheschemeoperatorbutthedemandoffuturebuyers. One feature that our model of Ponzi schemes shares with models of bubbles and pyramidschemesisthatwhenallagentsarerational,gainsfromtradecanbeaprecondition fortheseschemestoexist. Barlevy(2025)argues that in models of bubbleswhereagents 31

are rational, an inefficient initial allocation is typically necessary for a bubble to occur, whetheritisbecausetradingtheassetcreatesprivatevalue(asinsymmetricinformation models)orbecausetradebetweenspeculatorscutsintothesurplusthatagentscancreate whentheytrade(asinasymmetricinformationmodels). Amodelofpyramidschemesin whichagentsareunsureifthegoodinquestionisvaluablewouldalsoinvolvegainsfrom trade in certain states of the world. Proposition 11 shows that the same feature plays a crucialroleinourmodelaswell: Ponziequilibriaexistonlywhentheno-tradeoutcomeis inefficient. 10 Conclusion This paper developed an asymmetric information model of Ponzi schemes that allows ustoincorporateanelementofmisrepresentation,akeyfeatureofPonzifraudslikethe oneCharlesPonzioriginallyperpetrated. Ourframeworkallowsustoexaminequestions suchaswhenaPonzischemecanbeanequilibrium,howtheschemeevolvesovertime, whatfeaturescanpotentiallypreventit,andwhattheseschemestellusaboutwelfare. Therearevariouselementsthatouranalysisoverlooks. Forexample,ourbenchmark model does not incorporate uncertainty in either earnings y or the return 1+R . As L such,wecannotuseourmodeltomakeinferenceonwhenandwhyPonzischemesfail. Empirically, Ponzi schemes appear to be more likely to be exposed during recessions. Presumably, this is due to the decline in inflows into these schemes during recessions aswellaslowerreturnsthatmakeitdifficultforimposterstoremainsolvent. Uncertain endowments may also allow for schemes that end in a run. Capturing this formally is beyondthescopeofourcurrentsetup. Wealsoassumedshort-livedagentswhowithdraw all of their savings after one period. In practice, agents often reinvest some of their earningsratherthanwithdrawitall. Ignoringthissimplifiesouranalysisbutmayoverlook an important facet of such schemes. We similarly ignore recruiting fees, which recent empirical work has suggested is an important feature of many real-world Ponzi frauds. Finally,ourmodelignoresthepossibilityofheterogeneityacrossinvestors. Forexample, agents might have access to different rates of return R or might differ in their initial L beliefs about whether the long-lived agent is genuine. That would replace the indifference condition with conditions that govern the marginal investor. We leave these issues for futurework. 32

Appendix A: Proofs ProofofLemma 1: Thestatementisnecessarilytrueif(1+R )y+y<(1+R )y,since L H then the imposter would be insolvent even if they saved all of the resources x they 0 received. Thenon-trivialargumentisforthecasewherethisconditiondoesnothold. We first show that there exists a finite date atwhich the imposter must default. Let w∗ t denotethemaximalwealth theimpostercanhaveattheendof datet,i.e., iftheimposter consumesnothing. Sincex =y,thenw∗=y. Next,supposew∗≤yforsomedatet. Then 0 0 t w∗ = (1+R )w∗+y−(1+R )y t+1 L t H = w∗−R y+R w∗ t H L t ≤ w∗−(R −R )y t H L Since w∗ falls below w∗ by a discrete amount, the maximum wealth the imposter can t+1 t havewillceasetobepositiveinfinitetime. Inthatcase,theimposterwillhavelessthan (1+R )yandwillbeforcedtodefault. H Next, let T denote the earliest date at which default occurs with probability 1, i.e., eitherσ =1orS =0. SupposeT ≥2. Inthiscase,T−1andT−2arebothnonnegative. t T SinceT istheearliestdateatwhichdefaultoccurswithprobability1,theimpostermust be solvent at date T −1, i.e., S = 1. Given the definition of T, the imposter must T−1 weaklypreferdefaultingatdateT overdefaultingatdateT −1. Iftheimposterdefaults atdateT −1,theircontinuationpayoffwouldbe w +y T−1 IftheywaittodefaultatdateT,thebesttheycandoisimmediatelyconsumeanyresources theyhave leftover atT −1andthen eatany resourcesthey receiveatdate T anddefault. Thisgivesthemautilityof w +y−(1+R )y+β(1−µ)y T−1 H whereweusethefactthatinvestmentatdateT −2isy. Thisexpressionisstrictlybelow w +y. The imposter should therefore default at date T −1, which contradicts the T−1 assumptionthat T istheearliestdateatwhichdefaultoccurswithprobability1,andso waitinguntildateT todefaultwithcertaintycouldnotbeoptimal. (cid:4) Proofof Proposition1: Short-livedagentsreceiveatleastφ(1+R )iftheyinvestwith H long-termagentsatdate0. Ifφ >z,thiswouldexceedthereturn1+R onsavings,and L sox =y. SinceΦ isincreasingovertimeaslongasthelong-livedagentdoesnotdefault, 0 t 33

