CommitmentversusFlexibilitywithCostlyVericationMarinaHalacYaleUniversi - PDF document

CommitmentversusFlexibilitywithCostlyVericationMarinaHalacYaleUniversi
CommitmentversusFlexibilitywithCostlyVericationMarinaHalacYaleUniversi

CommitmentversusFlexibilitywithCostlyVericationMarinaHalacYaleUniversi - Description


thepresenceofinformationcostsandmisalignedincentivesInasurveyofmanufacturingrmsRoss1986observesthattopcorporatemanagementisoftentoobusyandpreoccupiedwithotherresponsibilitiestohavethetimeandresources ID: 897343 Download Pdf

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1 CommitmentversusFlexibilitywithCostlyVer
CommitmentversusFlexibilitywithCostlyVeriȚcationMarinaHalacYaleUniversityandCenterforEconomicPolicyResearchPierreYaredColumbiaUniversityandNationalBureauofEconomicResearchAprincipalfacesanagentwhoisbetterinformedbutbiasedtowardhigheractions.Shecanverifytheagentsinformationandspecifyhispermissibleactions.WeshowthatiftheveriȚcationcostissmallenough,athresholdwithanescapeclause(TEC)isoptimal:theagenteitherchoosesanactionbelowathresholdorrequestsveriȚcationandtheef-Țcientactionabovethethreshold.Forhighercosts,however,theprinci-palmayrequireveriȚcationonlyforintermediateactions,dividingthedelegationset.TECisalwaysoptimaliftheprincipalcannotcommittoinefȚcientallocationsfollowingtheveriȚcationdecisionandresult.I.IntroductionOrganizationsestablishcapitalbudgetingprocedurestoensuretherightallocationofcapitaltoprojects.BudgetingdecisionsarecomplicatedbyWethankManuelAmador,KyleBagwell,HeskiBar-Isaac,RolandBenabou,V.V.Chari,Yeon-KooChe,StephenCoate,EddieDekel,WouterDessein,SimoneGalperti,Johannesrner,NavinKartik,AndreasKleiner,JanKnoepße,FrancineLafontaine,BartLipman,AndreyMalenko,MarcoOttaviani,AlessandroPavan,DebrajRay,KenShotts,PaoloSico-nolȚ,JoelSobel,JeffZwiebel,andvariousseminarandconferenceaudiencesforhelpfulcomments.WealsothankSebastianDiTellaandNikoMatouschekforvaluablediscussionsofthepaper.WeijieZhongprovidedexcellentresearchassistance.ElectronicallypublishedOctober23,20202020JournalofPoliticalEconomy,2020,vol.128,no.12]2020byTheUniversityofChicago.Allrightsreserved.0022-3808/2020/12812-0004$10.00 thepresenceofinformationcostsandmisalignedincentives.InasurveyofmanufacturingȚrms,Ross(1986)observesthattopcorporatemanage-mentisoftentoobusyandpreoccupiedwithotherresponsibilitiestohavethetimeandresourcestoevaluateeveryinvestmentopportunity.Atthesametime,lower-leveldivisionheadswhopossesstherequiredin-formationfordecision-makingarepronetooverstatetheirinvestmentopportunitiesandspendexcessively(e.g.,BrealeyandMyers1981;Don-aldso

2 n1984).Infact,thecombinationofthesefacto
n1984).Infact,thecombinationofthesefactorsisseenasamainreasonwhytopmanagementoftenimposescapitalrationingontheȚrmdivisions.Analogousproblemsofinformationandincentivesariseinotherappli-cations.IntheoperationofȚscalpolicy,citizensdonothavethecapacitytoevaluateeverybudgetallocationthatmustbemade,whereaselectedofȚcialswhoknowthevalueofoutlaysarebiasedtowardoverspendingbecauseofpoliticalinterests(e.g.,AguiarandAmador2011;HalacandYared2014,2019a).Inthecontextofinternationaltrade,theWorldTradeOrganizationcannotassesstheappropriatenessofeverytariffresponsetodumping,whereasthegovernmentsthatunderstandthetruecircum-stancesarebiasedtowardimposinghightariffstoprotecttheirdomesticindustries(e.g.,AmadorandBagwell2013;BeshkarandBond2017).StartingwiththeworkofHolmstrm(1977,1984),decision-makingintheseenvironmentshasbeenformallyanalyzedasadelegationproblem.Thecanonicalsettingconsistsofaprincipalwhofacesabetterinformedbutbiasedagentandcannotrelyontransfers.Importantly,itisassumedthattheprincipalscostofverifyingtheagentsinformationisprohibi-tivelyhigh,soalltheprincipalcandoisspecifyasetofallowableactionsfromwhichtheagentcanselect.Theoptimaldelegationsetisshapedbyafundamentaltrade-offbetweencommitmentandßexibility:anarrowsetlimitsbiaseddecisionsbytheagent,whereasawidesetletstheagentutilizehisprivateinformationabouttheefȚcientaction.Amaininsightfromtheliteratureisthatunderweakconditions,thistrade-offisopti-mallyresolvedbythresholddelegation.Thatis,theprescriptionistoUsingsurveydata,PruittandGitman(1987)Țndthatseniormanagersareawarethatjuniormanagersoverstateestimatedprojectrevenues.TheyalsoȚndthatthisoverstate-mentistoalargeextentconsideredintentional.SeeMukherjeeandRahahleh(2011).Astheauthorsdescribe,alargeliteratureȚndsevidencethatȚrmsoperateundercapitalconstraintsand,moreover,thattheseconstraintsareimposedinternallybyseniormanagersratherthanexternallybythesuppliersoffunds.Asanotherexample,inaretailsetting,itistoocostlyformanagerstoscruti

3 nizethebestsalesstrategyforeveryclient,w
nizethebestsalesstrategyforeveryclient,whereasthesalesrepresentativeswhoareabletodosogenerallyoffertoomanydiscounts(e.g.,Loetal.2016).Invariousapplications,likethosedescribedabove,(contingent)transfersmayberuledoutbecauseofinstitutionalreasonsorethicalconsiderations.Intheirstudyofcapitalbudgetingpractices,MukherjeeandRahahleh(2011)reportthatdirectlyrewardingem-ployeesforproposinggoodinvestmentsisuncommoninlargeȚrms.4524journalofpoliticaleconomy setbudgetcapsformanagersinorganizations,deȚcitlimitsinthecon-textofȚscalpolicy,andtariffcapsaspartoftradeagreements.AkeylimitationofexistinganalysesofdelegationconcernstheuseofveriȚcation.Inpractice,whileitiscostlyforprincipalstoverifyagentsformation,thiscostisnotashighastoruleoutveriȚcationaltogether.Real-worlddelegationrulestypicallyfeatureacaponallowableactions,asinthecanonicalmodel,togetherwithreviewandapprovalproceduresforrequeststhatexceedthecap.Inorganizations,[s]mallerprojectscantypicallybeapprovedbydivisionheads,andthus,withinthebudgetlim-its,decision-makingfortheseprojectsiscompletelydecentralized....Largerprojects,bycontrast,mustbeapprovedbyacentralinvestmentcommitteeoreventheboardofdirectors(Taggart1987,18).Becausethecommitteeinchargeofapprovalmustspendcostlytimegatheringin-formation,evaluatingcashßowprojections,anddeliberatingontheap-propriateinvestment,onlycertainprojectsareveriȚed:Giventheirprior-ities,topmanagementoftencopeswithproductivityimprovementbyallocatingsmallȚxedsumstodivisionsandplants.Thatleavesthemthetimetocarefullyanalyzethelargeprojects(Ross1986,21).Similarproce-duresareusedinthecontextofȚscalandtradepolicy,whererulesspecifydeȚcitortariffcapstogetherwithescapeclauseanddisputesettlementprovisionsforbreachingthesecapsunderveriȚedspecialconditions.Motivatedbytheseapplications,westudyageneraldelegationframe-workinwhichveriȚcationiscostlybutfeasible.WemodelcostlyveriȚca-tionasintheseminalworkofTownsend(1979)andexplorehowitaffectsoptimaldelegation.Howdoest

4 heprincipalchoosethedelegationsettooptim
heprincipalchoosethedelegationsettooptimallyresolvethetrade-offbetweencommitmentandßexibilitywhileatthesametimeminimizingveriȚcationcosts?WeȚndthatoptimalrulescantakecomplicatedforms,asveriȚcationeffectivelyallowstheprincipaltorelaxincentiveconstraintsbydividingthedelegationsetintosubsets.Yetweshowthatundercertainconditions,anoptimalruletakesthesim-pleformofathresholdwithanescapeclause(TEC).WedeȚneTECasaruleinwhichtheagenteitherselectsanactionbelowathresholdorrequestsveriȚcationandtheefȚcientactionabovethethresholdbytriggeringtheescapeclause.Asnotedabove,rulesofthisformarecommonlyobservedinapplications.OurpaperprovidesatheoreticalfoundationforthebroaduseofTECandshowshowitsoptimalitydependsontheprincipalscostofveriȚcationandhercommitmentpower.Ourmodelfeaturesanagentwhoisbiasedtowardhigherspendingrel-ativetotheprincipal.Theagentsprivateinformation,or,concernsSeealsoBowerandLesard(1973),Ross(1986),MukherjeeandHenderson(1987),GitmanandVandenberg(2000),andGrinsteinandTolkowsky(2004),amongothers.See,e.g.,Schaechteretal.(2012),Lledetal.(2017),andCoateandMilton(2019)onȚscalrulesandBeshkarandBond(2017)ontradeagreements.commitmentversusexibilitywithcostlyverication4525 thevalueofspending;ahighertypecorrespondstoahighermarginalvalueofspendingforboththeprincipalandtheagent.Followingthelit-erature,webuilduponasettinginwhich,absentveriȚcation,anoptimaldelegationrulewouldbeathreshold,allowingtheagenttochooseanyspendinguptoamaximumlevel.Wedepartfrompriorworkbylettingtheprincipalverifyandperfectlylearntheagentstype.VeriȚcationen-tailsanadditivecostfortheprincipal,whichmayalsobepartiallybornbytheagent.Webeginouranalysisbyassumingthattheprincipalcanfullycommittoadelegationrule.Theproblemcanbeviewedinthreesteps:Țrst,theprin-cipalchoosesamappingfromtheagentsveriȚcationdecisionandresulttoasetofallowablespending;second,theagentdecideswhethertoseekveriȚcation;third,theagentchoosesaspendinglevelfromtheallowableset.Formally,adelegationruleisapai

5 rofschedulesspecifying,foreachagenttype,
rofschedulesspecifying,foreachagenttype,whetherheisveriȚedandhisspendinglevel.AdelegationruleisoptimalifitmaximizestheprincipalsexpectedwelfaresubjecttotheincentivecompatibilityconstraintthateachagenttypepreferhisveriȚca-tionassignmentandspendingleveltothoseofanyothertype.Inpartic-ular,eachtypemustpreferhisallocationtothatofanyothertypewhoisnotprescribedveriȚcation.DeviationstotypeswhoareveriȚedcanbetriv-iallydeterredastheygetrevealedbytheprincipalsveriȚcation.Asanim-plication,theuseofveriȚcationcanmakelocalincentivecompatibilityconstraintsslackwhilenonlocalconstraintsbind;ouranalysismakesuseofperturbationmethodstoaddresstheseissues.OurȚrstmainresultshowsthatTECisoptimalifthecostofveriȚcationissufȚcientlysmall.Importantly,wealsoshowthatverifyingallagenttypesisneveroptimal;hence,nomatterhowsmalltheveriȚcationcostis,anoptimalruleprescribesnoveriȚcationforsometypes.TheintuitionwhyTECisoptimalisthatverifyinganupperregionofagenttypesnotonlyallowstheprincipaltoimprovetheirspendingallocationbutalsoisanefȚcientmeansofimposingdisciplineonloweragenttypeswhoarenotveriȚed;thesetypesselectfromasetoflowerspendinglevelsandcannotmimichighertypeswhoareveriȚed.Toprovetheresult,weshowthatanyrulewithdecreasingveriȚcationprescribingveriȚcationforasetofagenttypesandnoveriȚcationforasetofhighertypescanbedominated.DecreasingveriȚcationisexpensivefortheprincipalbe-causeitrequiresincentivizingtypesintheveriȚcationregiontoseekveri-Țcationratherthanmimicahighertypeinano-veriȚcationregionabovethem,andthisinturnrequiresinducingsigniȚcantoverspendingintheno-veriȚcationregion.WeshowthatwhentheveriȚcationcostissmallWerestrictattentiontodeterministicrules(seesec.VIforadiscussion)andprovethatarevelationprincipleintermsofpayoffsholdsinoursetting.4526journalofpoliticaleconomy enough,aperturbationthatveriȚesalltypesinthedecreasingveriȚcationregionincreasestheprincipalswelfare.TEC,however,maynotbeoptimaliftheveriȚcationcostisrelativelylarger.Oursecondmainres

6 ultshowsthatdecreasingveriȚcationcanbest
ultshowsthatdecreasingveriȚcationcanbestrictlyoptimalinthiscase.Forexample,arulethatveriȚesonlyanintermediatesetoftypescanyieldtheprincipalhigherwelfarerelativetonotverifyinganytypeaswellasrelativetousingaTECrule.ThemainreasonwhyverifyingonlyintermediatetypescandominatenotverifyinganytypeisthattheveriȚcationregionservestodisciplinetypesintheno-veriȚcationregionbelow.ThemainreasonwhyverifyingonlyintermediatetypescandominateTECisthatitallowstheprincipaltosaveonveriȚca-tioncosts.WeshowthatthesebeneȚtscanoutweighthecostofoverspend-ingthatisneededtoincentivizeintermediatetypestobeveriȚed.Thus,arulethatinvolvesdecreasingveriȚcationcanbeoptimalwhentheveriȚca-tioncostisnotsmall(andnotlarge)enough.TheoptimalityofdecreasingveriȚcationdoesnotrelyonanysortofasymmetryinthepayoffordistributionfunctions.Asnoted,oursettingisoneinwhichthresholdrulesarealwaysoptimalabsentveriȚcation,andinfactweprovetheresultbytakingthewidelystudiedcaseofqua-draticpreferencesandauniformdistributionoftypes.Aninteriorveri-ȚcationregioncanbebeneȚcialbecauseitallowstheprincipaltodividethedelegationsetwhilekeepingveriȚcationataminimum.Observethattheagentwantstooverspendrelativetotheprincipalbuthispreferredspendingleveldependsonhistype.Consequently,requiringveriȚcationforintermediatespendinglevelsmaysufȚcetolimitthespendingofrel-ativelylowtypes:thesetypesareunabletojustifyincreasingtheirspend-ingtoanintermediatelevelviaveriȚcation,andincreasingtheirspend-ingfurtherwouldnotbeattractivetothem.Theaboveresultraisesthequestionofwhyruleswithdecreasingveri-Țcationarerarelyobservedinreality.Weprovideananswerbasedonapracticalconsideration:implementingsucharulerequiresstrongcom-mitmentpowerfromtheprincipal,strongerthanwhatmaybefeasibleinapplications.Taketheaforementionedruleinwhichtheprincipalver-iȚesonlyanintermediatesetoftypes.Underthisrule,theprincipalcom-mitstoanallocationthatmaybeinefȚcientexpost,followingtheveriȚ-cationdecisionandresult.Inparticular,therulemayassig

