Demagogues and the Fragility of Democracy

DemagoguesandtheFragilityofDemocracy*

DanBernhardt† StefanKrasa‡ MehdiShadmehr§

July27,2021

Abstract

WeinvestigatethesusceptibilityofDemocracytodemagogues,studyingtensionsbetweenrepresentativeswhoguardvoters’long-runinterestsanddemagogueswhocater tovoters’short-rundesires.Partiesproposeconsumptionandinvestment.Votersbase choicesoncurrent-periodconsumptionandvalenceshocks.Younger/poorereconomies andeconomically-disadvantagedvotersareattractedtothedemagogue’sdis-investment policies,forcingfar-sightedrepresentativestomimicthem.Thiselectoralcompetitioncan destroydemocracy:ifcapitalfallsbelowacriticallevel,adeathspiralensueswithcapital stocksfallingthereafter.Weidentifywheneconomicdevelopmentmitigatesthisriskand characterizehowthedeath-spiralriskdeclinesascapitalgrowslarge.

*WearegratefulforcommentsofOdilonCamara,WiolettaDziuda,andtheEditor,RolandBenabou,and seminarparticipantsatNorthwesternKellogg,UniversityofVienna,UniversityofBonn,and2019APSA.

†UniversityofIllinoisandUniversityofWarwick,danber@illinois.edu

‡UniversityofIllinois,skrasa@illinois.edu

§UniversityofNorthCarolinaatChapelHill,mshadmeh@gmail.com

“Therepublicanprinciple,”wroteHamiltoninFederalistNo.71,“doesnotrequireanunqualifiedcomplaisancetoeverysuddenbreezeofpassion,ortoeverytransientimpulsewhich thepeoplemayreceivefromtheartsofmen,whoflattertheirprejudicestobetraytheirinterests.”Tothecontrary,Hamiltonargued,when“theinterestsofthepeopleareatvariancewith theirinclinations,itisthedutyofthepersonswhomtheyhaveappointedtobetheguardiansof thoseinterests,towithstandthetemporarydelusion....conductofthiskindhassavedthepeople fromveryfatalconsequencesoftheirownmistakes,andprocured...theirgratitudetothemen whohadcourageandmagnanimityenoughtoservethemattheperiloftheirdispleasure.”Still, ifsuchmagnanimousrepresentativescausetoomuchdispleasure,theywouldloseelectionto thosewhowillimplementthose“temporarydelusions”,paying“obsequiouscourttothepeople; commencingdemagogues,andendingtyrants.”

OurpaperstudiesthetensionhighlightedbyHamiltonbetweenfar-sighted,magnanimous representativeswhoguardthelong-runinterestsofvotersandoffice-seekingdemagogueswho catertovoters’short-rundesires.1 Wecharacterizethelong-runoutcomesofdemocracyina countrypopulatedbyashort-sightedmajority.Demagoguesandshort-sightedvotershavelong beenconsideredinter-relatedvicesofrepublicangovernments.Forexample,“Madison’s[belief]aboutdemocracywasbasedon[one]abouthumanbeings:man,bynature,preferredto followhispassionratherthanhisreason;heinvariablychoseshort-termoverlong-terminterests”(Middlekauff (2007),p.678).Indeed,researchersdefinedemagoguesbythischaracteristic.AccordingtoGuisoetal.(2018),partiesledbydemagogues“championshort-termpolicies whilehidingtheirlong-termcosts.”2 Whatisnotwell-understoodishowdemagoguesdistort thebehavioroffar-sightedparties,andhowademocraticcountryemergesfromthelong-run confrontationbetweenselfishdemagoguesandsociallybenevolent,butpragmaticparties.

Weanalyzethedynamicpoliticalcompetitionbetweenafar-sighted,benevolentpartythat seekstomaximizevoterwelfare,andanoffice-motivateddemagoguewhoonlycaresabout winning.Thetwopartiesandarepresentativeshort-sightedmedianvoterinteractovertimein aninfinitehorizonsetting.Tocapturetensionsbetweenshort-andlong-termconsiderations, wemodelthepoliticaldecisionprocessasaninvestmentprobleminwhichpartiesproposehow toallocateexistingresourcesbetweencurrentconsumptionandinvestmentincapitalthatfacilitatesfutureconsumption,wheredis-investmentofuptosomefractionofthecurrentcapital stockisfeasible.Voterscareaboutaparty’sinvestmentpolicyanditsvalence,whichcaptures

1AsBurmanetal.(2010)observe,“Thebasicproblemisthatpolicymakerswanttomakepeoplehappy,which meansmorespendingandlowertaxes...Politicalleadersperceivethattheirreelectiondependsonshort-term results,eveniftheshort-termexpedientsmaybedisastrousoverthelongterm.”

2Historically,populistswerereferredtoasdemagoguesbutnowthesetermsareoftenusedinterchangeably.

theparty’snon-policyattributes.Thebenevolentparty’snetvalenceiseitherhighorlow—even ifthebenevolentpartyisfarmorelikelytohaveahighervalence,thereisstillachancethatthe demagoguehasthehighervalenceandwinselection.Themyopicmedianvoterassessesparties basedsolelyoncurrentperiodutility.Giventheproposedinvestmentsandrealizedvalences, thevoterpicksthewinnerwhoimplementsitsinvestmentpolicy.

Absentademagogue,thebenevolentpartyactslikeasocialplanner,internalizingvoters’ utilityfromfutureconsumption,andcapitalstocksgrowwithoutboundovertime.Demagogues,eventhoselikelytohavelowvalences,changethis.Demagoguesdesigntheirpolicies toappealmaximallytoshort-sightedvoters.Thebenevolentparty’sdilemmaisthatifitignoresthedemagogueinitspolicychoices,itimperilsitselectoralsupport,while“tryingto beata...populistinsurgencybybecomingone...turnsouttobeafool’serrand...[asit]hasa huge...economiccost.”3 ThepolicychoicesofHueyLongandPresidentRooseveltillustrate thisdilemma.InthemidstoftheGreatDepression,Longproposedahighprogressivetax,and distributingtherevenuetoeveryAmericanfamily,5,000dollarseach—supposedlyenoughfor ahome,acar,andaradio—plusshorterworkinghours,pensionsandmanyotherbenefits.In response,FDRproposedaSecondNewDealthatincludedaWealthTaxActdesigned“tosave oursystem”fromthe“crackpotideas”ofdemagogues.FDRknewthathisproposalwasbad fortheeconomy,but“IamfightingCommunism,HueyLongism,...,”hesaid,indicatingthat theconsequencesoflosingwereworse(Kennedy(1999),p.241-7).

WefocusonCRRAutilitywithcoefficientsofrelativeriskaversionthatexceedone.These preferenceshavethefeaturethatwhencapitalfalls,theutilitydifferencesinimpliedconsumptionsassociatedwithfixedinvestmentratesrise.IntheMarkovperfectequilibrium,thedemagogueproposestopropupshort-termconsumptionbyproposingtodisinvestasmuchaspossible,whilethebenevolentpartychoosesaninvestmentlevelthatensuresthatthemedianvoter supportsitwhenitsvalenceishigh.

Asabenchmark,weanalyzeascenarioinwhichthereisexogenousstochasticselectionof thewinner,sothatpolicychoicesdonotaffectwhowins.Wecontrastthissettingwiththat wherethewinnerisendogenouslydetermined.Weshowthatthebenevolentpartyproposesto investlessthanitwouldinthebenchmarkifandonlyifthecapitalstockisbelowathreshold. Whencapitalisabovethatthreshold,thebenevolentpartyproposestoinvestevenmorethanit wouldinthebenchmark,convergingtothebenchmarkinvestmentlevelfromaboveascapital stockgrowslarger.Thatis,thebenevolentparty’sproposedinvestment,asashareofcapital,is 3NYtimesOpEd,https://www.nytimes.com/2018/07/16/opinion/to-defeat-far-right-nationalists-dont-try-toimitate-them.html

non-monotoneinthelevelofcapital.Inparticular,whenthecapitalstockisbelowathreshold, thebenevolentparty’sproposedinvestment,asashareofcapital,isincreasingincapitalstock.4 Incontrast,whencapitalstocksaresufficientlyhigh,thebenevolentparty’soptimalinvestment, asashareofcapital,trendsdownwardwithcapital.

Thebenevolentpartyunderstandsthatitneedstowininordertoinvest,andthatworse outcomeswouldobtainwerethedemagoguetowin.Whencapital,andhenceproposedconsumption,islow,thesalienceofdifferencesinproposedconsumptionsishighrelativetothatin candidatevalences—whenevervotershaverelativeriskaversionsaboveone.Inessence,when votersarehungry,valencebecomesmoreofasecondaryconcern,raisingtherelativeattraction ofthedemagogue’sdis-investmentpolicy,makingpoorereconomiesmorepronetotheconsequencesofdemagoguery.Whencapitalislow,thebenevolentpartyproposesinvestmentrates justlowenoughtoavoidlosingwhenithashighvalence.Incontrast,oncecapitalissufficiently high,thebenevolentpartycanproposethebenchmarkinvestmentwithoutlosingwhenithas highvalence.Despitethis,itproposesevenhigherinvestmentstoinsureagainsttheriskthat thedemagoguemaywinandreducecapitalstockstowardlevelswhereitselectoralcompetition severelyconstrainsinvestment.Onlywhencapitallevelsaresohighthatsuchelectoralthreats couldonlyariseinthedistantfuturedoitsproposedinvestmentsconvergetothebenchmark.

Oursecondsetofresultsidentifythepossibilityofdeathspiralsfordemocracy:Weidentify acriticalcutoff oncapital(thepointofnoreturn)belowwhichcapitalstocksshrinkatacceleratingratestowardzero.Ifcapitaleverfallsbelowthislevel,thenthebenevolentparty,itself,must proposetodis-investtopreserveachanceofwinning.Asaresult,capitalstocksshrink,forcing thebenevolentpartytoincreasinglymimicthedemagogue.Ascapitalstocksspiraldownward, thebenevolentparty’sdisinvestmentconvergestowardthedemagogue’s,albeitneverreaching it.Thisdeathspiralisselfenforcing,leadingtozerocapitalandconsumptioninthelongrun.

Thisdownwardcycleisnotapovertytrapinwhichpeoplearetoopoortoinvest,thereby perpetuatingeconomicmisery.Rather,regardlessoftheeconomy’sintrinsicproductivity,the deathspiralisdrivenbytheheightenedelectoralpressurefromdemagogueswhencapitalis low.Moreover,ademagoguemaximizeshischancesofwinninganelectionbymaximizing spending,reducingfuturecapital.But,reducingfuturecapitalthenamplifiestheutilitydifferencebetweenthedemagogue’sdis-investmentpolicyandanyfixedinvestmentpolicy.Thus,by damagingcapitalstocksanddestroyingsocialcapitalassociatedwithinstitutionsandproperty rights,ademagogueraisesthefuturerelativeattractionofitspoliciestovoters.

4Thus,whencapitalstockislow,e.g.,duringaseveredepression,thebenevolentparty’soptimalinvestment impliesaformofconsumptionsmoothing,orcounter-cyclicalgovernment“spending”.

Wethencharacterizetheriskthatdemocracyentersadeathspiral.Evenwhencapitalstocks exceedthecriticaldeathspiralthreshold,thedemagoguealwayshasachanceofwinning,and hencecapitalstockscanalwaysfall.Thus,aneconomyisalwaysjustsomebaddrawsaway fromhavingcapitaldropbelowthedeathspiralthresholdandhencefromaninevitablemeltdown,underscoringtheshockstotherealeconomygeneratedbyelectionoutcomesthattake theformofavictorybyademagoguethatreducescapital.5

Thisdeathspiralriskcanbedescribedasagambler’sruinproblem.Inthisproblem,(i)one gamblerstartswithfinitewealthandtheothergamblerisinfinitelywealthy,(ii)thesizesofthe stepsupanddownneednotbeequal,and(iii)ruin(zerowealth)correspondstohittingacapital threshold.Weprovideconditionsunderwhichdemocracyisnotdoomed,i.e.,thereisachance ofneverenteringadeathspiral.Wederiveupperandlowerboundsontheprobabilityofadeath spiralasafunctionofthecurrentcapitalstock.Weshowthattherateatwhichtheprobability ofdeathspiralvanishesascapitalgrowsverylargeisslowerthanexponential;infact,thetail followsapowerlaw.Thisheavytailpropertymeansthatthepresenceofademagogueintroducessignificantdisasterrisktotheeconomy.Weshowthatdeathspiralstendtobelesslikely whencapitalstocksarehigherattheoutset,thebenevolentpartyselectshighvalencecandidates withhigherlikelihood,economicproductivityishigher,andtheinstitutionalconstraintsona demagogue’sdis-savingaretighter.

