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CommitmentversusFlexibilitywithCostlyVericationMarinaHalacYaleUniversi CommitmentversusFlexibilitywithCostlyVericationMarinaHalacYaleUniversi

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thepresenceofinformationcostsandmisalignedincentivesInasurveyofmanufacturingrmsRoss1986observesthattopcorporatemanagementisoftentoobusyandpreoccupiedwithotherresponsibilitiestohavethetimeandresources ID: 897343

econ thatis 1987 2017 thatis econ 2017 1987 2012 2013 2014 infact forexample 2019 1986 1979 2008 acontradiction 2018

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1 CommitmentversusFlexibilitywithCostlyVer
CommitmentversusFlexibilitywithCostlyVeriÞcationMarinaHalacYaleUniversityandCenterforEconomicPolicyResearchPierreYaredColumbiaUniversityandNationalBureauofEconomicResearchAprincipalfacesanagentwhoisbetterinformedbutbiasedtowardhigheractions.Shecanverifytheagentsinformationandspecifyhispermissibleactions.WeshowthatiftheveriÞcationcostissmallenough,athresholdwithanescapeclause(TEC)isoptimal:theagenteitherchoosesanactionbelowathresholdorrequestsveriÞcationandtheef-Þcientactionabovethethreshold.Forhighercosts,however,theprinci-palmayrequireveriÞcationonlyforintermediateactions,dividingthedelegationset.TECisalwaysoptimaliftheprincipalcannotcommittoinefÞcientallocationsfollowingtheveriÞcationdecisionandresult.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-nolÞ,JoelSobel,JeffZwiebel,andvariousseminarandconferenceaudiencesforhelpfulcomments.WealsothankSebastianDiTellaandNikoMatouschekforvaluablediscussionsofthepaper.WeijieZhongprovidedexcellentresearchassistance.ElectronicallypublishedOctober23,20202020JournalofPoliticalEconomy,2020,vol.128,no.12]2020byTheUniversityofChicago.Allrightsreserved.0022-3808/2020/12812-0004$10.00 thepresenceofinformationcostsandmisalignedincentives.InasurveyofmanufacturingÞrms,Ross(1986)observesthattopcorporatemanage-mentisoftentoobusyandpreoccupiedwithotherresponsibilitiestohavethetimeandresourcestoevaluateeveryinvestmentopportunity.Atthesametime,lower-leveldivisionheadswhopossesstherequiredin-formationfordecision-makingarepronetooverstatetheirinvestmentopportunitiesandspendexcessively(e.g.,BrealeyandMyers1981;Don-aldso

2 n1984).Infact,thecombinationofthesefacto
n1984).Infact,thecombinationofthesefactorsisseenasamainreasonwhytopmanagementoftenimposescapitalrationingontheÞrmdivisions.Analogousproblemsofinformationandincentivesariseinotherappli-cations.IntheoperationofÞscalpolicy,citizensdonothavethecapacitytoevaluateeverybudgetallocationthatmustbemade,whereaselectedofÞcialswhoknowthevalueofoutlaysarebiasedtowardoverspendingbecauseofpoliticalinterests(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,whereasawidesetletstheagentutilizehisprivateinformationabouttheefÞcientaction.Amaininsightfromtheliteratureisthatunderweakconditions,thistrade-offisopti-mallyresolvedbythresholddelegation.Thatis,theprescriptionistoUsingsurveydata,PruittandGitman(1987)Þndthatseniormanagersareawarethatjuniormanagersoverstateestimatedprojectrevenues.TheyalsoÞndthatthisoverstate-mentistoalargeextentconsideredintentional.SeeMukherjeeandRahahleh(2011).Astheauthorsdescribe,alargeliteratureÞndsevidencethatÞrmsoperateundercapitalconstraintsand,moreover,thattheseconstraintsareimposedinternallybyseniormanagersratherthanexternallybythesuppliersoffunds.Asanotherexample,inaretailsetting,itistoocostlyformanagerstoscruti

3 nizethebestsalesstrategyforeveryclient,w
nizethebestsalesstrategyforeveryclient,whereasthesalesrepresentativeswhoareabletodosogenerallyoffertoomanydiscounts(e.g.,Loetal.2016).Invariousapplications,likethosedescribedabove,(contingent)transfersmayberuledoutbecauseofinstitutionalreasonsorethicalconsiderations.Intheirstudyofcapitalbudgetingpractices,MukherjeeandRahahleh(2011)reportthatdirectlyrewardingem-ployeesforproposinggoodinvestmentsisuncommoninlargeÞrms.4524journalofpoliticaleconomy setbudgetcapsformanagersinorganizations,deÞcitlimitsinthecon-textofÞscalpolicy,andtariffcapsaspartoftradeagreements.AkeylimitationofexistinganalysesofdelegationconcernstheuseofveriÞcation.Inpractice,whileitiscostlyforprincipalstoverifyagentsformation,thiscostisnotashighastoruleoutveriÞcationaltogether.Real-worlddelegationrulestypicallyfeatureacaponallowableactions,asinthecanonicalmodel,togetherwithreviewandapprovalproceduresforrequeststhatexceedthecap.Inorganizations,[s]mallerprojectscantypicallybeapprovedbydivisionheads,andthus,withinthebudgetlim-its,decision-makingfortheseprojectsiscompletelydecentralized....Largerprojects,bycontrast,mustbeapprovedbyacentralinvestmentcommitteeoreventheboardofdirectors(Taggart1987,18).Becausethecommitteeinchargeofapprovalmustspendcostlytimegatheringin-formation,evaluatingcashßowprojections,anddeliberatingontheap-propriateinvestment,onlycertainprojectsareveriÞed:Giventheirprior-ities,topmanagementoftencopeswithproductivityimprovementbyallocatingsmallÞxedsumstodivisionsandplants.Thatleavesthemthetimetocarefullyanalyzethelargeprojects(Ross1986,21).Similarproce-duresareusedinthecontextofÞscalandtradepolicy,whererulesspecifydeÞcitortariffcapstogetherwithescapeclauseanddisputesettlementprovisionsforbreachingthesecapsunderveriÞedspecialconditions.Motivatedbytheseapplications,westudyageneraldelegationframe-workinwhichveriÞcationiscostlybutfeasible.WemodelcostlyveriÞca-tionasintheseminalworkofTownsend(1979)andexplorehowitaffectsoptimaldelegation.Howdoest

4 heprincipalchoosethedelegationsettooptim
heprincipalchoosethedelegationsettooptimallyresolvethetrade-offbetweencommitmentandßexibilitywhileatthesametimeminimizingveriÞcationcosts?WeÞndthatoptimalrulescantakecomplicatedforms,asveriÞcationeffectivelyallowstheprincipaltorelaxincentiveconstraintsbydividingthedelegationsetintosubsets.Yetweshowthatundercertainconditions,anoptimalruletakesthesim-pleformofathresholdwithanescapeclause(TEC).WedeÞneTECasaruleinwhichtheagenteitherselectsanactionbelowathresholdorrequestsveriÞcationandtheefÞcientactionabovethethresholdbytriggeringtheescapeclause.Asnotedabove,rulesofthisformarecommonlyobservedinapplications.OurpaperprovidesatheoreticalfoundationforthebroaduseofTECandshowshowitsoptimalitydependsontheprincipalscostofveriÞcationandhercommitmentpower.Ourmodelfeaturesanagentwhoisbiasedtowardhigherspendingrel-ativetotheprincipal.Theagentsprivateinformation,or,concernsSeealsoBowerandLesard(1973),Ross(1986),MukherjeeandHenderson(1987),GitmanandVandenberg(2000),andGrinsteinandTolkowsky(2004),amongothers.See,e.g.,Schaechteretal.(2012),Lledetal.(2017),andCoateandMilton(2019)onÞscalrulesandBeshkarandBond(2017)ontradeagreements.commitmentversusexibilitywithcostlyverication4525 thevalueofspending;ahighertypecorrespondstoahighermarginalvalueofspendingforboththeprincipalandtheagent.Followingthelit-erature,webuilduponasettinginwhich,absentveriÞcation,anoptimaldelegationrulewouldbeathreshold,allowingtheagenttochooseanyspendinguptoamaximumlevel.Wedepartfrompriorworkbylettingtheprincipalverifyandperfectlylearntheagentstype.VeriÞcationen-tailsanadditivecostfortheprincipal,whichmayalsobepartiallybornbytheagent.Webeginouranalysisbyassumingthattheprincipalcanfullycommittoadelegationrule.Theproblemcanbeviewedinthreesteps:Þrst,theprin-cipalchoosesamappingfromtheagentsveriÞcationdecisionandresulttoasetofallowablespending;second,theagentdecideswhethertoseekveriÞcation;third,theagentchoosesaspendinglevelfromtheallowableset.Formally,adelegationruleisapai

5 rofschedulesspecifying,foreachagenttype,
rofschedulesspecifying,foreachagenttype,whetherheisveriÞedandhisspendinglevel.AdelegationruleisoptimalifitmaximizestheprincipalsexpectedwelfaresubjecttotheincentivecompatibilityconstraintthateachagenttypepreferhisveriÞca-tionassignmentandspendingleveltothoseofanyothertype.Inpartic-ular,eachtypemustpreferhisallocationtothatofanyothertypewhoisnotprescribedveriÞcation.DeviationstotypeswhoareveriÞedcanbetriv-iallydeterredastheygetrevealedbytheprincipalsveriÞcation.Asanim-plication,theuseofveriÞcationcanmakelocalincentivecompatibilityconstraintsslackwhilenonlocalconstraintsbind;ouranalysismakesuseofperturbationmethodstoaddresstheseissues.OurÞrstmainresultshowsthatTECisoptimalifthecostofveriÞcationissufÞcientlysmall.Importantly,wealsoshowthatverifyingallagenttypesisneveroptimal;hence,nomatterhowsmalltheveriÞcationcostis,anoptimalruleprescribesnoveriÞcationforsometypes.TheintuitionwhyTECisoptimalisthatverifyinganupperregionofagenttypesnotonlyallowstheprincipaltoimprovetheirspendingallocationbutalsoisanefÞcientmeansofimposingdisciplineonloweragenttypeswhoarenotveriÞed;thesetypesselectfromasetoflowerspendinglevelsandcannotmimichighertypeswhoareveriÞed.Toprovetheresult,weshowthatanyrulewithdecreasingveriÞcationprescribingveriÞcationforasetofagenttypesandnoveriÞcationforasetofhighertypescanbedominated.DecreasingveriÞcationisexpensivefortheprincipalbe-causeitrequiresincentivizingtypesintheveriÞcationregiontoseekveri-Þcationratherthanmimicahighertypeinano-veriÞcationregionabovethem,andthisinturnrequiresinducingsigniÞcantoverspendingintheno-veriÞcationregion.WeshowthatwhentheveriÞcationcostissmallWerestrictattentiontodeterministicrules(seesec.VIforadiscussion)andprovethatarevelationprincipleintermsofpayoffsholdsinoursetting.4526journalofpoliticaleconomy enough,aperturbationthatveriÞesalltypesinthedecreasingveriÞcationregionincreasestheprincipalswelfare.TEC,however,maynotbeoptimaliftheveriÞcationcostisrelativelylarger.Oursecondmainres

6 ultshowsthatdecreasingveriÞcationcanbest
ultshowsthatdecreasingveriÞcationcanbestrictlyoptimalinthiscase.Forexample,arulethatveriÞesonlyanintermediatesetoftypescanyieldtheprincipalhigherwelfarerelativetonotverifyinganytypeaswellasrelativetousingaTECrule.ThemainreasonwhyverifyingonlyintermediatetypescandominatenotverifyinganytypeisthattheveriÞcationregionservestodisciplinetypesintheno-veriÞcationregionbelow.ThemainreasonwhyverifyingonlyintermediatetypescandominateTECisthatitallowstheprincipaltosaveonveriÞca-tioncosts.WeshowthatthesebeneÞtscanoutweighthecostofoverspend-ingthatisneededtoincentivizeintermediatetypestobeveriÞed.Thus,arulethatinvolvesdecreasingveriÞcationcanbeoptimalwhentheveriÞca-tioncostisnotsmall(andnotlarge)enough.TheoptimalityofdecreasingveriÞcationdoesnotrelyonanysortofasymmetryinthepayoffordistributionfunctions.Asnoted,oursettingisoneinwhichthresholdrulesarealwaysoptimalabsentveriÞcation,andinfactweprovetheresultbytakingthewidelystudiedcaseofqua-draticpreferencesandauniformdistributionoftypes.Aninteriorveri-ÞcationregioncanbebeneÞcialbecauseitallowstheprincipaltodividethedelegationsetwhilekeepingveriÞcationataminimum.Observethattheagentwantstooverspendrelativetotheprincipalbuthispreferredspendingleveldependsonhistype.Consequently,requiringveriÞcationforintermediatespendinglevelsmaysufÞcetolimitthespendingofrel-ativelylowtypes:thesetypesareunabletojustifyincreasingtheirspend-ingtoanintermediatelevelviaveriÞcation,andincreasingtheirspend-ingfurtherwouldnotbeattractivetothem.Theaboveresultraisesthequestionofwhyruleswithdecreasingveri-Þcationarerarelyobservedinreality.Weprovideananswerbasedonapracticalconsideration:implementingsucharulerequiresstrongcom-mitmentpowerfromtheprincipal,strongerthanwhatmaybefeasibleinapplications.Taketheaforementionedruleinwhichtheprincipalver-iÞesonlyanintermediatesetoftypes.Underthisrule,theprincipalcom-mitstoanallocationthatmaybeinefÞcientexpost,followingtheveriÞ-cationdecisionandresult.Inparticular,therulemayassig