7 naninefȚcientspendinglevelaftertheagents
naninefȚcientspendinglevelaftertheagentstypeisveriȚedbothinthecasethatthesseekingveriȚcationisonpathaswellaswhenthisveriȚcationispartofadeviation.Moreover,therulemayinduceanallocationaftertheagentdecidesnottoseekveriȚcationthatisinefȚcientconditionalonnoveriȚcation,thatis,whenignoringtheincentivesofveriȚedtypes.WhathappensiftheprincipalisunabletocommitexantetotheseexpostinefȚcientallocations?Inorganizations,forexample,eveniftopmanagementspeciȚescertainbudgetsandrequirementsexante,itiscommitmentversusexibilitywithcostlyverication4527 commonforthesetobechangedexpost.Ross(1986)documentsthatinȚrmswhosebudgetingproceduresresembleTEC,budgetapprovalsdonotconformtopreannouncedcriteriabutdependonthediscretionoftopmanagement.InvestmentcommitteesdecidethescopeofprojectsthatarebroughtupforveriȚcationandapprovalaswellasthebudgetcapforprojectsthathavenotbeensubmittedforreview.InrelatedworkonchiefexecutiveofȚcersandcorporateboards,GrinsteinandTolkow-sky(2004)ȚndthatcorporateboardsexertsigniȚcantdiscretioninre-viewingandapprovingannualbudgetsandlargecapitalrequestsmadebythechiefexecutiveofȚcer.Ourthirdmainresultshowsthatiftheprincipalscommitmentpowerislimited,thenTECisoptimalwheneververiȚcationisoptimal.Intermsofthethree-steptimingdescribedpreviously,limitedcommitmentpowermeansthattheprincipalnowrevisestheagentsallowablespendingsetfollowingtheagentsveriȚcationdecisionandresult.Weprovethatinthiscase,anyincentive-compatiblerulemusthaveweaklyincreasingver-iȚcationeverywhere.ThereasonisthatinducingdecreasingveriȚcationrequiresincentivizingveriȚedtypesnottodeviateandchooseahigherspendinglevelinano-veriȚcationregionabovethem,andunderlimitedcommitmentpoweritalsorequiresincentivizingnonveriȚedtypesnottoseekaveriȚcationthatguaranteesthemefȚcientspending.Whenunabletofullycommittoaruleexante,theprincipalcannotimplementthespendinglevelsthatwouldbeneededtomakethesedeviationsunattrac-tive,andthusdecreasingveriȚcationisnotfeasible.Consequently,weob-t

8 ainthatunderlimitedcommitmentpower,anyop
ainthatunderlimitedcommitmentpower,anyoptimalrulefeaturingveriȚcationmustbeTEC.Altogether,ourresultsprovideatheoreticaljustiȚcationforthepreva-lenceofTECintherealworld.WhenveriȚcationcostsaresmallenough,evenaprincipalwhocancommittoanyclassofdelegationrulewillȚnditoptimaltochooseonewiththesimpleformofTEC.WhenveriȚcationcostsarelarger,morecomplexrulesmayperformbetter,butTECre-mainstheprincipalsoptimalruleifhercommitmentpowerislimited.Animplicationisthatlimitationstocommitmentpower,aswehavecon-sidered,appeartobeprevalentinapplicationsandanimportantreasonbehindthebroaduseofTECrules.RelatedliteratureOurpaperisrelatedtoseveralliteratures.First,wecontributetotheliteratureonoptimaldelegationandself-control,start-ingwithHolmstrm(1977,1984).MainreferencesincludeMelumadandShibano(1991)andAlonsoandMatouschek(2008)ondelegationSeealsoBowerandLesard(1973)andTaggart(1987,18).Thelatternotesthat[i]fasworthwhileprojectsexceeditsbudget,topmanagementmaybewilingtorene-Additionally,MukherjeeandHenderson(1987)observethatȚrmscriteriaaresometimesunclear,astheydependontheclassofproject.Thisgivestopman-agementmorediscretiontomakedecisionsexpost.4528journalofpoliticaleconomy underquadraticpreferences;Amador,Werning,andAngeletos(2006)onconsumption-savingsproblemswithhyperbolicpreferences;andAmadorandBagwell(2013),whichconsidersageneralframeworkthatwetakeasourbaseline.Asinthisliterature,westudyaprincipal-agentenvironmentwithnotransfersinwhichtheagentisbetterinformedabouttheefȚcientactionbutbiasedrelativetotheprincipal.Incontrasttothisliterature,weallowtheprincipaltoverifytheagentsinformationatacost.Byintroducingthisadditionaltool,weareabletoexplorehowescapeclausesareoptimallyusedandhowoptimaldelegationdependsontheextentoftheprincipalscommitmentpower.Second,wecontributetotheliteratureoncostlyveriȚcation,startingwithTownsend(1979).BoththatpaperandothersthatfolloweditcludingGaleandHellwig(1985),BorderandSobel(1987),andMook-herjeeandPng(1989)analyzesetti

9 ngswithtransfers,whichweruleout.Morerece
ngswithtransfers,whichweruleout.Morerecently,Ben-Porath,Dekel,andLipman(2014)andErlansonandKleiner(2015)considercostlyveriȚcationinone-goodandcollectiveallocationproblemswithouttransfers,andGlazerandRubinstein(2004,2006)andMylovanovandZapechelnyuk(2017)studyrelatedquestionsusingdifferentveriȚcationtechnologies.Ourmaindeparturefromthisliterature(inadditiontootherdifferencesspeciȚctoeachpaper)isthatwestudyadelegationsettinginwhichweallowfordifferentdegreesofbiasbytheagentrelativetotheprincipal.ThisisalsoamaindistinctionwithrespecttoHarrisandRaviv(1996)andthedynamicversioninMalenko(2019)whoconsidercostlyveriȚcationinadelegationmodelwheretheagentalwaysbeneȚtsfromhigheractions.Suchanextremebiasassumptionimpliesthatgrantingtheagentßexibilityhasnovaluetotheprincipal.Weinsteadbuildonacanonicaldelegationframeworkinwhichßexibilityisvaluable;thatis,theagentsmostpreferredactionSeealsoAthey,Atkeson,andKehoe(2005),AmbrusandEgorov(2013,2017),HalacandYared(2014,2018,2019a,2019b),andAmador,Bagwell,andFrankel(2018).AusterandPavoni(2017)studyadelegationproblemwithlimitedawarenessthatgivesrisetoanonintervaldelegationset.WestudytheeffectsoftheprincipalnotbeingabletocommittonotchangingthesallowablespendingsetfollowingtheveriȚcationdecisionandresult.Adifferentquestionthataliteratureonauditinghasinvestigatedconcernsaprincipalsabilitytocom-mittoanauditstrategy;see,e.g.,ReinganumandWilde(1986),Banks(1989),andChat-terjee,Morton,andMukherji(2008).Workondelegationandself-controlhasalsostudiedlackofcommitmenttorules;thisincludesBernheim,Ray,andYeltekin(2015),DovisandKirpalani(2017),andHalacandYared(2019a).Morebroadly,thereisaliteratureonmechanismdesignandimplementationwithevidence,includingGreenandLaffont(1986),BullandWatson(2007),DeneckereandSeverinov(2008),Ben-PorathandLipman(2012),andKartikandTercieux(2012).ThemodelinHarrisandRaviv(1996)alsodiffersfromoursinotherrespects:thereareonlythreeagenttypes,theagentreceivesanoncontingenttransferfromthepri

10 ncipal,andtheprincipalcanchoosetoverifyt
ncipal,andtheprincipalcanchoosetoverifytheagentwithaninteriorprobability.HarrisandRaviv(1998)consideranextensioninwhichcapitalisallocatedacrossmultipleprojects.Malenko(2019)analyzesadynamicversioninwhichprojectsofindependentandidenti-callydistributedqualityarrivestochasticallyovertime.commitmentversusexibilitywithcostlyverication4529 ishigherthantheprincipalsbutnotnecessarilythehighestpossibleac-tion.OurpaperprovidestheȚrststudyofoptimaldelegationandveriȚ-cationinasettinginwhichtheprincipalfacesacommitmentversusßex-ibilitytrade-off.Weshowthatthistrade-offintroducesnewconceptualissuesintoourmechanismdesignproblemandshapestheprincipalsop-timalrule.Finally,ourpaperisalsorelatedtoaliteraturethatstudiespolicyruleswithescapeclausesinmacroeconomicmodels.BuildingontheseminalworkofRogoff(1985)oncommitmentversusßexibility,FloodandIsard(1988)andLohmann(1992)considertheuseofescapeclausesinmon-etarypolicy.Obstfeld(1997)discussesthemeritsofescapeclausesinthecontextofȚxedexchangeratesystems,wheremembercountriesareal-lowedtorealigninthefaceofsevereshocks.BeshkarandBond(2017)analyzetradeagreementswithintheclassoftariffcapswithescapeclauses,wheretherelianceonveriȚcationrelativetotariffoverhangoptimallydependsonthelevelofinternationalexternality.CoateandMilton(2019)considertheoptimaldesignofȚscallimitsforapoliticianwhoisallowedtooverridethelimitandselecthispreferredactionwiththecitizensproval.Wesharewiththisliteratureourmotivationofexaminingtheroleofescapeclauses.OurmaindepartureisthatweusemechanismdesigntostudyoptimalruleswithveriȚcationwithoutrestrictingtheirII.ModelOurbaselinemodelofdelegationisthesamegeneralprincipal-agentenvironmentofAmadorandBagwell(2013),wherewefocusonthecaseinwhichtheagentsbiasistowardhigheractions.Weexpandthisdele-gationmodelbyallowingforcostlystateveriȚcation,followingTown-send(1979).A.EnvironmentThereareaprincipalandanagent.Thestateis g,gfor withcontinuousdensity0forall.Thecorrespondingdistribu-tionfunctionis

11 ).Thelevelofspendingisdenotedby Theprinc
).Thelevelofspendingisdenotedby Theprincipalswelfareis),twicecontinuouslydifferentia-blewith0.Weassumethattheprincipalsoptimum,argmax,isinterior,andwerefertoitastheefȚcientlevelofspending.Weimposethefollowingsingle-crossingcondition: Thus,theefȚcientlevelofspendingisincreasinginthestate:4530journalofpoliticaleconomy Theagentswelfareis,with)twicecontinu-ouslydifferentiableand0.Weassumethattheagentsoptimum,argmax,isinterior,andwerefertoitastheofspending.NotethattheagentswelfaresatisȚesthesingle-crossingWeconsideranagentwhoisbiasedto-wardhigherspendingrelativetotheprincipal.SpeciȚcally,weaddthefollowingassumptiontothesettingofAmadorandBagwell(2013): UAg,pp� Condition(2)saysthattheagentnotonlybeneȚtsfromincreasingspend-ingwhenevertheprincipaldoesbutalsobeneȚtsfromanyspendingincreasemorethantheprincipal.NotethatsincewetooktheefȚcientandßexiblespendinglevelstobeinteriorandthepartiesutilitiesfromspendingtobestrictlyconcave,animplicationofthisconditionisthattheßexiblelevelofspendingalwaysexceedstheefȚcientlevel.Thatis,ourassumptionsyield UAg,ppp5pPđgȚ� UPg,ppp5pPđgȚ505 which,given0,impliesforallThestateisprivateinformationtotheagent,thatis,theagentTheprincipalcanperfectlyverifybypayinganadditivecost0.ThescostofveriȚcationis0,1.ThisformulationallowsustocoversituationsinwhichtheagentpaysnoveriȚcationcost(0)aswellassituationsinwhichhepaysacostnolargerthantheprincipal0,1).Onecouldalsoallowfortheagenttopayahighercostthantheprincipals.OurresultsinsectionIVcontinuetoholdunderprovidedthattheagentsbiasissufȚcientlylarge;ourresultsinsectionVholdindependentlyofthevalueofByfeaturingbothabiasandprivateinformationbytheagent,ouren-vironmentgivesrisetoacommitmentversusßexibilitytrade-off.Iftheagentwerenotbiasedrelativetotheprincipal,theprincipalcouldim-plementtheefȚcientlevelofspendingbyprovidingfullßexibilitytotheagent(whowouldinthiscasechoose).Similarly,ifthestatewerenottheagentsprivateinformation,theprincipalcouldimpl

12 ementtheefȚcientlevelofspendingbycommitt
ementtheefȚcientlevelofspendingbycommittingtheagenttoafullycontingentspendingplan.Inthepresenceofbothabiasandpri-vateinformation,however,theprincipalcannotimplementefȚcient)forallwithoutveriȚcation,andshefacesanontrivialtrade-offbetweencommitmentandßexibility.Forboththeprincipalandtheagentspreferences,wewillrefertosinglecrossingasthe(stronger)supermodularityconditionthatwehaveassumedthesepreferencessatisfy.commitmentversusexibilitywithcostlyverication4531 SpecialcasesThemodelofdelegationdescribedaboveencompassesspeciȚccasescommonlystudiedintheliterature.Oneexampleisthecaseofquadraticpreferenceswithaconstantbias(whichwewillrefertoassimplyquadraticpreferences),examinedbyMelumadandShibano(1991)andAlonsoandMatouschek(2008)andusedextensivelyinappliedwork.Underthesepreferences,theprincipalswelfareis2andtheagentswelfareis2forsome0representingtheagentsbias.Thisformulationisequivalenttolettingandfor2andisthere-foreaspecialcaseofourmodel.Wewillusethequadraticpreferencescasetoillustratesomeofourresults.Anotherexampleisthemodelofconsumptionunderhyperbolicpref-erences,analyzedbyAmador,Werning,andAngeletos(2006)andHalacandYared(2014,2018,2019a).Theprincipalswelfareinthiscaseisandtheagentswelfareis,whereareutilityfunctions;representconsumptionandexoge-nousincome,respectively;and0,1capturesthedegreeofpresentbiasbytheagent.Thisformulationisequivalenttoletting,with,andisthusalsoencompassedbyourmodel.B.TimingTheorderofeventsisasfollows:1.Theprincipalsetsarule,whichmapsaveriȚcationdecisionandresultintoanallowablespendingset2.TheagentchooseswhethertoseekveriȚcation,0,1,andtheprincipalperfectlyveriȚeshistype3.TheagentchoosesaspendinglevelfromtheallowablesetTheabovetimingassumesthattheagentlearnshistypebeforetheprincipalsetsaruleinstep1.Ouranalysisisunchangedifinsteadtheagentlearnshistypeaftertherulehasbeenset,thatis,atthebeginningofstep2.C.DelegationRulesGiventhegameformdescribedabove,wecananalyzetheprincipalproblemasthatofchoosingadelegatio