thesamelogicimpliesthatx equalsyfort >0untilthereisadefault. ByLemma1,the t impostermustdefaultinperiod1. (cid:4) Proof of Proposition 2: Suppose a Ponzi equilibrium existed. Given Lemma 1, we would need x <y for a Ponzi equilibrium to exist. Short-lived agents at date t must t 0 0 thereforebeindifferentbetweeninvestingwiththelong-livedagentandsavingontheir own. Short-livedagentsatdatet areindifferentifΦ +(1−Φ )(1−µ)(1−σ )=z. 0 t t t +1 0 0 0 SinceΦ =φ =z,thisconditionrequiresthatσ =1. Thatis,theprobabilitythatthe t t +1 0 0 agentswhoinvestinperiodt arerepaidifthelong-livedagentisanimpostermustbe0. 0 ButthenthiscannotbeaPonziequilibrium,whichrequiresσ <1. (cid:4) t +1 0 ProofofProposition3: SupposeaPonziequilibriumexisted. Forshort-livedagentstobe willingtoinvestatdatet ,itmustbethecasethatΦ +(1−Φ )(1−µ)(1−σ )≥z. 0 t t t +1 0 0 0 ForΦ =φ,wehave(1−µ)(1−σ )≥ z−φ . Ifµ > 1−z,then1−µ <1− 1−z = z−φ . t 0 t 0 +1 1−φ 1−φ 1−φ 1−φ Inthatcase,wewouldneed1−σ >1toensureshort-livedagentsarewillingtoinvest, t +1 0 whichmeansσ <0. Sinceσ ≥0,itfollowsthatshort-livedagentsatdatet would t +1 t +1 0 0 0 bebetteroffsavingontheirown. Butthiscontradictsourassumptionthatthereexistsa Ponziequilibrium. (cid:4) ProofofProposition4: Toconfirmthatnotradeisanequilibrium,weneedtoverifythat eachcohortofiswillingnottoinvestgiventhestrategiesofallotheragents. We start at date 0 and proceed inductively. Agents who save on their own at date 0 earn a return of 1+R . If they expect that all other agents will save in period 0, which L impliesx =0,thentheyknowthattherewillbenoinvestigationregardlessofwhatthey 0 dosinceeachagentisinfinitesimal. Iftheyfurtherthatexpectx (h1)=0forh1 ={0,0}, 1 then they knowthat a long-lived agent who isan imposter will be insolvent in period 1. Theexpectedreturnfrominvestingwiththelong-livedagentwillthenequaltoφ(1+R ). H Whenφ ≤z,thisexpressionwillnotexceed(1+R ). Short-livedagentsatdate0would L thereforebewillingtosavegiventheirexpectationsofthestrategiesofotheragents. Next, suppose there is no investment before date t. Since x = ··· = x = 0, the 0 t−1 imposter will not have any resources from past investments. If an agent expects all othershort-livedagentsatdatet tonotinvest,meaningx (ht)=0forht ={0,0,0},and t they expect no short-lived agent will invest in periodt+1, meaning x (ht+1)=0 for t+1 ht+1 ={0,0,0},thenbythesameargumentasforperiod0,theywouldbewillingtosave. Hence, when there is no trade at any date, all short-lived agents behave optimally, confirmingthatthisallocationisindeedanequilibrium. (cid:4) Proof of Lemma 2: Suppose x (ht)>0 for ht ={x(cid:48) ,0,0}. The proof is by contradict t−1 tion. 34

Suppose that x(cid:48) = 0. Since x (ht) > 0, short-lived agents know that an imposter t−1 t wouldbesolventandabletorepayinfullifonlytheydeviatedandinvested. Moreover, the imposter would strictly prefer to repay: Defaulting would yield infinitesimal benefits, butrevealingtheirtypewouldpreventthemfromstealingapositivemeasureofinvestment inthefuture,whichtheremustbegivenΦ increasesovertimeandmustexceedzinfinite t time. Theexpectedreturnfrominvestingwiththelong-livedagentatdatet isthus [Φ +(1−Φ )(1−µ)](1+R ) t t H SinceΦ isincreasingwitht,thisreturnisboundedbelowby t [φ+(1−φ)(1−µ)](1+R ) H The latter expression exceeds 1+R if φ +(1−φ)(1−µ)>z. The latter condition is L directlyimpliedby µ < 1−z. Ifwestartwithanequilibriumwherex(cid:48) =0,agentswill 1−φ t−1 haveanincentivetounilaterallydeviate. Applyingthesameargumentbyinduction,we havethatx(cid:48) >0forall j=0,...,t−1. (cid:4) j ProofofLemma3: SupposethatΦ ∈[φ,z)andσ ∈[0,1)fort =0,...,T −2forsome t t T ≥2. WewanttoshowthatΦ >Φ forallt =0,...,T −2andthat0<σ <1. t+1 t T−1 First,equation(18)impliesthat Φ t Φ = t+1 Φ +(1−Φ )(1−σ )(1−µ) t t t Since µ >0, the denominator is a weighted average of Φ times 1 and 1−Φ times an t t expressionstrictlybelow1. AslongasΦ ∈[0,1),thenΦ >Φ . SinceΦ <z<1for t t+1 t t t =0,...,T −2,wehavethatΦ >Φ fort =0,...,T −2. t+1 t Toshowthatσ ∈[0,1),weproceedbyinduction. Bydefinition,σ =0. Evaluating T−1 0 (17)fort =0andusingthefactthatΦ =φ implies 0 (cid:18) (cid:19) 1 z−φ 1−σ = 1 1−µ 1−φ Rearrangingtheinequality µ < 1−z yields 1 < 1−φ . Thisimplies 1−φ 1−µ z−φ (cid:18) (cid:19) 1 z−φ 1−σ = ∈(0,1) 1 1−µ 1−φ whichimpliesσ ∈(0,1). 1 Next, suppose that σ ∈[0,1) and Φ ∈[φ,z) for t =0,...,t∗. Let us divide (17) for t t 35

t =t∗ by(17)fort =t∗−1. Thisyields 1−σ t∗+1 (z−Φ t∗ )/(1−Φ t∗ ) = (24) 1−σ t∗−1 (z−Φ t∗−1 )/(1−Φ t∗−1 ) SinceΦ t∗−1 andΦ t∗ arebothbelowz,theRHSof(24)ispositive. Sinceσ t∗−1 ∈[0,1),it followsthatσ t∗+1 <1. Next, theexpression z−Φt isdecreasing inΦ , sincethederivativeof thisexpression 1−Φt t withrespecttoΦ t isequalto− (1− 1− Φ z t)2 <0. SinceΦ t∗ >Φ t∗−1 ,equation(24)implies 1−σ t∗+1 <1 1−σ t∗−1 or,uponrearranging,σ t∗+1 >σ t∗−1 . Sinceσ t∗−1 ≥0,thenσ t∗+1 >0. The final step is to show that there exists a T such that Φ ≥ z. Suppose to the T−1 contrarythatΦ <zforallt. Inthatcase,Φ +(1−Φ )(1−µ)≤z+(1−z)(1−µ)which t t t is bounded away from 1. This would mean Φt+1 > 1 and so lim Φ =∞, Φt z+(1−z)(1−µ) t→∞ t whichisacontradiction. Hence,T mustbefinite. DefineT∗ asthevalueofT toestablish thelemma. (cid:4) Proof of Proposition 5: First, suppose there was an equilibrium that solves (13), (14), (15), and (16) for some T where T > T∗. From Lemma 1, we know that σ ≥ 1 for t t = T∗ < T. Since probabilties are less than 1, this must mean σ T∗ = 1. But then the scheme must end at date T∗ which is before date T. The Ponzi equilibrium cannot last untildateT. Next,supposetherewasanequilibriumthatsolves(13),(14),(15),and(16)forsome T where T <T∗. For this to be an equilibrium that can last until date T, we must have either σ = 1 or S = 0 for some T ≤ T∗−1. Either way, short-lived agents at date T T T −1≤T∗−2 know they will not be repaid if the long-lived agent is an imposter. The returntheyexpecttoreceiveisthusΦ (1+R ). ButLemma1tellsusthatΦ <zfor T−1 L t t ≤T∗−2. This means these agents at date T −1 would not invest with the long-lived asset. ButthentherewouldbenoobligationtodefaultonatdateT. (cid:4) Proof of Proposition 6: To confirm that the path for {x }T∗ is an equilibrium, we first t t=0 need to verify that investors are indifferent between saving on their own and investing withthelong-livedagentinperiodst =0,...,T∗−1. ThisfollowsfromLemma3andthe factthatΦ T∗−1 =z. Second,weneedtoverifythelong-livedagentisindifferentbetween defaultinginanyperiodt =1,...,T∗−1andpreferstodefaultatdateT∗. Theindifference before date T∗ follows from the discussion in the text that before date T∗, the imposter willbesolventwithouthavingtosave,andthatwhenthereisnoneedtosave,theimposter 36