Ourresultshighlightcloseconnectionsbetweendemocracyanddevelopment.Democracies indevelopingeconomieswithlesscapitalaremoresusceptibletomeltdowns,becausefewer shocksareneededtodropcapitalbelowthepointwheremeltdownsbecomeinevitable.Similarly,youngdemocraciestendtohavelesssocialcapitalintheformoftrustininstitutions,makingthemmoresusceptibletonegativeshocksintheformofdemagogues.Lowerproductivity, lessinstitutionalconstraintsondemagogicpolicies,andineffectivepartiesthatfailtofrequently selecthighvalencecandidatesallexacerbatetheproblem.Ourresultsalsopointtothevalueof goodleadersinyoungdemocracieswhocanbuildenoughofacushionofcapitalstockthata countrycanwithstandthenegativeshockofaperhapsrare,butinevitablefuturedemagogue.

Finally,ouranalysispointstothevalueofinvestmentin(physicalandsocial)stockingood timeswhencapitalstockisalreadyhigh.Itisexactlyatsuchtimesthatthecountrycanbuilda cushionofcapitaltopreparefortheinevitableriseoffuturedemagogues.Hardtimes,incontrast,callforcompromiseandsomemimicryofdemagogicpoliciesasthelesserevil.Thus,for example,itisexactlyduringeconomicboomorperiodsofculturaltolerancewhendemagogues

5Addingmacroeconomicshockssuchasrealbusinesscycleshocksorwarscanalsohelpdrivecapitalbelow thedeathspiralthreshold.

areleastappealingandhencemayseemleastdangerousorrelevantthatdemocracybuildsits resilienceagainstsystemicdemagogicrisk.

Ourbaseanalysisfocusesonarepresentative“median”voter,becausethisvoter’spreferencessufficetodeterminethestrategicpolicychoicesoftheparties.Thatsaid,weshowhow ourmodelcanaccountforcross-sectionalpatternsinvoting,andhowthesepatternsvarywith economicdevelopment.Todothis,weenrichourbasemodelsothat(1)somevotersreceive moreoftheeconomicpiethanothers,and(2)thefractionofvoterswhoviewthebenevolent partytobethehighvalencecandidateisstochastic,eitherhighorlow.Thelogicofourbase modelextends.Morespecifically,whencapitalstocksarelow,itsinvestmentpolicydrawsthe supportofjustenoughvoterstowinwhenamajorityviewittobehighvalence.Moreover, itspolicyisdesignedtoappealtotheeconomicelites,whilemoreeconomically-disadvantaged voterssupportthedemagogueregardlessofvalence.Wethenshowhoweconomicdevelopment mitigatesademagogue’sappeal.Specifically,oncethebenevolentparty’spolicychoicescease tobeconstrainedbyelectoralcompetition,highercapitallevelsreduce(1)thedemagogue’s voteshare;and(2)theabilityofthedemagoguetoswayeconomically-disadvantagedvoters withitsshort-sightedpoliciesshrinks,withvalencedeterminingthechoicesofmorevoters.We alsotouchonhowotherrealworldconsiderationsaffectoutcomes,allowingforbusinesscycle fluctuationsintheformofshockstotheproductivityofcapital,aswellasthepossibilitythata demagoguewhowinscanalterinstitutionstoimproveitsfutureelectoralchances.

Toplaceourwork,itisusefultodescribethetechnicalchallengesandhowweaddress them.WecharacterizeMarkovperfectequilibriathatsatisfya“nouseofdominatedactionsin asubgame”refinement.Weshowthatthebenevolentparty’sequilibriumstrategysolvesawelldefinedconstrainedoptimizationproblem(ProblemP).However,challengearisesbecausethe “beat-the-demagogue”constraintsinProblemPisnon-linear,indeed,non-convex.Weproceed insteps.Wefirstconsiderthehypotheticalbenchmarkproblem(ProblemBP)wherewedrop theconstraintimposedbyelectoralcompetition,sothatrotationofoffice-holdersisexogenous, asinAguiarandAmador(2011).WeshowthatProblemBPisscalableincapital,permittingexplicitsolution.Inthishypotheticalscenario,deathspiralsneverarise—underscoringthe importanceofexplicitlymodelingelectoralcompetition.Wethenconsideramodifiedproblem (ProblemMP)inwhichthe“beat-the-demagogue”constraintsarereplacedbylinearconstraints. Withtheselinearconstraints,ProblemMPisalsoscalableincapital,althoughnottimeinvariant. WethenrelatethesolutionsofProblemsMPandBP.Finally,wechoosethelinearconstraint parametersofProblemMPtomapittoaversionofProblemPwithmorerelaxedconstraints. ThismappingletsusextendourcharacterizationofProblemMP’ssolutiontoProblemP.

RelatedLiterature. Anearlierdraft(Bernhardtetal.,2019a)analyzedasettinginwhich partiescouldnotdis-invest,butcapitaldepreciated,sothatcapitalwouldfallabsentinvestment. Thatframeworkalsofeaturedacontinuousvalenceshockonasupportthatcontainszero,so thateachcandidatehadachanceofhavingthehighervalence.Thebenevolentparty’sproposed capitalchoicetradedoff onthemarginbetweenthegainsfromthegreaterinvestmentwhenit wonandthecostoftheincreasedprobabilitythatthedemagoguewins.Qualitativelysimilar resultsobtain,butfocusingonabinaryvalenceshockpermitsmoredetailedandexplicitcharacterizations.Thatdraftalsoconsideredlogpreferencesoverconsumption,whichpermitexplicit solutionevenwhenvoterswerelessmyopic,anditcharacterizednon-Markovequilibria.

Bisinetal.(2015)buildathree-datemodelwithvoterswhousehyperbolicdiscounting. Votersunderstandtheirself-controlissues,andcanuseilliquidassetstopreventoverspendingat datetwo.Theyshowhowtwooffice-motivatedcandidatescanundothiscommitmentdevicevia excessivedebtaccumulation,hurtingvoters.Incontrast,votersinourmodelareunawareofhow investmentaffectsfutureconsumption.Ifweonlyhadoffice-motivatedcandidatesasintheir model,noinvestmentswouldbemade.OurpaperfocusesontheextenttowhichHamilton’s notionofagoodpoliticalleadercanbeeffectiveinthepresenceofademagogueandashortsightedmajority,howthiseffectivenesshingesontheeconomy’sstateofdevelopment,andthe roleofgoodleadershipattheoutsetofadevelopingeconomyforitslong-termprospects.

Guisoetal.(2018)defineapartyaspopulistifitchampionsshort-termpolicieswhilehiding theirlong-termcosts,andshowempirically,thathardeconomictimesleadtoincreasedsupport forpopulistsandpopulistpolicies,andthatestablishmentparties’policiesgrowmorepopulist innature.Ourtheoreticalmodelassumesthatdemagoguescanhidethelong-termconsequences ofeconomicpoliciesfromvoters,asintheirpaper.Consistentwiththeirfindings,weshowthat followinghardtimes,theneedtomimicpopulistpoliciestoappealtovotersrises,causingestablishedpartiestobecomemorepopulist.BacciniandSattler(2021)provideadditionalevidence, usingadifference-in-differenceanalysistoshowthat“austerityincreasessupportforpopulist partiesineconomicallyvulnerableregions,butausterityhaslittleeffectonvotingineconomicallylessvulnerableregions.”Withheterogeneousvoters,weshowthatthisresultobtains inourmodel—thedemagoguereceivesdifferentialsupportfromeconomically-disadvantaged votersespeciallywheneconomicconditionsarebad,whilethebenevolentpartyappealstothe elites,andthedemagogue’spolicieshavelittleimpactonvoterchoicesinadvancedeconomies.

Levyetal.(2021)developamodelinwhichpoliciesbasedonmisspecifiedmodelsofvoters areimplementedperiodicallyinperpetuity.Thesuccessesofbetter,sophisticatedpoliciescause voterswithmisspecifiedmodelsoftheworldtoinferthattheirpreferredpolicieswouldperform

evenbetter,increasingtheirturnout.Acemogluetal.(2013)buildamodelinwhichalobbycan trytobribepoliticianstoselectpoliciesthatfavorthewealthy.Intheirmodel,populistsarenot susceptibletobribesandsignalthisbychoosingextremeleft-wingpolicies.Incontrast,consistentwiththefindingsinGuisoetal.(2018),demagoguesinourmodelmaximizetheirelectoral chancesbychampioningshort-termpoliciesthatappealtoshort-sightedvoters.Wethencharacterizethelong-termconsequenceoftheelectoralcompetitionbetweensuchdemagoguesand strategic,far-sightedbenevolentpartiesthataimtomaximizevoterwelfare.

Ournotionofa“deathspiral”differsfromintertemporaldissavingcausedbyalowreturn onassetsrelativetoaconsumer’sdiscountfactor.Forexample,consideraninfinitely-livedconsumerwithCRRAutilityandalinearsavingstechnology.Thenconsumptionisalwaysafixed fractionofcurrentassets,andifthediscountfactortimesthegrossrateofreturnislessthanone, theconsumerdis-savesataconstantratesoassetsdeclineovertime,convergingtozero.This declinewouldbeacceleratediftheconsumerispresentbiased.AguiarandAmador(2011)show howexogenousprobabilisticreplacementofanoffice-holderwhocaresmoreaboutcitizenpayoffswheninofficethanwhenoutcanbeformulatedasaself-controlproblemofasinglepresentbiasedagent.HalacandYared(2014)buildonthispresent-biasedsettinginamodelwhere theoffice-holderisprivatelyinformedaboutthevalueofcurrentconsumption,whichevolves accordingtoabinaryMarkovprocess.Theycharacterizetheoptimalmechanismdesignunder differentlevelsofcommitment.Withoutex-antecommitment,thestochasticnaturegivesriseto aprecautionarysavingsmotive,whichsomewhatdiminishestheincentivetorunsavingsdown. However,ifthepresentbiasissufficientlystrong,capitalandsavingsagaingotozero.Thedynamicsareanalogoustoourbenchmarksettingwithoutelectoralcompetition,wherethereare stochasticfluctuations,butstrategiesandlong-runoutcomesdonothingeonthecapitalstateof theeconomy.Incontrast,deathspiralsinourmodelaredrivenbyintensifyingcompetitionfrom thedemagogueascapitalstocksgolow,andtheycanoccurregardlessofthediscountfactor andtheproductivityoftheeconomy.Whilethereisalwaysariskofadeathspiral,theyarenot inevitable,andtheprobabilityofafuturedeathspiraldependsonthecurrentlevelofcapital.

AsmallmacroeconomicsliteratureembedsprobabilisticvotingàlaPerssonandTabellini (2000)inadynamicmodel(e.g.,PerssonandSvensson(1989),CukiermanandMeltzer(1989), AlesinaandTabellini(1990),Songetal.(2012),Battaglini(2014)).Thesemodelsfeaturepolicy convergence:thepoliticalprocessgeneratesex-antedistortionswhenpartieschooseplatforms, butwithconvergenceitdoesnotmatterwhowins.BattagliniandCoate(2008)analyzetheeffect oflegislativebargainingongovernmentdebtandpublicgoodprovisionwhendebtconstrains publicgoodinvestmentandporkprovision,andeachdistrictisrepresentedbyalegislator.A

firstmoveradvantageinbargainingmeansthattheproposer’sidentitydetermineswhichdistrict receivesthemostpork,butpublicgoodinvestmentanddebtlevelsareunaffected.Whilethese formulationsofpoliticalcompetitionfacilitatemanyinsights,whowinstheactualelectionis irrelevant.Incontrast,inourmodeltheelectionitselfgeneratesuncertaintyanddynamiceconomicdistortions,capturingtheobservationthatelectionresults do matter(e.g.,Kellyetal. (2016)analyzetheimpactofelectoraloutcomesonforward-lookingfinancialmarkets).