7 naninefÞcientspendinglevelaftertheagents
naninefÞcientspendinglevelaftertheagentstypeisveriÞedbothinthecasethatthesseekingveriÞcationisonpathaswellaswhenthisveriÞcationispartofadeviation.Moreover,therulemayinduceanallocationaftertheagentdecidesnottoseekveriÞcationthatisinefÞcientconditionalonnoveriÞcation,thatis,whenignoringtheincentivesofveriÞedtypes.WhathappensiftheprincipalisunabletocommitexantetotheseexpostinefÞcientallocations?Inorganizations,forexample,eveniftopmanagementspeciÞescertainbudgetsandrequirementsexante,itiscommitmentversusexibilitywithcostlyverication4527 commonforthesetobechangedexpost.Ross(1986)documentsthatinÞrmswhosebudgetingproceduresresembleTEC,budgetapprovalsdonotconformtopreannouncedcriteriabutdependonthediscretionoftopmanagement.InvestmentcommitteesdecidethescopeofprojectsthatarebroughtupforveriÞcationandapprovalaswellasthebudgetcapforprojectsthathavenotbeensubmittedforreview.InrelatedworkonchiefexecutiveofÞcersandcorporateboards,GrinsteinandTolkow-sky(2004)ÞndthatcorporateboardsexertsigniÞcantdiscretioninre-viewingandapprovingannualbudgetsandlargecapitalrequestsmadebythechiefexecutiveofÞcer.Ourthirdmainresultshowsthatiftheprincipalscommitmentpowerislimited,thenTECisoptimalwheneververiÞcationisoptimal.Intermsofthethree-steptimingdescribedpreviously,limitedcommitmentpowermeansthattheprincipalnowrevisestheagentsallowablespendingsetfollowingtheagentsveriÞcationdecisionandresult.Weprovethatinthiscase,anyincentive-compatiblerulemusthaveweaklyincreasingver-iÞcationeverywhere.ThereasonisthatinducingdecreasingveriÞcationrequiresincentivizingveriÞedtypesnottodeviateandchooseahigherspendinglevelinano-veriÞcationregionabovethem,andunderlimitedcommitmentpoweritalsorequiresincentivizingnonveriÞedtypesnottoseekaveriÞcationthatguaranteesthemefÞcientspending.Whenunabletofullycommittoaruleexante,theprincipalcannotimplementthespendinglevelsthatwouldbeneededtomakethesedeviationsunattrac-tive,andthusdecreasingveriÞcationisnotfeasible.Consequently,weob-t

8 ainthatunderlimitedcommitmentpower,anyop
ainthatunderlimitedcommitmentpower,anyoptimalrulefeaturingveriÞcationmustbeTEC.Altogether,ourresultsprovideatheoreticaljustiÞcationforthepreva-lenceofTECintherealworld.WhenveriÞcationcostsaresmallenough,evenaprincipalwhocancommittoanyclassofdelegationrulewillÞnditoptimaltochooseonewiththesimpleformofTEC.WhenveriÞcationcostsarelarger,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)observethatÞrmscriteriaaresometimesunclear,astheydependontheclassofproject.Thisgivestopman-agementmorediscretiontomakedecisionsexpost.4528journalofpoliticaleconomy underquadraticpreferences;Amador,Werning,andAngeletos(2006)onconsumption-savingsproblemswithhyperbolicpreferences;andAmadorandBagwell(2013),whichconsidersageneralframeworkthatwetakeasourbaseline.Asinthisliterature,westudyaprincipal-agentenvironmentwithnotransfersinwhichtheagentisbetterinformedabouttheefÞcientactionbutbiasedrelativetotheprincipal.Incontrasttothisliterature,weallowtheprincipaltoverifytheagentsinformationatacost.Byintroducingthisadditionaltool,weareabletoexplorehowescapeclausesareoptimallyusedandhowoptimaldelegationdependsontheextentoftheprincipalscommitmentpower.Second,wecontributetotheliteratureoncostlyveriÞcation,startingwithTownsend(1979).BoththatpaperandothersthatfolloweditcludingGaleandHellwig(1985),BorderandSobel(1987),andMook-herjeeandPng(1989)analyzesetti

9 ngswithtransfers,whichweruleout.Morerece
ngswithtransfers,whichweruleout.Morerecently,Ben-Porath,Dekel,andLipman(2014)andErlansonandKleiner(2015)considercostlyveriÞcationinone-goodandcollectiveallocationproblemswithouttransfers,andGlazerandRubinstein(2004,2006)andMylovanovandZapechelnyuk(2017)studyrelatedquestionsusingdifferentveriÞcationtechnologies.Ourmaindeparturefromthisliterature(inadditiontootherdifferencesspeciÞctoeachpaper)isthatwestudyadelegationsettinginwhichweallowfordifferentdegreesofbiasbytheagentrelativetotheprincipal.ThisisalsoamaindistinctionwithrespecttoHarrisandRaviv(1996)andthedynamicversioninMalenko(2019)whoconsidercostlyveriÞcationinadelegationmodelwheretheagentalwaysbeneÞtsfromhigheractions.Suchanextremebiasassumptionimpliesthatgrantingtheagentßexibilityhasnovaluetotheprincipal.Weinsteadbuildonacanonicaldelegationframeworkinwhichßexibilityisvaluable;thatis,theagentsmostpreferredactionSeealsoAthey,Atkeson,andKehoe(2005),AmbrusandEgorov(2013,2017),HalacandYared(2014,2018,2019a,2019b),andAmador,Bagwell,andFrankel(2018).AusterandPavoni(2017)studyadelegationproblemwithlimitedawarenessthatgivesrisetoanonintervaldelegationset.WestudytheeffectsoftheprincipalnotbeingabletocommittonotchangingthesallowablespendingsetfollowingtheveriÞcationdecisionandresult.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.OurpaperprovidestheÞrststudyofoptimaldelegationandveriÞ-cationinasettinginwhichtheprincipalfacesacommitmentversusßex-ibilitytrade-off.Weshowthatthistrade-offintroducesnewconceptualissuesintoourmechanismdesignproblemandshapestheprincipalsop-timalrule.Finally,ourpaperisalsorelatedtoaliteraturethatstudiespolicyruleswithescapeclausesinmacroeconomicmodels.BuildingontheseminalworkofRogoff(1985)oncommitmentversusßexibility,FloodandIsard(1988)andLohmann(1992)considertheuseofescapeclausesinmon-etarypolicy.Obstfeld(1997)discussesthemeritsofescapeclausesinthecontextofÞxedexchangeratesystems,wheremembercountriesareal-lowedtorealigninthefaceofsevereshocks.BeshkarandBond(2017)analyzetradeagreementswithintheclassoftariffcapswithescapeclauses,wheretherelianceonveriÞcationrelativetotariffoverhangoptimallydependsonthelevelofinternationalexternality.CoateandMilton(2019)considertheoptimaldesignofÞscallimitsforapoliticianwhoisallowedtooverridethelimitandselecthispreferredactionwiththecitizensproval.Wesharewiththisliteratureourmotivationofexaminingtheroleofescapeclauses.OurmaindepartureisthatweusemechanismdesigntostudyoptimalruleswithveriÞcationwithoutrestrictingtheirII.ModelOurbaselinemodelofdelegationisthesamegeneralprincipal-agentenvironmentofAmadorandBagwell(2013),wherewefocusonthecaseinwhichtheagentsbiasistowardhigheractions.Weexpandthisdele-gationmodelbyallowingforcostlystateveriÞcation,followingTown-send(1979).A.EnvironmentThereareaprincipalandanagent.Thestateis g,gfor withcontinuousdensity0forall.Thecorrespondingdistribu-tionfunctionis

11 ).Thelevelofspendingisdenotedby Theprinc
).Thelevelofspendingisdenotedby Theprincipalswelfareis),twicecontinuouslydifferentia-blewith0.Weassumethattheprincipalsoptimum,argmax,isinterior,andwerefertoitastheefÞcientlevelofspending.Weimposethefollowingsingle-crossingcondition: Thus,theefÞcientlevelofspendingisincreasinginthestate:4530journalofpoliticaleconomy Theagentswelfareis,with)twicecontinu-ouslydifferentiableand0.Weassumethattheagentsoptimum,argmax,isinterior,andwerefertoitastheofspending.NotethattheagentswelfaresatisÞesthesingle-crossingWeconsideranagentwhoisbiasedto-wardhigherspendingrelativetotheprincipal.SpeciÞcally,weaddthefollowingassumptiontothesettingofAmadorandBagwell(2013): UAg,pp� Condition(2)saysthattheagentnotonlybeneÞtsfromincreasingspend-ingwhenevertheprincipaldoesbutalsobeneÞtsfromanyspendingincreasemorethantheprincipal.NotethatsincewetooktheefÞcientandßexiblespendinglevelstobeinteriorandthepartiesutilitiesfromspendingtobestrictlyconcave,animplicationofthisconditionisthattheßexiblelevelofspendingalwaysexceedstheefÞcientlevel.Thatis,ourassumptionsyield UAg,ppp5pPðgÞ� UPg,ppp5pPðgÞ505 which,given0,impliesforallThestateisprivateinformationtotheagent,thatis,theagentTheprincipalcanperfectlyverifybypayinganadditivecost0.ThescostofveriÞcationis0,1.ThisformulationallowsustocoversituationsinwhichtheagentpaysnoveriÞcationcost(0)aswellassituationsinwhichhepaysacostnolargerthantheprincipal0,1).Onecouldalsoallowfortheagenttopayahighercostthantheprincipals.OurresultsinsectionIVcontinuetoholdunderprovidedthattheagentsbiasissufÞcientlylarge;ourresultsinsectionVholdindependentlyofthevalueofByfeaturingbothabiasandprivateinformationbytheagent,ouren-vironmentgivesrisetoacommitmentversusßexibilitytrade-off.Iftheagentwerenotbiasedrelativetotheprincipal,theprincipalcouldim-plementtheefÞcientlevelofspendingbyprovidingfullßexibilitytotheagent(whowouldinthiscasechoose).Similarly,ifthestatewerenottheagentsprivateinformation,theprincipalcouldimpl

12 ementtheefÞcientlevelofspendingbycommitt
ementtheefÞcientlevelofspendingbycommittingtheagenttoafullycontingentspendingplan.Inthepresenceofbothabiasandpri-vateinformation,however,theprincipalcannotimplementefÞcient)forallwithoutveriÞcation,andshefacesanontrivialtrade-offbetweencommitmentandßexibility.Forboththeprincipalandtheagentspreferences,wewillrefertosinglecrossingasthe(stronger)supermodularityconditionthatwehaveassumedthesepreferencessatisfy.commitmentversusexibilitywithcostlyverication4531 SpecialcasesThemodelofdelegationdescribedaboveencompassesspeciÞccasescommonlystudiedintheliterature.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,whichmapsaveriÞcationdecisionandresultintoanallowablespendingset2.TheagentchooseswhethertoseekveriÞcation,0,1,andtheprincipalperfectlyveriÞeshistype3.TheagentchoosesaspendinglevelfromtheallowablesetTheabovetimingassumesthattheagentlearnshistypebeforetheprincipalsetsaruleinstep1.Ouranalysisisunchangedifinsteadtheagentlearnshistypeaftertherulehasbeenset,thatis,atthebeginningofstep2.C.DelegationRulesGiventhegameformdescribedabove,wecananalyzetheprincipalproblemasthatofchoosingadelegatio

13 nrulethatconsistsofapairHalacandYared(20
nrulethatconsistsofapairHalacandYared(2014,2018,2019a)usethismodeltostudyÞscalrules,whereagov-sdeÞcitbiasmayemergefromtheaggregationofheterogeneous,time-consistentpreferences(JacksonandYariv2015,2014)orfromturnoverinapoliticalecon-omysetting(AguiarandAmador2011;AlesinaandPassalacqua2016).SeeYared(2019)forabroaddiscussionofthisapplication.4532journalofpoliticaleconomy ofschedules,specifyingaveriÞcationdecisionandspend-inglevelforeachtype.Theprincipalchoosesaruletomaximizeherexpectedwelfare: Theobjective(3)istheprincipalsexpectedwelfareunderagivenrule,takingintoaccounttheadditiveveriÞcationcosts.Theconstraint(4)isanincentivecompatibility(ortruth-telling)constraint:itguaranteesthatanagentoftypeprefershisassignedveriÞcationdecisionandspending)and),toadifferentallocationforsometypewhoisnotveriÞed(i.e.,with0).NotethatitissufÞcienttocon-siderdeviationstononveriÞedtypes:sinceadeviationinwhichanagentoftypemimicsaveriÞedtypewouldbedetectedbytheprincipal(asveriÞcationrevealsthetruetype)andtheprincipalcanarbitrarilypunishtheagent(throughthespendingallocation)whenshelearnsthathehasdeviated,wedonotneedtoconsidersuchadeviation.Wealsonotethattheformulationabovedoesnotruleoutmixedstrat-egiesbytheagent.IftheagentwerewillingtomixoververiÞcationandnoveriÞcationoroverdifferentspendinglevels,hewouldbeindifferentovertheseallocations,andthustheprincipalcanselectoneofthesethatmaximizesherexpectedwelfare.Infact,buildingonthisobservation,wecanshowthatourresultsarenotlimitedtothegameforminsectionII.Bbutcontinuetoholdwhenallowingforanyindirectmechanismspecify-ingamessagespacefortheagentandadeterministicallocationfunctiontowhichtheprincipalcommits.Suchamechanisminducesagameinwhichtheagentsendsamessage,iseitherveriÞedornotasafunctionofthemessage,andisassignedaspendinglevelasafunctionofthemes-sageandveriÞcationresult.WeshowinappendixB(availableonline)thataversionoftherevelationprincipleintermsofpayoffsholdsinourset-ting,implyingthattostudytheoptimalde