β(1−µ) willbeindifferentaboutdefaultingatdatet iffx = x . Bythesameargument, t−1 1+R H t+1 theimposterwillstrictlyprefer todefaultinperiodT∗ thanwaittodefaultin periodt+1. (cid:4) ProofofProposition7: Weconjectureaparticularinvestmentpath{x }T∗ andconfirm t t=0 thatthereexistsacutoffz∗ suchthatourconjectureisindeedanequilibriumforz≥z∗. Weconjecturethatthepath{x }T∗−2 impliedby(23)satisfiesthefollowingproperties: t t=0 (i) IfT∗−t iseven: x = β(1−µ) x t 1+R H t+2 (ii) If T∗−t isoddandx ≥β(1−µ)x : x = β(1−µ) x t+1 t+2 t 1+R H t+2 (cid:16) (cid:17) (iii) If T∗−t isoddandx <β(1−µ)x : x =α x t+1 +(1−α) β(1−µ) x t+1 t+2 t 1+R H 1+R H t+2 This path coincides with the path (19) from Proposition 6 for periodst where T∗−t is even,butallowsfordifferentvaluesofx inperiodst whereT∗−t isodd. t When T∗ =2, this path implies x = β(1−µ) y, x =x =y. Since x 1 >1+R , the 0 1+R H 1 2 x 0 H imposter will be solvent at date 1. When T∗ =2, this is the only date where solvency matters (there is no debt to default on in period 0 and the imposter will default with certaintyinperiod2). SoaPonziequilibriumexistsinthiscase. Consideradatet whereT∗−t iseven(andT∗−t ≥2). From(ii)and(iii),weknow x satisfies(23). Thatensuresthatifw ≥s = (1+R H )xt−x t−1,theimposterwillbe t−1 t−1 t−1 1+R L indifferentbetweendefaultinginperiodt andperiodt+1. Next,consideradatet whereT∗−t isodd(andT∗−t ≥3). From(i),weknowthat β(1−µ) x = x . The imposter will be indifferent between defaulting in periodt and t−1 1+R H t+1 periodt+1iftheydonotneedtosavetoavoiddefaultatdatet,i.e.,ifx ≥(1+R )x . t H t−1 The argument that they will not need to save to avoid default at datest where T∗−t is oddisbyinductionovertheoddwholenumbers. First, when T∗−t = 1, we have x T∗−1 = y > β(1−µ)y = (1+R H )x T∗−2 . So the imposterdoesnotneedtosaveinordertoavoiddefaultatdateT∗−1. Next, suppose the for some odd number k, we have x ≥(1+R )x . We T∗−k H T∗−k−1 needtoshowthatx ≥(1+R )x . Therearetwooptionsforx : T∗−k−2 H T∗−k−3 T∗−k−2 (cid:16) (cid:17) • x T∗−k−2 =α x 1 T + ∗− R k H −1 +(1−α) β 1 ( + 1− R H µ) x T∗−k . Inthiscase,(iii)impliesthatx t−k−1 < x x β(1−µ)x . Combining the two conditions implies x > t−k−1 = t−k−3 ≥ t−k t−k−2 1+R H β(1−µ) (1+R )x . H T∗−k−3 • x = β(1−µ) x . In this case, then since T∗−k−3 must be even, (i) T∗−k−2 1+R H T∗−k β(1−µ) implies that x = x . Since x ≥ (1+R )x is true T∗−k−3 1+R H T∗−k−1 T∗−k H T∗−k−1 37

β(1−µ) by assumption we can multiply both sides by to confirm that x ≥ 1+R H T∗−k−2 (1+R )x . H T∗−k−3 To recap, if w exceeds the amount the imposter needs to save, the conjectured path t−1 leavestheimposterindifferentbetweendefaultingatdatet anddatet+1ateachdatet. To confirm that the imposer can afford to save when T∗−t is odd, it will suffice to showthatx ≥s inthosedateswheres >0. Inthatcase,wehave t−1 t−1 t−1 (1+R )x −x 1−α H t−1 t s = = (β(1−µ)x −x ) t−1 t+1 t 1+R 1+R L L Evaluatingx −s ,wehave t−1 t−1 (cid:20) (cid:21) (cid:20) (cid:21) α 1−α 1−α 1−α x −s = + x + − β(1−µ)x (25) t−1 t−1 t t+1 1+R 1+R 1+R 1+R H L H L (cid:20) (cid:21) x 1 β(1−µ) t ≥ + 1− (1−α) x (26) t+1 1+R z 1+R H H Thisexpressionwillturnpositiveasz→1. Foranyfinitesequence,wecanfindaminimum valueofzthatensurestheimposterremainssolventatalldates. Denotethisvaluebyz∗. Thisprovestheclaim. (cid:4) Proof of Proposition 8: Here, we build on Proposition 10 which shows that T∗ is decreasinginφ holdingotherparametersfixed. Asφ →z,thevalueofT∗ convergesto2: Itwilltakelesstimetoreachzifwestartclosetoz,andtheprobabilityσ onlyincreases t inφ. Likewise,thevalueofT∗ musttendto∞asφ →0: Itwilltakemoretimetoreachz ifwestartcloserto0,andthetheprobabilityσ onlydecreasesasφ becomessmaller. For t any2<T <∞,thismeansthatΦT−1 musttransitionfromΦT−1 >ztoΦT−1 <zasφ increases. Bycontinuity,theremustexistaφ suchthatΦT−1 =z. (cid:4) ProofofProposition9: FromLemma2,weknowthatx >0fort =0,...,T −1. t Next, we argue x <y fort =0,...,T −2 if β <β for some β ∈(0,1). For suppose t therewasadatet ∈{0,...,T−2}forwhichx =y. Afterraisingyinperiodt,theimposter t mustchoosewhethertodefaultinperiodt ornot. Theyhavethreeoptions: (i)defaultontheirobligationof(1+R )x ,whichwouldyieldacontinuationutility H t of w +y t (ii)Waitoneperiodandthendefault,whichwouldyieldacontinuationutilityof w +y−(1+R )x +β(1−µ)x t H t−1 t+1 38