Demagoguesarepoliticianswhoappealtothepeoplesolelytowinpowerforthemselves. Thetermpopulistisoftenusedinterchangeably,butcontainsaspectsthatwedonotmodel.For example,accordingtoMüller(2017),populistsclaimtorepresentthetruepeopleagainstan elitewhocontrolstheleversofgovernmentattheexpenseofthetruepeople.Asaresult,populistsbelieveitislegitimatetomoveawayfrompluralisticdemocracy,becausethey,andonly they,arethelegitimaterepresentativesofthepeople.Incontrast,inourmodel,majorityruleis alwayspreserved.Wereademagogue,instead,abletochangetherulesunderlyingfairpolitical competition,makingitharderforabenevolentpartytoregainpower,itwouldstrengthenour results.6 Inparticular,becauseitscostsoflosingwouldrise,thebenevolentpartywouldhave evenstrongerincentivestomimicpopulistpolicies.

Inourmodel,increasingcurrentconsumptioncomesattheexpenseofreducingfutureconsumption.Onemayarguethatcitizensshouldbeabletocometounderstandthislink.In reality,thislinkislessclearbecausegovernmentscanborrowforlongperiodsoftimewithout discernibleimpactsonconsumption.Wemodelthisbyassumingthatcitizensareshort-sighted. OtherpapersthatmodelvotersinthiswayincludeBaronandDiermeier(2001)andDalBo etal.(2017).Ourhistoricalcompanionpaper(Bernhardtetal.,2019b)documentstheextensive concernsoffoundingfathersofAmericanDemocracyaboutpreciselythisshort-sightedness.

1Model

Themodelextendsoverinfinitelymanytimeperiods, t = 0, 1, 2,... Thereisaconsumption goodandacapitalgood.Iftheperiod t capitalstockis kt,thenoutputis φkt,i.e.,therateof returnoncapitalis φ> 0.Outputandcapitaltogethercanbeturnedintoconsumption ct and investment it.Thatis, it canbenegative,butwelimitthisdis-investmenttonomorethana fraction δ> 0ofthecurrentcapitalstock.Thus,capitalevolvesaccordingto kt+1 = kt + it, wherethebudgetconstraintis ct + it ≤ φkt and it ≥−δkt

6Onecanalsointerpretademagogue’seffortstoweakendemocraticinstitutionsandnormsinordertoachieve short-termobjectivesasaformofreducedinvestmentinsocialcapital,withadversefutureconsequences.

Therearetwoparties,abenevolentpartyandademagogue,labeled b and d,respectively.A party’spolicyatdate t isaproposedfeasibleinvestment it.Themedianvoter’sdate t utilityis givenby

u(ct) + vP,t, where u (ct) > 0and u (ct) < 0,and vP,t isavalenceshockthatmeasurestheutilitythevoter derivesifparty P = b, d isinpower.Weinterpret vP,t asmeasuringthenon-economicpolicy aspectsassociatedwithcandidate P thatenteravoter’sutilitythatareoutofaparty’scontrol. Withoutlossofgenerality,weset vd,t = 0andwrite vt insteadof vb,t.Weassumethatthere aretwopossiblevalencerealizations, vH and vL,with vH > 0 >vL thatoccurwithprobabilities pH and pL,respectively.Thus,eithercandidatecanhavethenethighervalence.Weallow forgeneralvoterpreferencesoverconsumptioninourexistenceandgeneralcharacterizationin Section2.1.Furthercharacterizationsfocusonconstantrelativeriskaversionpreferences.

Weconsidermyopicvoterswhobaseelectoraldecisionssolelyonperiodutility.Thiscapturestheideathatvotersareunsophisticatedanddonotunderstandthelong-termimpactsof economicpolicy(Guisoetal.,2018).Ourworkingpapershowshowqualitativeresultsextend ifvotersjustunderweightfuturepayoffs(Bernhardtetal.,2019a).

Incontrasttovoters,partiesaresophisticatedandforwardlooking.Partiesdiscountfuture payoffsby β,where β ∈ (0, 1)isassumedtosatisfy β(1 + φ) > 1,sothatabsentthedemagogue, someinvestmentwouldalwaysbeoptimal.Wewillalsoimposeconditionsthatensureexpected lifetimepayoffsarefiniteandhencewelldefined.Thedemagogue, d,onlycaresaboutwinning; itreceivesaperiodpayoff of1ifitwins,and0,otherwise.Thebenevolentparty, b,ispolicy motivated,receivingthesameperiodutilityasthemedianvoter.Thisframeworknestsasetting inwhichmultiplebenevolentpartiescompete:Eachwouldofferthesameeconomicpolicy,and whenparty d loses,thebenevolentpartywiththehighestvalencewouldbeelected.

Aparty’spolicychoicecanbedescribedbytheproposedcapitallevelforthenextperiod, becauseitdeterminesthelevelsofinvestmentandconsumption.Thatis,if k isthecurrent capitalstockand k j isthecapitalstockproposedbyparty j = d, b,theninvestmentwouldbe i j = k j k,andconsumptionwouldbe c j = φk i j = (1 + φ)k k j.Thus,ineachperiodthe gameevolvesasfollows:

1. Bothpartiesproposecapitalstocksforthenextperiod.

2. Thevalenceshockisrealized.

3. Themedianvoterselectsthewinningparty,whichimplementsitsproposedpolicy.

2Equilibrium

2.1BasicPropertiesofEquilibria

WefocusonMarkovperfectequilibriaofthegame.Inparticular,thismeansthataparty’sstrategyonlydependsonthecurrentlevelofcapital, k.Aparty’sstrategydeterminesnextperiod’s capitallevel k asafunctionof k.Thecapitalchoicemustbefeasible,andcapitaldisinvestment cannotexceedfraction δ.Thus,aMarkovstrategyinourgameisdefinedasfollows.

Definition1 Partyj’sMarkovstrategyisgivenbys j : R+ → R+ suchthat (1 δ)k ≤ s j(k) ≤ (1 + φ)k,forallk ≥ 0.

Ourfirstresult,Proposition1,detailspropertiesofallequilibriathatsatisfyasimplerefinementcriterion.Therefinement,whichwenowdetail,hasamotivationsimilartothatforweak dominance.ConsiderMarkovstrategies s j, j ∈{b, d},andanysubgamethatstartswithsome capitallevel k attime t.Thesestrategiesinduceaone-shotgameinperiod t.Inthisgame, partiessimultaneouslyproposecapitallevelsforperiod t + 1,andfollowstrategies s j, j ∈{b, d}, inallperiodsafter t + 1.Ourequilibriumrefinementrequiresthattheproposedcapitalchoices notbeweaklydominatedinanyoftheseinducedgames.Moreformally,wehave:

Definition2 AMarkovperfectequilibriums j,j ∈{b, d},ofthedynamicgameusesweakly dominatedactionsifandonlyifthefollowingholds:

Thereexistsasubgamestartingatsomeperiodt,acapitallevelk,andacapitalchoice k j s j(k) forapartyj ∈{b, d} suchthatk j givesjautilitythatisatleastashighass j(k) against anycapitalchoiceofitsrival,andastrictlyhigherutilityforatleastonecapitalchoiceofits rival,assumingthatbothpartiescontinuewithstrategiess j,j ∈{b, d},inallfutureperiods.

Toseehowtherefinementworks,consideraninfinitely-repeatedgamewithastagegameof:

Playing(T, M)isanequilibriumofthestagegame,andplaying(T, M)inallperiodsisaMarkov perfectequilibrium.Supposeplayer1deviatesto M inthecurrentperiod.Thisdoesnotaffect futurepayoffs,becausebothplayersareassumedtocontinuewiththeirequilibriumstrategies.

Player2

However,inthecurrentperiodplayer1’spayoff remainsthesameifplayer2chooses M or R butitisstrictlyhigherifplayer2chooses L.Thus, T isaweaklydominatedactionforplayer1.7 Wenextprovideageneralcharacterizationofequilibriaofourelectoralgame.

Proposition1 SupposethereexistsapurestrategyMarkovperfectequilibriumthatdoesnot useweaklydominatedactions.Theninperiodt,

1. Thebenevolentpartywinselectionifandonlyifthemedianvoter’spreferenceshockis vH

2. Thedemagogue’sstrategyissd (k) = (1 δ)k.

3. Thebenevolentparty’sstrategysb(k) satisfiesu((1 + φ)k sb(k)) + vH ≥ u (φ + δ)k .If thisconditionholdsasanequality,thenthemedianvoterelectsthebenevolentparty.

Tounderstandtheintuitionfortheseresults,notethataMarkovstrategyinasubgamecan onlydependonthecurrentcapitalstock.Thus,thebenevolentpartycanalwaysensurethatit winswhenitsvalenceishighbyproposingthesameinvestmentlevelasthedemagogue.The medianvoterprefersparty b duetoitshighvalence,andparty b isalsobetteroff becauseit hasthesameperiodutilityasthemedianvoter.However,thisdeviationdoesnotaffectfuture payoffs,becausenext-period’scapitalwouldbeexactlythesameaswhenthedemagoguewins. Therefore,futurepayoffsareunaffected.Thestrictincreaseincurrentpayoff makesthisdeviationprofitableforthebenevolentparty.Thus,party b mustwinwhenitsvalenceis vH .Ananalogousargumentyieldsthatthedemagoguemustwinwhenthebenevolentparty’svalenceislow. Nextobservethatitisaweaklydominantactionforthedemagoguetomaximizethecurrent consumptionofferedtovoters,whichitdoesbychoosingthelowestpossiblecapitallevelinthe nextperiod,i.e., sd (k) = (1 δ)k.Bydoingthis,thedemagogueensuresvictoryforthelargest possiblesetofproposedinvestmentlevelsbyhisrival.Further,changingtheactioninthe currentperioddoesnotaffectthedemagogue’sfuturepayoffs—asestablishedabove,thedemagoguewinsinfutureperiodsifandonlyifthevalenceis vL andisthereforeindependentofcapital.Therefore,anyotherfuturecapitalchoicebythedemagogueisaweaklydominatedaction. Havingestablishedthat(i)thedemagoguealwaysproposestodisinvestmaximally,and(ii) thebenevolentpartymustwinwhenitsvalenceis vH ,thethirdstatementoftheProposition

7Notethatastrategyofplaying T inallperiodsindependentofhistoriesisnotaweaklydominatedstrategy. Toshowthis,itsufficestofinda(possiblynon-Markov)strategyforplayer2suchthatanystrategyotherthan alwaysplaying T makesplayer1strictlyworseoff.Forexample,player2coulduseatriggerstrategy,choosing action M aslongasplayer1chooses T ,butselecting R ifplayer1chooses M or B inanypreviousperiod.Then player1’sex-anteutilitystrictlydecreaseswhenswitchingtoanyotherstrategy.

followsdirectly.Inparticular,thetotaloutputavailableinperiod t is φk.If k isthecapitallevel chosenbythewinner,theninvestmentis k k,soconsumptionis φk (k k) = (1 + φ)k k .

Thevoter’sutilityfromparty b whenithashighvalenceistherefore u((1+φ)k sb(k))+vH .The voter’sutilityfromthedemagogueis u((1+φ)k sd (k)) = u((1+φ)k (1 δ)k) = u((φ+δ)k).Thus, u((1+φ)k sb(k))+vH ≥ u (φ+δ)k mustholdinorderforparty b towinwhenitsvalenceishigh.

2.2TheBenevolentParty’sStrategyandEquilibriumExistence

Proposition1showsthat,inequilibrium,thedemagogueproposestodis-investby δkt inany period t.Ouranalysisproceedsbydeterminingthebenevolentparty’sbestresponsetothis strategy.WeshowthatthisbestresponseisMarkovian,i.e.,itdependsonlyonthecurrent capitalstock kt.Inturn,thismeansthatalonganequilibriumpath,theextantcapitallevelonly dependsonthehistoryofvalencerealizations.Itfollowsthatthehistoryofvalencerealizations isasufficientstatisticfortheequilibriumpathofcapital.