14 terministicmechanismfortheprincipal,itis
terministicmechanismfortheprincipal,itiswithoutlosstorestrictattentiontodeterministicdirectTheprincipalcanpunishadeviationofatypeinwhichhemimicsatype1byassigningfollowingveriÞcationsomespendinglevelsuchthat.Itisclearthatsuchaspendinglevelexists;infact,settingwouldbeasufÞcientpunishment.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,withassociatedveriÞcationandspend-ingschedules,suchthatforallandstrictlyforsome.Althoughmultiplesolutionscaninprinciplecontinuetoexistunderthiscondition,thiscriterionturnsouttobesufÞcientforourcharacterization.III.NoVeriÞcationBenchmarkBeforeanalyzingtheoptimaldelegationrulewithveriÞcation,wereviewtheresultsoftheliteraturebyconsideringtheoptimalruleintheab-senceofveriÞcation.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-tion1issatisÞedun

15 dercommonlyuseddistributionfunctions,inc
dercommonlyuseddistributionfunctions,including4534journalofpoliticaleconomy exponential,lognormal,andanynondecreasingdensity.Givenassump-tion1,theresultsinAmadorandBagwell(2013)yieldthefollowing:1(OptimalruleundernopossibilityofveriÞcation).Takethedistributionoftruncatedfromaboveby.Iftheprin-cipalisconstrainedto0forall ,anoptimalruleisasuchthat UndernoveriÞcation,anoptimalruleisathresholdsuchthatallspendattheirßexiblelevelandalltypesarebunchedattheßexiblespendinglevelof.Theprincipalcanimplementthisrulebysettingaspendinglimitandallowingtheagenttochooseanyspendingleveluptothislimit.Figure1illustratesanoptimalruleundernoveriÞcationforthecaseofquadraticpreferences.Thelevelofspendingisontheverticalaxisandtheagentstypeonthehorizontalaxis.Inthissimpleexample,bothef-Þcientandßexiblespendingareincreasinglinearfunctionsofthestate,andßexiblespendingexceedsefÞcientspendingbyaconstantamount .1.OptimalruleundernopossibilityofveriÞcation.TheÞgureisdrawnforthequadraticpreferencescase(seesec.II.A),wherewelet 12,and)uniform.Wealsonotethatassumption1ontheoriginaldistributionimpliesthattheassump-tionissatisÞedforalltruncationsfromaboveiftheconditionsinproposition2ofAmadorandBagwell(2013)hold.commitmentversusexibilitywithcostlyverication4535 representingtheagentsbias.Therulecharacterizedinproposition1speciÞesaspendinglevelthatcoincideswiththeagentsßexiblelevelforandequalsAkeyinsightbehindtheresultinproposition1isthatholesaresubop-timal.Moreprecisely,theprincipalcanalwaysimproveuponaruleasthatdepictedinÞgure2,whichdoesnotallowtheagenttochoosespend-ingforsomeinteriorbutdoesallowtheagenttochoosespendingimmediatelybelowandimmediatelyabove.Thehole[]impliesthatanagentoftypeforwhomisnotallowedtospendathisßexiblelevel.Suchanagentspendsatthelowerlimitoftheholeifhistypeisrelativelylow,buthespendsattheupperlimitoftheholeifhistypeishigher.Theroleofassumption1istoguaranteethatiftheprincipalremovesthehole,thebeneÞtofreducingoverspendingforthetypesthatbun

16 chwouldoutweighanypotentialcostsofincrea
chwouldoutweighanypotentialcostsofincreasingspendingforthetypesthatbunchatIV.OptimalRuleWenowturntothestudyofoptimaldelegationwhencostlyveriÞcationispossible.Thefollowingclassofruleswillplayanimportantroleinour1.AruleisTECifitconsistsof,with suchthat .2.RulewithoutveriÞcationwithahole[].ParametersarethesameasinÞgure1.4536journalofpoliticaleconomy i.(threshold)if0andii.(escapeclause)if1andFigure3illustratesaTECruleusingthequadraticpreferencesexam-ple.UnderTEC,typesarenotveriÞedandspendattheirßexiblelevel,typesarenotveriÞedandarebunchedattheßexiblespendinglevelof,andtypesareveriÞedandareassignedtheirefÞcientspendinglevel.Theprincipalcanimplementthisrulebyallow-ingtheagenttoeitherchooseaspendingleveluptoalimitorrequestveriÞcationbytriggeringanescapeclause.WhentheagentisveriÞed,heisassignedhisefÞcientspendinglevelprovidedthatitisaboveaspeciÞedlevel(andisotherwisepunished).AnimportantfeatureofTECisthattheveriÞcationfunction)isweaklyincreasing;thatis,thereisnodecreasingveriÞcation:2.ArulefeaturesdecreasingveriÞcationatjumpsfrom1to0at;thatis,either(i)0andlimsup1or(ii)1andliminf0.Arulefeaturesweaklyincreas-ingveriÞcationatifneitherinoriiholds.Notethatwewillrefertodecreasing/increasingveriÞcationinthestrictsense,andwewillclarifywheneverweusedecreasing/increasingveriÞcationintheweaksense.Figure4depictsanexampleofarulewithdecreasingveriÞcation.ThisrulespeciÞesveriÞcationonlyfortypesbetweentwointeriorcutoffs,.Typesaboveandbelowthis .3.TECrule.ParametersarethesameasinÞgure1,with008and0.SolidlinedepictstheallocationofnonveriÞedtypes;dashedlinecorrespondstoveriÞedtypes.commitmentversusexibilitywithcostlyverication4537 regionarenotveriÞed,andhencetherulefeaturesdecreasingveriÞca-tionat.WewillreturntothisexampleinsectionIV.C.AnotherfeatureofTECisthatitspeciÞesveriÞcationforsomeagenttypesbutnotforall.WebeginbyshowinginsectionIV.AthatinducingnoveriÞcationforsometypesisinfactapropertyofanyoptimalrule.Furthermore,buildingonthisresult,weshow

17 thatTECisoptimalwhen-everoptimalveriÞcat
thatTECisoptimalwhen-everoptimalveriÞcationiseverywhereweaklyincreasing.Weconsiderasimpleextreme-biascaseinsectionIV.BandprovideananalysisforourgeneralsettinginsectionIV.C.A.PreliminariesThenextlemmashowsthatverifyingallagenttypesisneveroptimalfortheprincipal:1.Arulewith1forallisnotoptimal.Thelogicissimple.SupposethatarulethatveriÞesalltypesisoptimal.SucharulemusttriviallyassignefÞcientspendingtoalltypes.Nowcon-sideraperturbationinwhichtheprincipalallowstheagenttochoose withoutveriÞcation.Undertheperturbedrule,asetoftypes[ ]for willprefer overbeingveriÞedandassignedefÞcientMoreover,sincetheagentisbiasedtowardhigherspending .4.RulewithdecreasingveriÞcation.ParametersarethesameasinÞgure3.SolidlinedepictstheallocationofnonveriÞedtypes;dashedlinecorrespondstoveriÞedtypes.Notethatif ,yetourargumentapplies,givenouroptimalitycondition(5).Intheappendix,wealsoprovideanalternativeproofforthecaseof0thatdoesnotrelyoncondition(5).4538journalofpoliticaleconomy andpaysaveriÞcationcostnolargerthantheprincipals,itmustbethattheprincipalisstrictlybetteroffbynotverifyingthesetypes.Hence,weÞndthatincentivizinglowtypestonotoverspendischeaperthanverify-ingthem,andthusverifyingalltypescannotbeoptimal.Givenlemma1,weestablishthefollowing:2.IfanoptimalrulefeaturesveriÞcationthatisweaklyin-creasingeverywhere,thenTECisoptimal.Sinceverifyingallagenttypesissuboptimal,anoptimalrulewithveriÞ-cationthatisweaklyincreasingeverywheremustfeatureano-veriÞcationregionfollowedbyaveriÞcationregion;thatis,theremustbeatypesuchthat0forand1for.Considerarulethatoptimizesovereachoftheseregionsseparately.Conditionalonthestypebeingintheno-veriÞcationregion,anoptimalruleisathresh-(byproposition1).ConditionalontheagentstypebeingintheveriÞcationregion,anoptimalruleassignsefÞcientspendingtoalltypes.Toprovelemma2,weshowthattherulethatresultsfromoptimizingovereachregionseparatelyisincentivecompatibleandthereforeopti-maloverthewholesetoftypes.SpeciÞcally,weestablishthatnotype

18 whoisprescribedveriÞ-cationunderthepropo
whoisprescribedveriÞ-cationundertheproposedrulewouldhaveanincentivetodeviatetotheno-veriÞcationregion.Notethatanoptimalrulefortheno-veriÞcationregionsetsamaximumallowablespendinglevel.More-over,byoptimalityof,theprincipalpreferstopaythecostofverifyingtoassignhim)ratherthanbunchhimat.SincetheagentisbiasedtowardhigherspendingandpaysaveriÞcationcostnolargerthantheprincipals,itfollowsthattypesalsoprefertobeveriÞedratherthandeviateto.Thisprovesthatthepro-posedruleisincentivecompatible,whichimpliesthatitisalsooptimal,andbyconstructionthisruleisTEC.B.ExtremeBiasBeforeturningtoourmainresults,weconsiderasettinginwhichthesbiasisextreme.Suppose0forall ,sothattheswelfareissimply.Theagentinthiscasealwayspre-fershigherlevelsofspending:hisßexiblespendinglevelisAsmentionedintheintroduction,suchanextremebiascor-respondstowhatisassumedinothermodelsofcostlyveriÞcation,in-cludingtheseminalworkofTownsend(1979),thedelegationmodelByproposition1,notypewhoisprescribednoveriÞcationwouldhaveanin-centivetodeviateeither.Asassumedinsec.II.A,weareprimarilyinterestedinthecaseinwhich)isinte-riorratherthanacorner;however,weÞnditisinstructivetostudythiscornercaseÞrst.commitmentversusexibilitywithcostlyverication4539 ofHarrisandRaviv(1996,1998),andmorerecentcontributions,suchasBen-Porath,Dekel,andLipman(2014).AnextremebiasimpliesthatiftheagentisnotveriÞed,hewillchoosethehighestallowablelevelofspending,regardlessofhistype.Moreover,theagentwillseekveriÞcationonlyifthatallowshimtospendmorethanundernoveriÞcation.TheanalysisthereforeissigniÞcantlysimpliÞed.Theonlyincentive-compatibleruleforanagentwithanextremebiasin-volvesbunchingallnonveriÞedtypesatonespendinglevel;thatis,ßex-ibilityhasnovalueinthissetting.Furthermore,anytypethatisveriÞedmustbeassignedahigherspendinglevelthanthatatwhichnonveriÞedtypesarebunched.Asaresult,wehavethefollowing:2(Optimalruleunderextremebias).Suppose0forall .ThenifveriÞcationisoptimal,TECisoptimal.Whentheagentsbiasisextremeandverif

19 yingsometypesisoptimal,anoptimalruleisTE
yingsometypesisoptimal,anoptimalruleisTEC,withnonveriÞedtypesbunchedandawardednoßexibilityandveriÞedtypesspendingattheirefÞ-cientlevel.TheoptimalityofTECfollowsfromtheoptimalityofweaklyincreasingveriÞcation.SupposebycontradictionthatanoptimalrulefeatureddecreasingveriÞcation.TaketobeamarginalnonveriÞedtypesplittingaveriÞcationregionandahigherno-veriÞcationregion,0and1forsome0arbitrarilysmall.LetbethelevelofspendingatwhichnonveriÞedtypesarebunched.Theoptimalityofverifying,(6)where,asnoted,incentivecompatibilityrequires,and0,(6)yields.TheoptimalityofnotverifyingthenimpliesHowever,(6)and(7)togetherwithviolatethesingle-crossingcondition(1),yieldingacontradiction.Intuitively,theprincipalcanimproveuponarulewithdecreasingveriÞcationbyverifyingahigheragenttypeinsteadofalowertype,asthemarginalbeneÞtoflettingthehighertypespendmoreishigher.NotethatsuchaperturbationisalwaysincentivecompatiblefortheagentbecauseallnonveriÞedtypesarebunchedatthesamespendinglevel,which(byincentivecompatibility)islowerthanthespendinglevelassignedtoanyveriÞedtype.Thisfeatureisofcourseduetotheagentsbiasbeingextreme.Proposition2,aswellaspropositions3and5,describesanoptimalrulewhenveriÞ-cationisoptimal.Clearly,veriÞcationisoptimalifandonlyiftheveriÞcationcostisnottoohigh.4540journalofpoliticaleconomy C.OptimalRulewithVeriÞcationWenextstudyoptimaldelegationwithveriÞcationinourgeneralsettinginwhichtheagentsbiasisnotextreme.Tothisend,itisusefultocon-siderarelaxedversionoftheproblemin(3)and(4),inwhichweas-sumethattheagentpaysnoveriÞcationcost( Sincetheoriginalincentivecompatibilityconstraint(4)istighterthantherelaxedconstraint(9),ifasolutionto(8)and(9)satisÞes(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)satisÞestheoriginalconstraint(4),weestablishthatanyagen