(iii)Waittodefaultafterdatet+1,whichwouldyieldacontinuationutilityofatmost β(1−µ) w +y−(1+R )x + y t H t−1 1−β(1−µ) The latter expression is due to the fact that there are y resources available each period, so the best the agent can do after not defaulting is consuming y as long as they are not exposed. Wefirstclaimthatw ≤(1+R )x forallt. Theargumentisbyinduction. Atdate t L t−1 1,theimposter’swealthw ≤(1+R )x ,sincew =0andthemosttheycansaveisthe 1 L 0 0 amounttheyreceiveatdate0atareturnof1+R . L Next, suppose the imposter’s wealth w ≤(1+R )x at date t. At date t+1, the t L t−1 imposter’swealthmustsatisfy w ≤ (1+R )(w +x −(1+R )x ) t+1 L t t H t−1 ≤ (1+R )(x −(R −R )x ) L t H L t−1 ≤ (1+R )x L t Since x > 0 in any Ponzi equilibrium, we knowt hat (1+R )x > (1+R )x t H t−1 L t−1 foranydatet =1,...,T −1. Combiningthiswiththefactthatw ≤(1+R )x ,wehave t L t w ≤(1+R )x forallt,wehave t L t−1 w +y−(1+R )x <y t H t−1 Denote w +y−(1+R )x at date t by y−ε . In the limit as β → 0, the imposter t H t−1 t willprefertodefaultimmediatelytowaitingtodefaultuntilafterdatet+1. Hence,for sufficiently small β, the imposter would prefer to either default at date t or t+1 than towaitbeyonddatet+1,whichisinconsistentwithaPonziequilibriumthatlastsuntil periodT. Note that the imposter will not necessarily default for any value of x , since for x t t smallwecanalwayshave x t+1 ≥ 1 . Butitwillnotbepossibleforinvestmenttogrow xt β(1−µ) enoughtokeeptheimposterinterestedwhenx isalreadylarge. t Thevalueofβ thatensurestheimposterwoulddefaultatdatet ifx =ydependson t ε . Toobtainasinglevalueβ >0,weneedtomakesurethatinf{ε }∞ >0forallPonzi t t t=0 equilibria. Here, we use the fact that all Ponzi equilibria end by some finite date T. To seethis, notethatΦ ≥ Φt >Φ wherethe lastinequalityusesthefact that t+1 Φt+(1−Φt)(1−µ) t µ >0. Hence, there exists some finite T such that Φ ≥z fort ≥T. From that date on, t short-lived agents invest y in every period. But the imposter will not be able to sustain this scheme: Each period, he will have to finance a growing shortfall given it adds an 39

additionalobligationofatleast(R −R )yeachperiod,untileventuallytheshortfallwill H L exceedyandtheimposterwillbeinsolvent. Hence,aPonzischememustendbyafinite dateT. Defineβ asthesmallestvalueofβ thatensurestheimposterwouldrathergety nowthanwaitandearny−ε . (cid:4) t ProofofProposition10: Westartwithalemmaforthespecialcasewhere µ =0. Intermediate Lemma: If µ =0, the solution to (17) and (18) features σ =0 and 2k Φ =Φ fork=0,1,2,... 2k+1 2k ProofofIntermediateLemma: Theproofisbyinduction. Sincethereisnothingto default on at date 0, we have σ =0. The belief Φ is equal to the prior probability φ. 0 0 From(18),wecansolveforΦ = φ =φ =Φ . Sothestatementholdsfork=0. 1 φ+(1−φ) 0 Next, suppose the statement holds for k. We need to show it also holds for k+1. Evaluating(17)whent =2k andt =2k+1andcombiningthem,wehave (1−σ )(1−σ ) z−Φ 1−Φ 2k+1 2k+2 2k+1 2k = · (27) (1−σ )(1−σ ) 1−Φ z−Φ 2k 2k+1 2k+1 2k SinceΦ =Φ ,theRHSof(27)reducesto1. Fromthis,itfollowsthatσ =σ . 2k 2k+1 2k+2 2k Butthelatterisequalto0. Hence,σ =0. 2(k+1) Using(18)evaluatedatt =2k+2,wehave Φ 2k+2 Φ = (28) 2k+3 Φ +(1−Φ )(1−σ ) 2k+2 2k+2 2k+2 Sinceσ =0,itfollowsthatΦ =Φ ,i.e.,Φ =Φ . (cid:4) 2k+2 2k+3 2k+2 2(k+1)+1 2(k+1) Usingthelemma,wecanreducethesystemofequationsgivenby(17)and(18)sothat itiseasiertoworkwith. Settingt =2k in(17)andusingthefactthatσ =0,wehave 2k Φ +(1−Φ )(1−σ )=z (29) 2k 2k 2k+1 Next,settingt =2k+1in(18)andusingthefactthatΦ =Φ ,wehave 2k 2k+1 Φ 2k Φ +(1−Φ )(1−σ )= (30) 2k 2k 2k+1 Φ 2k+2 Sinceσ =0andΦ =Φ ,wecansolvefortherelevantequilibriumobjectsusing 2k 2k+1 2k thesystemofequationsdefinedoverthevariables{Φ ,σ }∞ . 2k 2k+1 k=0 LetΦ (z)andσ (z)denotethesolutionto(29)and(30)givenavalueforz. We 2k 2k+1 picktwovaluesz(cid:48)(cid:48) >z(cid:48) thatsatisfy µ < 1−z(cid:48)(cid:48) < 1−z(cid:48) . 1−φ 1−φ Fork=0,wehaveΦ (z)=φ regardlessofz. Trivially,then,Φ (z(cid:48)(cid:48))≤Φ (z(cid:48)). Turning 0 0 0 40