13 nrulethatconsistsofapairHalacandYared(20
nrulethatconsistsofapairHalacandYared(2014,2018,2019a)usethismodeltostudyȚscalrules,whereagov-sdeȚcitbiasmayemergefromtheaggregationofheterogeneous,time-consistentpreferences(JacksonandYariv2015,2014)orfromturnoverinapoliticalecon-omysetting(AguiarandAmador2011;AlesinaandPassalacqua2016).SeeYared(2019)forabroaddiscussionofthisapplication.4532journalofpoliticaleconomy ofschedules,specifyingaveriȚcationdecisionandspend-inglevelforeachtype.Theprincipalchoosesaruletomaximizeherexpectedwelfare: Theobjective(3)istheprincipalsexpectedwelfareunderagivenrule,takingintoaccounttheadditiveveriȚcationcosts.Theconstraint(4)isanincentivecompatibility(ortruth-telling)constraint:itguaranteesthatanagentoftypeprefershisassignedveriȚcationdecisionandspending)and),toadifferentallocationforsometypewhoisnotveriȚed(i.e.,with0).NotethatitissufȚcienttocon-siderdeviationstononveriȚedtypes:sinceadeviationinwhichanagentoftypemimicsaveriȚedtypewouldbedetectedbytheprincipal(asveriȚcationrevealsthetruetype)andtheprincipalcanarbitrarilypunishtheagent(throughthespendingallocation)whenshelearnsthathehasdeviated,wedonotneedtoconsidersuchadeviation.Wealsonotethattheformulationabovedoesnotruleoutmixedstrat-egiesbytheagent.IftheagentwerewillingtomixoververiȚcationandnoveriȚcationoroverdifferentspendinglevels,hewouldbeindifferentovertheseallocations,andthustheprincipalcanselectoneofthesethatmaximizesherexpectedwelfare.Infact,buildingonthisobservation,wecanshowthatourresultsarenotlimitedtothegameforminsectionII.Bbutcontinuetoholdwhenallowingforanyindirectmechanismspecify-ingamessagespacefortheagentandadeterministicallocationfunctiontowhichtheprincipalcommits.Suchamechanisminducesagameinwhichtheagentsendsamessage,iseitherveriȚedornotasafunctionofthemessage,andisassignedaspendinglevelasafunctionofthemes-sageandveriȚcationresult.WeshowinappendixB(availableonline)thataversionoftherevelationprincipleintermsofpayoffsholdsinourset-ting,implyingthattostudytheoptimalde

14 terministicmechanismfortheprincipal,itis
terministicmechanismfortheprincipal,itiswithoutlosstorestrictattentiontodeterministicdirectTheprincipalcanpunishadeviationofatypeinwhichhemimicsatype1byassigningfollowingveriȚcationsomespendinglevelsuchthat.Itisclearthatsuchaspendinglevelexists;infact,settingwouldbeasufȚcientpunishment.Whilethisselectionrelaxestheprincipalsproblem,itisnotusedundertheoptimalruledescribedinourmainresultinproposition3,whichinducesauniquebestresponsebytheagent.Hence,theresultdoesnotrelyonselectionofequilibriaofthegameinsec.II.B.commitmentversusexibilitywithcostlyverication4533 mechanisms(i.e.,wherethemessagespacecoincideswiththeagentstypespace)thatinducetruthfulreportingbytheagent,asconsideredintheprogramin(3)and(4)above.Becausethereisacontinuumoftypes,itispossiblethattheproblemin(3)and(4)admitsmultiplesolutionsthatareidenticaleverywhereex-ceptforameasurezerosetoftypes.Asameansofselectingtheoptimuminsuchasituation,wesaythataruleifitsolves(3)and(4)andthereisnoothersolution,withassociatedveriȚcationandspend-ingschedules,suchthatforallandstrictlyforsome.Althoughmultiplesolutionscaninprinciplecontinuetoexistunderthiscondition,thiscriterionturnsouttobesufȚcientforourcharacterization.III.NoVeriȚcationBenchmarkBeforeanalyzingtheoptimaldelegationrulewithveriȚcation,wereviewtheresultsoftheliteraturebyconsideringtheoptimalruleintheab-senceofveriȚcation.Considertheprincipalsproblemin(3)and(4)subjecttotheadditionalconstraintthat0forall(sothatcon-straint(4)becomesforall).AmadorandBagwell(2013)studythisproblem.Tosolveit,theymakethefollowingassumption1onthedistributionof;weextendthisassumptiontoanytruncationfromabove,withsupport[ ]for,densityanddistributionfunction1.Takethedistributionoftruncatedfromaboveby.Foreachsuchtruncateddistribution,thereexistssuchthat isnondecreasingforall ;and UP~g,pAg*p forall,withequalityatOnecanverifythatforthespecialcasestypicallystudiedintheliter-ature,suchasthosewithquadraticorhyperbolicpreferences,assump-tion1issatisȚedun

15 dercommonlyuseddistributionfunctions,inc
dercommonlyuseddistributionfunctions,including4534journalofpoliticaleconomy exponential,lognormal,andanynondecreasingdensity.Givenassump-tion1,theresultsinAmadorandBagwell(2013)yieldthefollowing:1(OptimalruleundernopossibilityofveriȚcation).Takethedistributionoftruncatedfromaboveby.Iftheprin-cipalisconstrainedto0forall ,anoptimalruleisasuchthat UndernoveriȚcation,anoptimalruleisathresholdsuchthatallspendattheirßexiblelevelandalltypesarebunchedattheßexiblespendinglevelof.Theprincipalcanimplementthisrulebysettingaspendinglimitandallowingtheagenttochooseanyspendingleveluptothislimit.Figure1illustratesanoptimalruleundernoveriȚcationforthecaseofquadraticpreferences.Thelevelofspendingisontheverticalaxisandtheagentstypeonthehorizontalaxis.Inthissimpleexample,bothef-Țcientandßexiblespendingareincreasinglinearfunctionsofthestate,andßexiblespendingexceedsefȚcientspendingbyaconstantamount .1.OptimalruleundernopossibilityofveriȚcation.TheȚgureisdrawnforthequadraticpreferencescase(seesec.II.A),wherewelet 12,and)uniform.Wealsonotethatassumption1ontheoriginaldistributionimpliesthattheassump-tionissatisȚedforalltruncationsfromaboveiftheconditionsinproposition2ofAmadorandBagwell(2013)hold.commitmentversusexibilitywithcostlyverication4535 representingtheagentsbias.Therulecharacterizedinproposition1speciȚesaspendinglevelthatcoincideswiththeagentsßexiblelevelforandequalsAkeyinsightbehindtheresultinproposition1isthatholesaresubop-timal.Moreprecisely,theprincipalcanalwaysimproveuponaruleasthatdepictedinȚgure2,whichdoesnotallowtheagenttochoosespend-ingforsomeinteriorbutdoesallowtheagenttochoosespendingimmediatelybelowandimmediatelyabove.Thehole[]impliesthatanagentoftypeforwhomisnotallowedtospendathisßexiblelevel.Suchanagentspendsatthelowerlimitoftheholeifhistypeisrelativelylow,buthespendsattheupperlimitoftheholeifhistypeishigher.Theroleofassumption1istoguaranteethatiftheprincipalremovesthehole,thebeneȚtofreducingoverspendingforthetypesthatbun

16 chwouldoutweighanypotentialcostsofincrea
chwouldoutweighanypotentialcostsofincreasingspendingforthetypesthatbunchatIV.OptimalRuleWenowturntothestudyofoptimaldelegationwhencostlyveriȚcationispossible.Thefollowingclassofruleswillplayanimportantroleinour1.AruleisTECifitconsistsof,with suchthat .2.RulewithoutveriȚcationwithahole[].ParametersarethesameasinȚgure1.4536journalofpoliticaleconomy i.(threshold)if0andii.(escapeclause)if1andFigure3illustratesaTECruleusingthequadraticpreferencesexam-ple.UnderTEC,typesarenotveriȚedandspendattheirßexiblelevel,typesarenotveriȚedandarebunchedattheßexiblespendinglevelof,andtypesareveriȚedandareassignedtheirefȚcientspendinglevel.Theprincipalcanimplementthisrulebyallow-ingtheagenttoeitherchooseaspendingleveluptoalimitorrequestveriȚcationbytriggeringanescapeclause.WhentheagentisveriȚed,heisassignedhisefȚcientspendinglevelprovidedthatitisaboveaspeciȚedlevel(andisotherwisepunished).AnimportantfeatureofTECisthattheveriȚcationfunction)isweaklyincreasing;thatis,thereisnodecreasingveriȚcation:2.ArulefeaturesdecreasingveriȚcationatjumpsfrom1to0at;thatis,either(i)0andlimsup1or(ii)1andliminf0.Arulefeaturesweaklyincreas-ingveriȚcationatifneitherinoriiholds.Notethatwewillrefertodecreasing/increasingveriȚcationinthestrictsense,andwewillclarifywheneverweusedecreasing/increasingveriȚcationintheweaksense.Figure4depictsanexampleofarulewithdecreasingveriȚcation.ThisrulespeciȚesveriȚcationonlyfortypesbetweentwointeriorcutoffs,.Typesaboveandbelowthis .3.TECrule.ParametersarethesameasinȚgure1,with008and0.SolidlinedepictstheallocationofnonveriȚedtypes;dashedlinecorrespondstoveriȚedtypes.commitmentversusexibilitywithcostlyverication4537 regionarenotveriȚed,andhencetherulefeaturesdecreasingveriȚca-tionat.WewillreturntothisexampleinsectionIV.C.AnotherfeatureofTECisthatitspeciȚesveriȚcationforsomeagenttypesbutnotforall.WebeginbyshowinginsectionIV.AthatinducingnoveriȚcationforsometypesisinfactapropertyofanyoptimalrule.Furthermore,buildingonthisresult,weshow

17 thatTECisoptimalwhen-everoptimalveriȚcat
thatTECisoptimalwhen-everoptimalveriȚcationiseverywhereweaklyincreasing.Weconsiderasimpleextreme-biascaseinsectionIV.BandprovideananalysisforourgeneralsettinginsectionIV.C.A.PreliminariesThenextlemmashowsthatverifyingallagenttypesisneveroptimalfortheprincipal:1.Arulewith1forallisnotoptimal.Thelogicissimple.SupposethatarulethatveriȚesalltypesisoptimal.SucharulemusttriviallyassignefȚcientspendingtoalltypes.Nowcon-sideraperturbationinwhichtheprincipalallowstheagenttochoose withoutveriȚcation.Undertheperturbedrule,asetoftypes[ ]for willprefer overbeingveriȚedandassignedefȚcientMoreover,sincetheagentisbiasedtowardhigherspending .4.RulewithdecreasingveriȚcation.ParametersarethesameasinȚgure3.SolidlinedepictstheallocationofnonveriȚedtypes;dashedlinecorrespondstoveriȚedtypes.Notethatif ,yetourargumentapplies,givenouroptimalitycondition(5).Intheappendix,wealsoprovideanalternativeproofforthecaseof0thatdoesnotrelyoncondition(5).4538journalofpoliticaleconomy andpaysaveriȚcationcostnolargerthantheprincipals,itmustbethattheprincipalisstrictlybetteroffbynotverifyingthesetypes.Hence,weȚndthatincentivizinglowtypestonotoverspendischeaperthanverify-ingthem,andthusverifyingalltypescannotbeoptimal.Givenlemma1,weestablishthefollowing:2.IfanoptimalrulefeaturesveriȚcationthatisweaklyin-creasingeverywhere,thenTECisoptimal.Sinceverifyingallagenttypesissuboptimal,anoptimalrulewithveriȚ-cationthatisweaklyincreasingeverywheremustfeatureano-veriȚcationregionfollowedbyaveriȚcationregion;thatis,theremustbeatypesuchthat0forand1for.Considerarulethatoptimizesovereachoftheseregionsseparately.Conditionalonthestypebeingintheno-veriȚcationregion,anoptimalruleisathresh-(byproposition1).ConditionalontheagentstypebeingintheveriȚcationregion,anoptimalruleassignsefȚcientspendingtoalltypes.Toprovelemma2,weshowthattherulethatresultsfromoptimizingovereachregionseparatelyisincentivecompatibleandthereforeopti-maloverthewholesetoftypes.SpeciȚcally,weestablishthatnotype

18 whoisprescribedveriȚ-cationunderthepropo
whoisprescribedveriȚ-cationundertheproposedrulewouldhaveanincentivetodeviatetotheno-veriȚcationregion.Notethatanoptimalrulefortheno-veriȚcationregionsetsamaximumallowablespendinglevel.More-over,byoptimalityof,theprincipalpreferstopaythecostofverifyingtoassignhim)ratherthanbunchhimat.SincetheagentisbiasedtowardhigherspendingandpaysaveriȚcationcostnolargerthantheprincipals,itfollowsthattypesalsoprefertobeveriȚedratherthandeviateto.Thisprovesthatthepro-posedruleisincentivecompatible,whichimpliesthatitisalsooptimal,andbyconstructionthisruleisTEC.B.ExtremeBiasBeforeturningtoourmainresults,weconsiderasettinginwhichthesbiasisextreme.Suppose0forall ,sothattheswelfareissimply.Theagentinthiscasealwayspre-fershigherlevelsofspending:hisßexiblespendinglevelisAsmentionedintheintroduction,suchanextremebiascor-respondstowhatisassumedinothermodelsofcostlyveriȚcation,in-cludingtheseminalworkofTownsend(1979),thedelegationmodelByproposition1,notypewhoisprescribednoveriȚcationwouldhaveanin-centivetodeviateeither.Asassumedinsec.II.A,weareprimarilyinterestedinthecaseinwhich)isinte-riorratherthanacorner;however,weȚnditisinstructivetostudythiscornercaseȚrst.commitmentversusexibilitywithcostlyverication4539 ofHarrisandRaviv(1996,1998),andmorerecentcontributions,suchasBen-Porath,Dekel,andLipman(2014).AnextremebiasimpliesthatiftheagentisnotveriȚed,hewillchoosethehighestallowablelevelofspending,regardlessofhistype.Moreover,theagentwillseekveriȚcationonlyifthatallowshimtospendmorethanundernoveriȚcation.TheanalysisthereforeissigniȚcantlysimpliȚed.Theonlyincentive-compatibleruleforanagentwithanextremebiasin-volvesbunchingallnonveriȚedtypesatonespendinglevel;thatis,ßex-ibilityhasnovalueinthissetting.Furthermore,anytypethatisveriȚedmustbeassignedahigherspendinglevelthanthatatwhichnonveriȚedtypesarebunched.Asaresult,wehavethefollowing:2(Optimalruleunderextremebias).Suppose0forall .ThenifveriȚcationisoptimal,TECisoptimal.Whentheagentsbiasisextremeandverif

