AnEconomicTheoryofLeadershipTurnover :mgallego@:Pitchik@,1998AbstractInanin nitehorizonmodel,aleaderofagroupofcitizensexertse ortineachperiodtomaintainapublicgoodthatenhancesthepro ,,we ndthat(1)leadershipturnoveroccurswhenthekingmakers'expectedearningsarelow;(2)leadershipturnoverdeclineswithdurationino ce;(3)leadershipturnoverdeclinesasthetechnologyforprovidingthepublicgoodimproves;(4)leadershipturnoverincreasesasthenumberofkingmakersincreases. ThispapergrewoutofGallego(1995,chapter2).WethankJe reyBanks,NancyGallini,EdGreen,ArthurHosios,YehudaKotowitz,,MikePeters,AnatolRapoport,AloysiusSiow,.(JELD72,C73)yPitchikgratefullyacknowledges nancialsupportfromtheSocialSciencesandHumanititesResearchCouncil.
'sfallenpoliticalleadersarenotdefeatedinpopularelec-tions(Blondel(1980),,Table1).Of1,028leaderswhoreachedhigho cebetween1945and1976,169remainedino cein1976;ofthe859whohadfallenfrompower,only5%,20% ceatelection,theattentiondevotedtomodelingcoupsd' etatandothernon-democraticprocesseshasbeensmallcomparedwiththatdevotedto\democratic": ce(BienenandvandeWalle(1991)).(LondreganandPoole(1990,1992)).(Gallego(1996)).(BienenandvandeWalle(1991)).(Blondel(1980),230).Westudyanin nite-horizonmodelofaneconomyinwhichtheleaderismain-tainedinpowerbythesupportofa nitesubgroupofeconomicagents,called2
,theleader rstreceivesa xedpayment,thenob-servestherealizationofanexogenousshock,and nallyexertscostlye toeachkingmakerdependspositivelyonboththeleveloftheshockandtheleader'se 'se ort,,anewleaderischosenatrandomfromamongthekingmakers,,3,4,'stypevarieswiththecostofexertinge ortisconsistentwiththe -agentmodelsofbothpoliticalcompe-titionanddictatorshipthatarebasedonGreenandPorter1(1984).Themodelsinthisliteratureassume,aswedo,thattheleaderofagroupprovidescostlye : Subsequentleadersarechosenatrandomfromthesubgroupwhosesupportisnecessaryfortheleader'ssurvivalsothatleadershipturnoverconfersabene ,itisassumedthatsubsequentleadersarechosenfromoutsideofthesubgroupsothat,intheabsenceofvariableleaderability,thesubgroupisindi erentbetweenanoldandnewleader. Thesubgroupis nitesothateachmemberofthesubgrouphasapositivechanceofbeingchosenthenewleaderandthereforemayderiveabene -agentmodelsofpoliticalcompetition,seeFerejohn(1986),CukiermanandMeltzer(1986),Rogo andSibert(1988),Austen-SmithandBanks(1989),Banks(1990),Rogo (1990)andBanksandSundaram(1991).Formodelsofdictatorship,seeforexample,GrossmanandNoh(1990),Olson(1991),O'Flaherty(1991)andGrosman(1991).OurmodeldrawsupontheworkofGreenandPorter(1984),ShapiroandStiglitz(1984),Ferejohn(1986),Olson(1991),andO'Flaherty(1991).3
,e ,supportfortheagentisneverwithdrawnbytheprincipalsincethee cientlevelofe ectour , nitesubgroupasinourmodel,-agentmodelsinthepoliticaleconomyliterature,
owofbene , tothemembersofthesubgroupinanyperioddependspositivelyontheleveloftheexogenousshockandonthee levelinaperiodismaintainedonlyifthee ,thekingmakerscomparethe
owof4
bene
owofbene tstoakingmakerofmaintainingthestatusquoinaperiodislargerthantheexpectedcapitalgainofachangeinstatusiftheleader'se ,theleader'se ,thesubgroup'sreservationstandardofe ,theleader'spayo inaperioddecreaseswithe orttodaywiththebene tofremainingasleadertomorrowdependsonthekingmakers',thebene tofremainingasleadertomorrowis 'sreservationvalueofe ortis ,theleadermeetsthesubgroup'svariablereservationstandardofe ortprovideditisbelowtheleader',theleaderexertszeroe ,thesubgroup'sequilibriumreservationstandardofe ortislargerthantheleader',theleaderexertszeroe 'Flaherty(1991),adictator'sstrengthino ,O'Flaherty ,heobtainsthatdictatorsareeitheroustedinthe ,O'Flaherty'sturnoverratedecreasesfromthe ,weprovideamodelinwhicha2O'