20 toftypeprefertopaytheveriÞcationcostands
toftypeprefertopaytheveriÞcationcostandspendathisefÞcientlevel)ratherthanpaynoveriÞcationcostandchoosethethresholdßexi-blespendinglevel.Thelogicissimilartothatbehindlemma2,whereweshowthattheoptimalityofverifyingtypefortheprin-cipalimpliesincentivecompatibilityofthisveriÞcationfortheagent.Hence,weobtainthattostudywhetherTECisoptimal,itiswithoutlosstofocusontherelaxedproblemin(8)and(9).Weanalyzethisprob-lemfortheremainderofthissection.Thefollowingtwolemmasestablishusefulpropertiesofanysolution:4.Ifasolutionto(8)and(9)prescribesveriÞcationfortypeithas.If(9)doesnotbindfor,then5.Inanysolutionto(8)and(9),)isweaklyincreasing.Lemma4statesthatifatypeisveriÞed,hisassignedspendinglevelis(weakly)betweenhisefÞcientlevelandhisßexiblelevel.Theargumentisstraightforward.IfassignedspendingfortypeiseitherbelowefÞcientoraboveßexible,theneitherincreasingordecreasingthisspending,re-spectively,makestheprincipalbetteroffandisincentivecompatiblefortheagent.Sincetheprincipalmaximizesherexpectedwelfaresubjecttoincentivecompatibility,ifaveriÞedtypesincentivecompatibilitycon-straintisslack,theprincipalassignsthistypeefÞcientspending.Wemaintainouroptimalityconditionin(5)toselectasolution.commitmentversusexibilitywithcostlyverication4541 Lemma5showsthattheprincipalassignsaspendinglevelthatisweaklyincreasingintheagentstype.WhencomparingtwoagenttypesthatarenotveriÞed,theresultnaturallyfollowsfromincentivecompatibility:atypecannotbeassignedhigherspendingthanahighertype,asatleastoneofthemwouldhaveanincentivetodeviate,giventhatprefer-encessatisfysinglecrossing.Whencomparingtwoagenttypessuchthat(atleast)oneofthemisveriÞed,theresultfollowsfromoptimality:ifatypeisassignedhigherspendingthanahighertype,theprincipalcanimprovewelfarebyswappingthesetypesspendinglevelsandveriÞca-tionassignments,andifincentivecompatibilitywasinitiallysatisÞed,itwillcontinuetobesatisÞedaftertheswap,givensinglecrossing.BydeÞnitionofTECandlemma2,whetheraTECruleisoptimalde-pendsonwheth

21 ertheprincipalcaninsteadbeneÞtfrominduci
ertheprincipalcaninsteadbeneÞtfrominducingde-creasingveriÞcation,namely,asituationinwhichasetoftypesisveriÞedandasetofhighertypesisnotveriÞed.Usinglemmas4and5,wenextshowthatanyrulefeaturingdecreasingveriÞcationmustinducesignif-icantoverspending,limitingthewelfarethatsucharulecanprovidetotheprincipal:6.Supposeasolutionto(8)and(9)featuresdecreasingver-iÞcationat.ThenthesolutionsatisÞes ðÞÞ ðÞÞIfanoptimalrulefeaturesdecreasingveriÞcationataninteriorpoint,thentheprincipalsexpectedwelfarefromtypesabovethispointisstrictlyboundedawayfromthatunderefÞcientspending.Forintuition,considerÞrsttheexampleinÞgure4,wheretheprincipalinducesveriÞcationonlyforaninteriorsetoftypes[].TheprincipalmustincentivizethesetypestoseekveriÞcationratherthandeviateandmimicatypeintheno-veriÞcationregionabove.Intheexample,theprincipalachievesthisbyassigningtypesimmediatelyabovetheirßexiblespendinglevelswhileassigningveriÞedtypesimmediatelybelowthespendinglevelsthatmakethemindifferentoverdeviatingto)undernoveriÞcation.Asaconsequence,however,theprincipalinducesoverspendingbyapos-itivemassoftypesabove.Infact,alltypesspendabovetheirefÞcientlevelintheexampleofÞgure4.Moregenerally,foranyoptimalrulewithdecreasingveriÞcationata,lemma6showsthattheprincipalsexpectedwelfareabove4542journalofpoliticaleconomy islowerthanefÞcientwelfare,withthedifferencebeingnosmaller)in(11).Thebound)capturestheminimumoverspend-ingabovethatisneededtodeterdeviationsbyveriÞedtypesbelowSpeciÞcally,let1andthus0for0arbitrarilysmall.Bylemma5,weknowthatalltypesabovespendmorethanthosebelow,andbylemma4,weknowthatveriÞedtypesspendnomorethantheirßexibleamount).Thus,fortypesintheveriÞcationre-gionbelownottodeviatetotheno-veriÞcationregionabove,itmustbethat;infact,byoptimality,thisinequalitymustbeGiventhatbylemma5alltypesspendweaklyabove,itfollowsthatalltypesspendstrictlyabove),whichexceedsefÞcientspending)forall,minThisyieldstheboundin(11).Importantly,theboundidentiÞedin(11)isindependentoftheveri

22 -Þcationcost.ThisallowsustoestablishourÞ
-Þcationcost.ThisallowsustoestablishourÞrstmainresult.Inwhatfollows,let3(OptimalruleundersmallveriÞcationcost).Let0.IfandveriÞcationisoptimal,TECisoptimal.Theideaisasfollows.Bylemma6,anyoptimalrulewithdecreasingveriÞcationimpliesawelfarelossduetooverspendinginthedecreasingveriÞcationregion.WeshowthatiftheprincipalsveriÞcationcostissmallrelativetotheminimumsuchloss,thenshecanraiseherwelfarebyverifyingalltypesinthedecreasingveriÞcationregionandreducingtheirspendingtotheefÞcientlevel.ItfollowsthatanoptimalrulemustinduceweaklyincreasingveriÞcationeverywhere,andthereforeTECisoptimalbylemma2.Formally,supposebycontradictionthatanoptimalruleinducesde-creasingveriÞcationatsomepoint,andletbethelowestveriÞedtypeunderthisrule.Weconsideraglobalperturbation:theprincipalveriÞesalltypesandassignsthemefÞcientspending)whilesolvingforanoptimalrulewithoutveriÞcationfortypes.Byproposition1,anoptimalrulefortheno-veriÞcationregionisathreshold,and(byoptimalityof)and0,itiseasytoverifythattheperturbedruleisincentivecompatible.Toshowthattheperturbationstrictlyraisestheprincipalswelfare,noteÞrstthatexpectedwelfareconditionalonweaklyincreasesbecauseitisnowmaximizedsubjecttofewerconstraints:undertheper-turbedrule,typescannotmimicatype.Thus,allweneedtoshowisthatexpectedwelfareconditionalonincreasesstrictly,namely,thatthe(allocative)beneÞtofverifyingthesetypesisstrictly,incentivecompatibilityrequires,butthentheprincipalcanimproveupontherulebysetting0whilekeepingeverythingelsecommitmentversusexibilitywithcostlyverication4543 greaterthantheadditionalveriÞcationcosttheprincipalincurs.BecauseveriÞedtypesareassignedefÞcientspending,thebeneÞtofver-ifyingthemisweaklypositive.Moreover,notethatbythecontradictionassumption,thereexistsatypeaboveatwhichtheoriginalrulefea-turesdecreasingveriÞcation.Thus,ifisthelowestsuchtype,lemma6impliesthatthebeneÞtofverifyingtypesisboundedfrombelowby,where)isdeÞnedin(11).Theclaimthenfollowsinthiscasefromthefactthat,given,theadd

23 itionalcostofverifyingtypesisstrictlysma
itionalcostofverifyingtypesisstrictlysmallerthanandhencestrictlysmallerthanthebeneÞtofver-ifyingthesetypes.IfthelowesttypeaboveatwhichtheoriginalrulefeaturesdecreasingveriÞcationis,ananalogousargumentap-plies,sinceinthiscasetheoriginalruleinducesstrictoverspendingbyandthebeneÞtofverifyingthistypeisnosmallerthanFigure5illustratestheresultinproposition3inasettingwithqua-draticpreferences(seesec.II.A)andauniformdistributionoftypes.Inthissetting,weobtainaclosed-formexpressionforthecutoffthusfortherangeofveriÞcationcosts,,underwhichTECisshowntobeoptimalunderveriÞcation.WeÞndthatisincreasingintheagentanddecreasingintherangeoftypes .Intuitively,iftheagentbiastowardhigherspendingislarge,theninducingdecreasingveriÞca-tionisveryexpensivefortheprincipal,asshemustallowhighoverspend-ingaboveanyinteriorveriÞcationregiontodeterdeviationsfromveriÞedtypes.Inthiscase,thebeneÞtofverifyingalltypesabovetheveriÞcationregionislarge,andthusTECispreferredeveniftheprincipalsveriÞca-tioncostisrelativelyhigh.Similarly,iftherangeoftypes issmall,thenthemassoftypesaboveanyinteriorveriÞcationregionisalsosmall,andthereforethecostofverifyingallsuchtypesinaTECruleislowevenisrelativelyhigh.Figure5providesanillustrationusingtheexampleofÞgure3.TheÞguredepictsvaluesof underwhichtheissatisÞed(shadedareas)aswellasthesubsetofthosevaluesunderwhichveriÞcationandthusTECisoptimal(darkgrayshadedarea).Wederivethiscutoffinapp.B.Weobtain b36 1g2 gifbg2 g, 16g2 g23bg2 g3b2b�g2 Inthisexample,theoptimalTECrulebunchesallnonveriÞedtypesatonespendinglevel,andasaresulttheoptimalityofveriÞcationgivenisindependentof.This,however,isnotageneralfeature.4544journalofpoliticaleconomy Theresultinproposition3providesajustiÞcationforthebroaduseofTECrulesinapplications.Asdescribedintheintroduction,capitalbud-getingstudies(e.g.,Ross1986;Taggart1987)reportthatTECiscommoninorganizations.Divisionmanagersarerequiredtoeitherabideabud-getarylimitorprovideprojectdocumentationtore

24 questarevisionoftheirbudgets.Schaechtere
questarevisionoftheirbudgets.Schaechteretal.(2012)andLledetal.(2017)ÞndthatÞscalrulesinmanycountriesalsotaketheformofTEC,namely,aspend-ingordeÞcitlimitwithescapeclauseprovisionsthatallowthegovernmenttobreakthelimitundercertaincircumstances.Additionally,TECrulesareusedininternationaltradeagreements,intheformofatariffcapwithanescapeclause(BeshkarandBond2017),andinpricedelegationin .5.OptimalityofTEC.ParametersarethesameasinÞgure3,withShadedareasindicatevaluesof underwhich.Darkgrayshadedareacor-respondstovaluesforwhich,inaddition,veriÞcationisoptimal.Comparativestaticsareasonewouldexpect.Inparticular,theloweris,otherthingsequal,andthelargeristheveriÞcationregion(i.e.,thesmalleris)intheoptimalTECrule.commitmentversusexibilitywithcostlyverication4545 Þrms,wheresalespeoplecanunilaterallyoffertheircustomersdiscountsuptoacertainpercentageoffthelistpricebutmustrequestasupervisorapprovalforlargerdiscounts(Loetal.2016).Proposition3provestheoptimalityofTECwhentheprincipalscostofveriÞcationissmallenough.WhathappensifthecostofveriÞcationislarger?Ournextresultshowsthatthereexistenvironmentsandveri-ÞcationcostsforwhichtheprincipalinducesveriÞcationbutnotintheformofTEC:4(OptimalruleunderintermediateveriÞcationcost).Thereexist{}forwhichanyoptimalrulefeaturesdecreasingToprovethisresult,weidentifyconditionsonparametersunderwhichverifyingonlyanintermediaterangeoftypes[]dominatesbothnotverifyinganytypeaswellasusingTEC.Themainreasonwhyverifyingonlyintermediatetypescandominatenotverifyinganytypeisthatanin-termediateveriÞcationregionimposesdisciplineontheno-veriÞcationregionbelow.Thatis,evenwhentheveriÞcationcostislargeenoughthattheprincipalwouldnotbeneÞtfromverifyingtypesin[]onlytoim-provetheirallocationrelativetoßexiblespending,shemaybeneÞtfromverifyingthesetypestodisciplinelowertypes:withtheintermediatever-iÞcationregion,typescannolongermimictypesin[].ThemainreasonwhyverifyingonlyintermediatetypescandominateusingaTECruleisthatitallowstheprin