toσ ,wecanuse(29)tosolveforσ ,i.e., 1 1 z−φ σ =1− (31) 1 1−φ This expression is decreasing in z. Since z(cid:48)(cid:48) >z(cid:48), we have σ (z(cid:48)(cid:48))<σ (z(cid:48)). Since µ < 1 1 1−z(cid:48)(cid:48) < 1−z(cid:48) ,itfollowsthatbothσ (z(cid:48))andσ (z(cid:48)(cid:48))arebetween0and1. 1−φ 1−φ 1 1 Wenowproceedbyinduction. Supposethatforsomeintegerk,wehave (i) φ ≤Φ (z(cid:48)(cid:48))≤Φ (z(cid:48))<z(cid:48) <z(cid:48)(cid:48) 2k 2k (ii) 0<σ (z(cid:48)(cid:48))<σ (z(cid:48))<1 2k+1 2k+1 We wantto show thattheseconditions alsohold forthe integerk+1, i.e.,that thesame conditionsholdforΦ (z)andσ (z). 2k+2 2k+3 WebeginwithΦ (z). From(30)toget 2k+2 Φ (z(cid:48)) Φ (z(cid:48)) = 2k 2k+2 Φ (z(cid:48))+[1−Φ (z(cid:48))][1−σ (z(cid:48))] 2k 2k 2k+1 1 = (32) (cid:16) (cid:17) 1+ 1 −1 [1−σ (z(cid:48))] Φ (z(cid:48)) 2k+1 2k Since0<Φ (z(cid:48)(cid:48))≤Φ (z(cid:48))<1and0<σ (z(cid:48)(cid:48))<σ (z(cid:48))<1,wehave 2k 2k 2k+1 2k+1 (cid:18) (cid:19) (cid:18) (cid:19) 1 1 −1 [1−σ (z(cid:48))]< −1 [1−σ (z(cid:48)(cid:48))] (33) Φ (z(cid:48)) 2k+1 Φ (z(cid:48)(cid:48)) 2k+1 2k 2k whichimpliesthatΦ (z(cid:48)(cid:48))≤Φ (z(cid:48))asclaimed. 2k+2 2k+2 Next,(29)implies z(cid:48)−Φ (z(cid:48)) 1−σ (z(cid:48)) = 2k+2 2k+3 1−Φ (z(cid:48)) 2k+2 z(cid:48)(cid:48)−Φ (z(cid:48)(cid:48)) 1−σ (z(cid:48)(cid:48)) = 2k+2 2k+3 1−Φ (z(cid:48)(cid:48)) 2k+2 Sincez(cid:48)(cid:48) >z(cid:48),wehave z(cid:48)(cid:48)−Φ (z(cid:48)(cid:48)) z(cid:48)−Φ (z(cid:48)(cid:48)) 2k+2 2k+2 ≥ 1−Φ (z(cid:48)(cid:48)) 1−Φ (z(cid:48)(cid:48)) 2k+2 2k+2 SincewejustshowedthatΦ (z(cid:48))≥Φ (z(cid:48)(cid:48))andtheexpression z−φ isdecreasingin 2k+2 2k+2 1−φ φ forφ <z,wehave z(cid:48)−Φ (z(cid:48)(cid:48)) z(cid:48)−Φ (z(cid:48)) 2k+2 2k+2 ≥ 1−Φ (z(cid:48)(cid:48)) 1−Φ (z(cid:48)) 2k+2 2k+2 41

Combininginequalitiesyields σ (z(cid:48)(cid:48))<σ (z(cid:48)) (34) 2k+3 2k+3 From Lemma 3, we have that T∗−2=sup{t :Φ (z)<z}. Since we just showed that t Φ (z)isdecreasinginzfromdate2on,itfollowsthatT∗ isweaklyincreasinginzwhen t µ =0. Bycontinuity,theclaimshouldholdfor µ closeto0aswell. We can use the same argument to show that for φ ∈(0,z), if 0<φ(cid:48) <φ(cid:48)(cid:48) <z, then T∗(φ(cid:48)(cid:48))>T∗(φ(cid:48)). (cid:4) Appendix B: Ponzi Schemes with Risky Investment InthisAppendix,weconsideravariationofthemodelinthetextinwhichtheimposter can undertake a risky but profitable investment. In contrast to our benchmark model in which the long-lived agent only benefits from stealing and will necessarily default on some cohort, adding investment will make it possible for the long-lived agent to avoid default. Nevertheless, the long-lived agent still preys on short-lived agents by pooling with a commitment type. If short-lived agents knew that the agent they invest with can onlyinvestinariskytechnology,theywouldrefusetoinvestwiththem. Investment Technology Formally,wemodifythemodeltoallowtheimpostertoinvestinariskytechnology. This isinadditiontotheoptionsofconsumingandsavingatthesamerateofreturn1+R that L short-livedagentscanachieveontheirown. Theassumptionsthatcharacterizetherisky technologyareasfollows: • The return on a risky investment initiated at date t is realized in period t+1. It equals 1+R with probability λ and 0 with probability 1−λ, where λ < z and R>R . H • If the long-lived agent invests in the risky technology and it yields a positive payoff, short-lived agents observe that the payoff was positive and that the investment was risky. Asuccessfulriskyinvestmentthusrevealsthelong-livedagent’stype. • There isa court thatcan verify whetherthe long-livedagent defaulted andwhich canseizetheproceedsfromanysuccessfulinvestmentofthelong-livedagentand usethemto payshort-livedinvestorsor thosewhoinheritthe unpaidobligationsof thepreviousgenerations. • Thecourtcannotpreventthelong-livedagentfromstealingtoconsume. Itcanonly seizetheproceedsfrominvestment. Along-livedagentwhodefaultswilltherefore 42

notbenefitfrominvesting,buttheycanbenefitfromstealingfundsandconsuming them. • The court cannot identify whether the technology that long-lived agent use is risky. It canpunishthelong-livedagentfordefaulting butnotforoperatingarisky technology. Under these assumptions, the long-lived agent will repay short-lived agents from the previous period if their investment succeeds. In addition, if they default, they will not investintheriskytechnologygiventheyexpectthecourttopunishthemfordefaulting. Timing Weintegratetheriskytechnologyintothetimelineofourmodelasfollows. In each period t ≥ 0, the imposter chooses at the end of each period whether to consume the resources they have at the end of the period, save them at rate 1+R , or L investthemintheriskytechnology. In each period t ≥1, if the imposter previously undertook the risky investment, its payoff is revealed at the start of the period, before the imposter might be exogenously exposed. Equilibrium when Long-Lived Agent type Revealed We now introduce a parametric assumption that ensures that if the long-run agent is exposedasanimposter,theywillbeunabletoprofitablytradewithshort-livedagents. (cid:16) (cid:17) Assumption1: λβ R−R L <1 λ The next argument establishes that under this condition, there will be no incentive for short-lived agents to trade with the long-lived agent if they arerevealed to onlyhave accesstotheriskytechnology. Claim 1: There are no gains from trade between short-lived agents and a known imposter. If the imposter is exposed at date t, then x = 0 for s ≥ 0 without loss of t+s generality. Proof: Let1+r denotethereturn oninvestment toshort-lived investorsfrom date t+1 t ifthelong-livedagentundertooktheriskyinvestmentatdatet anditpaysoutatt+1. Definer∗=R− 1 asthecutoffrateatwhichtheexpectedreturnafterpayinginvestors, λβ λβ(R−r∗), is equal to 1. If r > r∗, the long-lived agent will not initiate the risky t+1 investment. To see this, define V as the continuation value per unit invested in the t+1 risky technology for the exposed imposter at date t+1 when there is no evidence of a riskyinvestmentthatpaidoff. Sincethelong-livedagentalwayshastheoptionofdoing 43