dictator'-interactiveoptimizationmodeloftinpotdictatorship,Wintrobe(1990)assumesthattinpotdictatorsminimizethecostsofstayinginpowerinordertocontinuetoreapitsbene ' (1986),ano ,theaimsdi , nitehorizonasymmetricinformationmodelofelectoralcompetition,BanksandSundaram(1993)(1991),BanksandSundaramobtainthatundervariableleaderability,theprobabilityofre-electionincreaseswithanincumbent'slengthoftimeino ,weareinterestedinhowtheeconomya ectsofobservableexogenousshocksoncoupsd' ,ourmodelhasimplicationsforpoliticalpartyleadershipturnover6
\kingmakers"(1979,)arguesthatonepreconditionforacoupistheexistenceofasmall,educated,()thattheben-e ,decision-making, nitesupplyofkingmakers,nofwhomareactiveandownequalsharesinanexport ,,p,isindependentandidenticallydistributedeachperiodaccordingtoadistributionfunctionFwhosesupportis[0;1).Atthebeginningofeachperiod,thedictatorreceivesa xedrent,W,fromthecitizens,seestheprice,p,andchooseshowmuche ort,x2[0;1],toexertonapublicgoodthatpositivelya ectsthepro tsoftheexport 5toadictatorwhochoosesxisW C(x)whereC:[0;1]!<+isanincreasing,convex,non-negative,real-valuedfunctionwithC(0)= toadictatoristhediscountedsumoftheperiodpayo sfromtheperiodofinitialincumbencyuptoandincludingthe ,subsequenttoobservingthedictator'se ortandthepriceofexports,3Forexample,MargaretThatcher',theo ceholderseesarandomvariable andthendecideshowmuche orttoexertonthevoters' ceholderofexertinge ort is ( ).Theo ceholderreceivesW (a)iftheo ceholderexertse ortawhileino
ortexpendedbythedictator.(Alternatively,theexogenous
uctuationsinthekingmakers'payo scouldbeduetochangesinweatherortastes,forexample.)Ifthekingmakersstageacoup,thedictatorisreplacedbyarandomlychosenactivekingmakerwhois,inturn,replacedbyaninactivekingmaker;eachactivekingmaker'spayo ,eachactivekingmaker'speriodpayo 6ispY(x)=nwhereY:[0;1]!<+isanincreasing,concave,non-negative,real-valuedfunctionwithY(0)= ofanactivekingmakeristhediscountedsumoftheperiodpayo sthatthekingmakerreceiveswhileactive,plustheexpectedpro 2(0;1).,sothat,inessence,wehavetwoplayers,, stotheactiveplayersineachperiodaredeterminedsolelybytheobservedpriceandcurrentactionssothatthepastdoesnotin
,thedictatorchoosesalevelofe ortlevel, ectthefutureprice,welookforanequilibriumstrategypro ,welookforaMarkovstrategy6InFerejohn(1986),thevoterreceivesu(a; )whereaisthee ortexertedbytheo ceholderand
pro :[0;1)![0;1]:[0;1) [0;1]!f0;1gthattransformsthecurrentpriceandexertionlevelintoadecisiontostageacoup(denotedby1)ornottostageacoup(denotedby0),includingoneinwhichtherepresentativekingmakeralwayschoosestostageacoupandthedictatorexertszeroe -perfectsymmetricNashequilibriumoutcomeareconsistentwiththe
uencesMarkovstrategiesineachsub-game,,aMarkovsubgame-perfectequilibriumisasubgameperfectequilibriuminthegameinwhichplayersarenotrestrictedtoMarkovstrategies.(SeeFudenbergandTirole(1992,)formorediscussionofMarkovperfectequilibria.)AsstatedinAbreu(1988),inarepeatedgamewithdiscounting,astrategypro leisasubgameperfectequilibriumifandonlyifnoplayercangainbydeviatinginasingleperiodafteranyhistory.(Theideacomesfromtheprincipleofdynamicprogrammingcalled\unimprovability".Foraproofofthestatement,seeFudenbergandTirole(1992,,)orOsborneandRubinstein(1994,).),incheckingwhetherapair(X;S)ofMarkovstrategiesisasubgameperfectequilibriumwecanrestrictattentiontoalternativestrategiesthatdi erfromaplayer'