25 cipaltosaveonveriÞcationcosts.Specif-ica
cipaltosaveonveriÞcationcosts.Specif-ically,withintermediateveriÞcation,theprincipalmaybeabletoimposedisciplineontypeswithoutprescribingveriÞcationfortypesasshewouldunderaTECrule;thiswillbethecaseifhasnoincentivetodeviatetomimicatypeashighas.Insuchasituation,intermediateveriÞcationallowstheprincipaltosaveonthecostofveri-fyingtypesaboveTheseargumentsyieldthatarulewithdecreasingveriÞcationasthatdepictedinÞgure4candominateanyno-veriÞcationrule(asthatinÞg.1)andanyTECrule(asthatinÞg.3),providedthatthecostofveriÞ-cationisnotsmall(orlarge)enough.Weemphasizethatproposition4doesnotrelyonnonuniformityoftheprincipalsobjectiveacrosstypesoranyothersortofasymmetry;weprovetheresultbytakingthecaseofqua-draticpreferencesandauniformdistributionoftypes,asdepictedinourÞgures.WealsonotethatwhileourconstructionimpliestheoptimalityofdecreasingveriÞcationundersomeparameterswith,theoptimalruleinthiscasemaynottakethesimpleintermediate-veriÞcationstructurethatweconsidertoprovetheresult.Infact,wecanshowthatevenwhenrestrictingattentiontoquadraticpreferencesandauniformdistribution,Bylemma2,anyotherrulewithveriÞcationthatisweaklyincreasingeverywhereisthusalsodominated.Hence,theclaiminproposition4follows.4546journalofpoliticaleconomy thereexistparametersforwhichTEC,noveriÞcation,andintermediateveriÞcationarealldominatedbyarulefeaturingmultipleinteriorveriÞca-tionregions.Intuitively,intercalatingveriÞcationregionstofurtherdi-videthedelegationsetcanallowtheprincipaltoimprovedisciplinewhilekeepingveriÞcationcostsataminimum.Theimplicationsofproposition4forapplicationsareimmediate.Forexample,fororganizations,thisresulttellsusthatitcanbebeneÞcialtodeÞnedifferentcategoriesofinvestment.Seniormanagementcouldre-quiredivisionheadstoeithercomplywithalowbudgetarylimitmeantforrelativelysmallprojectsorchoosefromahigherrangeofinvestmentlev-elsmeantforlargeprojects;otherwise,documentationwouldbeneededtohaveintermediatelevelsofinvestmentapproved.SuchaveriÞcationrequire

26 mentmaysufÞcetodiscourageoverinvestmentb
mentmaysufÞcetodiscourageoverinvestmentbydivisionman-agerswithsmallprojects:thesemanagerslackprooftojustifyasmallin-creaseintheirbudgetandwouldnotwanttoincreasetheirinvestmentasmuchasforalargeproject.Nevertheless,whereasdelegationruleswithdecreasingveriÞcationcanbeoptimal,theydonotappeartobecommoninpractice,andouranalysismayhelpexplainwhy.OurconstructionshowsthatimplementingarulewithdecreasingveriÞcationdemandsstrongcommitmentpowerfromtheprincipal.Take,forexample,theruledepictedinÞgure4.Theprin-cipalassignsspendingstrictlyabovetheefÞcientleveltosomeagenttypeswhoareveriÞed.Bydoingthis,theprincipalincentivizesthosetypestobeveriÞed:iftheywereinsteadassignedefÞcientspendingfollow-ingveriÞcation,theywouldnotseekveriÞcationintheÞrstplace.TheprincipalmustbecommittedtoallowingthisinefÞcientspendingdespiteherlearningthetruetypeoftheagent.StrongcommitmentpowerisalsorequiredtoincentivizetypessufÞcientlyclosetotonotseekveriÞcation.IntheruleofÞgure4,thesetypesarepunishediftheyseekveriÞcation,eventhoughexpost,onceveriÞcationtookplace,boththeprincipalandtheagentwouldstrictlypreferefÞcientspendingtopunish-ment.Withoutthethreatofpunishment,theprincipalmaynotbeabletopreventanagentoftypesufÞcientlyclosetofromseekingveri-Þcation,asanefÞcientallocationfollowingveriÞcationwouldallowthisagenttoincreasehisspendingtowardhisßexiblelevel.Inpractice,principalsmaynothavesufÞcientcommitmentpowertoimplementallocationsthatareinefÞcientexpost.Weexploretheimpli-cationsoflimitedcommitmentpowerinsectionV.Inparticular,theruleconstructedinlemma9intheproofofproposition4inapp.Aisnotoptimalforsomeparametervaluessatisfyingtheassumptionsofthelemma.Forin-stance,takingtheexampleofÞg.3,andconsistentwithourintuitionbehindÞg.5,weÞndthatarulewithmultipleinteriorveriÞcationregionsbecomesoptimaliftherangeoftypes becomeslargeenough.Forthisreason,whendecreasingveriÞcationisoptimal,theoptimalruleisverysen-sitivetoparameters,suchasthevalueof commitmentversusexibilitywithcos

27 tlyverication4547 V.LimitedCommitmentWes
tlyverication4547 V.LimitedCommitmentWestudyasettinginwhichtheprincipalhaslimitedcommitmentpower.WemodifytheorderofeventsinsectionII.Basfollows:1.Theprincipalsetsarule,whichmapsaveriÞcationdecisionandresultintoanallowablespendingset2.TheagentchooseswhethertoseekveriÞcation,0,1,andtheprincipalperfectlyveriÞeshistype3.Theprincipalrevisestheallowablespendingset4.TheagentchoosesaspendinglevelfromtheallowablesetTheÞrsttwostepsarethesameasinourenvironmentwithfullcom-mitmentpower.Whatisnewisstep3:afterobservingtheagentsveriÞ-cationdecisionandtheresultifveriÞcationischosen,theprincipalnowrevisestheallowablespendingsetfortheagent.Thisisamildformoflimitedcommitment.Inparticular,instep2wemaintaintheassumptionthattheprincipalisabletocommittoaveriÞcationplan,sotheagenttypeisveriÞedifandonlyiftheagentrequestsveriÞcation.Moreover,instep4wemaintaintheassumptionthattheprincipalisabletocommittoallowingtheagenttochooseanyspendinglevelfromtheallowablespendingset,soourproblemisstilloneofdelegationratherthancheaptalk.Theonlyassumptionthatwerelaxisabouttheprincipalscommit-menttonotchangingtheallowablespendingsetfollowingtheveriÞca-tiondecisionandresult.Thisformoflimitedcommitmentisrelevanttoapplicationsofourmodel.Forexample,divisionmanagersinorganizationsmayrequestarevisionoftheirbudgetsforthenextperiod.Canseniormanagementcommittonotchangingtheirallocationexpostwhennorequestissub-mitted?Andinthecaseofarequest,canseniormanagementcommittoaninefÞcientbudgetafterverifyingthebeneÞtsofthedivisionsprojects?Asdiscussedintheintroduction,theanswerisoftenno.Seniormanage-mentmakesdecisionsonbudgetcapsandthescopeofprojectsbroughtupforreviewexpost,andthesedecisionsdonotalwayscoincidewithpre-announcedcriteria(seeBowerandLesard1973;Ross1986;TaggartWenotethatourresultsinthissectionarenotlimitedtotheexactgamedescribedbelow;analogoustoourclaimsinsec.II.C,ourÞndingscanbeextendedtovariationsofthisgamethatallowmessagesbetweentheprincipalandtheagent(whilekeepingour

28 assumptionsontheprincipalslimitedcommitm
assumptionsontheprincipalslimitedcommitment).Throughoutthissection,wemaintainouroptimalityconditionin(5).Asnotedinn.10,thereisaliteraturethatstudiesauditingwhentheprincipalcannotcommittoanauditstrategy.Inmanyoftheapplicationsofourproblem,however,weÞndthatthereareofteninstitutionsensuringthatprincipalscannotdenyveriÞcationonceithasbeenrequested.Inthissense,theagentcanalwayschoosetotriggerveriÞcation.Lackofcommitmentbytheprincipalinthisrespectwouldchangethenatureofourproblem;weleaveitsanalysisforfuturework.4548journalofpoliticaleconomy 1987).Infact,thesecriteriaaresometimesleftambiguous,astheyde-pendontheclassofproject,whichmaynotbewellspeciÞed(MukherjeeandHenderson1987).Thisgivesseniormanagementmorediscretiontomakebudgetarydecisions.Interestingly,intheirstudyofÞscalrulesacrosscountries,Schaechteretal.(2012)alsoobservethatescapeclausesaresometimesnotwellspeciÞed:inthepastescapeclauseprovisionshaveinseveralcaseslefttoolargearoomforinterpretation(Schaechteretal.2012,20).Inourmodel,limitedcommitmentonthesideoftheprincipalmattersfortworeasons.First,conditionalonnoveriÞcation,theprincipalmustchooseanallocationthatisoptimalforthenonveriÞedtypes.Thatis,theprincipalassignsspendinginthiscasetakingintoaccountthedistribu-tionofnonveriÞedtypesandignoringtheincentivesofveriÞedtypes.Second,conditionalonveriÞcation,theprincipallearnstheagentstrueandmustassigntheagenttheefÞcientspendinglevel).ThisistruebothwhentheagentsseekingveriÞcationisonpathaswellaswhenthisveriÞcationdecisionispartofadeviation.Hence,theagentcanalwayschoosetobeveriÞedtoguaranteehimselftheefÞcientlevelofspending.Asaresult,limitedcommitmentimpliescertainconditionsthatanyincentive-compatiblerulemustsatisfy.Inwhatfollows,werestrictattentiontostrategiesthatspecifypiecewisecontinuousmappings{)}.7.Underlimitedcommitment,anyincentive-compatiblerulesatisÞesthefollowing:i.IfthereisdecreasingveriÞcationat,then,(12)1.Moreover,ii.IfthereisincreasingveriÞcationat,then,(14)Partishow

29 sthatifsplitsaveriÞcationregionfromahigh
sthatifsplitsaveriÞcationregionfromahigherno-veriÞcationregion,thenmustbeindifferentbetweenbeingveriÞedandspendingattheefÞcientlevelversusnotbeingveriÞedandspend-ingat),asallowedintheno-veriÞcationregionabovethistype.Likewise,partiishowsthatifsplitsano-veriÞcationregionfromahigherveriÞcationregion,thenmustbeindifferentbetweenbeingver-iÞedandspendingattheefÞcientlevelversusnotbeingveriÞedandspendingat),asallowedintheno-veriÞcationregionbelowthistype.commitmentversusexibilitywithcostlyverication4549 ThisresultfollowsfromthefactthataprincipalwithlimitedcommitmentpowerassignsefÞcientspendingwhenevertheagentseeksveriÞcation.Therefore,ifthereisapointatwhichaveriÞcationregioneitherendsorstarts,themarginalveriÞedtypeatsuchpointmustweaklypreferver-iÞcationwithefÞcientspendingtonoveriÞcation,andthemarginalnonveriÞedtypemustweaklyprefernoveriÞcationtoveriÞcationwithef-Þcientspending.Themarginaltypemustthusbeindifferent.Lemma7alsoshowsthatfortypeasdeÞnedinthelemma,anincentive-compatiblerulemustset.ThisisrequiredtoindifferentbetweenveriÞcationandnoveriÞcation:ifthisin-equalityisnotsatisÞed,themarginalveriÞedtypewouldinsteadprefertodeviateandnotseekveriÞcation.Fortheremainderofouranalysis,werequirethefollowing:2.If,thenforallThisisasingle-crossingproperty:weassumethatifatypeweaklypre-fersveriÞcationwithefÞcientspending)tonoveriÞcationwithahigherspendinglevel,thenanylowertypestrictlypre-fersveriÞcationwithefÞcientspending)tonoveriÞcationwiththehigherspendinglevelThispropertyholdsinthecasescommonlystudiedintheliterature,suchasthosewithquadraticpreferencesorwithhyperbolicpreferencesundercommonparameterizations.Givenassumption2,weobtainthefollowing:5(Optimalruleunderlimitedcommitment).Underlim-itedcommitment,anyincentive-compatiblerulefeaturesweaklyincreas-ingveriÞcationeverywhere.Moreover,ifveriÞcationisoptimal,TECisUnderlimitedcommitment,decreasingveriÞcationisnotincentivecompatiblefortheprincipal.AswediscussedinsectionIV.C,decreasingv