Let ht denotethehistoryofvalencerealizationsuptobutnotincludingtimeperiod t.Thus, h0 = ∅.Let Ht bethesetofallsuchhistories.Then ht = (ht 1,vt 1) ∈Ht,whentheperiod t 1valencerealizationis vt 1.Let P(ht)betheprobabilityofhistory ht.Thisprobabilitycan bedefinedinductivelyby P(ht 1,v) = pvP(ht 1)for v ∈{vH ,vL},where P(h0) = 1.

Wedenotethecapitallevelattime t by kht tocaptureitsdependenceonthehistoryofvalencerealizations.Thishistoryfullydetermineselectoraloutcomesandpolicychoicesalong theequilibriumpathinallperiodspriorto t.Thebenevolentpartyproposesacapitallevel k for thenextperiod t + 1.Proposition1yieldsthatthispolicywillbeimplementedinperiod t ifand onlyifthevalencerealization vt = vH .Thus, kht ,vH = k .Ifthevalencerealizationis,instead, vt = vL,thenProposition1revealsthatthedemagoguewinsand kht ,vL = (1 δ)kht .Thecapital choicesdeterminethecurrentconsumptionlevel.Weusethisnotationanddropthetimeindex onthevalenceshockswherethecontextisclear.

Intheanalysisthatfollows,weassumethefollowingsufficientconditions.

Assumption1

Utilityfromconsumptionmustbewelldefinedwhendisinvestmentismaximal,i.e., ∞ t=0 β

u((φ+ δ)k(1 δ)t)mustconverge.Thisconditionissatisfiedifandonlyifcondition1ofAssumption1

holds.WithCRRApreferences, u(c) = c1 s 1 s ,itisequivalenttorequiring β(1 δ)1 s < 1.Inaddition,delayingconsumptiontoinfinitymustresultinzeroutility,i.e.,lim

Thisholdsifandonlyifcondition2inAssumption1holds.ForCRRAutilitywith s > 1this isautomaticallysatisfied,becauseutilityisboundedfromabovebyzero.

Lemma1 Partyb’sequilibriumstrategysolvesthefollowingoptimizationproblem:

ProblemP

Thisoptimizationproblemiswell-definedsincetheinfinitesumofutilitiesintheobjective convergebyAssumption1.Constraint(2)istheconditioninstatement3ofProposition1that party b winsifandonlyifitsvalenceishigh.Inparticular,thevotermustweaklypreferthe benevolentpartygivenitsproposedfuturecapitalstockwhenitsvalenceishightoademagogue thatproposestoincreasecurrentconsumptiontothemaximumextentpossible.Recallthatthe benevolentpartyreceivesthesameperiodutilityastheconsumer,whichincludesthevalence ofthewinningcandidate.Becausethebenevolentpartywinsifandonlyifitsvalenceishigh, thisconstraintmeansthatwecandropthesevalencetermsfromitsoptimizationproblem.

Constraint(3)reflectsstatement2ofProposition1,i.e.,thatthedemagoguemaximizes consumptioninthecurrentperiod,andhencedeterminesnextperiod’scapitalstockwhenthe valenceis vL.Constraint4reflectstheconditionthatcapitalcannotbereducedatarateexceeding(1 δ),and(5)isafeasibilityconstraintensuringthatconsumptionisnon-negative.Wewill provethatthelasttwoconstraintsareslack.

Thebenevolentparty’sobjectivefunctionisconcave.However,theconstraintsetinProblemPisnotnecessarilyconvexbecauseofconstraint(2).Thus,thesolutiontothisoptimization problemdoesnotneedtobeunique.Note,however,thatparty b’optimalinvestmentstrategy at t onlydependsonthecurrentcapitallevel kht .Thus,thefollowingresultobtains.

Proposition2 ThereexistsaMarkovperfectequilibriumthatdoesnotuseweaklydominated actions.Inthisequilibrium,thedemagoguemaximallydisinvestsineveryperiodandwinsif andonlyifthevalenceis vL.Thebenevolentparty’sstrategysolvesoptimizationproblemP.

Intheremaininganalysis,weassumethatreducingconsumptionbyafixedpercentagehas ahigherutilityimpactonconsumerswhenconsumptionislower,i.e., u(c) u(γc)isdecreasing in c for0 <γ< 1.Thisassumptionsaysthatitismoredifficulttoconvinceshort-sightedvoters tosaveinbadtimes;thismakesithardertosellausterityinbadtimes.Withconstantrelative riskaversionvoterpreferences, u(c) = c1 s 1 s ,thisconditionimpliesthat s > 1,consistentwith mostmacroeconomiccalibrations.8 Wenowfocusonthiscase.

2.3Benchmark:ExogenousWinningProbabilities

Webeginwithabenchmarkprobleminwhichthereisexogenousstochasticrotationofoffice holdersasinAguiarandAmador(2011).Thatis,thereisnoelectoralcompetition.Appliedto ourmodelthismeansdroppingconstraint(2)—thebenevolentpartyanddemagoguewinwith exogenousprobabilities pH and pL,respectively,regardlessofthemedianvoter’spreferences. WelabelthisbenchmarkproblemasProblemBP.Let VBP(k)denotethediscountedpayoff that thebenevolentpartyexpectsinthisbenchmarkproblemwhenthecurrentcapitalstockis k.We startbycharacterizingsolutionsofthisbenchmarkproblem.

Proposition3 Inthebenchmarkproblem,ProblemBP,

1. Partyb’svaluefunctiontakestheformVBP(k) = abk1 s/(1 s),whereab > 0 isaconstant.

2. Thebenevolentparty’scapitalchoiceforthenextperiodisgivenbyk = (1 + λ∗)k,where kisthecurrentcapital,and λ∗ solves

3. Partybalwaysproposestoincreasethecapitalstock: λ∗ ∈ (0,φ)

4. λ∗ decreasesintheprobabilitypH thatpartybwins,butincreasesintheextent δ towhich ademagoguecandis-invest,thediscountfactor β,andtheproductivity φ oftheeconomy.

Setting pH = 1yieldsthebenevolentparty’soptimalinvestmentpolicywhenitdoesnot facethethreatofademagogue.Withnopoliticalcompetitionfromademagogue,wehave λ∗ = (β(1 + φ)) 1 s 1 > 0.

8u(c) u(γc)decreasesin c if u (c) γu (γc) < 0.Thisconditionholdsif γu (γc)isstrictlydecreasingfor γ ∈ (0, 1).Differentiating γu (γc)withrespectto γ showsthatthisholdsifandonlyifrelativeriskaversionstrictly exceeds1.

InproblemBP,eventhoughthelikelihoodthatthebenevolentpartywinsisexogenous,the presenceofthedemagoguestillaffectstheparty b’schoiceof λ∗.Infact,themorelikelythat thedemagogueistowin,themorethebenevolentpartywillsaveinordertoinsureagainstthe possibilitythatthedemagoguewinsandspendsdowncapital.Withouttheelectoralconstraint (2),thebenevolentpartyisfreetoproposegreatersavingsinordertoprovidegreaterinsurance.

2.4CharacterizationoftheBenevolentParty’sEquilibriumStrategy

Wenowanalyzesolutionsofthebenevolentparty’soptimizationproblem(problemP),inwhich constraint(2)captureselectoralcompetitionfromthedemagogue.However,thisconstraintis non-linearandnon-concave,resultinginanon-convexconstraintset,whichsignificantlycomplicatestheanalysis.

Similarlytothebenchmarkproblem,wenowdescribetheinvestmentratebyparameters λ(k),where k = (1 + λ(k))k.Wesawthatinthebenchmarkproblemtheoptimal λ(k)isconstant.Incontrast,thepresenceofconstraint(2)willcausetheoptimalinvestmentratetodepend onthecapitalstock.ForCRRAutility,werewriteconstraint(2)as

Giventhecurrentcapitalstock k,let λc(k)betheproposedinvestmentrateatwhichtheconstraintbinds.Then

Notethat,when s > 1, λc(k)isstrictlyincreasingin k.Thatis,whencapitalstocksarehigher, thepresenceofthedemagogueislessbindingonthebenevolentparty’scurrentperiodinvestmentproposals.Wenowprovideageneralcharacterizationofparty b’sequilibriumstrategy.

Proposition4 Letkbethecurrentcapitalstock,andletk∗ solve λ

(k∗) = λ∗,where λc( ) is definedin (7).

1. IfpH < 1,thenthereexists ˆ k > k∗ suchthatthebenevolentpartyproposesfuturecapital stockk = (1 + λc(k))kifk ≤ ˆ k,anditproposesk > (1 + λ∗)kifk > k∗

2. IfpH = 1,thenthebenevolentpartyproposesfuturecapitalstockk = (1+min{

3. Theinvestmentshare λ(k) convergesto λ∗ ask →∞

k)})k.

Proposition4conveystheessenceofhowthedemagogue’spresenceaffectsthebenevolent party’sequilibriumpolicychoices.Whencapitallevelsarelow,voterscareprimarilyabout

policyvis-à-visvalence,sothedemagogue’swillingnesstoappealtoashort-sightedvoterinits policybindsonthebenevolentparty.Thisconstrainsthebenevolentparty,causingittomimic thedemagogue’spolicy,butbytheminimumamountneededtoensurethatitwins.Ascapital risestoward k∗,voterscarerelativelylessaboutpolicy,allowingthebenevolentpartytoreduce theextentbywhichitmimicsthedemagogue,increasinginvestment.

Oncecapitallevelsexceed k∗,thebenevolentpartycouldoffertheunconstrainedinvestment λ∗ associatedwiththebenchmarkprobleminwhichthereisnoelectoralcompetition.However, party b choosestoinvestevenmore.Infact,itturnsouttheelectoralcompetitionconstraint (2)stillbindsforsomecapitallevelsabove k∗.Thisover-investmentreflectsaprecautionary savingsmotive—party b imposesausterityinordertoensurethattheelectoralcompetitionconstraintwouldbesomewhatlessbindingifthedemagoguewinsinthefutureanddrivesdown capital.Thisprecautionarysavingsmotiveagainstthepossibilityoffutureelectoralconstraints isoverandabovethatinthebenchmarkproblem,where λ∗ alreadyreflectsinsuranceagainstthe exogenousprobabilitythatthedemagoguewillwinanddis-invest,drivingdowncapitalstocks.

Ascapitalgrowsarbitrarilyhigh,capitallevelsatwhichthecompetitionconstraintbinds canonlybereachedinthedistantfuture.Discountingensuresthatthevalueofprecautionary savingsgoestozero,andhencethatinvestmentconvergesto λ∗.Itfollowsthatthebenevolent party’sinvestmentpolicy,i.e., λ(k),evolvesnon-monotonically:If k < ˆ k then λ(k)isstrictly increasing(and λ( ˆ k) >λ∗),while λ(k)approaches λ∗ fromaboveas k →∞.

Intherestofthissectionweexplaintheconstructionoftheproofanditsintuition.Themain ideacanbegleanedmosteasilywhen pH = 1,sothatparty b alwayswins.Ignoringthelinear constraints,whichdonotaltertheintuition,party b’soptimizationproblemtakestheform

(8) where ht(kt)isstrictlyincreasingin kt.

0

Let {k∗ t }t∈N beasolutiontothisoptimizationproblem,andsupposethatsomeoftheconstraintsbind.Let W1(k1)bethecontinuationvalueinperiod1.Nextdefinetheinvestment shares λt = k∗ t+1/k∗ t 1foreachperiod.Thenifinvestmentsharesarefixedat λt forall t andwe startwithcapitallevel k1 at t = 1,thecontinuationvalueinperiod1isgivenby

Bydefinition, V1(k∗ 1) = W1(k∗ 1).Nowconsidertheexperimentofincreasingcapitalfromits initiallevel, k∗ 1.Theconstraintsinoptimizationproblem(8)arestillsatisfied,becauseeach ht is

increasingincurrentcapital.Thus, V1(k) ≤ W1(k)for k ≥ k∗ 1.Combining V1(k∗ 1) = W1(k∗ 1)and V1(k) ≤ W1(k)for k ≥ k∗ 1 yieldsthatif W isdifferentiablethen V1(k∗ 1) ≤ W1(k∗ 1).