'spayo inanyperioddependspositivelyonboththedictator'se ''sbestresponseistostageacoupforsomeexertionlevelxwhenfacingagivenpricep,thenthekingmaker''sbestresponseistostageacoupforsomepricepandexertionlevelx,thenthekingmaker',whenfacingapricepandastrategyXofthedictator,thereisareservationfunctionRk(X;p),withRknon-increasinginpsuchthatonlyforx<Rk(X;p)'soptimalreservationstandardwhenadictatorexertse ortxandthepriceisp?Iftheactivekingmakerschoosenottostageacoupandtherebymaintainthestatusquo,theneachactivekingmakerearnspY(x)=nfromexportsinthisperiodandexpectstoearn E(K)intheensuingperiodswhereKrepresentsthepresentvalueofanactivekingmaker'smaximumexpectedearningsandEdenotestheexpectationoperatoroverp2[0;1).Thebene tofremainingaskingmakerorequivalently,theopportunitycostofacoupistherefore(pY(x)=n)+ E(K).Iftheactivekingmakerschoosetostageacoupandtherebytakeanopportunitytochangestatus,,eachactivekingmakerremainsanactivekingmakerwithprobability(n 1)=nandchangesstatusbybecomingadictatorwithprobability1=,whentheactivekingmakerschoosetostageacoup,eachactivekingmakerearns[ E(K)(n 1)=n]+[ E(D)=n](whichrepresentstheexpectedcapitalgainofchangingstatusandsoisthebene tofacoup)whereDrepresentsthepresentvalueofadictator'(thebene tofacoup)isgreaterthanthebene tofremainingaskingmaker(theopportunitycostofacoup).Formally,anactivekingmaker's10
maximizationproblemisadynamicprogrammingproblemforwhichtheBellmanequationis()K(p)=maxY(x)p+n 1 E(K); E(K)+1 E(D)()nnnSubgameperfectionrequiresthateachactivekingmakerchoosetostageacoupwhenevertheopportunitycostofacoup(thebene tofremainingaskingmaker)islessthanthebene tofacoup(theexpectedcapitalgainofachangeinstatus)(<Y 1( [E(D) E(K)]=p))andtoavertacoupwhenevertheopportunitycostisgreaterthanthebene t(>Y 1( [E(D) E(K)]=p)).Thus,formally,abestresponse8ofanactivekingmakertothetripleX;x;pisthestrategy8< [E(D) E(K)]ifx<Y 1)S(p;x)=1(p:0(D) E(K)]ifx Y 1 [E()pWecallsuchastrategyareservationstrategywithreservationstandardfunctionRk(X;p)=Y 1( [E(D) E(K)]=p).Notethattheactivekingmakers'variablereservationstandarddecreasesinpsinceYisincreasingine ,thee 'variablereservationstandardisthelevelofe ortexertedbythedic-tatorthatmaintainsequalitybetweenthebene ortmaintainsequalitybetweenthevariableopportunitycostofacoupandthe xedbene -makers'reservationstandardisthequanityofe ortthatexactlycompensatesthe8Thekingmakerisindi erentbetweenstagingandavertingacoupwhenx=Y 1( [E(D) E(K)]=p).Consequently,thekingmaker'sbestresponseistostageacoupwithprobabilityq2[0;1]whenx=Rk(X;p).However,asisshownbelow,theonlycandidateforanequilibriumreservationstrategyisthatinwhichq=0whenx=Rk(X;p).11
,thecompensat-ingquantityofe ,weneedto (p)'spayo dependsnegativelyonthedictator'se 'sbestresponseiseithertoexertzeroe ,thekingmakers',adictator'sbestresponsetoadecreasingreservationstandardofthekingmakersistoexertzeroe ortunlessthereservationstandardislowenough,orequivalently,'variablereservationstandardislowenough(thatis,theexportpriceishighenough)thedictatorexertstheminimume ,adictator'sbestresponsetotherepresentativekingmaker'sreservationstrategywithreservationstandardrk(p)isareservationstrategywithreservationvalueRd(rk(p))oftheform(X0ifrk((p)=p)>Rd(rk(p))rk(p)ifrk(p) Rd(rk(p))-pretationofthedictator''spayo dependsonthe xedrentsW,thee ortexerted, ortexertedisgreaterthanorequaltothekingmakers'reservationstandard;otherwise,it12