30 eriÞcationrequiresthattheprincipalcommit
eriÞcationrequiresthattheprincipalcommittoallowingtheagenttospendatalevelthatisinefÞcientexpost,followingtheagentsveriÞca-tiondecisionandresult.Weprovethatwithoutthiscommitment,theOursingle-crossingconditionsonpreferencesimplythatifatypeweaklyprefersver-iÞcationwithefÞcientspending)tonoveriÞcationwithalowerspendinglevel,thenanyhighertypestrictlyprefersveriÞcationwithefÞcientspending)tonoveriÞcationwiththelowerspendinglevel.Assumption2requiresthatthispropertybemaintainedintheoppositedirectionaswell.Forexample,inthehyperbolicpreferencescase(seesec.II.A),assumption2holdsiftheutilityfunctionsforpresentandfutureconsumptionarethesameandeitherexponen-tialorconstantrelativeriskaversionwithacoefÞcientweaklygreaterthan1.4550journalofpoliticaleconomy principalcannotinducedecreasingveriÞcation,andhenceanyincentive-compatiblerulemustfeatureweaklyincreasingveriÞcationatalltypes.Analogousargumentstothosebehindlemmas1and2inourfull-commitmentenvironmentthenimplythatifverifyingsomeagenttypesisoptimal,aTECruleisoptimal.Asketchoftheproofofproposition5isasfollows.Supposebycontra-dictionthatthereisanincentive-compatiblerulethatinducesdecreas-ingveriÞcation,withbeingatypesplittingaveriÞcationregionfromahigherno-veriÞcationregion.Givenlimitedcommitment,veriÞedtypesimmediatelybelowareassignedefÞcientspending,andtypesdiatelyabovespendatalevelthatmakesindifferentbe-tweenveriÞcationandnoveriÞcation(cf.lemma7).Thismeansthattypesimmediatelyabovemustbestrictlyoverspending,infactspend-ingabovetheirßexiblelevel.Theheartoftheproofisshowingthattheprincipalcannotcommittoallowingsuchoverspending.ItisclearthatconditionalontheagentnotseekingveriÞcation,theprincipalwouldliketoreducetheoverspendingbytypesimmediately.Reducingthisoverspendingisexpostincentivecompatibleforthesetypes:havingchosennoveriÞcation,typeswouldprefer)to.Hence,theonlyreasontheprincipalwouldnotre-ducetheoverspendingimmediatelyabovefollowingnoveriÞcationisifdoingsowouldviolateincentivecompat

31 ibilityforsomeothernon-veriÞedtype.Sucha
ibilityforsomeothernon-veriÞedtype.SuchanonveriÞedtypemustbebelow;speciÞcally,theremustexistatypewhoisnotveriÞedandisexactlyindifferentbetweenhisassignedspendinglevel,callit,andthespendinglevel.Infact,becauseofsinglecrossing,thistypemustbethemarginaltyperightbelowtheveriÞcationregionthatendsat;thatis,therulemustinduceveriÞcationfortypesandnoveriÞcationfortypesimmediatelybelowandabovethisset.Anexampleistherulede-pictedinÞgure4.NowiftheprincipalinducessuchaninteriorveriÞcationregion[],thenbylemma7,typemustbeindifferentbetweennoveriÞca-tionwithspendingandveriÞcationwithefÞcientspending.SincewehavedeÞnedasbeingindifferentbetweenspendingatandspend-ingatundernoveriÞcation,bytransitivity,weobtainthatmustbeindifferentbetweennoveriÞcationwithspendingandveriÞcationwithefÞcientspending.However,recallthattypeisalsoindifferentbetweennoveriÞcationwithspendingandveriÞcationwithefÞcientspending.Hence,byassumption2,cannothold,andwemustThismeansthattheprincipalveriÞesasingletypeatthis,theindifferenceoftypebetweenveriÞcationwithefÞcientspendingandnoveriÞcationwithspendingwouldimplythatstrictlyprefersveriÞcationwithefÞ-cientspendingtonoveriÞcationwithspending,acontradiction.commitmentversusexibilitywithcostlyverication4551 pointwhoisindifferentbetweenveriÞcationwithefÞcientspending,noveriÞcationwithhigherspendingat,andnoveriÞcationwithlowerspendingat.ConditionalonnoveriÞcation,thisisthusanallocationinwhichtheagentfacesahole[];namely,heisnotallowedtochoosespendinginthissetbutcanchoosespendingimmediatelybelowandaboveit.ButouranalysisinsectionIIIshowsthatsuchaholeissub-optimalconditionalonnoveriÞcation;hence,followingnoveriÞcation,theprincipalwouldhaveastrictincentivetoclosethehole.ThisshowsthatarulewithdecreasingveriÞcationcannotbeincentivecompatiblewhentheprincipalhaslimitedcommitmentpower,allowingustoestab-lishthatTECisoptimalinthiscase.Recallthatinthefull-commitmentenvironment,TECisoptimaliftheprincipalscostofveriÞcationissmallenough(assh

32 owninproposi-tion3),butmorecomplexrulesm
owninproposi-tion3),butmorecomplexrulesmaybeoptimalotherwise(asshowninproposition4).Incontrast,proposition5tellsusthatTECisoptimalun-derlimitedcommitmentforanyveriÞcationcostforwhichveriÞcationisoptimal.GiventheprevalenceofTECrulesintherealworld,thesere-sultssuggestthatlimitationstocommitmentpowerarealsoprevalent.Moreover,theselimitationsmaybeanimportantreasonbehindthebroaduseofTECinapplications.AsaÞnalremark,itisworthnotingthatwhileTECisoptimalbothwhentheprincipalhasfullcommitmentpowerandasmallveriÞcationcostaswellaswhenshehaslimitedcommitmentpower,thespeciÞcde-tailsofanoptimalTECrulevarywitheachcase.Underfullcommitment,anoptimalTECruleissuchthattheprincipalpreferstoverifytoassignthemefÞcientspendingratherthanbunchthemwithoutveriÞcation,whereastheoppositeistruefortypes.Hence,theprincipalisindifferentbetweenverifyingandnotverifyingthethresholdtype;thatis,theincreaseinassignedspendingatexactlycompensatestheprincipalforthecostofveri-fyingthistype.Incontrast,underlimitedcommitment,itistheagentwhoisindifferentat:asimpliedbylemma7,typemustbeindifferentbetweenbeingveriÞedandassignedefÞcientspendingversusnotbeingveriÞedandassigned,andthusanyincreaseinassignedspendingmustexactlycompensatethistypeforhisveriÞcationcostVI.ConclusionThispaperhasstudiedthetrade-offbetweencommitmentandßexibilityinthepresenceofcostlystateveriÞcation.Wehaveexaminedageneraldelegationprobleminwhichaprincipaldelegatesdecision-makingtoAdditionally,asnotedinsec.II.A,thisresultappliesnotonlyto0,1butalsoto4552journalofpoliticaleconomy anagentwhohassuperiorinformationabouttheefÞcientactionbutisbiasedtowardhigheractions.Anovelelementofourframeworkisthattheprincipalcanverifytheagentsprivateinformation.BecauseveriÞca-tioniscostly,theprincipalwishestousethistechnologyselectivelyandinawaythatsupplementsdelegationandimproveshercommitmentversusßexibilitytrade-off.Ourresultsprovideinsightintohowtheprincipalachievesthisbyde-signinganoptimaldelegationrule.Wehaveshowntha

33 tunderfullcom-mitmentpowerandasmallenoug
tunderfullcom-mitmentpowerandasmallenoughveriÞcationcost,anoptimalruleisaTEC,allowingtheagenttofreelyselectanyactionuptoathresholdortorequestveriÞcationandtheefÞcientactionabovethethreshold.WhentheveriÞcationcostislarger,theprincipalmayinsteadprefertorequireveriÞcationonlyforintermediateactions,stillimposingsomedisciplineontheagentbutsavingonveriÞcationcosts.However,theoptimalityofTECisrestoredundermildlimitationstotheprincipalscommitmentpower.SpeciÞcally,iftheprincipalisunabletocommittonotchangingtheagentspermissibleactionsetfollowingtheveriÞcationdecisionandresult,TECisoptimalforanyveriÞcationcostforwhichveriÞcationisAswehavediscussed,thereareavarietyofapplicationswheredelega-tioniscentralandrulesmakeuseofveriÞcationbyspecifyingescapeclauses.OuranalysisshedslightontheoptimalstructureofescapeclausesandprovidesatheoreticalfoundationforthecommonuseofTECrules.Morebroadly,ourframeworkmayhelpinformtheempiricalanalysisofreal-worldrules.Dataondelegationpoliciesareincreasinglyavailableandofferanopportunitytoexplorethedesignoftheserulesinmoredetail.Forinstance,inthecontextofcapitalbudgeting,ithasbeenob-servedthattheextentofcapitalrationingandtheuseofveriÞcationvaryacrossÞrms(e.g.,Ross1986),andonecouldstudyhowthesedifferencesrelatetoÞrmsize,industry,andotherfactorsthatarelikelytoaffectse-niormanagementscostofverifyingthequalityofprojects.InthecontextofÞscalpolicy,countriesÞscalrulesvaryintheuseofescapeclausepro-visionsandtheirtriggerevents(Schaechteretal.2012),andthesemaycorrelatewithcountriesinstitutionalandmacroeconomicconditionsthataffectthecostofauditingagovernmentaswellastheneedforßex-ibilitytorespondtoshocks.Last,byuncoveringanewsetofissuesthatarisewhenveriÞcationisin-troducedtoasettinginwhichbothcommitmentandßexibilityarevalu-able,ourpaperopensthedoorforfurtherworkthatcanhelpunder-standtheoptimaljointdesignofdelegationandveriÞcation.Wehavefocusedonasimplemodelthatemphasizesthemainforcesatplaybutabstractsfromotherpotential

34 lyrelevantaspects,forinstance,associated
lyrelevantaspects,forinstance,associatedwithmorecomplexveriÞcationtechnologies.Weclosebydiscussingsomepossibleextensionsandvariationsofourwork.commitmentversusexibilitywithcostlyverication4553 RandomveriÞcationAsintheseminalworkofTownsend(1979),wehaveconsidereddeterministicveriÞcation;namely,weassumedthattheprincipalsruleassigns0,1toeachagenttype.Moregen-erally,onecouldallowformechanismsinwhichtheprincipalrandomizesovertheveriÞcationassignment,choosingaprobabilityofveriÞcationforeachtype.TheliteratureonÞnancialcontractingandtaxcollectionÞndsthatrandomveriÞcationcanyielddifferentresultscomparedwithdeter-ministicveriÞcation;seeBorderandSobel(1987)andMookherjeeandPng(1989).OurfocusondeterministicveriÞcationismotivatedbytheapplica-tionswestudy.Takecapitalbudgeting.Ascapturedbythegameformthatwehaveproposed,heretheagent(divisionhead)decideswhethertorequestveriÞcationtoobtainapprovaltochooseactionsthatarenotallowedbytheprincipal(seniormanagement)withoutveriÞcation.Theprincipalcommitstofollowingtheagentsrequest,andsoitistheagentchoicewhethertotriggertheveriÞcationprocess.Unlikeinotherappli-cationswhereveriÞcation/auditisusedtodetermineÞnesformisbehavior(e.g.,taxcollection),randomveriÞcationisnotnaturalinthesecontexts.UsingthetimingofsectionII.B,randomveriÞcationwouldmeanthattheagentchoosesinstep2notbetweenveriÞcationandno-veriÞcationbutratherbetweendifferentlotteriesoververiÞcation.Thisisrarelyobservedinpractice,possiblybecausecommittingtoanondegeneratelotterycanbedifÞcultforaprincipal.Thatsaid,inasettinginwhichtherearenolimitationstotheprinci-scommitmentpower,thestudyofrandomveriÞcationcouldbeaninterestingextensionofourwork.Asnotedintheaforementionedliter-ature,oneissueisthatanoptimalrandomizedmechanismwoulddependontheextenttowhichtheagentcanbepunishedfollowingveriÞcation,whichinturnwoulddependonpreferenceassumptionsinoursetting,giventhatpunishmentsareimposedthroughthespendingallocationonly.Importantly,thesepunishmentsmustbebo

35 unded;otherwise,theefÞcientallocationcan
unded;otherwise,theefÞcientallocationcanbeapproachedwitharulethatveriÞesallagenttypeswithverylowprobabilityandarbitrarilypunishestheagentwhenveriÞcationrevealsthathehasdeviated.Suchapossibilitynotonlyyieldsratherimplausiblepredictionsbutalsoimpliesthatanoptimalruleingeneralwillfailtoexistunlessaboundonpunishmentsisimposed.WhenthedecisionissimplyoververiÞcationornoveriÞcation,commitmenttotheveriÞcationpolicywouldinprinciplebefacilitatedbythefactthattheprincipalsexecu-tionoftheagentsrequestcanbeeasilymonitored.However,checkingthattheprincipalimplementsaspeciÞclotteryisharder,asitrequiresmonitoringoftherandomizationitselfratherthanitsoutcome.Inourgameform,arulethatapproachestheefÞcientallocationwouldbeimple-mentedbyinducingeachagenttypetochooseadifferentlotteryoververiÞcation.4554journalofpoliticaleconomy ImperfectveriÞcationAlsofollowingTownsend(1979),ouranalysisas-sumedthatveriÞcationrevealstheagentstypeperfectly.AnalternativewouldbetoconsiderimperfectveriÞcation,namely,veriÞcationthatprovidesonlyimperfectinformationabouttheagentstype.Forexample,inthecontextofcapitalbudgetinginorganizations,seniormanagementmayreviewinformationaboutthebeneÞtsofaprojectthatadivisionman-ageradvocates,buttheavailabledocumentationmaybeincompleteandfailtorevealthefullmeritsoftheproject.AsimplespeciÞcationthatmaybepossibletoaccommodatewithinourframeworkiswhenimperfectveriÞcationeitherrevealstheagentstypeperfectlyorprovidesnoinformation(i.e.,whentherearenofalsere-sults).Providedthattheprincipalcanseverelypunishtheagent(throughthespendingallocation),shewouldbeabletoprevent,atnocost,anydeviationinwhichanagenttypemimicsanothertypewhoisveriÞed,asistrueinourproblemwithperfectveriÞcation.Yetadifferenceintro-ducedbyimperfectveriÞcationisthattheprincipalmaynotobservethestypeandthusmaynotbeabletoassignatype-dependentspend-inglevelfollowingveriÞcation;theprincipalsrulemustspecifyaspend-ingallocationforthecaseofveriÞcationandnoinformation.Allowingforimperf