Thenextkeystepistoshowthatthemarginalutility V1(k∗)islargerthanthemarginalutility VBP(k∗)fromtheunconstrainedbenchmarkproblem.Becausewehavedroppedalllinear constraintsthebenchmarkproblemis(8)withouttheconstraints kt+1 ≤ ht(kt).AsProposition3 shows,suchunconstrainedproblemsarescalableincapital,whichimpliesthatthevaluefunctiontakestheform VBP(k) = aBPk1 s,where aBP isaconstant.Withlinearconstraints,itcan similarlybeshownthat V1(k) = aM k1 s,forsomeconstant aM .Iftheconstraintsbind,then V1(k∗ 1) < VBP(k∗ 1),i.e.,ahigherutilitycanbeachievedwithouttheconstraints.Because s > 1, thisimplies aM < aBP < 0,which,inturn,implies

Abovewehaveestablishedthat W1(k∗ 1) ≥ V1(k∗ 1).This,and(10)yield W1(k∗ 1) > VBP(k∗ 1),i.e., themarginalproductofcapitalinperiod1ishigherinproblemPthaninproblemBP.However, thismeansthatiftheconstraintintheinitialperiod t = 0isslack,thentheoptimalinvestment islargerthaninproblemBP: λ(k0) >λ∗.Thisobservationhastwokeyimplications.First,if theconstraintisslackthen λ(k0) >λ∗,i.e.,thereisover-investmentrelativetothebenchmark modelwithoutelectoralcompetition.Second,if λc(k0) <λ∗ thentheconstraintmustbind. Thus,investmentfor k < k∗,where k∗ isdefinedby λc(k∗) = λ∗ isdeterminedbytheconstraint. Bycontinuity,when pH < 1theconstraintcontinuestobinduptosomelevel ˆ k > k∗

Thisintuitionextendstoanytimeperiod t > 0,anditdoesnotrelyontheassumptionthat pH = 1.Todealwiththemoregeneralcase,weturn(9)intoaseparateoptimizationproblem. Inaddition,wecircumventassumingthat W isdifferentiablebyproceedingbywayofcontradiction.Thisisnecessarybecausestandardresultsthatestablishtheequivalenceoftherecursive approachfailbecauseutilityisnotboundedfrombelowandtheconstraintsetisnon-convex.In fact,thevaluefunctionisnotdifferentiableatthepointwheretheconstraintceasestobind.

2.5DeathSpirals

Wesaythatcapitalstocksexhibitadeathspiralifthereexistsacapitallevelbelowwhichcapitalstocksdeclinethereafterconvergingmonotonicallytozeroatanacceleratingrate.From (7), λc(k)isincreasingin k with λc(0) = δ< 0,andhencethereisacapitalstockthreshold k solving λc(k) = 0.Thismeansthatoncecapitalfallsbelow k,theneventhebenevolentparty willproposedis-investment: λc(k) < 0.Thus,regardlessofwhowinstheelection,capitalwill

continuetofallevenfurther.Moreover,because λc(k)decreasesincapital,thefallaccelerates ascapitalstocksshrinkfurther,resultinginadeathspiral.

Proposition5 Let ksatisfy λc(k) = 0,i.e.,

Thenadeathspiraloccurswithprobability1ifandonlyifeitherk

Thepropositionhighlightsthatoncecapitalfallsbelow k,thebenevolentpartymustproposetodis-saveinordertohaveachanceofbeatingthedemagogue.Asaresult,capitalstocks fallfurther.Thisdeclineforcesthebenevolentpartytoincreasinglymimicthedemagogue’s dis-investmentpolicyevenwhen pL issmall,i.e.,evenwhenthedemagogueistypicallyunpopularwithlittlechanceofwinning.Ascapitalstocksspiraldownwardfurthertowardzero,the benevolentparty’sinvestmentpolicyconvergestowardthedemagogue’s,albeitneverreaching it,ashavingahighervalencestillprovidesthebenevolentpartysomeadvantage.

Corollary1 Thedeathspiralcutoff kdeclineswithproductivity φ,butriseswiththedemagogue’sdis-savingcapacity δ.Moreover,themarginaleffectofhigherproductivityinreducing kishigherwhenthedemagoguehasahigherdis-savingcapacity: ∂2k ∂δ∂φ < 0.

Inmoreproductiveeconomies,capitalstockshavetofallfurtherbeforeadeathspiralensues.However,whenthereareweakerinstitutionalconstraintsonthedestructionofcapital stock,sothat δ ishigher,adeathspiraloccursatahighercapitalstock.Corollary1alsoshows thatthemarginaleffectofhigherproductivityinreducingthedeathspiralcriticalthresholdis higherwheninstitutionalconstraintsonthedestructionofcapitalstocksareweaker,sothat δ is higher.Animplicationisthattheeffectofproductivityshocks(e.g.,intheformofnewtechnologies)inpreventingdeathspiralsismorepronouncedincountrieswithweakerinstitutional constraintsonpolicies(e.g.,innewerdemocracieswithweakerjudiciaries).Inspectionof λc(k) indicatesthatitalwaysexceeds δ andthatithasthesamecomparativestaticsasthosefor k, implying,forexample,thatdeathspiralsproceedmoreslowlyinmoreproductiveeconomies.

2.6EconomicDevelopmentandtheRiskofDeathSpirals

Weshowedthatoncecapitalstockfallsbelowathreshold,thepresenceofademagogueleads toadeathspiral.Wenowinvestigatetheriskofenteringadeathspiralwhenthecapitalstock ishigher.Wefirstidentifyalowerboundontheprobabilityofenteringadeathspiralgiven

anycurrentcapitalstock k.Wethenidentifyconditionsunderwhich,evenwhenthebenevolent partyisabletogrowthecapitalstockarbitrarilyhigh,democracyisstilleventuallydoomedto enteradeathspiralwithprobabilityone.Onemightthinkthatbecausethedemagoguealways hasachanceofwinninganddrivingcapitalstocksdownbelowthecriticallevelthatadeath spiralmaybeinevitableinthelongrun.Weshowthatthisisnotso.Weidentifysufficient conditionsforeconomicdevelopmenttomitigatethethreatofademagogueandtheriskofa futureimplosionofdemocracy.Thatsaid,weshowthatsomerisktodemocracyalwaysremains.Specifically,weestablishthatascapitalgrowslarge,theprobabilityofadeathspiral onlydeclinesaccordingtoapowerlaw.

Proposition6 1. Ifk > k,thenadeathspiraloccurswithaprobabilityofatleastp1+α L , where α = log(k/k)/ log(1 δ).

2. If (1 δ)pL (1 + λ∗)pH < 1,thenadeathspiraloccurswithprobability1regardlessofthe currentcapitallevelk.

3. Suppose (1 δ)pL (1 + λ∗)pH > 1 andletQ(k) betheprobabilityofenteringadeathspiral startingfromk > k∗ > k.Fixanarbitrary > 0.ThereisaconstantC ∈ (0, 1) suchthat

where y istheuniquesolutionintheinterval (0, 1) to

Thefirstresultreflectsthatevenwhenthecurrentcapitalstock k exceeds k∗ > k,thedemagoguemaywinanygivenelectionwhen pL > 0.Ifthedemagoguewinsenoughtimes,thenthe demagogue’sdis-savingwilldrivecapitalstocksbelow k,atwhichpointadeathspiralensues. Thisresultrevealsthatpoordemocraciesaremorevulnerable,aswhen k islower,ademagogue needstowinfewertimesbeforeadeathspiralensues.Interpreting k asincludingwell-defined propertyrightsandthesocialcapitalassociatedwiththeinstitutionalnormsofdemocracy,this resultfurthersuggeststhatyoungerdemocraciesaremorevulnerable.

Thesecondandthirdresultsidentifyconditionsunderwhichadeathspiralisavoidableand characterizetherateatwhichtheprobabilityofdeathspiralvanishesascapitalgrowsverylarge. Twoconditionsdeterminewhetheradeathspiralisavoidable: k > k and(1 δ)pL (1 + λ∗)pH > 1.

Recallinghowtheenvironmentinfluences λ∗ fromProposition3,thisresultestablishedthat if (i)ademocracyisluckyenoughattheoutsettodrawgoodleaderswhogrowcapitaltosome

level k > k∗,(ii)demagoguesaresufficientlyunlikelytowin(pL = 1 pH issmall),(iii) productivity(φ)issufficientlyhigh,and(iv)thedemagogue’sabilitytodis-saveissufficiently limited(δ issmall),thentheprobabilityofadeathspiralisboundedawayfromone.Insuchcircumstances,when k islarge,electingademagoguecausesdamage,butthedemocracyislikely resilientenoughtorecover.Thatsaid,thedisasterriskofadeathspiralonlyfallsslowlyas capitalgrowslarge,decliningaccordingtoapowerlaw.Thatis,thepresenceofademagogue introducessignificanttailrisktotheeconomy.

Ourproofcaststheproblemasa“gambler’sruinproblem.”Theclassicformulation,which datesbacktoaletterfromBlaisePascaltoPierreFermatin1656,considerstwoplayerswho beginwithfixedstakesandapossiblybiasedrandomwalkthatdeterminesthedirectionofaunit transferfromoneplayertotheotherthatcontinuesuntiloneoftheplayersis“ruined”byreachingzero.Feller(1968)analyzesmoregeneralrandomwalksandallowsforoneplayertobe infinitelyrich.Wecharacterizethegeneralizedrandomwalkthatmovesup kH stepswithprobability pH andmovesdown kL stepswithprobability pL.Wethencharacterizetheprobability thatthecapitalstock k everfallstocapitalstock k (“ruin”).Thepowerlawtailarisebecauseof therandomproportionalgrowthnatureoftheevolutionofcapital.Inparticular, k

= (1 + x)kt, where x isarandomvariablethattakesthevalueof δ withprobability pL (whenthedemagogue wins)andthevalueof λ(kt)withprobability pH (whenthebenevolentpartywins),where λ(kt) ≈

λ∗ forlargecapital.AsGabaix(2016)highlights,proportionalgrowthmodelsoftenunderliethe emergenceofpowerlaws,andconsistentempiricalevidencehasbeenfoundinmanysettings.

Toillustratethisresult,supposethatthereexistsaninteger j,suchthat1 log(1+λ∗)/ log(1 δ) = 1 + 1/ j.Forexample,if(1 + λ∗)(1 δ) = 1then(13)reducestothequadraticequation pH y2 y + pL = 0,withsolution y = pL/pH .Thenwhen pL < pH ,(12)implies

where canbemadearbitrarilysmall.Forexample,if pH = 0 9and δ = 0 3thentheexponent isapproximately 6.16.Thus,a50%increaseincapitallowerstheboundontheprobabilityof fallingbelow k∗ by92%,substantiallyreducingthechanceofenteringadeathspiral.

Nowsupposethatproductivity φ isloweredtothepointwhere(1 + λ∗)2(1 δ) = 1.Then equation(13)becomes pH y3/2 y + pL = 0,withsolution y = pL 1 + pH + pL + 3pH pL 2p2 H .

Now,when pH = 0.9and δ = 0.3,theexponentbecomes 5.23.Thus,a50%increaseincapital reducestheboundontheprobabilityoffallingbelow k∗ by88%.

Proposition3providescomparativestaticsfor λ∗,forexample,that λ∗ isincreasinginthe productivity, φ.Thediscussionaboveshowsthatsuchaproductivityreductionmayinitially onlyhaveasmallimpactontheprobabilityofenteringadeathspiral,providedthattheestablisheddemocracyhasreachedahighlevelofcapital.However,ifaneconomysuffersa sufficientpermanentdecreaseofproductivity,thenthesecondstatementofProposition6shows thatadeathspiralbecomesinevitable.

Forexample,suppose pH = 0.9, δ = 0.3(asabove), s = 1.2,and β = 0.9.Iftheproductivity φ dropsto15%orlowerthenthesecondstatementofProposition6applies,anddeathspirals occurwithprobability1,butfor φ = 0.16,statement3applies.Inthislattercase, y = 0.983and a50%increaseincapitalreducestheprobabilityofdroppingbelow k∗ byonly2%,butdeath spiralsarenotinevitable.However,furtherincreasesinthereturn,evensmallones,reducethe probabilityofdeathspiralssubstantially:if φ = 17%thena50%increaseincapitalreduces theprobabilitybyalmost30%,andat φ = 18%thatreductionis45%.Thus,smallchangesin productivitycansometimeshaveverylargeimpactsonthelong-termstabilityofdemocracies.