,thereisamaximumlevelofe ,Rd(rk), ortrequiredtoavertacoupiftheminimume ortrequiredislessthanRd(rk).Thedictatorstrictlypreferstoexertzeroe ortandbedeposedratherthantoexerte ortgreaterthanRd(rk) erentbetweenexertingthemini-mume ortandzeroe ortiftheminimume ortequalsRd(rk);weassumethatthedictatorexertstheminimume ortinthiscase(thoughtheresultsremainifthedictatorrandomizesbetweentheminimume ortandzeroe ort).Wearenowinapositiontocalculatethedictator'soptimalreservationvalueRd(rk)inresponsetothekingmakers' ort, ortrk(p),acoupisaverted,andthedictatorearnsW C(rk(p)) ortwhenthekingmakers'variablereservationstandardislowerthanthemaximume ortthatthedictatoriswillingtomakegivenWandthefuturebene ,themaximizationproblemthatthedictatorfacesisastationarydynamicprogrammingproblemforwhichtheBellmanequationisD(p)=maxfW C(rk(p))+ E(D);Wg()whereE(D)istheexpectedpresentvalueofadictator'(rk)=C 1( E(Drk)):()Notethatthedictatorisindi erentbetweenavoidingacoup,ifitcanbedonebyexertinge ortRd(rk);andcausingacoupbyexertinge ,adictator'sbestresponseistoexertpositivee ortrk(p)whenrk(p)<C 1( E(Drk))andto13
exertzeroe ortwhenrk(p)>C 1( E(Drk)).Ifrk(p)=C 1( E(Drk))thenthedictatorisindi erentbetweenexertinge ortrk(p)and0andstrictlypreferseithertoanyothere ,theonlystrategypro lesthatarecandidatesforasubgameperfectMarkovequilibriumarepairsofreservationstrategiesinwhichthedictatorusesareservationstrategyforwhichthereservationvalueisrdandeachactivekingmakerusesareservationstrategyforwhichthevariablereservationstandardrk(p),anactivekingmaker'soptimalreservationstandardfunctionisY 1( [E(D) E(Krd)]=p),whereKrdisthepresentvalueoftherepresentativekingmaker' erentbetweenstagingacoupornotwhenthedictatorexertstherequirede ortY 1( [E(D) E(Krd)]=p)-atedwiththispairofreservationstrategiesisatriggerpriceT(rd)thatleaveseachactivekingmakerindi erentbetweenstagingacoupornotwhenthedic-tatorexertse ortrdandthepriceisT(rd).Byde nitionofthetriggerprice,theoptimalreservationvalueoftherepresentativekingmakermustequalrdwhenthedictator'sreservationvalueisrdandthepriceisT(rd).Thatis,itmustbethecasethatrd=Rk(Xrd;T(rd))sothatrd=Y 1( [E(D) E(Krd)]=T(rd)).ThisimpliesthatT(rd)=( [E(D) E(Krd)]=Y(rd)).ForpricesaboveT(rd),9Ifthekingmakers'reservationstrategyhasreservationvaluerk(p)andthekingmakersstageacoupwithprobabilityq2(0;1],whenx=rk(p)then,asabove,thedictatorstrictlypreferstoexertzeroe ortwhenrk(p)>C 1( E(Drk)).Whenrk(p)<C 1( E(Drk))thenthedictatorstrictlypreferstoavertacoupbyexertinge ortx2(rk(p);C 1( E(Drk))]ratherthanprecipitateacoupbyexertingzeroe ortore ortrk(p).Sincethedictator'spayo decreasesine ort,thedictatorstrictlypreferstoexertlesse ortthantoexertmoree ortamonge ,thereisnostrictbestresponseofthedictatortosuchareservationstrategyofthekingmakersandsoarepresentativekingmaker'scandidateequilibriumstrategyisareservationstrategy(withreservationvaluerk(p)thatisdecreasinginp)inwhichthekingmakerstagesacouponlywhenx<rk(p)sothatq=