19 yingsometypesisoptimal,anoptimalruleisTE
yingsometypesisoptimal,anoptimalruleisTEC,withnonveriȚedtypesbunchedandawardednoßexibilityandveriȚedtypesspendingattheirefȚ-cientlevel.TheoptimalityofTECfollowsfromtheoptimalityofweaklyincreasingveriȚcation.SupposebycontradictionthatanoptimalrulefeatureddecreasingveriȚcation.TaketobeamarginalnonveriȚedtypesplittingaveriȚcationregionandahigherno-veriȚcationregion,0and1forsome0arbitrarilysmall.LetbethelevelofspendingatwhichnonveriȚedtypesarebunched.Theoptimalityofverifying,(6)where,asnoted,incentivecompatibilityrequires,and0,(6)yields.TheoptimalityofnotverifyingthenimpliesHowever,(6)and(7)togetherwithviolatethesingle-crossingcondition(1),yieldingacontradiction.Intuitively,theprincipalcanimproveuponarulewithdecreasingveriȚcationbyverifyingahigheragenttypeinsteadofalowertype,asthemarginalbeneȚtoflettingthehighertypespendmoreishigher.NotethatsuchaperturbationisalwaysincentivecompatiblefortheagentbecauseallnonveriȚedtypesarebunchedatthesamespendinglevel,which(byincentivecompatibility)islowerthanthespendinglevelassignedtoanyveriȚedtype.Thisfeatureisofcourseduetotheagentsbiasbeingextreme.Proposition2,aswellaspropositions3and5,describesanoptimalrulewhenveriȚ-cationisoptimal.Clearly,veriȚcationisoptimalifandonlyiftheveriȚcationcostisnottoohigh.4540journalofpoliticaleconomy C.OptimalRulewithVeriȚcationWenextstudyoptimaldelegationwithveriȚcationinourgeneralsettinginwhichtheagentsbiasisnotextreme.Tothisend,itisusefultocon-siderarelaxedversionoftheproblemin(3)and(4),inwhichweas-sumethattheagentpaysnoveriȚcationcost( Sincetheoriginalincentivecompatibilityconstraint(4)istighterthantherelaxedconstraint(9),ifasolutionto(8)and(9)satisȚes(4),thenitisalsoasolutiontotheproblemin(3)and(4).Furthermore,wecanshowthatifasolutionto(8)and(9)isTEC,thenitwillindeedsatisfy(4),implyingthefollowing:3.IfaTECruleisasolutionto(8)and(9),itisalsoasolu-tionto(3)and(4).ToshowthataTECrulethatsolves(8)and(9)satisȚestheoriginalconstraint(4),weestablishthatanyagen

20 toftypeprefertopaytheveriȚcationcostands
toftypeprefertopaytheveriȚcationcostandspendathisefȚcientlevel)ratherthanpaynoveriȚcationcostandchoosethethresholdßexi-blespendinglevel.Thelogicissimilartothatbehindlemma2,whereweshowthattheoptimalityofverifyingtypefortheprin-cipalimpliesincentivecompatibilityofthisveriȚcationfortheagent.Hence,weobtainthattostudywhetherTECisoptimal,itiswithoutlosstofocusontherelaxedproblemin(8)and(9).Weanalyzethisprob-lemfortheremainderofthissection.Thefollowingtwolemmasestablishusefulpropertiesofanysolution:4.Ifasolutionto(8)and(9)prescribesveriȚcationfortypeithas.If(9)doesnotbindfor,then5.Inanysolutionto(8)and(9),)isweaklyincreasing.Lemma4statesthatifatypeisveriȚed,hisassignedspendinglevelis(weakly)betweenhisefȚcientlevelandhisßexiblelevel.Theargumentisstraightforward.IfassignedspendingfortypeiseitherbelowefȚcientoraboveßexible,theneitherincreasingordecreasingthisspending,re-spectively,makestheprincipalbetteroffandisincentivecompatiblefortheagent.Sincetheprincipalmaximizesherexpectedwelfaresubjecttoincentivecompatibility,ifaveriȚedtypesincentivecompatibilitycon-straintisslack,theprincipalassignsthistypeefȚcientspending.Wemaintainouroptimalityconditionin(5)toselectasolution.commitmentversusexibilitywithcostlyverication4541 Lemma5showsthattheprincipalassignsaspendinglevelthatisweaklyincreasingintheagentstype.WhencomparingtwoagenttypesthatarenotveriȚed,theresultnaturallyfollowsfromincentivecompatibility:atypecannotbeassignedhigherspendingthanahighertype,asatleastoneofthemwouldhaveanincentivetodeviate,giventhatprefer-encessatisfysinglecrossing.Whencomparingtwoagenttypessuchthat(atleast)oneofthemisveriȚed,theresultfollowsfromoptimality:ifatypeisassignedhigherspendingthanahighertype,theprincipalcanimprovewelfarebyswappingthesetypesspendinglevelsandveriȚca-tionassignments,andifincentivecompatibilitywasinitiallysatisȚed,itwillcontinuetobesatisȚedaftertheswap,givensinglecrossing.BydeȚnitionofTECandlemma2,whetheraTECruleisoptimalde-pendsonwheth

21 ertheprincipalcaninsteadbeneȚtfrominduci
ertheprincipalcaninsteadbeneȚtfrominducingde-creasingveriȚcation,namely,asituationinwhichasetoftypesisveriȚedandasetofhighertypesisnotveriȚed.Usinglemmas4and5,wenextshowthatanyrulefeaturingdecreasingveriȚcationmustinducesignif-icantoverspending,limitingthewelfarethatsucharulecanprovidetotheprincipal:6.Supposeasolutionto(8)and(9)featuresdecreasingver-iȚcationat.ThenthesolutionsatisȚes đȚȚ đȚȚIfanoptimalrulefeaturesdecreasingveriȚcationataninteriorpoint,thentheprincipalsexpectedwelfarefromtypesabovethispointisstrictlyboundedawayfromthatunderefȚcientspending.Forintuition,considerȚrsttheexampleinȚgure4,wheretheprincipalinducesveriȚcationonlyforaninteriorsetoftypes[].TheprincipalmustincentivizethesetypestoseekveriȚcationratherthandeviateandmimicatypeintheno-veriȚcationregionabove.Intheexample,theprincipalachievesthisbyassigningtypesimmediatelyabovetheirßexiblespendinglevelswhileassigningveriȚedtypesimmediatelybelowthespendinglevelsthatmakethemindifferentoverdeviatingto)undernoveriȚcation.Asaconsequence,however,theprincipalinducesoverspendingbyapos-itivemassoftypesabove.Infact,alltypesspendabovetheirefȚcientlevelintheexampleofȚgure4.Moregenerally,foranyoptimalrulewithdecreasingveriȚcationata,lemma6showsthattheprincipalsexpectedwelfareabove4542journalofpoliticaleconomy islowerthanefȚcientwelfare,withthedifferencebeingnosmaller)in(11).Thebound)capturestheminimumoverspend-ingabovethatisneededtodeterdeviationsbyveriȚedtypesbelowSpeciȚcally,let1andthus0for0arbitrarilysmall.Bylemma5,weknowthatalltypesabovespendmorethanthosebelow,andbylemma4,weknowthatveriȚedtypesspendnomorethantheirßexibleamount).Thus,fortypesintheveriȚcationre-gionbelownottodeviatetotheno-veriȚcationregionabove,itmustbethat;infact,byoptimality,thisinequalitymustbeGiventhatbylemma5alltypesspendweaklyabove,itfollowsthatalltypesspendstrictlyabove),whichexceedsefȚcientspending)forall,minThisyieldstheboundin(11).Importantly,theboundidentiȚedin(11)isindependentoftheveri

22 -Țcationcost.ThisallowsustoestablishourȚ
-Țcationcost.ThisallowsustoestablishourȚrstmainresult.Inwhatfollows,let3(OptimalruleundersmallveriȚcationcost).Let0.IfandveriȚcationisoptimal,TECisoptimal.Theideaisasfollows.Bylemma6,anyoptimalrulewithdecreasingveriȚcationimpliesawelfarelossduetooverspendinginthedecreasingveriȚcationregion.WeshowthatiftheprincipalsveriȚcationcostissmallrelativetotheminimumsuchloss,thenshecanraiseherwelfarebyverifyingalltypesinthedecreasingveriȚcationregionandreducingtheirspendingtotheefȚcientlevel.ItfollowsthatanoptimalrulemustinduceweaklyincreasingveriȚcationeverywhere,andthereforeTECisoptimalbylemma2.Formally,supposebycontradictionthatanoptimalruleinducesde-creasingveriȚcationatsomepoint,andletbethelowestveriȚedtypeunderthisrule.Weconsideraglobalperturbation:theprincipalveriȚesalltypesandassignsthemefȚcientspending)whilesolvingforanoptimalrulewithoutveriȚcationfortypes.Byproposition1,anoptimalrulefortheno-veriȚcationregionisathreshold,and(byoptimalityof)and0,itiseasytoverifythattheperturbedruleisincentivecompatible.Toshowthattheperturbationstrictlyraisestheprincipalswelfare,noteȚrstthatexpectedwelfareconditionalonweaklyincreasesbecauseitisnowmaximizedsubjecttofewerconstraints:undertheper-turbedrule,typescannotmimicatype.Thus,allweneedtoshowisthatexpectedwelfareconditionalonincreasesstrictly,namely,thatthe(allocative)beneȚtofverifyingthesetypesisstrictly,incentivecompatibilityrequires,butthentheprincipalcanimproveupontherulebysetting0whilekeepingeverythingelsecommitmentversusexibilitywithcostlyverication4543 greaterthantheadditionalveriȚcationcosttheprincipalincurs.BecauseveriȚedtypesareassignedefȚcientspending,thebeneȚtofver-ifyingthemisweaklypositive.Moreover,notethatbythecontradictionassumption,thereexistsatypeaboveatwhichtheoriginalrulefea-turesdecreasingveriȚcation.Thus,ifisthelowestsuchtype,lemma6impliesthatthebeneȚtofverifyingtypesisboundedfrombelowby,where)isdeȚnedin(11).Theclaimthenfollowsinthiscasefromthefactthat,given,theadd

23 itionalcostofverifyingtypesisstrictlysma
itionalcostofverifyingtypesisstrictlysmallerthanandhencestrictlysmallerthanthebeneȚtofver-ifyingthesetypes.IfthelowesttypeaboveatwhichtheoriginalrulefeaturesdecreasingveriȚcationis,ananalogousargumentap-plies,sinceinthiscasetheoriginalruleinducesstrictoverspendingbyandthebeneȚtofverifyingthistypeisnosmallerthanFigure5illustratestheresultinproposition3inasettingwithqua-draticpreferences(seesec.II.A)andauniformdistributionoftypes.Inthissetting,weobtainaclosed-formexpressionforthecutoffthusfortherangeofveriȚcationcosts,,underwhichTECisshowntobeoptimalunderveriȚcation.WeȚndthatisincreasingintheagentanddecreasingintherangeoftypes .Intuitively,iftheagentbiastowardhigherspendingislarge,theninducingdecreasingveriȚca-tionisveryexpensivefortheprincipal,asshemustallowhighoverspend-ingaboveanyinteriorveriȚcationregiontodeterdeviationsfromveriȚedtypes.Inthiscase,thebeneȚtofverifyingalltypesabovetheveriȚcationregionislarge,andthusTECispreferredeveniftheprincipalsveriȚca-tioncostisrelativelyhigh.Similarly,iftherangeoftypes issmall,thenthemassoftypesaboveanyinteriorveriȚcationregionisalsosmall,andthereforethecostofverifyingallsuchtypesinaTECruleislowevenisrelativelyhigh.Figure5providesanillustrationusingtheexampleofȚgure3.TheȚguredepictsvaluesof underwhichtheissatisȚed(shadedareas)aswellasthesubsetofthosevaluesunderwhichveriȚcationandthusTECisoptimal(darkgrayshadedarea).Wederivethiscutoffinapp.B.Weobtain b36 1g2 gifbg2 g, 16g2 g23bg2 g3b2b�g2 Inthisexample,theoptimalTECrulebunchesallnonveriȚedtypesatonespendinglevel,andasaresulttheoptimalityofveriȚcationgivenisindependentof.This,however,isnotageneralfeature.4544journalofpoliticaleconomy Theresultinproposition3providesajustiȚcationforthebroaduseofTECrulesinapplications.Asdescribedintheintroduction,capitalbud-getingstudies(e.g.,Ross1986;Taggart1987)reportthatTECiscommoninorganizations.Divisionmanagersarerequiredtoeitherabideabud-getarylimitorprovideprojectdocumentationtore

24 questarevisionoftheirbudgets.Schaechtere
questarevisionoftheirbudgets.Schaechteretal.(2012)andLledetal.(2017)ȚndthatȚscalrulesinmanycountriesalsotaketheformofTEC,namely,aspend-ingordeȚcitlimitwithescapeclauseprovisionsthatallowthegovernmenttobreakthelimitundercertaincircumstances.Additionally,TECrulesareusedininternationaltradeagreements,intheformofatariffcapwithanescapeclause(BeshkarandBond2017),andinpricedelegationin .5.OptimalityofTEC.ParametersarethesameasinȚgure3,withShadedareasindicatevaluesof underwhich.Darkgrayshadedareacor-respondstovaluesforwhich,inaddition,veriȚcationisoptimal.Comparativestaticsareasonewouldexpect.Inparticular,theloweris,otherthingsequal,andthelargeristheveriȚcationregion(i.e.,thesmalleris)intheoptimalTECrule.commitmentversusexibilitywithcostlyverication4545 Țrms,wheresalespeoplecanunilaterallyoffertheircustomersdiscountsuptoacertainpercentageoffthelistpricebutmustrequestasupervisorapprovalforlargerdiscounts(Loetal.2016).Proposition3provestheoptimalityofTECwhentheprincipalscostofveriȚcationissmallenough.WhathappensifthecostofveriȚcationislarger?Ournextresultshowsthatthereexistenvironmentsandveri-ȚcationcostsforwhichtheprincipalinducesveriȚcationbutnotintheformofTEC:4(OptimalruleunderintermediateveriȚcationcost).Thereexist{}forwhichanyoptimalrulefeaturesdecreasingToprovethisresult,weidentifyconditionsonparametersunderwhichverifyingonlyanintermediaterangeoftypes[]dominatesbothnotverifyinganytypeaswellasusingTEC.Themainreasonwhyverifyingonlyintermediatetypescandominatenotverifyinganytypeisthatanin-termediateveriȚcationregionimposesdisciplineontheno-veriȚcationregionbelow.Thatis,evenwhentheveriȚcationcostislargeenoughthattheprincipalwouldnotbeneȚtfromverifyingtypesin[]onlytoim-provetheirallocationrelativetoßexiblespending,shemaybeneȚtfromverifyingthesetypestodisciplinelowertypes:withtheintermediatever-iȚcationregion,typescannolongermimictypesin[].ThemainreasonwhyverifyingonlyintermediatetypescandominateusingaTECruleisthatitallowstheprin