36 ectveriÞcationthatmayproducefalseresults
ectveriÞcationthatmayproducefalseresultswouldnaturallyintroducefurtherissues,asnowpunishinganagenttypeformimickinganothertypewhoisveriÞedwouldrequireimposingpunishmentsonHowimperfectisimperfectveriÞcation?Atoneextreme,ifveriÞcationissufÞcientlyaccurate,weconjecturethatourqualitativeresultswouldremainvalid.Attheotherextreme,ifveriÞcationissufÞcientlyinaccu-rate,itwouldbecomeequivalenttomoneyburning,andtheresultsoftheliteratureonwhenmoneyburningisusedinanoptimaldelegationrulewouldthenapply(seeAmador,Werning,andAngeletos2006;AmadorandBagwell2013;AmbrusandEgorov2017).Moregenerally,itwouldbeofinteresttoexploretheroleofveriÞcationindelegationawayfromthesetwoextremes.VeriÞcationcostsWehaveconsideredveriÞcationcoststhatarebothtypeindependentandexogenous.Anextensionofourproblemcouldexploretheeffectsoftype-dependentveriÞcationcosts:theprincipalcostofverifyingtheagentsprivateinformationmaybeincreasinginhistype,forexample,becausemoreevidenceisneededtoverifylargerprojectbeneÞts,oronemaytaketheviewthatveriÞcationcostsareactu-allylowerforextremetypes,asthesestatesaremorevisible.OnepossibledifÞcultyisthatmonotonicityofthespendingallocation(asshowninlemma5)mayfailtoholdifveriÞcationcostsincreaseveryrapidlywiththeagentstype.ButiftheveriÞcationcostfunctionissuchthattheprin-cipalwouldstillprefertoswaptheveriÞcationandspendingallocationsoftwotypeswhenevertypehashigherspendingthancommitmentversusexibilitywithcostlyverication4555 monotonicitywillbesatisÞedandouranalysiscouldbeextendedtoallowfortype-dependentveriÞcationcosts.Anothervariationwouldbetoendogenize,sothattheprincipalcanaffecttheagentscostofveriÞcation.Ourresultswouldcontinuetoholdunderthisextension.SpeciÞcally,whentheprincipalhasfullcommit-mentpower,wehavederivedconditionsfortheoptimalityofaTECrulethatareindependentofthevalueof0,1,andclearlytheprincipalwelfareunderthisruledoesnotvarywitheither.Thus,theprincipalinthiscasewouldbeindifferentoverany0,1,whereas0wouldbepreferredbytheagen

37 t.Moregenerally,underfullcommitmentitisa
t.Moregenerally,underfullcommitmentitisal-waysoptimalfortheprincipaltoset0,asazeroveriÞcationcostfortheagentmaximallyrelaxestheagentsincentivecompatibilitycon-straint(4).ThingsaremoreinterestinginthesettingofsectionV,wheretheprincipalhaslimitedcommitmentpower.Inthiscase,theprincipalmaywanttosetastrictlypositiveveriÞcationcostfortheagent,asthatlimitsthesetofagenttypesthatmaywanttodemandveriÞcationandefÞcientspending.Inanycase,foranygiven0thattheprincipalwouldset,ouranalysisandtheoptimalityofTECapplywithoutchange.TransfersOurfocushasbeenonacanonicaldelegationprobleminwhichtransfersbetweentheprincipalandtheagentarenotfeasible.Therearevariouswaysinwhichtransferscouldbeintroducedinourframeworkandusedtoalterthefeasibilityandcostofinducingdifferentallocations.TransferscouldbecontingentontheagentsveriÞcationde-cisionand/ortheveriÞcationresult;moreover,theprincipalcouldofferdifferentallowablespendingsetsfortheagenttochoosefromandspec-ifytransfersassociatedwitheachset.Thesequestionsarebeyondthescopeofourpaper,andsoweleavethemforfutureresearch.AppendixAProofsA1.ProofofProposition1Theclaimfollowsfromproposition1(parta)inAmadorandBagwell(2013,A2.ProofofLemma1Supposebycontradictionthatarule1foralloptimal.Sincetheincentivecompatibilityconstraint(4)istriviallysatisÞedun-derthisrule,itmustbethatforall.DeÞneasthesolu-tionto 4556journalofpoliticaleconomy ifsuchasolutionexistsandotherwise.Considernowaperturbedrule,with0and .BysinglecrossingandthedeÞnitionofin(15),theper-turbedrulesatisÞestheincentivecompatibilityconstraint(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 thnoveriÞcationbelowstrictlydominatesthe
thnoveriÞcationbelowstrictlydominatestheoriginalrule,con-tradictingtheoptimalityofarulethatveriÞesalltypes..If0,then andtheperturbedrulewehaveconstructedincreasestheprincipalswelfarefromtype relativetoverifyingalltypes.Theclaimthereforefollowsinthiscase,givenouroptimalitycondition(5).More-over,when0,wecanalsoconsideradifferentperturbationtoprovetheclaimwithoutrelyingonthiscondition.SpeciÞcally,takeaperturbedrulethat0forall g, 0arbitrarilysmall,bunchingallattheiraverageefÞcientspendinglevel.Thisruleisincentivecompatibleandincreasestheprincipalswelfarerelativetoverifyingalltypes:givenenough,thewelfarelossfromnotassigningefÞcientspendingto g, issecondorder,whilethegainfromsavingonveriÞcationcostsisÞrstorder.A3.ProofofLemma2SupposethatanoptimalrulefeaturesveriÞcationthatisweaklyincreasingevery-where.Bylemma1,0forsome,andhencethisrulemustfeatureano-veriÞcationregionfollowedbyaveriÞcationregion.Thatis,theprincipalsolves(3)and(4)bychoosingathresholdsuchthat0for1foraswellasaspendingallocation)foreachNowconsiderarelaxedversionofthisprobleminwhichtheprincipalchoosesanoptimalallocationintheno-veriÞcationandveriÞcationregionsseparately,ignoringtheincentivesoftypesinoneregiontodeviatetotheotherregion.Tak-ingtheno-veriÞcationregiontobe[ ],itfollowsfromproposition1thatanoptimalallocationisathresholdsuchthatforeach .FortheveriÞcationregion(],sinceincentivecompat-ibilityistriviallysatisÞed,anoptimalallocationassigns)toeachNotethattheresultingruleforthewholesetisTEC.Moreover,becausethiscommitmentversusexibilitywithcostlyverication4557 rulesolvesarelaxedproblem,itissufÞcienttoshowthatitisincentivecompat-ibleoverthewholesettoproveitsoptimalityintheoriginalproblem.Toshowincentivecompatibility,noteÞrstthatincentivecompatibilitywithineachregionisguaranteedbyconstruction.Furthermore,since,asexplainedinsectionII.C,notypewouldhaveincentivestodeviatetomimicadifferenttypethatisveriÞed,incentivecompatibilityissatisÞedforall .Allthatislefttobeshowni

39 sthatnotypehasincentivestodeviatetomimic
sthatnotypehasincentivestodeviatetomimicatype Thesingle-crossingconditioninimpliesthatasufÞcientconditionfortheaboveinequalitytoholdisNownotethatoptimalityoffortheprincipalimpliesGiventheagentsbias(2)and0,1,(18)implies(17)if,or,equivalently,since0,ifWeprovethattheTECrulethatweconstructedsatisÞes(19).Theoptimalintheno-veriÞcationregionsolves TheÞrst-orderconditionyields Notethat0if,and0if.Hence,theÞrst-orderconditionforsome,implyingthat(19)musthold.A4.ProofofProposition20forall .Supposebycontradictionthatanoptimalrule1forsomebutTECisnotoptimal.Bylemma2,thisrulemustfeaturedecreasingveriÞcation.WeproceedbyshowingthatanoptimalrulecannotfeaturedecreasingveriÞcationatanyConsiderÞrstdecreasingveriÞcationatsome0,so1forsome0arbitrarilysmall.Asshowninthetext,theoptimalityofverifyingtypeimplies(6)and,whereastheoptimalityofnotverifyingimplies(7).However,thetwoequationstogetherwithviolatethesingle-crossingcondition(1),acontradiction.4558journalofpoliticaleconomy ConsidernextdecreasingveriÞcationatsome1,so0forsome0arbitrarilysmall.Analogousargumentstothoseaboveapplytothiscaseandyieldacontradiction.A5.ProofofLemma3SupposethataTECrulewithcutoffsisasolutionto(8)and(9).Notethatanyrulesatisfyingconstraint(4)willsatisfyconstraint(9).Hence,(8)and(9)arearelaxedversionof(3)and(4),implyingthatanysolutionto(8)and(9)thatsatisÞes(4)willalsobeasolutionto(3)and(4).Itfollowsthattoprovetheclaim,allweneedtoshowisthattheTECrulethatsolves(8)and(9)willsatisfyconstraint(4).Itisimmediatethatforany0,(9)beingsat-isÞedimpliesthat(4)willbesatisÞed.Nowconsider1.OptimalityofverifyingtypeunderaTECrulethatsolves(8)and(9)implies,(20)sinceaperturbationthatassignsnoveriÞcationandspendingleveltoaisincentivecompatible.Notethatbytheargumentsintheproofoflemma2,aTECrulethatsolves(8)and(9)satisÞesforall.Hence,combining(20)with(2)andthefactthat0,1Itfollowsthat(4)issatisÞedfortypeA6.ProofofLemma4Supposethatarulesolving(8)and(9)speciÞes1forToprovethattherulespeciÞes,supposeby

40 contradictionthat.Consideraperturbedrule
contradictionthat.Consideraperturbedrulethatsetswhilekeepingtheallocationunchangedforall.Thisperturbationstrictlyincreasestheprincipalswelfareconditionalon,leavestheprincipalswelfareconditionalonunchanged,andisincentivecom-patiblefortheagent.Similarly,toprovethattherulespeciÞes,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-ibilitywasinitiallysatisÞedandwhile(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,byincentivecompatibilitybeingsatisÞedundertheoriginalrule,wehaveCombiningthisequationwith(27)yieldsHowever,given,(28)violatesthesingle-crossingcondi-tionin,acontradiction.Case40and1.Bylemma4,,andhencegiven,incentivecompatibilityfortype.Consideraperturbedrulethatsets1andwhileleavingtheallocationfortypesunchanged.SincetheoriginalrulesatisÞesincentivecompatibilityfor,singlecrossingimpliesthatthisperturbationisincentivecompatiblefor.Optimalityoftheoriginalrulethenrequiresthisperturbationtonotstrictlyincreasetheprincipalswelfare,whichrequiresThesingle-crossingconditioninthenimpliesNowconsideradifferentperturbedrulethatsets0andwhileleavingtheallocationfortypesunchanged.Equa-tion(29)impliesthatsuchaperturbationwouldstrictlyincreasetheprincipalwelfare.Hence,optimalityoftheoriginalrulerequiresthatthisperturbationviolateincentivecompatibility;thatis,theremustexistwith0suchthatNotethatsince,wemusthave.Moreover,byincen-tivecompatibilitybeingsatisÞedundertheoriginalrule,wehaveCombiningthisequationwith(30)yieldscommitmentversusexibilitywithcostlyverication4561 However,given,(31)violatesthesingle-crossingcondi-tionin,acontradiction.A8.ProofofLemma6Supposethatarulesolves(8)and(9)andfeaturesdecreasingver-iÞcationatsome,with1.Then0forsome0arbi-trarilysmall.Supposethatitwerethecasethat.Thenoptimal-ityofthisrulewouldbeviolated,asaperturbedrulethatsets0andwhilekeepingtheallocationunchangedforwouldbeincentivecompatibleandstrictlyincreasetheprincipalswelfare(recall0).Itfollowsthat,andhencebylemma5,Moreover,bylemma4,,andthusincentivecompatibilityforwouldbeviolatedifitwerethecasethat.Itthereforefollowsthat0arbitrarilysmall.Lemma5thenimpliesforall,whichimpliesMoreover,bydeÞnition,Combining(33)and(34