therepresentativekingmaker'sreservationvalueY 1( [E(D) E(Krd)]=p)islessthanthedictator'sreservationvaluerd,sothatthedictatorexertstherequirede (rd),therepresentativeking-maker'sreservationvalueY 1( [E(D) E(Krd)]=p)isgreaterthanthedictator'sreservationvaluerd,sothatthedictatorexertszeroe xede 'reservationstandardmaintainsequalitybetweenthe xedbene tandthecostofacoup,thekingmakers'(rd),thevalueofthekingmakers'optimalreservationstandardfunctionislessthanorequaltothedictator'sreservationvaluesothatthedictatorexertsthee ,forpricesabovethetrigger,thekingmakersreservationstandardismetandthereforethethe
owofbene tsfromremainingaskingmaker()ishighrelativetothe xedbene t()(rd),thevalueofthekingmakers'optimalreservationstandardfunctionisgreaterthanthedictator'sreservationvaluesothatthedictatorexertszeroe ,forpricesbelowthetrigger,thekingmakers'reservationstandardisnotmetandthereforethe
owofbene 'sreservationvaluerd,thetriggerpriceassociatedtherepresentativekingmaker'sbestresponseisT(rd)= [E(D) E(Krd)]:()Y(rd)10Iftheprobabilityofsuccessofacoupattemptwerelessthan1thenthekingmaker',
lestobereser-vationstrategypro les11,'sequilibriumstrategyX (p)isareservationstrategywiththefollowingproperties:exerte ort0ifthepricep<T(r d),theequilibriumtriggerprice;exerte ortr k(p)(thekingmakers'equi-libriumreservationstandard),ifthepricep T(r d),'equilibriumstrategyS (p;x)isareservationstrategywiththefollowingproperties:stageacoupifthedictator'se ortx<r k(p),thekingmak-ers'equilibriumreservationstandard;maintainthestatusquoifthedictator'se ortx r k(p),thekingmakers',thekingmakers'reservationstandard,thedictator'sreservationvalue,andthelifetimeequilibriumpayo (uptoasetofmeasurezero)Markov,kingmaker-symmetric,subgameperfectequilibriumisthepairofreservationstrategies(X ;S )forwhichthedictator'sreservationvalueisr d=(C 1( E(D ))),thetriggerpriceisT(r d)= E(D )n(1 )=Y(C 1( E(D )))[n(1 )+ ],thekingmakers'reser-vationstandardisr k(p)=Y 1( [nE(D )(1 )]=p[n(1 )+ ])andE(D ),thepresentvalueofadictator'slifetimeequilibriumearnings,istheuniquesolutioninE(D)toZ1( !!)E(D)=W+ E(D) CY 1 E(D)(n(1 ))dF(p):T(C 1( E(D)))p[n(1 )+ ]Thepresentvalueofanactivekingmaker'sequilibriumlifetimeearningsisE(K )= E(D )n(1 )+: 11Wecannottiedownthedictator'sstrategyexactlyatpsuchthatrk(p)=Rd(rk(p))sinceanydistributionoverfrk(p);0gisabestresponseofthedictatorandthekingmakers'bestresponseisnotdependentonthedictator'
, ortrequiredtoavertacoupinsteadofexertingzeroe ortdependsonlyonthee ,thereisamaximumlevelofe ortthatthedictatoriswillingtoexertthatdependsonthegainstostayingonasdictatorrelativetothecostofexertinge 'sreservationlevelofe tofstagingacoupis tsdependpositivelyonthepriceofexportsandthee -vationstandardofe ortthatkeepseachkingmakerindi ,thekingmakers'reservationstandardofe ,thereservationstandardofe ,whenthepriceislowenough,thekingmakers'reservationstandardrequiredtoavertacoupisgreaterthanthedictator'sreservationlevel,themaximume ,thedictatorisnotwillingtoexerttherelativelyhighlevelofe ,theopportunitycostofstagingacoupislowerthanthe xedbene -ouspoliticalelectoralanddictatorshipmodelsofleadershipturnover,weassumethatthemembersofthesubgroup(kingmakers)whosesupportisnecessaryfor12Iftherent,W,dependsontheprice,p,ineachperiod,(W)
theleader(dictator)tostayinpowerhaveachanceatbecomingtheleader(thedictator)-petition,,kingmakersmightwithholdsupportinordertopunishadictatorfornotprovidinge , (C 1( E(D )))= E(D )n(1 )Y(C 1( E(D )))[n(1 )+: ]Therefore,,coupsfollowabinomialdistributionforwhichtheequilibriumprobabilitythatacoupoccursinanyperiodisF(T (C 1( E(D )))).Asthetriggerpriceincreases,
owofbene tsfromremainingaskingmakerincreases(),,asthekingmakers'payo sintheeventofacoup(Y)increasesorthecosttothedictatorofexertinge ort(C)decreases,the