25 cipaltosaveonveriȚcationcosts.Specif-ica
cipaltosaveonveriȚcationcosts.Specif-ically,withintermediateveriȚcation,theprincipalmaybeabletoimposedisciplineontypeswithoutprescribingveriȚcationfortypesasshewouldunderaTECrule;thiswillbethecaseifhasnoincentivetodeviatetomimicatypeashighas.Insuchasituation,intermediateveriȚcationallowstheprincipaltosaveonthecostofveri-fyingtypesaboveTheseargumentsyieldthatarulewithdecreasingveriȚcationasthatdepictedinȚgure4candominateanyno-veriȚcationrule(asthatinȚg.1)andanyTECrule(asthatinȚg.3),providedthatthecostofveriȚ-cationisnotsmall(orlarge)enough.Weemphasizethatproposition4doesnotrelyonnonuniformityoftheprincipalsobjectiveacrosstypesoranyothersortofasymmetry;weprovetheresultbytakingthecaseofqua-draticpreferencesandauniformdistributionoftypes,asdepictedinourȚgures.WealsonotethatwhileourconstructionimpliestheoptimalityofdecreasingveriȚcationundersomeparameterswith,theoptimalruleinthiscasemaynottakethesimpleintermediate-veriȚcationstructurethatweconsidertoprovetheresult.Infact,wecanshowthatevenwhenrestrictingattentiontoquadraticpreferencesandauniformdistribution,Bylemma2,anyotherrulewithveriȚcationthatisweaklyincreasingeverywhereisthusalsodominated.Hence,theclaiminproposition4follows.4546journalofpoliticaleconomy thereexistparametersforwhichTEC,noveriȚcation,andintermediateveriȚcationarealldominatedbyarulefeaturingmultipleinteriorveriȚca-tionregions.Intuitively,intercalatingveriȚcationregionstofurtherdi-videthedelegationsetcanallowtheprincipaltoimprovedisciplinewhilekeepingveriȚcationcostsataminimum.Theimplicationsofproposition4forapplicationsareimmediate.Forexample,fororganizations,thisresulttellsusthatitcanbebeneȚcialtodeȚnedifferentcategoriesofinvestment.Seniormanagementcouldre-quiredivisionheadstoeithercomplywithalowbudgetarylimitmeantforrelativelysmallprojectsorchoosefromahigherrangeofinvestmentlev-elsmeantforlargeprojects;otherwise,documentationwouldbeneededtohaveintermediatelevelsofinvestmentapproved.SuchaveriȚcationrequire

26 mentmaysufȚcetodiscourageoverinvestmentb
mentmaysufȚcetodiscourageoverinvestmentbydivisionman-agerswithsmallprojects:thesemanagerslackprooftojustifyasmallin-creaseintheirbudgetandwouldnotwanttoincreasetheirinvestmentasmuchasforalargeproject.Nevertheless,whereasdelegationruleswithdecreasingveriȚcationcanbeoptimal,theydonotappeartobecommoninpractice,andouranalysismayhelpexplainwhy.OurconstructionshowsthatimplementingarulewithdecreasingveriȚcationdemandsstrongcommitmentpowerfromtheprincipal.Take,forexample,theruledepictedinȚgure4.Theprin-cipalassignsspendingstrictlyabovetheefȚcientleveltosomeagenttypeswhoareveriȚed.Bydoingthis,theprincipalincentivizesthosetypestobeveriȚed:iftheywereinsteadassignedefȚcientspendingfollow-ingveriȚcation,theywouldnotseekveriȚcationintheȚrstplace.TheprincipalmustbecommittedtoallowingthisinefȚcientspendingdespiteherlearningthetruetypeoftheagent.StrongcommitmentpowerisalsorequiredtoincentivizetypessufȚcientlyclosetotonotseekveriȚcation.IntheruleofȚgure4,thesetypesarepunishediftheyseekveriȚcation,eventhoughexpost,onceveriȚcationtookplace,boththeprincipalandtheagentwouldstrictlypreferefȚcientspendingtopunish-ment.Withoutthethreatofpunishment,theprincipalmaynotbeabletopreventanagentoftypesufȚcientlyclosetofromseekingveri-Țcation,asanefȚcientallocationfollowingveriȚcationwouldallowthisagenttoincreasehisspendingtowardhisßexiblelevel.Inpractice,principalsmaynothavesufȚcientcommitmentpowertoimplementallocationsthatareinefȚcientexpost.Weexploretheimpli-cationsoflimitedcommitmentpowerinsectionV.Inparticular,theruleconstructedinlemma9intheproofofproposition4inapp.Aisnotoptimalforsomeparametervaluessatisfyingtheassumptionsofthelemma.Forin-stance,takingtheexampleofȚg.3,andconsistentwithourintuitionbehindȚg.5,weȚndthatarulewithmultipleinteriorveriȚcationregionsbecomesoptimaliftherangeoftypes becomeslargeenough.Forthisreason,whendecreasingveriȚcationisoptimal,theoptimalruleisverysen-sitivetoparameters,suchasthevalueof commitmentversusexibilitywithcos

27 tlyverication4547 V.LimitedCommitmentWes
tlyverication4547 V.LimitedCommitmentWestudyasettinginwhichtheprincipalhaslimitedcommitmentpower.WemodifytheorderofeventsinsectionII.Basfollows:1.Theprincipalsetsarule,whichmapsaveriȚcationdecisionandresultintoanallowablespendingset2.TheagentchooseswhethertoseekveriȚcation,0,1,andtheprincipalperfectlyveriȚeshistype3.Theprincipalrevisestheallowablespendingset4.TheagentchoosesaspendinglevelfromtheallowablesetTheȚrsttwostepsarethesameasinourenvironmentwithfullcom-mitmentpower.Whatisnewisstep3:afterobservingtheagentsveriȚ-cationdecisionandtheresultifveriȚcationischosen,theprincipalnowrevisestheallowablespendingsetfortheagent.Thisisamildformoflimitedcommitment.Inparticular,instep2wemaintaintheassumptionthattheprincipalisabletocommittoaveriȚcationplan,sotheagenttypeisveriȚedifandonlyiftheagentrequestsveriȚcation.Moreover,instep4wemaintaintheassumptionthattheprincipalisabletocommittoallowingtheagenttochooseanyspendinglevelfromtheallowablespendingset,soourproblemisstilloneofdelegationratherthancheaptalk.Theonlyassumptionthatwerelaxisabouttheprincipalscommit-menttonotchangingtheallowablespendingsetfollowingtheveriȚca-tiondecisionandresult.Thisformoflimitedcommitmentisrelevanttoapplicationsofourmodel.Forexample,divisionmanagersinorganizationsmayrequestarevisionoftheirbudgetsforthenextperiod.Canseniormanagementcommittonotchangingtheirallocationexpostwhennorequestissub-mitted?Andinthecaseofarequest,canseniormanagementcommittoaninefȚcientbudgetafterverifyingthebeneȚtsofthedivisionsprojects?Asdiscussedintheintroduction,theanswerisoftenno.Seniormanage-mentmakesdecisionsonbudgetcapsandthescopeofprojectsbroughtupforreviewexpost,andthesedecisionsdonotalwayscoincidewithpre-announcedcriteria(seeBowerandLesard1973;Ross1986;TaggartWenotethatourresultsinthissectionarenotlimitedtotheexactgamedescribedbelow;analogoustoourclaimsinsec.II.C,ourȚndingscanbeextendedtovariationsofthisgamethatallowmessagesbetweentheprincipalandtheagent(whilekeepingour

28 assumptionsontheprincipalslimitedcommitm
assumptionsontheprincipalslimitedcommitment).Throughoutthissection,wemaintainouroptimalityconditionin(5).Asnotedinn.10,thereisaliteraturethatstudiesauditingwhentheprincipalcannotcommittoanauditstrategy.Inmanyoftheapplicationsofourproblem,however,weȚndthatthereareofteninstitutionsensuringthatprincipalscannotdenyveriȚcationonceithasbeenrequested.Inthissense,theagentcanalwayschoosetotriggerveriȚcation.Lackofcommitmentbytheprincipalinthisrespectwouldchangethenatureofourproblem;weleaveitsanalysisforfuturework.4548journalofpoliticaleconomy 1987).Infact,thesecriteriaaresometimesleftambiguous,astheyde-pendontheclassofproject,whichmaynotbewellspeciȚed(MukherjeeandHenderson1987).Thisgivesseniormanagementmorediscretiontomakebudgetarydecisions.Interestingly,intheirstudyofȚscalrulesacrosscountries,Schaechteretal.(2012)alsoobservethatescapeclausesaresometimesnotwellspeciȚed:inthepastescapeclauseprovisionshaveinseveralcaseslefttoolargearoomforinterpretation(Schaechteretal.2012,20).Inourmodel,limitedcommitmentonthesideoftheprincipalmattersfortworeasons.First,conditionalonnoveriȚcation,theprincipalmustchooseanallocationthatisoptimalforthenonveriȚedtypes.Thatis,theprincipalassignsspendinginthiscasetakingintoaccountthedistribu-tionofnonveriȚedtypesandignoringtheincentivesofveriȚedtypes.Second,conditionalonveriȚcation,theprincipallearnstheagentstrueandmustassigntheagenttheefȚcientspendinglevel).ThisistruebothwhentheagentsseekingveriȚcationisonpathaswellaswhenthisveriȚcationdecisionispartofadeviation.Hence,theagentcanalwayschoosetobeveriȚedtoguaranteehimselftheefȚcientlevelofspending.Asaresult,limitedcommitmentimpliescertainconditionsthatanyincentive-compatiblerulemustsatisfy.Inwhatfollows,werestrictattentiontostrategiesthatspecifypiecewisecontinuousmappings{)}.7.Underlimitedcommitment,anyincentive-compatiblerulesatisȚesthefollowing:i.IfthereisdecreasingveriȚcationat,then,(12)1.Moreover,ii.IfthereisincreasingveriȚcationat,then,(14)Partishow

29 sthatifsplitsaveriȚcationregionfromahigh
sthatifsplitsaveriȚcationregionfromahigherno-veriȚcationregion,thenmustbeindifferentbetweenbeingveriȚedandspendingattheefȚcientlevelversusnotbeingveriȚedandspend-ingat),asallowedintheno-veriȚcationregionabovethistype.Likewise,partiishowsthatifsplitsano-veriȚcationregionfromahigherveriȚcationregion,thenmustbeindifferentbetweenbeingver-iȚedandspendingattheefȚcientlevelversusnotbeingveriȚedandspendingat),asallowedintheno-veriȚcationregionbelowthistype.commitmentversusexibilitywithcostlyverication4549 ThisresultfollowsfromthefactthataprincipalwithlimitedcommitmentpowerassignsefȚcientspendingwhenevertheagentseeksveriȚcation.Therefore,ifthereisapointatwhichaveriȚcationregioneitherendsorstarts,themarginalveriȚedtypeatsuchpointmustweaklypreferver-iȚcationwithefȚcientspendingtonoveriȚcation,andthemarginalnonveriȚedtypemustweaklyprefernoveriȚcationtoveriȚcationwithef-Țcientspending.Themarginaltypemustthusbeindifferent.Lemma7alsoshowsthatfortypeasdeȚnedinthelemma,anincentive-compatiblerulemustset.ThisisrequiredtoindifferentbetweenveriȚcationandnoveriȚcation:ifthisin-equalityisnotsatisȚed,themarginalveriȚedtypewouldinsteadprefertodeviateandnotseekveriȚcation.Fortheremainderofouranalysis,werequirethefollowing:2.If,thenforallThisisasingle-crossingproperty:weassumethatifatypeweaklypre-fersveriȚcationwithefȚcientspending)tonoveriȚcationwithahigherspendinglevel,thenanylowertypestrictlypre-fersveriȚcationwithefȚcientspending)tonoveriȚcationwiththehigherspendinglevelThispropertyholdsinthecasescommonlystudiedintheliterature,suchasthosewithquadraticpreferencesorwithhyperbolicpreferencesundercommonparameterizations.Givenassumption2,weobtainthefollowing:5(Optimalruleunderlimitedcommitment).Underlim-itedcommitment,anyincentive-compatiblerulefeaturesweaklyincreas-ingveriȚcationeverywhere.Moreover,ifveriȚcationisoptimal,TECisUnderlimitedcommitment,decreasingveriȚcationisnotincentivecompatiblefortheprincipal.AswediscussedinsectionIV.C,decreasingv

30 eriȚcationrequiresthattheprincipalcommit
eriȚcationrequiresthattheprincipalcommittoallowingtheagenttospendatalevelthatisinefȚcientexpost,followingtheagentsveriȚca-tiondecisionandresult.Weprovethatwithoutthiscommitment,theOursingle-crossingconditionsonpreferencesimplythatifatypeweaklyprefersver-iȚcationwithefȚcientspending)tonoveriȚcationwithalowerspendinglevel,thenanyhighertypestrictlyprefersveriȚcationwithefȚcientspending)tonoveriȚcationwiththelowerspendinglevel.Assumption2requiresthatthispropertybemaintainedintheoppositedirectionaswell.Forexample,inthehyperbolicpreferencescase(seesec.II.A),assumption2holdsiftheutilityfunctionsforpresentandfutureconsumptionarethesameandeitherexponen-tialorconstantrelativeriskaversionwithacoefȚcientweaklygreaterthan1.4550journalofpoliticaleconomy principalcannotinducedecreasingveriȚcation,andhenceanyincentive-compatiblerulemustfeatureweaklyincreasingveriȚcationatalltypes.Analogousargumentstothosebehindlemmas1and2inourfull-commitmentenvironmentthenimplythatifverifyingsomeagenttypesisoptimal,aTECruleisoptimal.Asketchoftheproofofproposition5isasfollows.Supposebycontra-dictionthatthereisanincentive-compatiblerulethatinducesdecreas-ingveriȚcation,withbeingatypesplittingaveriȚcationregionfromahigherno-veriȚcationregion.Givenlimitedcommitment,veriȚedtypesimmediatelybelowareassignedefȚcientspending,andtypesdiatelyabovespendatalevelthatmakesindifferentbe-tweenveriȚcationandnoveriȚcation(cf.lemma7).Thismeansthattypesimmediatelyabovemustbestrictlyoverspending,infactspend-ingabovetheirßexiblelevel.Theheartoftheproofisshowingthattheprincipalcannotcommittoallowingsuchoverspending.ItisclearthatconditionalontheagentnotseekingveriȚcation,theprincipalwouldliketoreducetheoverspendingbytypesimmediately.Reducingthisoverspendingisexpostincentivecompatibleforthesetypes:havingchosennoveriȚcation,typeswouldprefer)to.Hence,theonlyreasontheprincipalwouldnotre-ducetheoverspendingimmediatelyabovefollowingnoveriȚcationisifdoingsowouldviolateincentivecompat