nothing, V ≥0. If the exposed imposter chooses to undertake the risky investment, t+1 theirexpectedutilityperunitinvestedintheriskytechnologywillbe λβ(R−r )+(1−λ)βV < λβ(R−r∗)+(1−λ)βV t+1 t+1 t+1 ≤ 1+(1−λ)βV t+1 ≤ 1+βV t+1 Hence, if r > r∗, the long-lived agent will prefer to steal the funds they receive to t+1 investingthemintheriskytechnology. Assumption 1 implies that r∗ < R L. Let 1+r0 denote the return on investment to λ t short-livedinvestorsfromdatet ifthereisnopayofftoariskyinvestmentatdatet+1. To attractshort-livedinvestmentatdatet,thelong-livedagentmustofferanexpectedreturn ofatleast1+R . Hence,ifthelong-livedagentinvestsintheriskytechnology,wemust L have λr +(1−λ)r0 ≥R (35) t+1 t+1 L Rearranging,wehave R −λr r0 ≥ L t+1 (36) t+1 1−λ Forthelong-livedagenttobewillingtooperatetheriskytechnology,weneedr ≤r∗. t+1 Hence,ifthelong-livedagentinvests,thereturnwhentheinvestmentdoesnotpaymust bealeast R L −λr∗ ,whichunderAssumption1mustbestrictlypositive. 1−λ Supposethelong-livedagentborrowstoinvestintheriskytechnologyatdatet. Ifthe payoffontheriskyinvestmentwaszero,thelong-livedagentwouldhavetoborrowfrom newinvestorstopayofftheirpromisedreturnofr0 . Thereafter,theirdebtwouldgrow t+1 atarateboundedawayfromzerounlesstheyinvestedintheriskytechnologyanditpaid off. Otherwise,thelong-livedagentwouldhavetoborrowatarateof1+R iftheychose L nottoinvestgivenshort-livedagentswoulddemandthatasthesafereturn,ortheywould borrowataratethatexceeds R L −λr∗ >0iftheyinvestedintheriskyassetanditpaidzero. 1−λ As long as the long-lived agent failed to make a successful risky investment, their debt obligationwouldgrowwithoutbound. Since the debt obligation grows without bound, there exists some finite date t∗ in whichthelong-livedagentwillnotbeabletoborrowenoughfromnewinvestorstopay investorsfromdatet∗−1anexpectedreturnof1+R . Knowingthis,short-livedagents L will not agree to invest in periodt∗−1 if the investment was unsuccessful. That means thelong-livedagentcannotinvestintheriskytechnologyinperiodt∗−2,sinceweknow theyneedtoborrowresourcesinperiodt∗−1iftheirinvestmentisunsuccessfultooffer investors in period t∗−2 an expected return of 1+R . Repeating the same argument, L 44

thelong-livedagentcannotborrowtoinvestintheriskytechnologyatdatet,whichisa contradiction. (cid:4) Essentially, even when the long-lived agent’s type is known, short-lived agents still cannotmonitorthelong-livedagent. Ifthehighreturnontheprojectfailstomaterialize, theycannotverifywhetherthisisbecausetheinvestmentfailedorbecausetheimposter stolethefundsanddidn’tinvestanything. Thisimpliesthatthelong-livedagentcannot promisetoohighof areturnincasetheinvestment issuccessful: Iftheydid,theywould have an incentive to not invest at all. A low promised return if the risky investment is successful requires a high return if it is unsuccessful. But that could create a situation wherethe long-lived agentmay have tokeep rollingoverdebt forarbitrarilylong periods, whichcannothappenwhentheendowmentisconstant. Thatis,astrategyofrollingover debtuntilaninvestmentsucceedsandcanbeusedtorepaydebtisnotsustainable. Optimal Behavior when Investment Stops Growing WenowproceedtolookforaPonziequilibriumwhiletheimposter’stypeisuncertain. We beginwithtwoadditionalparametricassumptions. Wethencharacterizetheimposter’s optimalstrategyifx isequaltoyforallt. t Ourfirstassumptionisthatthelong-livedagentisimpatient: Assumption2: β < 1 1+R H Asbefore,thisassumptionimpliesthatβ < 1 sothelong-livedagentwouldprefer 1+R L consumingimmediatelytosavingandconsumingafteroneperiod. Ournextassumptionrules outthecasewherethelong-lived agentprefersstealingto investingintheriskytechnology. Assumption3: βλ(R−R )>1 H Iftheinequalityabovewerereversedandβλ(R−R )werelessthan1,thelong-lived H agent would strictly prefer to consume resources immediately to investing them in the riskytechnology: Themarginalutilityfromconsumingexceedsthemarginalutilityfrom investing, and the long-lived agent can do anything after consuming that it could after investing,whileiftheyinvestsuccessfullytheiroptionstocontinueraisingfundswould end. Assumption 3 is necessary but not sufficient for the imposter to invest in the risky technology. Before we turn to the question of whether the imposter chooses to undertake the riskyinvestment,weobtainaresultaboutwhattheimposterwoulddooncebeliefswere sufficientlyoptimistictoensureshort-livedagentswouldkeepinvestingyinevery period. Claim2: Supposex =yforalls≥0andthelong-livedagent’stypeisnotrevealed t+s bydatet. Theimposterwilldefaultbydatet+1. 45