owofbene tsfromremainingaskingmakerincreasesrelativetothe ,apredictionofthemodelisthatasrealGDPincreases,
,theequilibriumexpectedlifetimepayo softhedictator,E(D ),andoftherepresentativeking-maker,E(K ), xedasn,thesizeofthesubgroup,,thedictatormusttransferaportionofW(intheformofe ortthata ectsthelevelofthepublicgood),foreachprice,theportionofWrequiredtoavertacoupbynkingmakersincreasessothatitisnowmorelikelytobeabovethedictator'sreservationlevelofe ,theprobabilitythatacoupoccurs(alsoknownasthehazardrate)'scostofexertinge ortandthepriceineachperiodisindependentlydrawnfromacommondistribution,thelengthofadictator'stermino cecannota ectthetriggerpriceandsocannota ,thestylizedfactintheleadershipturnoverliteratureisthatthehazardratedeclineswiththelengthoftermino (proposedbyBienenandvandeWalle(1991))isvariableinnateability:ifleadersdi erinability,thenitwouldbeexpectedthatthoseoflowerabilitywouldbeweededout ,
that,foreachpotentialdictator,thecostofe ort(tomaintainthepublicgoodthatenhancesthewell-beingofthesubgroup)iseitherhighorlow.(Theresultsholdifweassumeany nitenumberorcontinuumofcostsofe ort.),inthiscase,thereexistsauniquesymmetricMarkovsubgameperfectequilibrium,,thekingmakers',thekingmakers''reservationstandardisthelevelofdictatoriale ortthato ortincreasesasthepricedecreasessothatthekingmakers'=1;2istomeettherepre-sentativekingmaker'sreservationvaluewheneverthisislowenoughrelativetothecostofexertinge ,foreachtypeofdictator,thereisatriggerpricesuchthatifthepricefallsbelowadictator'striggerprice,thecompensatingamountofe ortrequiredbythekingmakers'reservationstandardissohighthatnoe ortisexertedandthekingmakers'expectedcapitalgainfromachangeinstatusishigherthanthe
owofbene tsfrommaintainingthestatusquosothatacoupisstaged;ifthepricerisesaboveadictator'striggerprice,thecompensatingamountofe ortislowenoughthatthedictatorexertstherequirede 'scostofe ort,thehigheristhedictator'sreservationvalue,andsotheloweristhedictator',whenthepriceislowenoughinaperiod,neithertypeofdictatorexertstherequiredhighlevelofe ort,sothatbothtypesofdictatorareoustedfrompower;whentheexportpriceisinsomeintermediaterange,onlythedictatorwhofacesthehighcostofe ortexertslessthantheintermediatelevelofe ortrequiredandisousted;whenthepriceishighenough,bothtypesofdictatorexerttherequiredlowlevelofe orthasthelower20
'stermino ce,,'scostofe ort,theloweristhedictator' ort(andhasthelowerreservationvalue) ort(andhasthehigherreservationvalue)'stermino ce,'stermino ce,'stermino ce, , ortrequiredwheneveritislowenoughrelativetothecostofexertinge ,foreachtypeofdictator, ortincreasesacrossthedi 'stermino ce,(1991) ndthat21
thelengthoftimeino ceinverselya ,theevidenceinBienenandvandeWalle(1991) ndthatasthetechnologyforprovidingthepublicgoodimproves,(1990,1992) ,thenthisresultisconsistentwiththeevidenceinLondreganandPoole(1990,1992).Weshowthatcoupsoccurinthemodelwhenkingmakers'(1996) ,,thedictatorandkingmakersaresharinga ,eachindividualkingmaker'sshareofthe ,(1991,) ,(1980,,230) nesaslongleadership,aperiodino ,,thepercentageoftypesofleaderswhoexperiencelongleadershipareasfollows:68%ofmonarchs,58%offoundersofstates,41%ofcommunistleaders,19%ofmilitaryleaders,6%