Summarizingthecontentofthesefindings,ifattheoutset,acountryhasthegoodfortuneof electingbenevolentleaders,thoseleadersmaygrowthecapitalbysomuchthatthereisagood chanceofforestallingaretreatbelowthecurrenthighlevel.Concretely,enlightenedleadership byWashington,Adams,Madison,...maybeabletobuildenoughinstitutionalcapitaltoforestalltheadverseeffectsoflateroccasionallydrawingademagogue.If,instead,acountryhas themisfortuneattheoutsetofdrawingafewdemagogues,thenitmaybedoomedthereafter.

2.7HeterogeneousVoters

Tothispointouranalysishasfocusedonasingledecisivevoterbecausethisvoter’sbehavior fullypinsdownthestrategicbehaviorofthetwoparties.Inthissection,weintroducevoterheterogeneity,sothatsomevotersreceivemoreoftheeconomicpiethanothers.Thisletsusdraw insightsintowhichvotertypessupportthedemagogue,andwhichonessupportthebenevolent party,andhowthedegreeofdevelopmentintheeconomyaffectsthesupportofeachcandidate. Weconsideracontinuumofmeasureoneofvoters.Voter j obtainsaperiodpayoff of u(w j(φk i)) = (w j c)1 s 1 s whenthecurrentcapitalstockis k andinvestmentis i,where s > 1.Here, w j ∈ [a, b],with a > 0,isameasureofvoter j’sclaimtotheeconomicpie—avoterwithahigher w j receivesmore.Weassumethat w j isdistributedinthepopulationaccordingtoastrictly increasingcdf G.Wealsoallowvoterstodisagreeonthenetvalenceattachedtothebenevolentparty.Votersreceiveidiosyncratic,conditionally-independent,voter-specificshocks,where withprobability pH ameasure α> 1/2ofvotersassigns(net)valence vH > 0tothebenevolent

party,andwithprobability pL = 1 pH ,ameasure α> 1/2ofvotersassignsitavalence vL < 0.

Themultiplicativestructureofperiodpayoffsfromconsumptionmeansthatthebenevolentpartyhasthesameoptimalpolicychoiceforeachvoter.Thismeansthatitsobjectiveis unchangedfromourbasecasesetting.Ofnote,thismultiplicativestructurepreservesthescalabilityofProblemMP,allowingustouseourexistinganalysistocharacterizeindividualvoting choices.Thekeytocharacterizationthenreducestoidentifyingthedecisivevoterthatthe benevolentpartymusttargetinordertowinwhenamajorityofvotersfindittobehighvalence.

FromProposition2,thebenevolentpartywinsifandonlyifamajorityofvotersfindsitto behighvalence.Therelativepreferencesofdifferent wi votersoverparty b anddemagogueare orderedinthesamewayregardlessofthecapitallevel—higher wi voterswiththesamevalence shockhavearelativelyhigherpreferenceforparty b.Thefollowingresultisimmediate.

Lemma2 Voter wm isdecisive,where wm solves α(1 G(wm)) = 1/2.Partybwinswhenthe capitalstockiskht ifandonlyifamajorityofvotersfinditsvalencetobehighanditsproposed futurecapitalstockkht ,vH satisfies

Ifthedecisivevoter wm weaklypreferspartyb,thensodoallvoterswith wi >wm whoattach valence vH topartyb.Ifthedecisivevoterisindifferentbetweenthecandidates,thenallvoters with wi <wm preferthedemagogue.

Onecanpartitionvoterswhofindparty b tohavevalence vH intothosewithalargeenough share wi tosupportit,andthosewhoseshare wi istoosmall.Thus,themodelpredictsthatparty b drawsitssupportfromthe“economicelites,"whilerelativelyeconomically-disadvantaged voterssupportthedemagogue.Itfollowsthatparty b’sproposedcapitalchoicessolveProblem P,withconstraint(2)replacedbyconstraint(14).Thus,thereexistsa ˆ k suchthatfor k ≤ ˆ k, party b’scapitalchoicesolves(14)atequality,mimickingthedemagoguebytheminimumextentneededtodelivervictory,whilefor k > ˆ k,constraint14isslack,anditsvoteshareishigher.

Proposition7 Amongvoterswhofindthebenevolentpartytobehighvalence,

• Thosevoterswith wi ≥ wm alwaysselectthebenevolentparty.

• Fork ≤ ˆ k,onlythosevoterssupportthebenevolentparty.

• Fork > ˆ k,voterswith wi <wm butsufficientlycloseto wm selectthebenevolentparty.

• Ask →∞,allvoterswhoattachahighvalencetothebenevolentpartyselectit.

Theseresultsindicatethatthebenevolentpartydrawsitselectoralsupportfrom“elites”who obtainsufficientlylargesharesoftheeconomicpieandalsoattachavalence vH toparty b.In contrast,thedemagoguewinsallvoterswhofindparty b tobelowvalenceplusthosewhoattach ahighvalencetoparty b butreceivesufficientlysmallsharesofthepie.Thesefindingsdonot reflectthateconomically-disadvantagedvotersaremoreshort-sightedthantheelitesorthatthey donotmindthefactthatademagoguehasalowvalence.Rather,theyreflectthatdisadvantaged, low wi,voterscaremoreaboutthepoliciesproposed,andrelativelylessaboutvalence.

Whencapitalstocksaresmallenough,party b isconstrainedbytheneedtodesignitspolicy toappealtovoter wm whofindsthatithasahighvalence.Asaresult,thedemagoguewinssupportofallvoterswith wi <wm.However,ascapitalstocksgrowlarge,thedemagogue’spolicy appealsfadeinimportancerelativetotheconsiderationsthatvotersplaceonvalence.Asaresult,amongvoterswhofindparty b tobehighvalence,thedemagogueonlywinssupportdrawn fromthoseatthebottompercentilesoftheeconomicstrata.Further, wi isboundedawayfrom zero,sothatoncedevelopmentlevelsgrowhighenough,valenceconsiderationsratherthanpolicydeterminevotingchoicesofallcitizens,includingthepoor,limitingademagogue’sappeal.

ThefinalstatementofthePropositionindicatesthatthevoteshareofthedemagogueshrinks whencapitalbecomeslarge.Inparticular,forsufficientlyhighcapitallevelsallvoterswhoattachalowervalencetothedemagoguevoteforthebenevolentparty:

Corollary2 Ifcapitallevelsarelowenoughthatconstraint(14)binds,thendemagoguereceivesvoteshare 1/2 withprobabilitypH andshare α + (1 α)G(wm) withprobabilitypL. Ifcapitallevelsarehighenoughthatconstraint(14)doesnotbind,thenthedemagogue’svote shareisstrictlylower,andforsufficientlyhighcapitallevels,thedemagoguereceivesvoteshare 1 α withprobabilitypH andvoteshare α withprobabilitypL. 9

2.8TheRoleofRelativeRiskAversion s > 1

Acentralassumptionofourmodelisthat u(c) u(γc)decreasesin c for0 <γ< 1.WithCRRA preferences,thisimpliesthatthecoefficientofrelativeriskaversionsatisfies s > 1.Byraising thesalienceofinvestmentpolicydifferenceswhencapitalislow, s > 1makesithardertosell austerityinbadtimes.Tounderstandhowradicallyoutcomesareaffectedifwechangethis assumption,supposethat s < 1,andwritetheelectoralcompetitionconstraintonparty b as: (1 s)vH k1 s ≥ (φ + δ)1 s (φ λ)1 s (15)

9Whenconstraint(14)doesnotbind,asufficient,butnotnecessary,conditionforthedemagogue’svoteshare tobedecreasingincapitalisfor λ(k)tobedecreasingin k,whichwefindnumerically.

Nowinhardtimes,wherecapitalstocksgoverylow,theleft-handsidegoestoinfinity—policy choicesbyparty b ceasetoaffectthechoicesofvoters,leavingparty b completelyunconstrained initschoiceof λ(k).Inparticular,iftheeconomystartsatasmalllevelofcapital k,thenittakes manyperiodsuntiltheconstraintstartstobind.Asaconsequence, λ(k)iscloseto λ∗ forsmall k.This,inturn,meansthataslongas φ issufficientlyhighthat(1 + λ∗)pH (1 δ)pL > 1then k = 0isneveranabsorbingstate,insharpcontrasttowhen s > 1.Further,(15)implies

For k > k,thecompetitionconstraintimpliesthatparty b mustdis-investtowin.Thus, k becomesareflectingboundarywhen pH < 1,aslongascapitalstartsbelow ˜ k.Itfollowsthat whenever(1 δ)pL (1 + λ∗)pH > 1, capitallevelsmovestochasticallyinanintermediaterange, neverbecomingverysmallorverylarge.

Insum,when s < 1,deathspiralsneverarise,butconverselythepotentialforgrowingthe economyverylargeisdestroyedbytheelectoralcompetitionfromthedemagogue.Inparticular, s < 1isatoddswithrealworldobservation,asitimpliesthattheattractionofpopulistsis smallestwhencapitalislow,andhighestwhencapitalishigh.

2.9ChangingtheRules

Ouranalysispresumesthatademagoguewhowinsofficecannotalterthelegalrulesandinstitutionstoincreaseitsprobability pL ofwinning.Inpractice,thedemagoguemaybeable togerrymanderelectoraldistrictstofavorthosewhosupportthedemagogue,ortodisenfranchiseorraisethecostsofvotingtoindividualswhomightbepre-disposedtovotingagainstthe demagogue.Suchchangeswouldincreasethedemagogue’sfuturechancesofwinning.

Onecanincorporatesuchapossibilitybymodifyingourmodeltointroduceaprobability ζ> 0that,ifelected,thedemagoguecanincreaseitsprobabilityof‘drawingahighvalence’ from pL to¯pL.Aftersucharulechange,Lemma6andProposition4yieldthatparty b proposes increasedinvestmentwhenunconstrained,anditraisestheprobabilityofenteringadeathspiral. Further,priortosucharulechange,similarresultsfollowsdirectlyfromthefactthatthepossibilityofsuchafuturerulechangereducesthevaluefunctionofparty b, raising themarginal valueofincreasingfuturecapitalstocks.Inparticular,suchchangesincreasetheanalogueto λ∗,inducingparty b tosavemore.Intuitively,thebenevolentpartyinternalizesthatifthedemagoguedoeschangethelawsdowntheroad,thegreatercapitalmitigatestheconsequencesof theincreasedprobabilitythatthedemagoguewillwinanddrivecapitaldown.

2.10ProductivityFluctuationsandEconomicDownturns

Economicstagnationinourmodelcorrespondstoalowvalueof φ.Thecomparativestatics ofourmodelyieldthatboth λc(k)and λ∗ increasein φ.Theseresultsimplythateconomic stagnationisassociatedwithreducedinvestmentinfuturecapitalstocks,andhenceagreater probabilityofademocraticdeclineintheformofadeathspiral.

Attheexpenseofsignificantlymorenotation,onecanintroduceproductivityshockstothe economy,andshowthatthequalitativeresultsextend.Specifically,onecanallowfori.i.d.productivityshocks φ ∈{φL,φH },with0 <φL <φH where φ j occurswithprobability q j, j = L, H

Thetwopartiesseethecurrentproductivityshockbeforeproposingpolicy.Thesametechniquesstillapply,permittingasimilarcharacterization.Withi.i.d.shocks,thefuturevalueof agivenlevelofcapitaltoparty b doesnotdependonthecurrentproductivityshock,andhence neitherdoesitsmarginalvalue.Itfollowsthat(i)party b’sunconstrainedoptimalinvestment shareislowerafter φL than φH ,and(ii)electoralcompetitionfromthedemagogueplacesa lowerboundonparty b’sproposedinvestmentshareafter φL than φH .Thus,thebenevolent partyalwaysproposesreducedinvestmentsharesineconomicdownturns.

3Conclusion

Ourpaperinvestigatesthelong-runsusceptibilityofDemocracytodemagogues.Weanalyze thedynamicpoliticalcompetitionbetweenafar-sighted,benevolentpartythatseekstomaximizevoterwelfare,andanoffice-motivateddemagoguewhoonlycaresaboutwinning.Parties proposehowtoallocateexistingresourcesbetweencurrentconsumptionandinvestment.Myopicvotersbaseelectoralchoicesonthecurrentutilitiesderivedfrompoliciesandavalence shock,andthewinningcandidate’spolicyisimplemented.