Proof: When x =y, the imposter would increase their debt obligation by at least t+s the amount (R −R )y each period in which they fail to successfully invest and do not H L default. Their debt obligation would eventually exceed y, and the imposter would have to default if they were unsuccessful in investing. Let t+S denote the earliest date of defaultforanimposterwhosetyperemainsuncertain,meaningtheyneitherdefaultednor successfullyinvestedinthepast. Suppose S > 1. What could the imposter have done in period t+S−1 if it was behavingoptimally? Theywouldhavestartedperiodt+S−1withwealthw . Ifthey t+S−1 consumedtheresourcestheyhadaccesstoindatet+S−1afterdefaulting,theirutility wouldbe w +y t+S−1 Waitingtodefaultinperiodt+S withoutexposingtheirtypewouldrequirethemtopay theirobligationof(1+R )x =(1+R )y. Iftheyconsumedtheamountw + H t+S−2 H t+S−1 y−(1+R )y,theirutilitywouldbe H w +y−(1+R )y+β(1−µ)y t+S−1 H Thisis strictlyless thandefaulting andconsuming everythingat datet+S−1. Hence, if theimposterchosetowaituntildatet+StodefaultforS>1,theywouldnothavechosen to avoid default and consume the resources they had left at the end of period t+S−1. Avoidingdefaultandsavingtheresourcesleftattheendofperiodt+S−1wouldyield even lower utility. It follows that if the imposter first defaults in datet+S for S>1, it mustbebecausetheychosetoinvesttheirwealthintheriskyprojectatdatet+S−1. Fortheimpostertohaveresourcestoinvestintheriskyprojectatdatet+S−1without revealing their type, it must be the case that w +y−(1+R )y > 0. This means t+S−1 H that w > 0. For the agent to have started with this wealth required them to save t+S−1 w∗ = w t+S−1. Buttheimposterwouldhavebeenbetteroffinvestingw∗ attheend t+S−2 1+R L t+S−2 ofperiodt+S−2giventhatβ(1+R )<1. Hence,theimposterwouldnotwaittoinvest L beyonddatet andwouldnotdefaultbeyonddatet+1. (cid:4) Constructing a Ponzi Equilibrium In a Ponzi equilibrium where the imposter uses new funds to cover their obligation to previousinvestors, short-lived agents would revise their beliefs Φ upwards as longas the t long-livedagentdoesnotdefault. Eventually,Φ wouldbehighenoughthatallshort-lived t agents would choose to invest, implying x =y for all subsequentt. From Claim 2, we t knowthattheimposterwoulddefaultwithinoneperiod. Wenowlookforanequilibrium inwhichφ islowenoughsothatx canfallbelowyandthelong-livedagentiswillingto 0 46

postponedefaultforseveralperiodsandusenewfundstocovertheirobligations. Onceagain,welookforaPonziequilibriuminwhichx ∈(0,y)untiljustbeforethe t imposterdefaults. Thisrequiresthatshort-livedagentsbeindifferentbetweensavingon theirownandinvestingwiththelong-livedagentuntiljustbeforetheimposterdefaults. Tocharacterizesuchanequilibrium,weintroducesomenotation. LetΦ denotethe t beliefoftheshort-livedagentatthetimetheyinvestatdatet thatthelong-livedagentis the genuine type. Let σ denote the probability that a solvent imposter defaults at date t t. It will also be useful to define several indicator variables. Let I denote a dummy t variable that is equal to 1 if short-lived agents know at the time they invest at date t thatthelong-livedagentisanimposterand0otherwise. I canequal1ifthelong-lived t agent defaulted in the past, was exogenously exposed before short-lived agents invest at date t, or if the long-lived agent successfully invested in the risky technology in the pastandtherebyrevealeditstype. LetU denoteanindicatorvariablethatequals1ifthe t long-livedinvestor chooses toundertake theriskyproject attheend ofdatet. Finally, let S beanindicatorvariableofwhetherthe imposter issolventatdatet. Thatis,S =1if t t w +x ≥(1+R )x and0otherwise. t t H t−1 Ifshort-livedagentssaveontheirownatanydate,theywillearn1+R . L Ifthey investwiththelong-livedagentatdatet andthelong-livedagentis genuine, whichoccurswithprobabilityΦ ,thenthelong-livedagentwillpay1+R infull. t H Ifthelong-livedagentisanimposter,whichoccurswithprobability1−Φ ,anddoes t notundertaketheriskyprojectatdatet,i.e.,ifU =0,thenthelong-livedagentwillrepay t iftheyarenotexogenouslyexposednextperiod,iftheyaresolventnextperiod,andifthey donotdefault. Theprobabilityofrepaymentinthiscasewillbe(1−µ)S (1−σ ). t+1 t+1 If the long-lived agent is an imposter and does undertake the risky project at datet, i.e.,if U =1,thenthelong-livedagentwillrepayifeithertheirinvestmentissuccessful, t which occurs with probability λ, or if they are not exposed next period, if they are are solventnextperiod,andiftheydonotdefault. Theprobabilityofrepaymentinthiscase willbeλ +(1−λ)S (1−µ)(1−σ ). t+1 t+1 Theconditionthatleavesshort-livedagentsindifferentbetweenthetwoisgivenby  Φ +(1−Φ )[(1−µ)(1−σ )S ] ifU =0  t t t+1 t+1 t  z= (37)   Φ +(1−Φ )[λ +(1−λ)(1−µ)(1−σ )S ] ifU =1 t t t+1 t+1 t 47