amonarchandcommunistleaderissmallerthanthatsupportingademocraticallyelectedleader, ectsofnegativeobservableexogenousshocksoncoupsd' 1( [E(D) E(K)]=p) ortequaltothekingmakers'reservationstandardtheneachkingmakerisindi [ E(K)(n 1)=n]+[ E(D)=n],therepresentativekingmaker'soptimalreservationstandardislessthanthedictator'sreservationvaluesothatthedic-tatorexertse ortequaltotherepresentativekingmaker'sreservationstandard,acoupisaverted,andeachkingmakerearns[ E(K)(n 1)=n]+[ E(D)=n].Forpricesbelowthetriggerprice,therepresentativekingmaker'soptimalreservationstandardisgreaterthanthedictator'sreservationvaluesothatthedictatorexertszeroe ortandeachkingmakerearns[ E(K)(n 1)=n]+[ E(D)=n].Thus,the23
kingmakers'earningsareindependentofpriceandE(K)= E(K)+ ((E(D) E(K))nwhichimpliesE(K)= E(D).1)n(1 )+(A ,if(X;S)isanequilibrium,=Rd(r)=C 1( E(D))whereDsolvesthedictator' ndthatthetriggerpriceassociatedwiththisstrategypro le(X;S)isT(C 1( E(D)))= [(E(D) E(K)]=Y(C 1( E(D))).WecanthensubstituteforE(K)(C 1( E(D)))= E(D)n(1 )Y(C 1( E(D)))[n(1 )+ ](K)inY 1( [E(D) E(K)]=p)toob-tainthattherepresentativekingmaker'sreservationstandardisY 1( [nE(D)(1 )]=p[n(1 )+ ])ifE(D)isthepresentvalueofthedictator'(D)toE(D)=RT(C 1( E(D))0WdF(p)+R1 1 [E(D) E (K)]T(C 1( E(D))fW C(Y p)+ E(D)gdF(p)Theright-handsideofthisequationcanbesimpli edsothatZ1 E(D)=W+f E(D) C(Y 1 [E(D) E(K!)])gdF(p)()T(C 1( E(D))pWecanagainsubstituteforE(K)(D ),thepresentvalueofthedictator'sequilibriumlifetimeearnings,must24
bethesolution,inE(D),toZ1 !E1 E(D(D)=W+f E(D) C(Y )n(1 ))gdF(p)()T(C 1( E(D))p[n(1 )+ ],asrequired,(X ;S )isaMarkovsubgameperfectequilibriumthatgeneratesexpectedequilibriumlifetimeearningsE(D )andE(K ),(D).-handsideisR1n on (1 ) E(D)n(1 )T(C 1( E(D)) n(1 Y 10))+ (CF(p) p[n(1 )+d ]n o0 E(D) C(Y 1 E(D)n(1 )T(C 1( E(DfF(T(C 1(E(D))))(T C 10)(E(D))g)))[n(1 )+ ]However, E(D) C(Y 1( E(D)n(1 )=T(C 1( E(D)))[n(1 )+ ]))=0byde nitionofthetriggerprice,sothatthederivativeoftheright-handsidecanbesimpli edtoZ1( )0 E(D!))n(1 ) n (1(C Y 1)dF(pT(C 1( E(D))n(1 )+ )() p[n(1 )+ ] since(C Y 10),(withrespecttoE(D)),,thedictatorearnslessbyalwaysrunningawaywiththemoneythanbyrunningawaywiththemoneyonlywhenit',thatis,atE(D)=W,,earningWeachperiodwhileexertingzeroe ortisthehighestlifetimepayo (D)=W=(1 ),,E(D )
,wearguethatthetriggerpricefunctionshiftsdownasYorC ,thetriggerpricefunctionsatis esT(C 1( E(D)))= E(D)n(1 )Y(C 1( E(D)))[n(1 )+ ]TheequilibriumtriggerpricefunctionshiftsdownifYorC ,wearguethattheequilibriumexpectedearningsofthedictatorincreaseifYorC ' 1shiftsupsincethetriggerfunctionshiftsdownandY ,thedictator'searningsincreaseasYorC (D )increasesandYorC 1shiftsupinsuchawaythatc=Y C 1(c)shiftsdownforeachcwherec= E(D).Thus,F(T(C 1( E(D ))))decreasesasYorC 1shiftsupinsuchawaythatc=Y C 1(c) E(D)n(1 )=p[n(1 )+ ]increasesinnandC Y ,E(D ),(K )= E(D )=[n(1 )+ ],itsu ,theequilibriumtriggerpriceis E(D )n(1 )Y(C 1( E(D )))[n(1 )+ ]26
SinceE(D )decreasesinnandsincen(1 )=[n(1 )+ ]increasesinn;andsinceY C 1decreasesasE(D )decreases,the nalresultfollowsifY C 1(c),Y C 1isconcave,(0)=C(0)=0,c (Y C 1)0(c)=Y C 1(c)atc=,anypositiveincreasingconcavefunctiong(c)suchthatg(0)=0isinelastic13inc>,Y C 1isinelastic,=1;2facesanincreasing,convexcostofe ortCisuchthatC1(x)<C2(x)forallx2[0;1].Afteracoupoccurs,thenewdictator','stypethroughthelevelofe (K)andE(Di),i=1;2,representthepresentvalueofthelifetimeexpectedearningsofanactivekingmakerandadictatorknowntobeoftypei=1;(Du),pure,Markov,sub-gameperfectequilibriumisthetripleofreservationstrategies(Xi ;i=1;2;S )wherethedictator'sreservationvalueisr id=( E(D u)n(1 )Y(C 1( E(D i)))[n(1 )+; ]13Anincreasingfunctiong(c)isinelasticinc>0ifcg0(c)<g(c).Thelatterinequalityistrueifg(0)=,0g0(0)=g(0)(c)+cg00(c)<g0(c)sothatthederivativeoftheleft-handsideislessthanthatoftherightforc>