Demagoguesdesigntheirpoliciestoappealmaximallytoshort-sightedvoters,proposingto increaseconsumptionbysacrificingsomecapital.WeshowthatwhenvotershaveCRRApreferenceswithrelativeriskaversionsaboveone,thiselectoralcompetitionconstrainsabenevolent party’schoiceswhenevercapitallevelsaresufficientlysmall.Thisreflectsthatthesalienceof differencesinproposedconsumptionlevelsrisesrelativetodifferencesinvalenceswhencapital,andhenceconsumption,islower.Relatedly,weestablishthatthebenevolentpartydrawsits electoralsupportfrom“elites”whoobtainsufficientlylargesharesoftheeconomicpie,while thedemagoguedrawssupportfromtheeconomicallydisadvantagedwhocaremoreaboutthe policiesproposed,andrelativelylessaboutvalence.Weshowthatasdevelopmentlevelsincrease,valenceconsiderationsratherthanpolicydeterminethevotingchoicesofmorecitizens,

includingthepoor,limitingademagogue’sappeal.

Demagogueshaveachanceofwinningsothereisalwaysariskthatcapitalisreducedlow enoughthatabenevolentpartymustalsoproposetodis-investtowin.Whenthishappens,the economyentersadeathspiral,withcapitaldecliningtozero.Moreoptimistically,weidentify sufficientconditionsfordevelopmenttomitigatethelong-termthreatofademagogue.If(i)a democracydrawsgoodleadersattheoutsetwhogrowcapitalsufficiently,(ii)demagoguesare sufficientlyunlikelytowinandlimitedintheirabilitiestodis-invest,and(iii)productivityis highenoughthendeathspiralsbecomeextremelyunlikely.Insuchcircumstances,electinga demagoguecausesdamage,butthedemocracyistypicallyresilientenoughtorecover.

ProofofProposition1. Statement1. Supposethatparty d winswhenthepreferenceshockis vH .Then b canwinbymakingthesameoffer.Thefuturecapitalstockandhencetheactions inthesubgameisunaffected.However,party b wins,whichmakes b betteroff,becausehe receivestheadditionalutilityof vH .Thiscontradictsthepremiseofoptimizationby b.Now supposethatparty b winswhenthepreferenceshockis vL.Thenusingananalogousargument tothatforparty b,thedemagoguecanmatchtheofferandwininthecurrentperiod,without affectingfuturepayoffs,acontradictionofoptimizationby d. Statement2. Supposeparty d proposes id (k) > δk.Notethatdeviationsdonotaffect d’sfuturepayoffs,because d onlycaresaboutwinning,whichonlydependsonvalenceshocks(from Statement1).Ifparty d lowersitsinvestment,then d stillwinswhenthepreferenceshockis vL. Iftheshockis vH then id (k) = δk weaklydominatesallotheractions,because d winsifparty b choosesaninvestment ib(k)with u(φk ib(k)) + vH < u (φ + δ)k . Statement3. Becauseparty b winswhentheshockis vH ,wehave u

. Supposethattheconditionholdsatequality.Ifthemedianvoterdoesnotchooseparty b,then thefactthat ib(k) > δk impliesthat b canchooseamarginallysmallerinvestment,whichwould makethemedianvoterstrictlybetteroff.Thefutureactionsofparty d areunaffected(from Statement1).Thus,thisdeviationwouldmake b strictlybetteroff,acontradiction.Therefore, party b mustbeelectediftheconstraintholdswithequality.

ProofofProposition3. Statement1. Wefirstprovethatthevaluefunctionisscalableand thenderiveitsfunctionalform.Giveninitialcapital kh0 = k,party b’soptimizationproblemis:

ProblemBP

Let {kht }∞ t=0 beanoptimalsequenceforthisproblem.Now,multiplytheinitialcapitalby α> 0, sothattheinitialcapitalstockis ˆ k = αk,andconsiderthesequence { ˆ kht }∞ t=0,where ˆ kht = αkht . Thisnewsequence, { ˆ kht }∞ t=0,satisfiesalltheconstraintsbecause

ifand

onlyif δ ˆ

ˆ kht .Because VBP( ˆ k)reflectsoptimizationgiven ˆ k ratherthan k,

Because α and k arebotharbitrary,wecanuse 1 α insteadof α,and αk insteadof k toget

Supposetheinequalityin(17)werestrict.Then,substitutingtheright-handsideof(17)into (18)yields

acontradiction.Thus,wemusthave

Substituting k = 1and α = k yields

where VBP(1)dependson s.Itremainstoshowthat VBP(1)isfinite.First,considerthelower boundobtainedifparty b mimicsparty d.Thentheconsumptionpayoff becomes

as β(1 δ)1 s < 1,byAssumption1.Further,astrictupperboundon VBP(1)iszero.Thus, VBP(1)andthevaluefunctionsarewelldefined.

Statements2and3. Thebenevolentpartysolves

Assumingthattheconstraintisslack,thefirst-orderconditionis

Writing k = (1 + λ∗)k andnotingthat u ((1 + φ)k k ) = (

),equation(20)implies

Wecanapplytheenvelopetheoremto(19)toget

Re-arrangingandusingStatement1yields

Substituting(21)into(23)yields

Dividingby u (k)andmultiplyingby β and(φ λ∗)s yieldsthefirstordercondition.

Toestablishthat λ∗ > 0,weshowthatthederivativeat λ∗ = 0isstrictlypositive.Itsuffices toshowthattheleft-handsideof(24)isstrictlylessthanitsright-handsideat λ∗ = 0.At λ∗ = 0, multiplyingbothsidesof(24)by βφs/u (k)andre-arrangingyields

whichisequivalentto

Toshowthat(25)holds,itissufficienttoprovethat

Thentheright-handsideof(25)wouldbegreaterorequalto βpH (1+φ)+βpLφ>βφ,andbyassumption βφ ≥ 1.Toestablishthis,notethattheleft-handsideof(26)isstrictlyincreasingin δ when s > 1.Inparticular,thederivativeoftheleft-handsideof(26)withrespectto δ isgivenby (s 1)

Thus,itissufficienttoshowthat(26)holdsfor δ = 0,whichisimmediate.That λ∗ > 0also impliesthattheconstraint k ≥ (1 δ)k isslack.That λ∗ <φ followsfrom u(0) = −∞. Statement4. Thevaluefunctionisincreasingin pH ,becauseparty b canchoosetoimitatethe demagogue.FromStatement1and(19),party b’soptimizationproblemisstrictlyconcavewith uniquesolution k = (1 + λ∗)k.FromStatement2, λ∗ > 0.Thus,ahigher δ lowersthepayoff

whenthedemagoguewins.Thus,thevaluefunctionisdecreasingin δ.FromStatement1, VBP(k) = abk1 s/(1 s).Thus, ab isdecreasingin pH andincreasingin δ.Party b solves

withthefirst-ordercondition

Theright-handsideof(27)isincreasingin λ∗ andhence λ∗ isincreasingin ab.Thecomparative staticpropertiesfor pH and δ follow.

Toobtainthecomparativestaticsfor β and φ,weuse(6).Observethattheleft-handside of(6)isincreasingin λ∗,whiletheright-handsideisdecreasing.Nextobservethat,fixing λ∗ , increasesin β reducetheLHS,butincreaseRHS,implyingthat λ∗ mustrisewith β. Finally, φ doesnotappearontheLHSof(6),but,fixing λ∗,theRHSisincreasingin φ since (

increasesin φ.Thus, λ∗ increasesin φ.

ProofofProposition4. Weproceedintwosteps.First,weintroduceamodifiedproblem withlinearconstraints,whichwecallproblemMP.Thelinearityofconstraintsinthemodified problemMPallowsustocharacterizeitssolutionbycomparingitsvaluefunctionwiththatof thebenchmarkproblemBP.Inthesecondstep,wechoosethe(linear)constraintparametersof problemMPtomapittoaversionofproblemPwithmorerelaxedconstraints.Thismapping enablesustocarryoverourcharacterizationofproblemMP’ssolutiontoproblemP.

Step1. ThemodifiedproblemMPcorrespondsexactlytoproblemPexceptthatwemodify constraints(2):foreach t ∈ N wereplaceconstraints(2)withconstraintsthatarelinearin k. Wefixlinearparameters {λ(ht): ht ∈Ht, t ∈ N},where

ProblemMP

WenextusethelinearityoftheconstraintsinProblemMPtoprovethatProblemMPis scalableincapital.RecallthatthevaluefunctionforproblemBPisscalable,takingtheform VBP(k) = abk1 s/(1 s),where ab > 0isaconstant,andtheassociatedoptimalinvestmentstrategyisgivenby kht,vH = (1 + λ∗)kht .OnecansimilarlywriteProblemMPrecursively,withassociatedvaluefunction VMP(ht, k).Thedependenceonhistory ht indicatesthattheoptimization problemisnottimeinvariantbecauseparameters λ( )maydependonthehistoryofpastshocks.

Lemma3 ProblemMPhasauniquesolution.LetVMP(ht, k) bethecontinuationutilitygiven historyht andcurrentcapitallevelk,andletkMP(h0,vH ) betheoptimalcapitalchoicegiven history (h0,vH ).Forallk > 0,

1. VMP(ht, k) = am,ht k1 s/(1 s),wheream,ht > 0 isaconstant.

2. VMP(h0, k) ≥ VBP(k).Further,if λ(ht) λ∗,forsomet > 0,thenVMP(h0, k) > VBP(k).

3. If λ(h0) ≤ λ∗,thenconstraint (29) binds,i.e.,kMP(h0,vH ) = (1 + λ(h0))kh0

4. If λ(ht) λ∗,forsomet > 0,and λ(h0) >λ∗,thenkMP(h0,vH ) > (1 + λ∗)kh0 .

ProofofLemma3. ProblemMPhasauniquesolution,becausetheobjectiveisstrictlyconcaveandtheconstraintsetisconvex.Because λ( ) ≥−δ,thecondition β(1 δ)1 s < 1,implied byAssumption1,ensuresthatutilityisfinite.

Statement1. Let kh0 = k betheinitialcapitalstockandlet {kht }∞ t=0 beanoptimalsolutiontoproblemMP.Now,multiplytheinitialcapitalby α> 0,sothattheinitialcapitalstockis ˆ k = αk, andconsiderthesequence { ˆ kht }∞ t=0,where ˆ kht = αkht .Thisnewsequence { ˆ kht }∞ t=0 satisfiesallthe constraints.Thus,thesameproofforthescalabilityofthevaluefunctioninproblemBPapplies.

Statement2. ProblemMPismoreconstrainedthanproblemBP.Thus, VMP(h0, k) ≤ VBP(k) < 0 forall k.This,inturn,implies ab ≤ am,h0 ,whichimplies VBP(k) ≤ VMP(h0, k)forall k.If λ(ht) λ∗ foranyconstraintthatisreachedwithpositiveprobability,then VMP(h0, k) < VBP(k) < 0for all k.Thus, ab < am,h0 ,which,inturn,impliesthat VBP(k) < VMP(h0, k)forall k.

Statement3. FromPart1ofLemma3and(19),theobjectivefunctioninProblemBPisstrictly concave.Thus,fromPart2ofLemma3and(20),

Moreover,fromPart2, VBP(k) ≤ VMP(h0, k).Combiningthiswith(32),wehave

+ φ)

Thus,forProblemMP,wehave kMP(h0,vH ) = (1 + λ(h0))kh0 forany λ(h0) ≤ λ∗ .

Statement4. Because λ(ht) λ∗ , t > 0,Part2impliesthat VBP(k) < VMP(h0, k).Thus,wegeta strictinequalityin(33),whichimpliestheresult.

Step2. WeuseLemma3toprovethestatementsoftheProposition.