31 ibilityforsomeothernon-veriȚedtype.Sucha
ibilityforsomeothernon-veriȚedtype.SuchanonveriȚedtypemustbebelow;speciȚcally,theremustexistatypewhoisnotveriȚedandisexactlyindifferentbetweenhisassignedspendinglevel,callit,andthespendinglevel.Infact,becauseofsinglecrossing,thistypemustbethemarginaltyperightbelowtheveriȚcationregionthatendsat;thatis,therulemustinduceveriȚcationfortypesandnoveriȚcationfortypesimmediatelybelowandabovethisset.Anexampleistherulede-pictedinȚgure4.NowiftheprincipalinducessuchaninteriorveriȚcationregion[],thenbylemma7,typemustbeindifferentbetweennoveriȚca-tionwithspendingandveriȚcationwithefȚcientspending.SincewehavedeȚnedasbeingindifferentbetweenspendingatandspend-ingatundernoveriȚcation,bytransitivity,weobtainthatmustbeindifferentbetweennoveriȚcationwithspendingandveriȚcationwithefȚcientspending.However,recallthattypeisalsoindifferentbetweennoveriȚcationwithspendingandveriȚcationwithefȚcientspending.Hence,byassumption2,cannothold,andwemustThismeansthattheprincipalveriȚesasingletypeatthis,theindifferenceoftypebetweenveriȚcationwithefȚcientspendingandnoveriȚcationwithspendingwouldimplythatstrictlyprefersveriȚcationwithefȚ-cientspendingtonoveriȚcationwithspending,acontradiction.commitmentversusexibilitywithcostlyverication4551 pointwhoisindifferentbetweenveriȚcationwithefȚcientspending,noveriȚcationwithhigherspendingat,andnoveriȚcationwithlowerspendingat.ConditionalonnoveriȚcation,thisisthusanallocationinwhichtheagentfacesahole[];namely,heisnotallowedtochoosespendinginthissetbutcanchoosespendingimmediatelybelowandaboveit.ButouranalysisinsectionIIIshowsthatsuchaholeissub-optimalconditionalonnoveriȚcation;hence,followingnoveriȚcation,theprincipalwouldhaveastrictincentivetoclosethehole.ThisshowsthatarulewithdecreasingveriȚcationcannotbeincentivecompatiblewhentheprincipalhaslimitedcommitmentpower,allowingustoestab-lishthatTECisoptimalinthiscase.Recallthatinthefull-commitmentenvironment,TECisoptimaliftheprincipalscostofveriȚcationissmallenough(assh

32 owninproposi-tion3),butmorecomplexrulesm
owninproposi-tion3),butmorecomplexrulesmaybeoptimalotherwise(asshowninproposition4).Incontrast,proposition5tellsusthatTECisoptimalun-derlimitedcommitmentforanyveriȚcationcostforwhichveriȚcationisoptimal.GiventheprevalenceofTECrulesintherealworld,thesere-sultssuggestthatlimitationstocommitmentpowerarealsoprevalent.Moreover,theselimitationsmaybeanimportantreasonbehindthebroaduseofTECinapplications.AsaȚnalremark,itisworthnotingthatwhileTECisoptimalbothwhentheprincipalhasfullcommitmentpowerandasmallveriȚcationcostaswellaswhenshehaslimitedcommitmentpower,thespeciȚcde-tailsofanoptimalTECrulevarywitheachcase.Underfullcommitment,anoptimalTECruleissuchthattheprincipalpreferstoverifytoassignthemefȚcientspendingratherthanbunchthemwithoutveriȚcation,whereastheoppositeistruefortypes.Hence,theprincipalisindifferentbetweenverifyingandnotverifyingthethresholdtype;thatis,theincreaseinassignedspendingatexactlycompensatestheprincipalforthecostofveri-fyingthistype.Incontrast,underlimitedcommitment,itistheagentwhoisindifferentat:asimpliedbylemma7,typemustbeindifferentbetweenbeingveriȚedandassignedefȚcientspendingversusnotbeingveriȚedandassigned,andthusanyincreaseinassignedspendingmustexactlycompensatethistypeforhisveriȚcationcostVI.ConclusionThispaperhasstudiedthetrade-offbetweencommitmentandßexibilityinthepresenceofcostlystateveriȚcation.Wehaveexaminedageneraldelegationprobleminwhichaprincipaldelegatesdecision-makingtoAdditionally,asnotedinsec.II.A,thisresultappliesnotonlyto0,1butalsoto4552journalofpoliticaleconomy anagentwhohassuperiorinformationabouttheefȚcientactionbutisbiasedtowardhigheractions.Anovelelementofourframeworkisthattheprincipalcanverifytheagentsprivateinformation.BecauseveriȚca-tioniscostly,theprincipalwishestousethistechnologyselectivelyandinawaythatsupplementsdelegationandimproveshercommitmentversusßexibilitytrade-off.Ourresultsprovideinsightintohowtheprincipalachievesthisbyde-signinganoptimaldelegationrule.Wehaveshowntha

33 tunderfullcom-mitmentpowerandasmallenoug
tunderfullcom-mitmentpowerandasmallenoughveriȚcationcost,anoptimalruleisaTEC,allowingtheagenttofreelyselectanyactionuptoathresholdortorequestveriȚcationandtheefȚcientactionabovethethreshold.WhentheveriȚcationcostislarger,theprincipalmayinsteadprefertorequireveriȚcationonlyforintermediateactions,stillimposingsomedisciplineontheagentbutsavingonveriȚcationcosts.However,theoptimalityofTECisrestoredundermildlimitationstotheprincipalscommitmentpower.SpeciȚcally,iftheprincipalisunabletocommittonotchangingtheagentspermissibleactionsetfollowingtheveriȚcationdecisionandresult,TECisoptimalforanyveriȚcationcostforwhichveriȚcationisAswehavediscussed,thereareavarietyofapplicationswheredelega-tioniscentralandrulesmakeuseofveriȚcationbyspecifyingescapeclauses.OuranalysisshedslightontheoptimalstructureofescapeclausesandprovidesatheoreticalfoundationforthecommonuseofTECrules.Morebroadly,ourframeworkmayhelpinformtheempiricalanalysisofreal-worldrules.Dataondelegationpoliciesareincreasinglyavailableandofferanopportunitytoexplorethedesignoftheserulesinmoredetail.Forinstance,inthecontextofcapitalbudgeting,ithasbeenob-servedthattheextentofcapitalrationingandtheuseofveriȚcationvaryacrossȚrms(e.g.,Ross1986),andonecouldstudyhowthesedifferencesrelatetoȚrmsize,industry,andotherfactorsthatarelikelytoaffectse-niormanagementscostofverifyingthequalityofprojects.InthecontextofȚscalpolicy,countriesȚscalrulesvaryintheuseofescapeclausepro-visionsandtheirtriggerevents(Schaechteretal.2012),andthesemaycorrelatewithcountriesinstitutionalandmacroeconomicconditionsthataffectthecostofauditingagovernmentaswellastheneedforßex-ibilitytorespondtoshocks.Last,byuncoveringanewsetofissuesthatarisewhenveriȚcationisin-troducedtoasettinginwhichbothcommitmentandßexibilityarevalu-able,ourpaperopensthedoorforfurtherworkthatcanhelpunder-standtheoptimaljointdesignofdelegationandveriȚcation.Wehavefocusedonasimplemodelthatemphasizesthemainforcesatplaybutabstractsfromotherpotential

34 lyrelevantaspects,forinstance,associated
lyrelevantaspects,forinstance,associatedwithmorecomplexveriȚcationtechnologies.Weclosebydiscussingsomepossibleextensionsandvariationsofourwork.commitmentversusexibilitywithcostlyverication4553 RandomveriȚcationAsintheseminalworkofTownsend(1979),wehaveconsidereddeterministicveriȚcation;namely,weassumedthattheprincipalsruleassigns0,1toeachagenttype.Moregen-erally,onecouldallowformechanismsinwhichtheprincipalrandomizesovertheveriȚcationassignment,choosingaprobabilityofveriȚcationforeachtype.TheliteratureonȚnancialcontractingandtaxcollectionȚndsthatrandomveriȚcationcanyielddifferentresultscomparedwithdeter-ministicveriȚcation;seeBorderandSobel(1987)andMookherjeeandPng(1989).OurfocusondeterministicveriȚcationismotivatedbytheapplica-tionswestudy.Takecapitalbudgeting.Ascapturedbythegameformthatwehaveproposed,heretheagent(divisionhead)decideswhethertorequestveriȚcationtoobtainapprovaltochooseactionsthatarenotallowedbytheprincipal(seniormanagement)withoutveriȚcation.Theprincipalcommitstofollowingtheagentsrequest,andsoitistheagentchoicewhethertotriggertheveriȚcationprocess.Unlikeinotherappli-cationswhereveriȚcation/auditisusedtodetermineȚnesformisbehavior(e.g.,taxcollection),randomveriȚcationisnotnaturalinthesecontexts.UsingthetimingofsectionII.B,randomveriȚcationwouldmeanthattheagentchoosesinstep2notbetweenveriȚcationandno-veriȚcationbutratherbetweendifferentlotteriesoververiȚcation.Thisisrarelyobservedinpractice,possiblybecausecommittingtoanondegeneratelotterycanbedifȚcultforaprincipal.Thatsaid,inasettinginwhichtherearenolimitationstotheprinci-scommitmentpower,thestudyofrandomveriȚcationcouldbeaninterestingextensionofourwork.Asnotedintheaforementionedliter-ature,oneissueisthatanoptimalrandomizedmechanismwoulddependontheextenttowhichtheagentcanbepunishedfollowingveriȚcation,whichinturnwoulddependonpreferenceassumptionsinoursetting,giventhatpunishmentsareimposedthroughthespendingallocationonly.Importantly,thesepunishmentsmustbebo

35 unded;otherwise,theefȚcientallocationcan
unded;otherwise,theefȚcientallocationcanbeapproachedwitharulethatveriȚesallagenttypeswithverylowprobabilityandarbitrarilypunishestheagentwhenveriȚcationrevealsthathehasdeviated.Suchapossibilitynotonlyyieldsratherimplausiblepredictionsbutalsoimpliesthatanoptimalruleingeneralwillfailtoexistunlessaboundonpunishmentsisimposed.WhenthedecisionissimplyoververiȚcationornoveriȚcation,commitmenttotheveriȚcationpolicywouldinprinciplebefacilitatedbythefactthattheprincipalsexecu-tionoftheagentsrequestcanbeeasilymonitored.However,checkingthattheprincipalimplementsaspeciȚclotteryisharder,asitrequiresmonitoringoftherandomizationitselfratherthanitsoutcome.Inourgameform,arulethatapproachestheefȚcientallocationwouldbeimple-mentedbyinducingeachagenttypetochooseadifferentlotteryoververiȚcation.4554journalofpoliticaleconomy ImperfectveriȚcationAlsofollowingTownsend(1979),ouranalysisas-sumedthatveriȚcationrevealstheagentstypeperfectly.AnalternativewouldbetoconsiderimperfectveriȚcation,namely,veriȚcationthatprovidesonlyimperfectinformationabouttheagentstype.Forexample,inthecontextofcapitalbudgetinginorganizations,seniormanagementmayreviewinformationaboutthebeneȚtsofaprojectthatadivisionman-ageradvocates,buttheavailabledocumentationmaybeincompleteandfailtorevealthefullmeritsoftheproject.AsimplespeciȚcationthatmaybepossibletoaccommodatewithinourframeworkiswhenimperfectveriȚcationeitherrevealstheagentstypeperfectlyorprovidesnoinformation(i.e.,whentherearenofalsere-sults).Providedthattheprincipalcanseverelypunishtheagent(throughthespendingallocation),shewouldbeabletoprevent,atnocost,anydeviationinwhichanagenttypemimicsanothertypewhoisveriȚed,asistrueinourproblemwithperfectveriȚcation.Yetadifferenceintro-ducedbyimperfectveriȚcationisthattheprincipalmaynotobservethestypeandthusmaynotbeabletoassignatype-dependentspend-inglevelfollowingveriȚcation;theprincipalsrulemustspecifyaspend-ingallocationforthecaseofveriȚcationandnoinformation.Allowingforimperf

36 ectveriȚcationthatmayproducefalseresults
ectveriȚcationthatmayproducefalseresultswouldnaturallyintroducefurtherissues,asnowpunishinganagenttypeformimickinganothertypewhoisveriȚedwouldrequireimposingpunishmentsonHowimperfectisimperfectveriȚcation?Atoneextreme,ifveriȚcationissufȚcientlyaccurate,weconjecturethatourqualitativeresultswouldremainvalid.Attheotherextreme,ifveriȚcationissufȚcientlyinaccu-rate,itwouldbecomeequivalenttomoneyburning,andtheresultsoftheliteratureonwhenmoneyburningisusedinanoptimaldelegationrulewouldthenapply(seeAmador,Werning,andAngeletos2006;AmadorandBagwell2013;AmbrusandEgorov2017).Moregenerally,itwouldbeofinteresttoexploretheroleofveriȚcationindelegationawayfromthesetwoextremes.VeriȚcationcostsWehaveconsideredveriȚcationcoststhatarebothtypeindependentandexogenous.Anextensionofourproblemcouldexploretheeffectsoftype-dependentveriȚcationcosts:theprincipalcostofverifyingtheagentsprivateinformationmaybeincreasinginhistype,forexample,becausemoreevidenceisneededtoverifylargerprojectbeneȚts,oronemaytaketheviewthatveriȚcationcostsareactu-allylowerforextremetypes,asthesestatesaremorevisible.OnepossibledifȚcultyisthatmonotonicityofthespendingallocation(asshowninlemma5)mayfailtoholdifveriȚcationcostsincreaseveryrapidlywiththeagentstype.ButiftheveriȚcationcostfunctionissuchthattheprin-cipalwouldstillprefertoswaptheveriȚcationandspendingallocationsoftwotypeswhenevertypehashigherspendingthancommitmentversusexibilitywithcostlyverication4555 monotonicitywillbesatisȚedandouranalysiscouldbeextendedtoallowfortype-dependentveriȚcationcosts.Anothervariationwouldbetoendogenize,sothattheprincipalcanaffecttheagentscostofveriȚcation.Ourresultswouldcontinuetoholdunderthisextension.SpeciȚcally,whentheprincipalhasfullcommit-mentpower,wehavederivedconditionsfortheoptimalityofaTECrulethatareindependentofthevalueof0,1,andclearlytheprincipalwelfareunderthisruledoesnotvarywitheither.Thus,theprincipalinthiscasewouldbeindifferentoverany0,1,whereas0wouldbepreferredbytheagen