Turningtothebeliefsoftheagent,Short-livedagentswillupdatetheirbeliefsasfollows:  Φ  t ifI =0andU =0     Φ t +(1−Φ t )(1−σ t )(1−µ) t+1 t      Φ Φ t+1 = t ifI =0andU =1 (38) t+1 t  Φ +(1−Φ )(1−σ )(1−λ)(1−µ)  t t t         0 ifI =1 t+1 ThesystemofdifferenceequationshastheboundaryconditionΦ =φ andσ =0. 0 0 As in the benchmark model, the probability Φ will grow while I = 0, i.e., while t t thelong-livedagent’styperemainsuncertain. Ifφ < z−λ,theuniquepath{Φ ,σ }that 1−λ t t solves(37)and(38)willsatisfythefollowingproperties: 1. ThereexistsafiniteT ≥2suchthatσ <1forallt <T andσ ≥1 t T 2. Φ < z−λ fort =0,1,...T −1andΦ ≥ z−λ t 1−λ T−1 1−λ 3. Φ andσ arebothincreasingint fort =1,...,T t t Incontrasttoourbenchmarkmodel,thereisnoguaranteethatthevalueofσ thatsolves 1 this systemof equationswill be positive whenever φ > z−λ. Forparameters thatimply 1−λ σ > 0, we know that σ will be between 0 and 1 for t = 1,...,T −1. An equilibrium 1 t withthesevaluesforσ requirestheimpostertobeindifferentbetweendefaultingandnot t defaulting. Wethusneedtocheckthatthereexistsapath{x }∞ thatleavestheimposterindiffert t=0 entaboutdefaultingineveryperiod. Wenowturntotheimposter’sdecision. A Ponzi Equilibrium with T = 2 Forsimplicity,wefocusonthecasewhereT =2. Givenremainingparametervalues,we canchooseφ toensurethatσ ∈(0,1)andσ ≥1bysettingφ justbelow z−λ. Wefocus 1 2 1−λ onthe caseinwhich thevalueofσ thatsolvesthesystem ofequationsdefined by(37) 2 and(38)isstrictlygreaterthan1. Thatisthegenericcase;whenT =2,thenσ ≥1and 2 is exactly equal to 1 for exactly one value of φ. If σ >1, then in equilibrium we will 2 havex =x =y. Theonlyequilibriumvaluewewouldneedtosolveforisx . 1 2 0 FromClaim2,weknowthatatdatet =2,theimpostershoulddefaultiftheirtypeis notrevealed. Inthatcase,theywillraisex innewfundsandshouldimmediatelyconsume 2 themtogetherwithanywealthw theyhaveatthestartoftheperiod. 2 Next, atdatet =1, theimposter mustchoose whetherto default. Ifthey default, they should consume the amount w +x : This is better than saving given their impatience, 1 1 48

and operating the risky technology is unprofitable since their proceeds will be seized if theinvestmentissuccessful. Theirpayoffasofdate1inthiscasewillthusbe w +x (39) 1 1 Iftheimposterrepaystheirobligation(1+R )x ,theymustdecidebetweenconsuming, H 0 saving,andinvestingthew +x −(1+R )x theyhaveaccessto. Giventheyintendto 1 1 H 0 defaultinperiod2,consumingdominatessaving. Iftheyconsumetheiravailableresources atdate1,theirexpectedpayoffwillbe w +x −(1+R )x +β(1−µ)x (40) 1 1 H 0 2 Iftheyinvestintheriskytechnology,theirexpectedpayoffwillbe βλ[(R−R )x +(1+R)(w −(1+R )x )]+β(1−λ)(1−µ)x (41) H 1 1 H 0 2 Finally,atdatet =0thereisnodefaultdecision. Theimpostermustchoose between consuming x , saving x and earning the riskless return 1+R , and investing x in the 0 0 L 0 riskytechnologywhosereturnisstochastic. For0<σ <1tobeoptimal,weneedtheimpostertobeindifferentbetweendefaulting 1 inperiod1andchoosingwhateveractionisoptimalwhennotdefaulting. If they intend to default in period 1, the imposter should consume x and default in 0 period1iftheyarenotexposed,yieldinganexpectedpayoffof x +β(1−µ)x (42) 0 1 Iftheydonotintendtodefaultinperiod1,theymustchoosewhethertosave,consume, orinvestinperiod0. Iftheyconsume,theywillreceivex inutilityinperiod0andstart 0 period1withw =0inwealth. Iftheysave,theywillreceive0inutilityinperiod0and 1 start period1 withw =(1+R )x in wealth. If theyinvest, they willreceive 0in utility 1 L 0 inperiod 0andstart period1witheither (R−R )x iftheir investment issuccessful and H 0 w =0iftheirinvestmentisunsuccessful. Thecontinuationutilitywouldbeβ timesthe 1 expressionsin(40)or(41),dependingonwhattheimposterchooses. To ensure that the imposter is indifferent between defaulting and not defaulting in period1,theexpressionin(42)mustequalthemaximalvalueofconsuming,investing,or savinginperiod0andthenchoosingeither(40)or(41). As an example, supposewe look for the value ofx that leaves the imposterindiffer- 0 ent between defaulting and investing in period 1 after investing in period 0 (and being 49

unsuccessfulsothatanotherchoicecanbemade). Thenx willsolve 0 y=βλ[(R−R )y−(1+R)(1+R )x ]+β(1−λ)(1−µ)y (43) H H 0 Numerical Example Considerthefollowingparametervalues: R = 0.10 φ = 0.99 H R = 0.09 β = 0.80 L R = 65.0 µ = 0.40 λ = 0.02 WecanverifythattheseparameterssatisfyAssumptions1,2,and3. Given these values, the value of x that solves (43) is given by x =0.438y. At this 0 0 value,iftheimposterhasnowealthatdate1,theywillbeindifferentbetweendefaulting atdate1ontheirinvestorsfromperiod0andconsumingthenewinflowoffundsx =yon 1 theonehandandpayingtheinvestorsfromdate0theamount(1+R )x =0.482ythatis H 0 owedtothematdate1andtheninvestingtheremaining0.518yintheriskytechnology. Theexpectedex-antepayofftotheimposterfromfollowingeitherofthesestrategiesafter undertakingtheriskyinvestmentinperiod0is0.925y. Wecanverifythattheexpectedpayofftootherstrategiesislowerthan0.925y. Todo this,weneedtoidentifythestrategiestheimpostercanfollow. Sincethereisnoobligation in0, defaultis nota considerationat date0. Instead, theimposter mustchoose between investing, consuming, or saving. Since we know from Claim 2 that they will default in period2,thereisnobenefittosavinginperiod1. Thatmeansthatinperiod1,theimposter willeitherinvest,consume,ordefault. Atdate2,theimposterwilldefaultiftheyhaven’t already. Therearethusninestrategiestoconsider.8 Thestrategywiththesecondhighest ex-antepayoffisfortheimpostertoinvestinperiod0andthenconsumex −(1+R )x 1 H 0 if they fail, which yields an ex-ante expected payoff of 0.924y. The next best strategies involveconsumingx inperiod0andtheneitherinvestingordefaultinginperiod1since 0 bothgivethesamepayoff. Thisstrategyyieldsanexpectedpayoffof0.918y. Thepayoff tosavinginperiod0islowerstill. We can further verify that there exists no other Ponzi equilibrium in which an optimizingimposterisindifferentbetweendefaultingandnotdefaultinginperiod1. Thatis, wecanlookatthevalueofx thatleavestheimposterindifferentbetweendefaultingin 0 8Theimpostercaninprinciplemixbetweenactions,buttheywoulddothisonlyiftheyareindifferent, inwhichcasethepayoffwillbethesameastothepurestrategy. 50

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