Statement1. Let k betheinitialcapitallevelandlet kP(ht)betheoptimalcapitallevelgiven history ht inProblemP.Suppose k ≤ k∗ and kP(h0,vH ) < (1+λc(k))k.Usingthispositedoptimal solution,wenowconstructparametersfortheconstraintsofProblemMPthatweusetoderive acontradiction.ConsiderProblemMP,startingat t = 0withinitialcapital k andtheconstraints givenby λ(h0) = λc(k)and λ(ht) = (kP(ht,vH )/kP(ht)) 1,for t > 0.ByPart3ofLemma3,the optimalcapitallevelat t = 1is kMP(h0,vH ) = (1+λc(k))k,andhence kMP(h0,vH ) > kP(h0,vH ).By construction,theoptimalchoicesinProblemP, {kP(ht): ht ∈Ht},satisfytheconstraintsofProblemMP.Thus,thecapitallevels {kMP(ht): ht ∈Ht} generateastrictlyhighervaluefortheobjectivefunctionofProblemMPthan {kP(ht): ht ∈Ht} thatwepositedsolveProblemP.Because theobjectivefunctionsofProblemMPandProblemPareidentical, {kMP(ht): ht ∈Ht} also generateastrictlyhighervaluefortheobjectivefunctionofProblemPthan {kP(ht): ht ∈Ht}. Toobtainacontradiction,weshowthat {kMP(ht): ht ∈Ht} isfeasibleinProblemP.

Toseethis,notethat kMP(h0,vH ) > kP(h0,vH ).Thisandconstraint(30)implythatthecapital level kMP(ht)thatsolvesProblemMPexceedsthe kP(ht)thatsolvesProblemP.Thisimplies

kMP(ht,vH ) kMP(ht) = 1 + λ(ht) = kP(ht,vH ) kP(ht) ≤ 1 + λc(kP

wherethelastinequalityfollowsbecause λc(k)isincreasingin k

Theaboveargumentusesstatement2ofLemma3,whichestablishesthestrictinequalitybetweenthemarginalproductsofcapitalforproblemsMPandBP,respectively.Thus,by continuitytheargumentimmediatelyextendstocapitallevels k thatarenottoofarabove k∗

Theprooffor k > ˆ k isanalogous,exceptthatweusePart4ofLemma3.Inparticular, because k > k∗,wehave λ(h0) = λc(k) >λ∗.Because pH < 1,thedemagoguesometimeswins andhencethecapitalsometimeswillfall.Thus, λc(kP(ht)) <λ∗ forsome t > 0.Thus,byPart4 ofLemma3, kMP(h0,vH ) > (1 + λ∗)k.Therestoftheproofisidenticaltoabove.

Statement2. If k ≤ k∗,Part1implies k = (1 + λc(k))k.Next,suppose k > k∗.Because pH = 1, thedemagogueneverwinsinequilibrium.Thus,thesolutiontoProblemBP(i.e., k = (1+λ∗)k) isalwaysfeasibleinProblemP.

Statement3. Let k > k∗.Define n(k) = min{n ∈ N|(1 δ)nk > k∗}.Ifwestartwithcapital k, thenwecaneliminateconstraint(2)intheoptimizationproblemforthefirst n periods.

Fromthesecondstatementitfollowsthattheresultisimmediateif pH = 1;andfromthe firststatementfor pH < 1,wehave λ(k) >λ∗ for k > k∗.Supposebywayofcontradictionthat liminfk→∞ λ(k) >λ∗.ConsiderProblemBP,whichdoesnothaveconstraint(2).Thenthestrict optimalityof λ∗ impliesthatparty b’sutilityunder λ∗ exceedsthatfrominvestments λ(k)byat leastsomeamount ε> 0.Further,thereexistsatimeperiod T suchthat,foranyinvestment strategyoftheinfinitehorizonmodel,theinvestmentstrategyrestrictedtoamodelwithfinite timehorizon T resultsinautilitylevelthatdiffersfromthatovertheinfinitehorizonbyatmost ε/2.Thus,theutilityfromusing λ∗ for T periodsifconstraint(2)isslackforthoseperiods exceedsthatfromusing λ(k)byatleast ε/2.However,if k islargethenconstraint(2)isslack for T periods.Thiscontradictsthepositedoptimalityof λ(k).

ProofofProposition6. Statement1. FromProposition5,deathspiraloccurswithprobability1 ifcapitaldropsbelow k.Startingwithacapitallevel, k,wereach k ifwehave α consecutivelow valencerealizations,where(1 δ)αk ≤ k,i.e., α = log(k/k)/ log(1 δ).If α isnotaninteger,we needoneadditionallowvalencerealization.Thus,theprobabilityofreaching k isatleast pα+1 L . Statements2and3. Supposethatwhenparty b winsandcurrentcapitalis k,itinvests λ∗k Wewillusethistofindboundsontheprobabilityofadeathspiral.Consideralog-scale,so thatwithprobability pH ,log(k ) = log(k) + log(1 + λ∗);andwithprobability pL,log(k ) = log(k) −| log(1 δ)|.Let kH = log(1 + λ∗)and kL = | log(1 δ)|.Thus,wehavearandomwalk withtwopotentiallyunequalsteps:ineachperiod,withprobability pH ,thelocationmovesupa stepofsize kH ;andwithprobability pL,thelocationmovesdownastepofsize kL.Westartfrom locationlog(k),with k > k∗,andweareinterestedintheprobabilitythatthelocationfallsto log(k∗)(orbelow)atanyfutureperiod.Changingtheorigin,thisisequivalenttotheprobability that,startingfromlog(k/k∗),thelocationbecomesnon-positive(≤ 0)atanyfutureperiod. Thisprocesscorrespondstoagambler’sruinprobleminwhichoneplayerisinfinitelyrich, analyzed,forexample,inChapter14.8ofFeller(1968).First,supposethereexistsaupper absorbinglocation a > 0,sothatifthelocation z weaklyexceeds a,thentheprocessends. Ouranalysiscorrespondstothelimitas a →∞.Let Q(z)betheprobabilitythatthelocation becomesnon-positiveatanyfutureperiodwhentheprocessstartsfrom z attime0.Then,

(z) = pH Q(z + kH ) + pL Q(z kL), for0 < z < a, (34)

withboundaryconditions

Thecharacteristicequationassociatedwithequation(34)is

Thisequationalwayshasasolutionat σ = 1.Moreover, L(σ)isstrictlyconvexin σ ∈ (0, ∞), withlimσ→0 L(σ) = limσ→∞ L(σ) = ∞.Supposetheprocessdoesnothaveazeromean,i.e.,

pL( kL) + pH (kH ) 0.Then, L (1) 0andequation(36)hasexactlyoneotherpositivesolution besides1:mirroringFeller(1968)’sanalysisonp.366,forthepurposeoffindingboundson theprobabilityofruin,wedonotneedtoconsidernegativesolutionstoequation(36).Callthis solution σ1.If L (1) < 0,then σ1 > 1.If,instead, L (1) > 0,then σ1 < 1.Then, Q(z) = A+ Bσz 1, where A and B areconstants,satisfiesequation(34)forsome A and B

Next,observethatifwechoose A = A and B = B suchthat Q(z = 0) ≡ Q(z = 0; A = A, B = B) = 1and Q(z = a + kH ) ≡ Q(z

0,then

Thus, A + Bσz 1 Q(z)satisfiesthedifferenceequation(34)withnon-negativeboundaryvalues (37).Thus, A + Bσz 1 ≥ Q(z).This,willbeanupperboundon Q(z).Tofind A and B,observethat Q(0) = A + B = 1, and Q(a + kH ) = A + Bσ a+kH 1

Thus,

whichimplies

Wecanfindalowerboundfor Q(z)inasimilarmanner.Choose A = A and B = B,sothat Q(z = kL) ≡ Q(

) = 0. Then,

Thus,

whichimplies

Combining(38)and(39)yields

(40)

If L (σ = 1) < 0,sothat σ1 > 1,then,as a →∞,boththelowerandtheupperboundsin (40)convergeto1,andhencesodoes Q(z).

If,instead, L (σ = 1) > 0,sothat σ1 < 1,then,from(40),inthelimitwhen a →∞,wehave

(41)

wherewerecallthat z = log(k/k∗)and σ1 < 1istheuniquepositivesolution,otherthanone,of thecharacteristicequation(36).

Thus,wehaveproventhefollowing.

Result. Startingfromk > k∗ andassumingthatpartybalwaysinvests λ∗ whenitwins,the probabilityoffallingtok∗ orbelowinanyfutureperiod,denotedbyP(k; k∗,λ∗),issuchthat:

1. If pH log(1 + λ∗) + pL log(1 δ) < 0,then P(k; k∗,λ∗) = 1.

2. If pH log(1 + λ∗) + pL log(1 δ) > 0,then

where σ1 < 1 istheuniquepositivesolutiontopH σlog(1+λ∗) + pLσlog(1 δ) = 1.Using y = σ log(1 δ),thisequationisequivalenttopH y 1 log(1+λ∗) log(1 δ) y + pL = 0.

Wenowapplytheseresultstooursetting,inwhichthebenevolentparty’sinvestmentdecisiondependsonthecapitalstock k.Let Q(k)betheprobabilitythat,startingfromacapital level k,theeconomyentersadeathspiral(i.e.,thecapitalstockfallsbelow k)insomeperiod. FromProposition4,when k > k∗,theequilibriuminvestmentis λ(k) ≥ k∗.Thisimpliesthat thestepupexceeds λ∗.Moreover,recallthat k∗ > k.Thus,theupperboundontheprobability ofruininStatement2oftheResultdirectlyapplies:If pH log(1 + λ∗) + pL log(1 δ) > 0, then Q(k) ≤ σlog(k/k∗) 1 , where σ1 ∈ (0, 1)solves pH σlog(1+λ∗) + pLσlog(1 δ) = 1.Substituting y = σ log(1 δ) yieldsthetheupperbound.

Next,observethatStatement1oftheResultdoesnotdependon k or k∗.However,itassumesthatparty b investsafraction λ∗ whenitwins,implyingastepupofsizelog(1 + λ∗).

FromProposition4,as k growslarge, λ(k)convergesto λ∗.Thus,thereexistsa ˆ k suchthatfor all k > ˆ k, λ(k)iscloseenoughto λ∗ that pH log(1 + λ(k)) + pL log(1 δ) < 0.Now,replace λ∗ withsupk> ˆ k λ(k)and ˆ k insteadof k∗.Then,startingfrom k > ˆ k,theprobabilityoffallingbelow ˆ k

is1.FromStatement1ofthisProposition,startingfrom k ≤ ˆ k,thereisapositiveprobabilityof goingbelow k beforegoingabove ˆ k.Letˆp = p1+α L ,with α = log(k/ ˆ k)/ log(1 δ).Moreover,as wejustshowed,ifwegoabove ˆ k,thenwithprobability1,wegobelow ˆ k again.Thus,starting from k > ˆ k,theprobabilitythatwenevergobelow k doesnotexceedlimn→∞(1 ˆ p)n = 0.Thus, if pH log(1 + λ∗) + pL log(1 δ) < 0,then k fallsbelow k withprobability1.

Weuseasimilarargumenttofindalowerboundfor Q(k)when k issufficientlylarge.Let λm = supk≥k λ(k).FromStatement2oftheResult,ifwereplace λ∗ with λm,thentheprobability offallingfrom k > ˜ k to ˜ k, P(k; ˜ k,λm),islargerthan˜σlog(k/k)+| log(1

)| 1 ,where˜σ1 < 1istheunique positivesolutionto pH

= 1.Itisstraightforwardtoshowthat σ1

Thus, P(k; ˜

,λ

Becauselimk→∞ λ(k) = λ∗,itfollowsthatforevery ε> 0thereexists k suchthat,forall k > k, 1 ε< ˜ σ1/σ1 ≤ 1.

First,suppose k ≤ ˜ k.Theprobabilityofdroppingbelow k isboundedawayfromzerofor k ≤ k.Moreover,thetermsontheleft-handsideoftheinequalityinthePropositioniscontinuous onthecompactset[k, ˜ k].Thus,thereexistsa C > 0suchthatthelowerinequalityissatisfied.

Next,suppose k > ˜ k.Let C1( ˜ k) = σ| log(1 δ)| 1 .Then

where C1( ˜ k) > 0isindependentof k.

Now,startingfrom k,theprobabilityoffallingto k isaconstant,whichisstrictlybetween 0and1.Callit C2( ˜ k) ∈ (0, 1).Thus,for k > ˜ k > k,

σ1) ε , (42) and C(k) = C1(k) · C2(k) · k

isboundedawayfrom0and1.

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