37 t.Moregenerally,underfullcommitmentitisa
t.Moregenerally,underfullcommitmentitisal-waysoptimalfortheprincipaltoset0,asazeroveriȚcationcostfortheagentmaximallyrelaxestheagentsincentivecompatibilitycon-straint(4).ThingsaremoreinterestinginthesettingofsectionV,wheretheprincipalhaslimitedcommitmentpower.Inthiscase,theprincipalmaywanttosetastrictlypositiveveriȚcationcostfortheagent,asthatlimitsthesetofagenttypesthatmaywanttodemandveriȚcationandefȚcientspending.Inanycase,foranygiven0thattheprincipalwouldset,ouranalysisandtheoptimalityofTECapplywithoutchange.TransfersOurfocushasbeenonacanonicaldelegationprobleminwhichtransfersbetweentheprincipalandtheagentarenotfeasible.Therearevariouswaysinwhichtransferscouldbeintroducedinourframeworkandusedtoalterthefeasibilityandcostofinducingdifferentallocations.TransferscouldbecontingentontheagentsveriȚcationde-cisionand/ortheveriȚcationresult;moreover,theprincipalcouldofferdifferentallowablespendingsetsfortheagenttochoosefromandspec-ifytransfersassociatedwitheachset.Thesequestionsarebeyondthescopeofourpaper,andsoweleavethemforfutureresearch.AppendixAProofsA1.ProofofProposition1Theclaimfollowsfromproposition1(parta)inAmadorandBagwell(2013,A2.ProofofLemma1Supposebycontradictionthatarule1foralloptimal.Sincetheincentivecompatibilityconstraint(4)istriviallysatisȚedun-derthisrule,itmustbethatforall.DeȚneasthesolu-tionto 4556journalofpoliticaleconomy ifsuchasolutionexistsandotherwise.Considernowaperturbedrule,with0and .BysinglecrossingandthedeȚnitionofin(15),theper-turbedrulesatisȚestheincentivecompatibilityconstraint(4).Conditionalon,thisruleyieldsthesameexpectedwelfaretotheprincipalandtheagentastheoriginalrule.However,conditionalon,theperturbedruleyieldstheagentahigherwelfarethantheoriginalone,sinceby(15), forall.Moreover,notethat(2)implies gUPg,pPgUPg,pP forall .Hence,using(16)andthefactthat0,1 forall.Conditionalon,theprincipalisthereforestrictlybetteroffundertheperturbedrulethanundertheoriginalrule.Itfollowsthattheper-turbedrulewi

38 thnoveriȚcationbelowstrictlydominatesthe
thnoveriȚcationbelowstrictlydominatestheoriginalrule,con-tradictingtheoptimalityofarulethatveriȚesalltypes..If0,then andtheperturbedrulewehaveconstructedincreasestheprincipalswelfarefromtype relativetoverifyingalltypes.Theclaimthereforefollowsinthiscase,givenouroptimalitycondition(5).More-over,when0,wecanalsoconsideradifferentperturbationtoprovetheclaimwithoutrelyingonthiscondition.SpeciȚcally,takeaperturbedrulethat0forall g, 0arbitrarilysmall,bunchingallattheiraverageefȚcientspendinglevel.Thisruleisincentivecompatibleandincreasestheprincipalswelfarerelativetoverifyingalltypes:givenenough,thewelfarelossfromnotassigningefȚcientspendingto g, issecondorder,whilethegainfromsavingonveriȚcationcostsisȚrstorder.A3.ProofofLemma2SupposethatanoptimalrulefeaturesveriȚcationthatisweaklyincreasingevery-where.Bylemma1,0forsome,andhencethisrulemustfeatureano-veriȚcationregionfollowedbyaveriȚcationregion.Thatis,theprincipalsolves(3)and(4)bychoosingathresholdsuchthat0for1foraswellasaspendingallocation)foreachNowconsiderarelaxedversionofthisprobleminwhichtheprincipalchoosesanoptimalallocationintheno-veriȚcationandveriȚcationregionsseparately,ignoringtheincentivesoftypesinoneregiontodeviatetotheotherregion.Tak-ingtheno-veriȚcationregiontobe[ ],itfollowsfromproposition1thatanoptimalallocationisathresholdsuchthatforeach .FortheveriȚcationregion(],sinceincentivecompat-ibilityistriviallysatisȚed,anoptimalallocationassigns)toeachNotethattheresultingruleforthewholesetisTEC.Moreover,becausethiscommitmentversusexibilitywithcostlyverication4557 rulesolvesarelaxedproblem,itissufȚcienttoshowthatitisincentivecompat-ibleoverthewholesettoproveitsoptimalityintheoriginalproblem.Toshowincentivecompatibility,noteȚrstthatincentivecompatibilitywithineachregionisguaranteedbyconstruction.Furthermore,since,asexplainedinsectionII.C,notypewouldhaveincentivestodeviatetomimicadifferenttypethatisveriȚed,incentivecompatibilityissatisȚedforall .Allthatislefttobeshowni

39 sthatnotypehasincentivestodeviatetomimic
sthatnotypehasincentivestodeviatetomimicatype Thesingle-crossingconditioninimpliesthatasufȚcientconditionfortheaboveinequalitytoholdisNownotethatoptimalityoffortheprincipalimpliesGiventheagentsbias(2)and0,1,(18)implies(17)if,or,equivalently,since0,ifWeprovethattheTECrulethatweconstructedsatisȚes(19).Theoptimalintheno-veriȚcationregionsolves TheȚrst-orderconditionyields Notethat0if,and0if.Hence,theȚrst-orderconditionforsome,implyingthat(19)musthold.A4.ProofofProposition20forall .Supposebycontradictionthatanoptimalrule1forsomebutTECisnotoptimal.Bylemma2,thisrulemustfeaturedecreasingveriȚcation.WeproceedbyshowingthatanoptimalrulecannotfeaturedecreasingveriȚcationatanyConsiderȚrstdecreasingveriȚcationatsome0,so1forsome0arbitrarilysmall.Asshowninthetext,theoptimalityofverifyingtypeimplies(6)and,whereastheoptimalityofnotverifyingimplies(7).However,thetwoequationstogetherwithviolatethesingle-crossingcondition(1),acontradiction.4558journalofpoliticaleconomy ConsidernextdecreasingveriȚcationatsome1,so0forsome0arbitrarilysmall.Analogousargumentstothoseaboveapplytothiscaseandyieldacontradiction.A5.ProofofLemma3SupposethataTECrulewithcutoffsisasolutionto(8)and(9).Notethatanyrulesatisfyingconstraint(4)willsatisfyconstraint(9).Hence,(8)and(9)arearelaxedversionof(3)and(4),implyingthatanysolutionto(8)and(9)thatsatisȚes(4)willalsobeasolutionto(3)and(4).Itfollowsthattoprovetheclaim,allweneedtoshowisthattheTECrulethatsolves(8)and(9)willsatisfyconstraint(4).Itisimmediatethatforany0,(9)beingsat-isȚedimpliesthat(4)willbesatisȚed.Nowconsider1.OptimalityofverifyingtypeunderaTECrulethatsolves(8)and(9)implies,(20)sinceaperturbationthatassignsnoveriȚcationandspendingleveltoaisincentivecompatible.Notethatbytheargumentsintheproofoflemma2,aTECrulethatsolves(8)and(9)satisȚesforall.Hence,combining(20)with(2)andthefactthat0,1Itfollowsthat(4)issatisȚedfortypeA6.ProofofLemma4Supposethatarulesolving(8)and(9)speciȚes1forToprovethattherulespeciȚes,supposeby

40 contradictionthat.Consideraperturbedrule
contradictionthat.Consideraperturbedrulethatsetswhilekeepingtheallocationunchangedforall.Thisperturbationstrictlyincreasestheprincipalswelfareconditionalon,leavestheprincipalswelfareconditionalonunchanged,andisincentivecom-patiblefortheagent.Similarly,toprovethattherulespeciȚes,supposebycontradic-tionthat.Consideraperturbedrulethatsets1andwhilekeepingtheallocationunchangedforall.Thisperturbationstrictlyincreasestheprincipalswelfareconditionalon,leavestheprincipalswelfareconditionalonunchanged,andisincentivecom-patiblefortheagent.Finally,weprovethattherulemustspecifyif(9)doesnotbind.Supposebycontradictionthat(9)doesnotbindforBytheclaimabove,,andthustherulemustset.Butthenaperturbedrulethatsets1and0arbitrarilysmallwhilekeepingtheallocationunchangedforallstrictlyincreasestheprincipalswelfareconditionalon,leavestheprincipalswel-fareconditionalonunchanged,andisincentivecompatiblefortheagent.commitmentversusexibilitywithcostlyverication4559 A7.ProofofLemma5Supposebycontradictionthatarulethatsolves(8)and(9)spec-forsome.Weconsiderfourcasesseparately.Case10.Then(9)forwhichtogetherimplyHowever,given,(21)violatesthesingle-crossingcondi-tionin,acontradiction.Case21.Bylemma4,,andthus.Usinglemma4again,itthenfol-lowsthat(9)bindsfor;thatis,thereexists0suchthatFurthermore,notethatwemusthave,sincestrictlyconcave.Incentivecompatibilityforwhich,combinedwiththeobservationthat,(23)Combining(22)and(24)yieldsHowever,given,(25)violatesthesingle-crossingcondi-tionin,acontradiction.Case31and0.Notethat(23)musthold.Thenconsideraperturbedrulethatsets1andwhileleavingtheallocationfortypesunchanged.Sinceincentivecompat-ibilitywasinitiallysatisȚedandwhile(23)holds,thisperturbationisin-centivecompatible.Optimalityoftheoriginalrulethereforere-quiresthisperturbationtonotstrictlyincreasetheprincipalswelfare,whichThesingle-crossingconditioninthenimplies4560journalofpoliticaleconomy Nowconsideradifferentperturbedrulethatsets0andwhileleavingtheallocationfortype

41 sunchanged.Equa-tion(26)impliesthatthisp
sunchanged.Equa-tion(26)impliesthatthisperturbationwouldstrictlyincreasetheprincipalwelfare.Hence,optimalityoftheoriginalrulerequiresthatthisperturbationviolateincentivecompatibility;thatis,theremustexistwith0suchthatNotethatsince,wemusthave.Moreover,byincentivecompatibilitybeingsatisȚedundertheoriginalrule,wehaveCombiningthisequationwith(27)yieldsHowever,given,(28)violatesthesingle-crossingcondi-tionin,acontradiction.Case40and1.Bylemma4,,andhencegiven,incentivecompatibilityfortype.Consideraperturbedrulethatsets1andwhileleavingtheallocationfortypesunchanged.SincetheoriginalrulesatisȚesincentivecompatibilityfor,singlecrossingimpliesthatthisperturbationisincentivecompatiblefor.Optimalityoftheoriginalrulethenrequiresthisperturbationtonotstrictlyincreasetheprincipalswelfare,whichrequiresThesingle-crossingconditioninthenimpliesNowconsideradifferentperturbedrulethatsets0andwhileleavingtheallocationfortypesunchanged.Equa-tion(29)impliesthatsuchaperturbationwouldstrictlyincreasetheprincipalwelfare.Hence,optimalityoftheoriginalrulerequiresthatthisperturbationviolateincentivecompatibility;thatis,theremustexistwith0suchthatNotethatsince,wemusthave.Moreover,byincen-tivecompatibilitybeingsatisȚedundertheoriginalrule,wehaveCombiningthisequationwith(30)yieldscommitmentversusexibilitywithcostlyverication4561 However,given,(31)violatesthesingle-crossingcondi-tionin,acontradiction.A8.ProofofLemma6Supposethatarulesolves(8)and(9)andfeaturesdecreasingver-iȚcationatsome,with1.Then0forsome0arbi-trarilysmall.Supposethatitwerethecasethat.Thenoptimal-ityofthisrulewouldbeviolated,asaperturbedrulethatsets0andwhilekeepingtheallocationunchangedforwouldbeincentivecompatibleandstrictlyincreasetheprincipalswelfare(recall0).Itfollowsthat,andhencebylemma5,Moreover,bylemma4,,andthusincentivecompatibilityforwouldbeviolatedifitwerethecasethat.Itthereforefollowsthat0arbitrarilysmall.Lemma5thenimpliesforall,whichimpliesMoreover,bydeȚnition,Combining(33)and(34

42 )andtakingintoaccountthat10yields(10).Su
)andtakingintoaccountthat10yields(10).Supposenextthatarulesolves(8)and(9)andfeaturesde-creasingveriȚcationatsome,with0.Then1forsome0arbitrarilysmall,andargumentsanalogoustothoseaboveyield(10).A9.ProofofProposition3Theargumentsintheproofsoflemmas1and2applytotherelaxedproblem,implyingthatifasolutionto(8)and(9)involvesverifyingsometype,thissolutioniseitheraTECruleorarulethatfeaturesdecreasingveriȚcationat.ToprovetheoptimalityofTECfor,wethusproceedbyshow-ingthatforanysuchveriȚcationcost,arulefeaturingdecreasingveriȚcationcannotbeasolutionto(8)and(9).Supposethatarulesolves(8)and(9)andfeaturesdecreasingveriȚcation.DenotebytheinȚmumofthelowestveriȚcationregionunderthisrule.Nowconsideraperturbedrulethatsets0for,and1for.If0,letasdeȚnedinproposition1under.If1,let.Bytheargumentsintheproofoflemma2,thisruleisincentivecompatiblefortypesprescribednoveriȚcationandsets.Moreover,giventhisinequalityandthefactthat0,itfollowsthat4562journalofpoliticaleconomy theruleisalsoincentivecompatiblefortypesprescribedveriȚcation.Wenowshowthatthisrulestrictlyincreasestheprincipalsexpectedwelfareforcontradictingtheoptimalityoftheoriginalrule.DenotebythelowesttypefeaturingdecreasingveriȚcationintheoriginalrule.Thenthechangeintheprincipalsexpectedwelfarefromusingtheperturbedruleinsteadoftheoriginalruleis ,minđȚȚđȚȚNotethatsincealltypesaboveareveriȚed,theprincipalswelfarecondi-tionalontheagentstypebeingintheno-veriȚcationregionoftheperturbedruleisoptimizedsubjecttofewerincentivecompatibilityconstraintsinthisrulecomparedwiththeoriginalrule.Hence,theȚrsttermin(35)isweaklypositive.Toevaluatethesecondandthirdtermsin(35),supposeȚrstthat.Thenbylemma6,thesecondtermin(35)satisȚesđȚȚđȚȚMoreover,thethirdtermin(35)satisȚesTogether,(36)and(37)implythattheperturbationstrictlyincreaseswelfare.Supposenextthat.Analogousargumentstothoseaboveimplythattheperturbationmakestheprincipalweaklybetteroffconditionalon.Toeval-uatethechangeinwelfareconditionalon,notethatinthiscasewe