ifiisthedictator'stype;thekingmakers'reservationstandardisr k(p)=Y 1( [nE(D u)(1 )];p[n(1 )+ ])thepresentvalueofequilibriumlifetimeearnings,E(D i),ofadictatoroftypei=1;2istheuniquesolutioninE(Di)toZiE)(n(1 )(Di)=W1+f E(Di) C(Y 1 E(D(F(p);T(C 1(E(Di)))p[n(1 )+))gd ]E(D u)istheaverageofE(D 1)andE(D 2);andthepresentvalueofeachactivekingmaker'sequilibriumlifetimeearningsisE(K )= E(D u)=[n(1 )+ ]:,therepresentativekingmaker' ,thekingmaker' 1(p)=maxpY(x)f+ E(Ksn); E(Ku)+1 E(Du)g()nnnwherethesuperscripts,tands,belongtothesetf1;2;; ortissuchthat(pY(x)=n)+ E(Ks)<[(n 1) E(Ku)+ E(Du)]='sreservationstandardisrk(p)suchthat(pY(rk(p))=n)+ E(Ks)=[(n 1) E(Ku)+ E(Du)]=,whenacoupisstaged,eachkingmakerearns[(n 1) E(Ku)+1n E(Du)]=,eachkingmaker'searningsareatleastaslargeas[(n 1) E(Ku)+ E(Du)]=nanddependonthee ,analogoustotheargumentinthesimplermodel,
exertsrk(p)whenthisissmallenoughandotherwiseexertszeroe ortsothateachkingmakerearns[(n 1) E(Ku)+ E(Du)]=,therepresentativekingmaker'sexpectedpro ts,E(Kj),equal[(n 1) E(Ku)+1n E(Du)]=,inequilibrium,E(K)= E(Du)n(1 )+() whereE(Du),representstheexpectedlifetimeearningsofadictatorofunknowntype,andE(K),asinthesimplermodel,therepresentativekingmaker'sequilibriumstrategyistostageacoupwheneverx<Y 1( [nE(Du)(1 )]=p[n(1 )+ ]).Themaximiza-tionproblemthatadictatoroftypei=1;,theuniqueMarkov,subgameperfectequilibriumis(Xi ;i=1;2;S )andtheexpectedequilibriumlifetimeearningsequalE(K );E(D i);i=1;((Ci 1)( E(D i))= E(D u)n(1 )Y(C 1( E(D i)))[n(1 )+ ]denotetheequilibriumtriggerpriceforadictatoroftypei=1;(E(D1);E(D2))betheaverageoftheequilibriumconditionaltimestoacoupF(T((C1) 1( E(D1))))andF(T((C2) 1( E(D2)))).Acoupoccurswheneverthee ortrequiredtoavertacoupislargerthanthemaximume ortthatthedictatoriswillingtoexert,.,Y 1( [nE(D u)(1 1)]=p[n(1 )+ ])>(Ci )( E(D i)),acoupoccurswheneverp< E(D u)n(1 )Y(C 1( E(D i)))[n(1 )+ ]29
sothatF( E(D u)n(1 )=Y(C 1( E(D i)))[n(1 )+ ])istheprobabilitythatacoupoccursinanyperiodifthedictatorisoftypei=1;((C2) 1( E(D 2))))<Y((C1) 1( E(D 1)))).Thisinequalityholdsif(C2) 1( E(D 2)))<(C1) 1( E(D 1))). rstisthatsinceadictatoroftype1faceslowercoststhanthosefacedbyoneoftype2,thepayo 'equilibriumstrategysothatE(D 1) E(D 2).ThesecondreasonisthatsinceCi;i=1;2isincreasingandconvexsuchthatC2(x)>C1(x)itmustbethat(C1) 1(x)>(C2) 1(x).Thesetwofactsimplythat(C2) 1( E(D 2)))<(C1) 1( E(D 1))),whenadictatorisnewlyinstalled,theexpectedtimetoacoupisA(E(D 1);E(D 2)).IfthepricefallsbelowT((C2) 1( E(D 2)))insomeperiod,adictatorwhoexertsthee ,theprobabilitythatacoupoccursinthenextperiodisF(T((C1) 1( E(D 1)))).ThatthisisgreaterthanA(E(D 1);E(D 2)),ifthepricefallsbelowT((C2) 1( E(D 2)))insomeperiodpriortoperiodn,theexpectedtimetoacoupislowerthaninthedictator's ((C2) 1( E(D 2)))giventhatthepriceisaboveT((C1) 1( E(D 1)))ineachofthenperiodsis R1!nT((C2) 1( E(D 2)))pdF(p)R1:T((C1) 1( E(D 1)))pdf(p)Thus,ifadictatorhasbeeninpowernperiods,theconditionalprobabilitythatthepriceisbelowT((C2) 1( E(D 2)))inatleastoneofthenperiodsgiventhatthepriceisaboveT((C1) 1( E(D 1)))ineachofthenperiodsis R1!nT((C2) 1( E(D 2)))pdF(p)1 R1:T((C1) 1( E(D 1)))pdf(p)30
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