43 must0and1for0arbitrarilysmall.Analogousa
must0and1for0arbitrarilysmall.Analogousargumentstothoseintheproofoflemma6thenimply.Moreover,by(11),,(38)wherewehaveappealedtothedeȚnitionof.Itthusfollowsfrom(38)thattheperturbationstrictlyincreasestheprincipalswelfareconditionalonA10.ProofofProposition4Considerthefollowingquadratic-uniformsetting:preferencessatisfy2and2for0,and1forall.Inthissetting,theefȚcientandßexiblespendinglevelsaregivenbycommitmentversusexibilitywithcostlyverication4563 ,respectively.Let0,sothattheagentpaysnoveriȚcationcost.WeȚrstestablishthatinthissetting,iftheveriȚcationcostsatisȚesTECissuboptimal,asitisdominatedbyarulewithoutveriȚcation.8.Considerthequadratic-uniformsettingwith0.IfthenTECisnotoptimal.Proof.Takethequadratic-uniformsettingwith0and2.Considerthefollowingproblem: NotethatthesolutiontothisprogramcoincideswitharulewithoutveriȚcationifitsets,anditcoincideswitharulethatveriȚesalltypesifitsets BythedeȚnitionofTEC,anecessaryconditionforaTECruletobeoptimalisthatthesolutiontoprogram(39)specify .Weshowthatthiscannotbesat-isȚedwhenTheȚrst-orderconditionfor,givenourassumptionsonpreferencesandthedistributionof,implies ,(40)wherewehavetakenintoaccountthefactthatmaybelowerthan .Iftheso-lutionto(39)setsstrictlyinterior,thentheȚrst-orderconditionforimplies g*1b21 Substitutingwith(40)andrearrangingtermsyields g**2max Notethatif ,(41)implies2,contradictingtheassumptionthat2.Therefore, ,(42)andthus(41)implies Substitutingbackinto(40),weobtain However,combinedwith(42),equation(43)implies2,contradictingtheassumptionthat2.Therefore,thesolutionto(39)cannotsetinteriorwhen2.QED4564journalofpoliticaleconomy Wenextshowthatthereexists2underwhicharulewithveriȚcationis9.Considerthequadratic-uniformsettingwith0.If3and6 ,thenarulewithveriȚcationisoptimal..Takethequadratic-uniformsettingwith3,and .AnoptimalrulewithoutveriȚcationsetswhereusing(40)(with)andthefactthat g14b� ,wehaveWeconstructaperturbedrulethatfeaturesveriȚcationandyieldstheprincipal

44 strictlyhigherexpectedwelfarethanthisopt
strictlyhigherexpectedwelfarethanthisoptimalrulewith-outveriȚcation.Foranygiven,deȚneasthesolutiontowhichaftersomealgebrayieldsTakesufȚcientlyclosetosothat (notethattheassumptionthat6 ensuresthatsuchaexists).TypeisdeȚnedsothatheisindifferentbetweentheßexiblespendinglevelofandtheopti-malspendinglimitundernoveriȚcationforadistributiontruncatedat(whichisgivenby).Nowconstructtheperturbedruleasfollows:if,then0and;if,then0and;andif,then1andwhichaftersomealgebrayieldsNotethatgiventhedeȚnitionof,thisruleisincentivecompatible.Theper-turbationchangestheprincipalswelfareonlyfortypes.ThechangeinwelfareisequaltođȚȚđȚȚAftersomealgebraandsubstitutionof(44),usingourassumptionsonprefer-encesandthedistributionof,thissimpliȚesto g2gH13b2dg2đgHgH22b gH2g2b2dg2đgHgH22bfdg1đgHgH24b Simplifyingfurtheryieldsthatthechangeinwelfareisequalto commitmentversusexibilitywithcostlyverication4565 wheretheinequalityfollowsfromtheassumptionthat3.Therefore,theperturbedrulewithveriȚcationstrictlyincreasestheprincipalsexpectedwel-farerelativetonoveriȚcation.QEDItfollowsfromlemmas8and9thatinaquadratic-uniformsettingwith3,and6 ,veriȚcationisoptimalbutTECisnot.Bylemma2,anyoptimalrulemustthereforefeaturedecreasingveriȚcation.A11.ProofofLemma7PartiSupposethatanincentive-compatibleruleinducesdecreasingveriȚca-tionat.ConsiderȚrstthecaseinwhich0andthus0arbitrarilysmall.Incentivecompatibilityfortype,(45)canchoosetobeveriȚedandguaranteehimselftheefȚcientlevelofspending.Incentivecompatibilityfortype,(46)canchoosenottobeveriȚedandspendat).Giventheconti-nuityofintheirrespectivearguments,wecantakethelimitofbothsidesof(46)asapproaches0toobtainCombining(45)and(47)yields(12).Considernextthecaseinwhich1andthus0forarbitrarilysmall.Analogousargumentstothoseaboveimplythefollowingincen-tivecompatibilityconstraintsfor,respectively:,(48)Sincetheruleispiecewisecontinuous,limexistsandcanbedeȚned).Takingthelimitofbothsidesof(48)and(49)asgoesto0yields(47)and(45),andcombiningthesetwoin

45 equalitiesyields(12).Tocompletetheproofo
equalitiesyields(12).Tocompletetheproofofparti,weshowthatmusthold.Notethatby(12),either.Forthepurposeofcon-tradiction,supposeitwerethecasethat.Considertheincentivecompatibilityconstraintoftype0arbitrarilysmall.TakeȚrstthecaseinwhich1.ThenmustweaklypreferveriȚcationtonoveriȚcation,whichrequires,(50)impliesCombining(12)and(51)yields4566journalofpoliticaleconomy ,thisinequalityviolatesthesingle-crossingconditioninthusyieldingacontradiction.Considernextthecaseinwhich0.GivendecreasingveriȚcation,inthiscasewemusthave1and0for0arbi-trarilysmall.Moreover,givenourdeȚnitionof.Byincentivecompatibility,typemustweaklypre-ferveriȚcationtonoveriȚcation,whichrequires,(52)whereastypemustweaklyprefernoveriȚcationtoveriȚcation,whichCombining(52)and(53)andusingthefactthatapproaching0,thisinequalityvio-latesthesingle-crossingconditionin,thusagainyieldingacontradiction.Therefore,weobtainthatcannothold,andwemustthushavePartiiSupposeanincentive-compatibleruleinducesincreasingveriȚcation.Thenanalogousargumentstothoseusedtoproveparticanbeappliedtoshowthat(14)mustholdat.Sincethestepsareanalogous,weomitthedetails.A12.ProofofProposition5Toprovethisresult,weȚrstestablishthefollowinglemmas.10.Underlimitedcommitment,ifanincentive-compatiblerulefea-turesincreasingveriȚcationat,then,(54).Supposeanincentive-compatiblerulefeaturesincreasingveriȚcationat.Byequation(14)inlemma7,either.Forthepurposeofcontradiction,supposeholds.TakeȚrstthecasein1,sothat0for0arbitrarilysmalland,givenourdeȚnitionof),lim.Byincentivecompatibility,mustweaklyprefernoveriȚcationtoveriȚcation,whichrequiresHowever,(14)and(55)togetherwiththefactthatimplythatas-sumption2isviolated,yieldingacontradiction.commitmentversusexibilitywithcostlyverication4567 Considernextthecaseinwhich0,sothat1for0ar-bitrarilysmall.Byincentivecompatibility,typemustweaklypreferveriȚca-tiontonoveriȚcation,whichrequiresNotethatinthiscase,However,(14)and(56)togetherwithimplythatassump-tion2isviolated,yieldingagaina

46 contradiction.Therefore,weobtainthatcann
contradiction.Therefore,weobtainthatcannothold,andwemustthushave.QED11.Underlimitedcommitment,ifanincentive-compatiblerulefea-turesdecreasingveriȚcationat,thenthereexists1,andeither1,lim0,andProof.Supposeanincentive-compatiblerulefeaturesdecreasingveriȚcationat.Bycondition(13)inlemma7,.ConsidertheproblemoftheprincipalaftertheveriȚcationdecision)hasbeenmadeandtheveriȚcationresult(incaseofveriȚcation)hasbeenobtained: 1,(59)ThisprogramtakesintoaccountthattheprincipalwillassigntheefȚcientspend-ingleveltoanyagenttypewhochoosestobeveriȚed,andshewillignoretheincentivesofveriȚedtypeswhendecidingthespendingallocationoftypeswhochoosenottobeveriȚed.Wenowconsidertheoptimallevelof)givendecreasingveriȚcationatandtheconditionsthatarenecessaryfortheprin-cipaltochooseStep1ConsiderthespendingallocationconditionalonnoveriȚcation.Notethatanalogousargumentstothoseusedintheproofoflemma5implythatmustbeweaklyincreasingfornonveriȚedtypes.ForeachnonveriȚedtypedenoteby thespendinglevelclosestto)frombelowintheallowablespendingsetfornonveriȚedtypes(i.e.,amongallspendinglevelsassignedtotypeswhochoosenoveriȚcation).Analogously,denotebytheclosestspend-inglevelto)fromaboveintheallowablespendingsetfornonveriȚedtypes.Clearly,if)isinthisallowablespendingset,then .Theincentivecompatibilityconstraint(60)togetherwiththeconcavityofrequiresthatif0,thenargmax Step2Asnoted,givendecreasingveriȚcationat,therulemustset.Weshowthatasaresult,therulemustinduce0and4568journalofpoliticaleconomy foralltypes.Toseewhy,noteȚrstthatby(61)andthesingle-crossingconditionin,anytypewhoisnotveriȚednecessarilychoosesspending.Therefore,itissufȚcienttoshowthatanytypemusthave0.Supposebycontradic-tionthatthiswerenotthecase.Thenincentivecompatibilityforatypewith1requiresthatthistypeweaklypreferveriȚcationtonoveriȚcation,whichrequiresHowever,(12)and(62)togetherwiththefactthatolateassumption2.Theclaimthereforefollows.Step3Weshowthatinanincentive-compatiblerule,constraint(60)cannotbeuniform

47 lyslackforall,whererecallbyde-creasingve
lyslackforall,whererecallbyde-creasingveriȚcationat.Supposebycontradictionthatthisistrue.Notethatfromstep2,(60)isthenuniformlyslackforallwhere0forallsuch.Nowconsiderthefollowingperturbation:for0arbitrarilysmallandall,set;forall,set;andforallothertypes,leavethespendingallocationunchanged.Thisperturbationstrictlyincreasestheprincipalswelfareasitreducesoverspendingbytypes.Moreover,since(bythecontradictionassumption)(60)wasuni-formlyslackbeforetheperturbationforall,itisstillsatisȚedaftertheper-turbation,andincentivecompatibilityforalltypesisguaranteedastheper-turbationsatisȚes(61).Therefore,weobtainthatif(60)isuniformlyslackforall,theprincipalcanstrictlyimproveupontheoriginalrulebyreducing)aftertheveriȚcationdecisionhasbeenmade,andhencetheorig-inalruleviolatesincentivecompatibilityfortheprincipal.Theclaimfollows.Step4Bystep3,inanyincentive-compatiblerulewithdecreasingveriȚca-tionat,thereexistssatisfying(57).Moreover,sincedecreasingveriȚ-cationat,thisrequires.Thisprovesthelemma.QED12.Underlimitedcommitment,ifanincentive-compatiblerulefea-turesdecreasingveriȚcationat,thenthereexistsatwhichtherulefea-turesincreasingveriȚcation.Moreover,1forallfor1and.Supposeanincentive-compatiblerulefeaturesdecreasingveriȚcationat.Bylemma11,thereexistsatypesatisfying(57)eitherwithoratwhichthereisincreasingveriȚcation.Wecanestablishthatsuchatypeisunique.Supposebycontradictionthattherearetwotypes,satisfyingtheconditioninlemma11.Then,(64)Incentivecompatibilityrequirescommitmentversusexibilitywithcostlyverication4569 ,(66)Combining(64)(67)yieldsbydecreasingveriȚcationatandlemma11,thisinequalityviolatesthesingle-crossingconditioninyieldingacontradiction.Therefore,thereexistsauniquetypebelowforwhich(57)holds,anddenotingthistypebyyields(63).Next,weshowthat1forall.NoteȚrstthataspendinglevelcannotbeallowedbytheruleundernoveriȚcation,sinceoth-erwisetypewouldhaveastrictincentivetodeviatetosuchaspendinglevel.Considertherelevantcaseinwhichandsupposebycontr

48 adictionthat0forsometype.Letdenotethehig
adictionthat0forsometype.Letdenotethehighestsuchtype.Since,asnoted,spendinglevelsstrictlybetweennotallowed,itfollowsfrom(63)andthattherulemustsetMoreover,sincebyconstructiontherulefeaturesincreasingveriȚcationatcondition(14)inlemma7impliesHowever,given(12)and(13),equation(68)violatesassumption2.Itfollows1forall.QEDWecannowprovetheproposition.WebeginbyrulingoutdecreasingveriȚca-tion.Supposebycontradictionthatanincentive-compatiblerulefeaturesdecreas-ingveriȚcationatsome.Bylemma12,theremustexistatypesatisfyingtheconditionsinthelemma.Weproceedintwosteps.Step1.Thenitfollowsfrom(14)and(63)thatHowever,(12)and(69)togetherwiththefactthat(by[13])implythatassumption2isviolated,acontradiction.Step2Bystep1,anyincentive-compatiblerulewithdecreasingveriȚcationmusthaveateachpointatwhichthereisdecreasingveriȚcation.Nowconsidertheprincipalsproblem(58)(60).Letbethehighestnon-veriȚedtype.SincethetypeswithdecreasingveriȚcationareatomisticandtheruleispiecewisecontinuous,followingadecisionofnoveriȚcationtheprincipal Byproposition1,thesolutionassignsfor andsome.However,inthiscase,conditions(13)and(54)(whichrequire4570journalofpoliticaleconomy and,respectively)cannotbesatisȚedatapoint atwhichthereisdecreasingveriȚcationandthuscontradiction.Theclaimsaboveshowthatunderlimitedcommitment,anyincentive-compatiblerulefeaturesweaklyincreasingveriȚcationeverywhere.Analogousargumentstothoseintheproofsoflemmas1and2canthenbeappliedtoshowthataTECruleisoptimalifarulewithveriȚcationthatisweaklyincreasingeverywhereisoptimal.Therefore,underlimitedcommitment,ifveriȚcationisoptimal,TECisoptimal.Aguiar,Mark,andManuelAmador.2011.GrowthundertheShadowofExpro-Q.J.E.Alesina,Alberto,andAndreaPassalacqua.2016.ThePoliticalEconomyofGov-ernmentDebt.HandbookofMacroeconomics,vol.2,editedbyJ.B.TaylorandHaraldUhlig,2599651.NewYork:North-Holland.Alonso,Ricardo,andNikoMatouschek.2008.OptimalDelegation.Rev.Econ.Amador,Manuel,andKyleBagwell.2013.TheTheoryofDelegationwithanApplicat

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