42 )andtakingintoaccountthat10yields(10).Su
)andtakingintoaccountthat10yields(10).Supposenextthatarulesolves(8)and(9)andfeaturesde-creasingveriÞcationatsome,with0.Then1forsome0arbitrarilysmall,andargumentsanalogoustothoseaboveyield(10).A9.ProofofProposition3Theargumentsintheproofsoflemmas1and2applytotherelaxedproblem,implyingthatifasolutionto(8)and(9)involvesverifyingsometype,thissolutioniseitheraTECruleorarulethatfeaturesdecreasingveriÞcationat.ToprovetheoptimalityofTECfor,wethusproceedbyshow-ingthatforanysuchveriÞcationcost,arulefeaturingdecreasingveriÞcationcannotbeasolutionto(8)and(9).Supposethatarulesolves(8)and(9)andfeaturesdecreasingveriÞcation.DenotebytheinÞmumofthelowestveriÞcationregionunderthisrule.Nowconsideraperturbedrulethatsets0for,and1for.If0,letasdeÞnedinproposition1under.If1,let.Bytheargumentsintheproofoflemma2,thisruleisincentivecompatiblefortypesprescribednoveriÞcationandsets.Moreover,giventhisinequalityandthefactthat0,itfollowsthat4562journalofpoliticaleconomy theruleisalsoincentivecompatiblefortypesprescribedveriÞcation.Wenowshowthatthisrulestrictlyincreasestheprincipalsexpectedwelfareforcontradictingtheoptimalityoftheoriginalrule.DenotebythelowesttypefeaturingdecreasingveriÞcationintheoriginalrule.Thenthechangeintheprincipalsexpectedwelfarefromusingtheperturbedruleinsteadoftheoriginalruleis ,minðÞÞðÞÞNotethatsincealltypesaboveareveriÞed,theprincipalswelfarecondi-tionalontheagentstypebeingintheno-veriÞcationregionoftheperturbedruleisoptimizedsubjecttofewerincentivecompatibilityconstraintsinthisrulecomparedwiththeoriginalrule.Hence,theÞrsttermin(35)isweaklypositive.Toevaluatethesecondandthirdtermsin(35),supposeÞrstthat.Thenbylemma6,thesecondtermin(35)satisÞesðÞÞðÞÞMoreover,thethirdtermin(35)satisÞesTogether,(36)and(37)implythattheperturbationstrictlyincreaseswelfare.Supposenextthat.Analogousargumentstothoseaboveimplythattheperturbationmakestheprincipalweaklybetteroffconditionalon.Toeval-uatethechangeinwelfareconditionalon,notethatinthiscasewe

43 must0and1for0arbitrarilysmall.Analogousa
must0and1for0arbitrarilysmall.Analogousargumentstothoseintheproofoflemma6thenimply.Moreover,by(11),,(38)wherewehaveappealedtothedeÞnitionof.Itthusfollowsfrom(38)thattheperturbationstrictlyincreasestheprincipalswelfareconditionalonA10.ProofofProposition4Considerthefollowingquadratic-uniformsetting:preferencessatisfy2and2for0,and1forall.Inthissetting,theefÞcientandßexiblespendinglevelsaregivenbycommitmentversusexibilitywithcostlyverication4563 ,respectively.Let0,sothattheagentpaysnoveriÞcationcost.WeÞrstestablishthatinthissetting,iftheveriÞcationcostsatisÞesTECissuboptimal,asitisdominatedbyarulewithoutveriÞcation.8.Considerthequadratic-uniformsettingwith0.IfthenTECisnotoptimal.Proof.Takethequadratic-uniformsettingwith0and2.Considerthefollowingproblem: NotethatthesolutiontothisprogramcoincideswitharulewithoutveriÞcationifitsets,anditcoincideswitharulethatveriÞesalltypesifitsets BythedeÞnitionofTEC,anecessaryconditionforaTECruletobeoptimalisthatthesolutiontoprogram(39)specify .Weshowthatthiscannotbesat-isÞedwhenTheÞrst-orderconditionfor,givenourassumptionsonpreferencesandthedistributionof,implies ,(40)wherewehavetakenintoaccountthefactthatmaybelowerthan .Iftheso-lutionto(39)setsstrictlyinterior,thentheÞrst-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 Wenextshowthatthereexists2underwhicharulewithveriÞcationis9.Considerthequadratic-uniformsettingwith0.If3and6 ,thenarulewithveriÞcationisoptimal..Takethequadratic-uniformsettingwith3,and .AnoptimalrulewithoutveriÞcationsetswhereusing(40)(with)andthefactthat g14b� ,wehaveWeconstructaperturbedrulethatfeaturesveriÞcationandyieldstheprincipal

44 strictlyhigherexpectedwelfarethanthisopt
strictlyhigherexpectedwelfarethanthisoptimalrulewith-outveriÞcation.Foranygiven,deÞneasthesolutiontowhichaftersomealgebrayieldsTakesufÞcientlyclosetosothat (notethattheassumptionthat6 ensuresthatsuchaexists).TypeisdeÞnedsothatheisindifferentbetweentheßexiblespendinglevelofandtheopti-malspendinglimitundernoveriÞcationforadistributiontruncatedat(whichisgivenby).Nowconstructtheperturbedruleasfollows:if,then0and;if,then0and;andif,then1andwhichaftersomealgebrayieldsNotethatgiventhedeÞnitionof,thisruleisincentivecompatible.Theper-turbationchangestheprincipalswelfareonlyfortypes.ThechangeinwelfareisequaltoðÞÞðÞÞAftersomealgebraandsubstitutionof(44),usingourassumptionsonprefer-encesandthedistributionof,thissimpliÞesto g2gH13b2dg2ðgHgH22b gH2g2b2dg2ðgHgH22bfdg1ðgHgH24b Simplifyingfurtheryieldsthatthechangeinwelfareisequalto commitmentversusexibilitywithcostlyverication4565 wheretheinequalityfollowsfromtheassumptionthat3.Therefore,theperturbedrulewithveriÞcationstrictlyincreasestheprincipalsexpectedwel-farerelativetonoveriÞcation.QEDItfollowsfromlemmas8and9thatinaquadratic-uniformsettingwith3,and6 ,veriÞcationisoptimalbutTECisnot.Bylemma2,anyoptimalrulemustthereforefeaturedecreasingveriÞcation.A11.ProofofLemma7PartiSupposethatanincentive-compatibleruleinducesdecreasingveriÞca-tionat.ConsiderÞrstthecaseinwhich0andthus0arbitrarilysmall.Incentivecompatibilityfortype,(45)canchoosetobeveriÞedandguaranteehimselftheefÞcientlevelofspending.Incentivecompatibilityfortype,(46)canchoosenottobeveriÞedandspendat).Giventheconti-nuityofintheirrespectivearguments,wecantakethelimitofbothsidesof(46)asapproaches0toobtainCombining(45)and(47)yields(12).Considernextthecaseinwhich1andthus0forarbitrarilysmall.Analogousargumentstothoseaboveimplythefollowingincen-tivecompatibilityconstraintsfor,respectively:,(48)Sincetheruleispiecewisecontinuous,limexistsandcanbedeÞned).Takingthelimitofbothsidesof(48)and(49)asgoesto0yields(47)and(45),andcombiningthesetwoin

45 equalitiesyields(12).Tocompletetheproofo
equalitiesyields(12).Tocompletetheproofofparti,weshowthatmusthold.Notethatby(12),either.Forthepurposeofcon-tradiction,supposeitwerethecasethat.Considertheincentivecompatibilityconstraintoftype0arbitrarilysmall.TakeÞrstthecaseinwhich1.ThenmustweaklypreferveriÞcationtonoveriÞcation,whichrequires,(50)impliesCombining(12)and(51)yields4566journalofpoliticaleconomy ,thisinequalityviolatesthesingle-crossingconditioninthusyieldingacontradiction.Considernextthecaseinwhich0.GivendecreasingveriÞcation,inthiscasewemusthave1and0for0arbi-trarilysmall.Moreover,givenourdeÞnitionof.Byincentivecompatibility,typemustweaklypre-ferveriÞcationtonoveriÞcation,whichrequires,(52)whereastypemustweaklyprefernoveriÞcationtoveriÞcation,whichCombining(52)and(53)andusingthefactthatapproaching0,thisinequalityvio-latesthesingle-crossingconditionin,thusagainyieldingacontradiction.Therefore,weobtainthatcannothold,andwemustthushavePartiiSupposeanincentive-compatibleruleinducesincreasingveriÞcation.Thenanalogousargumentstothoseusedtoproveparticanbeappliedtoshowthat(14)mustholdat.Sincethestepsareanalogous,weomitthedetails.A12.ProofofProposition5Toprovethisresult,weÞrstestablishthefollowinglemmas.10.Underlimitedcommitment,ifanincentive-compatiblerulefea-turesincreasingveriÞcationat,then,(54).Supposeanincentive-compatiblerulefeaturesincreasingveriÞcationat.Byequation(14)inlemma7,either.Forthepurposeofcontradiction,supposeholds.TakeÞrstthecasein1,sothat0for0arbitrarilysmalland,givenourdeÞnitionof),lim.Byincentivecompatibility,mustweaklyprefernoveriÞcationtoveriÞcation,whichrequiresHowever,(14)and(55)togetherwiththefactthatimplythatas-sumption2isviolated,yieldingacontradiction.commitmentversusexibilitywithcostlyverication4567 Considernextthecaseinwhich0,sothat1for0ar-bitrarilysmall.Byincentivecompatibility,typemustweaklypreferveriÞca-tiontonoveriÞcation,whichrequiresNotethatinthiscase,However,(14)and(56)togetherwithimplythatassump-tion2isviolated,yieldingagaina

46 contradiction.Therefore,weobtainthatcann
contradiction.Therefore,weobtainthatcannothold,andwemustthushave.QED11.Underlimitedcommitment,ifanincentive-compatiblerulefea-turesdecreasingveriÞcationat,thenthereexists1,andeither1,lim0,andProof.Supposeanincentive-compatiblerulefeaturesdecreasingveriÞcationat.Bycondition(13)inlemma7,.ConsidertheproblemoftheprincipalaftertheveriÞcationdecision)hasbeenmadeandtheveriÞcationresult(incaseofveriÞcation)hasbeenobtained: 1,(59)ThisprogramtakesintoaccountthattheprincipalwillassigntheefÞcientspend-ingleveltoanyagenttypewhochoosestobeveriÞed,andshewillignoretheincentivesofveriÞedtypeswhendecidingthespendingallocationoftypeswhochoosenottobeveriÞed.Wenowconsidertheoptimallevelof)givendecreasingveriÞcationatandtheconditionsthatarenecessaryfortheprin-cipaltochooseStep1ConsiderthespendingallocationconditionalonnoveriÞcation.Notethatanalogousargumentstothoseusedintheproofoflemma5implythatmustbeweaklyincreasingfornonveriÞedtypes.ForeachnonveriÞedtypedenoteby thespendinglevelclosestto)frombelowintheallowablespendingsetfornonveriÞedtypes(i.e.,amongallspendinglevelsassignedtotypeswhochoosenoveriÞcation).Analogously,denotebytheclosestspend-inglevelto)fromaboveintheallowablespendingsetfornonveriÞedtypes.Clearly,if)isinthisallowablespendingset,then .Theincentivecompatibilityconstraint(60)togetherwiththeconcavityofrequiresthatif0,thenargmax Step2Asnoted,givendecreasingveriÞcationat,therulemustset.Weshowthatasaresult,therulemustinduce0and4568journalofpoliticaleconomy foralltypes.Toseewhy,noteÞrstthatby(61)andthesingle-crossingconditionin,anytypewhoisnotveriÞednecessarilychoosesspending.Therefore,itissufÞcienttoshowthatanytypemusthave0.Supposebycontradic-tionthatthiswerenotthecase.Thenincentivecompatibilityforatypewith1requiresthatthistypeweaklypreferveriÞcationtonoveriÞcation,whichrequiresHowever,(12)and(62)togetherwiththefactthatolateassumption2.Theclaimthereforefollows.Step3Weshowthatinanincentive-compatiblerule,constraint(60)cannotbeuniform

47 lyslackforall,whererecallbyde-creasingve
lyslackforall,whererecallbyde-creasingveriÞcationat.Supposebycontradictionthatthisistrue.Notethatfromstep2,(60)isthenuniformlyslackforallwhere0forallsuch.Nowconsiderthefollowingperturbation:for0arbitrarilysmallandall,set;forall,set;andforallothertypes,leavethespendingallocationunchanged.Thisperturbationstrictlyincreasestheprincipalswelfareasitreducesoverspendingbytypes.Moreover,since(bythecontradictionassumption)(60)wasuni-formlyslackbeforetheperturbationforall,itisstillsatisÞedaftertheper-turbation,andincentivecompatibilityforalltypesisguaranteedastheper-turbationsatisÞes(61).Therefore,weobtainthatif(60)isuniformlyslackforall,theprincipalcanstrictlyimproveupontheoriginalrulebyreducing)aftertheveriÞcationdecisionhasbeenmade,andhencetheorig-inalruleviolatesincentivecompatibilityfortheprincipal.Theclaimfollows.Step4Bystep3,inanyincentive-compatiblerulewithdecreasingveriÞca-tionat,thereexistssatisfying(57).Moreover,sincedecreasingveriÞ-cationat,thisrequires.Thisprovesthelemma.QED12.Underlimitedcommitment,ifanincentive-compatiblerulefea-turesdecreasingveriÞcationat,thenthereexistsatwhichtherulefea-turesincreasingveriÞcation.Moreover,1forallfor1and.Supposeanincentive-compatiblerulefeaturesdecreasingveriÞcationat.Bylemma11,thereexistsatypesatisfying(57)eitherwithoratwhichthereisincreasingveriÞcation.Wecanestablishthatsuchatypeisunique.Supposebycontradictionthattherearetwotypes,satisfyingtheconditioninlemma11.Then,(64)Incentivecompatibilityrequirescommitmentversusexibilitywithcostlyverication4569 ,(66)Combining(64)(67)yieldsbydecreasingveriÞcationatandlemma11,thisinequalityviolatesthesingle-crossingconditioninyieldingacontradiction.Therefore,thereexistsauniquetypebelowforwhich(57)holds,anddenotingthistypebyyields(63).Next,weshowthat1forall.NoteÞrstthataspendinglevelcannotbeallowedbytheruleundernoveriÞcation,sinceoth-erwisetypewouldhaveastrictincentivetodeviatetosuchaspendinglevel.Considertherelevantcaseinwhichandsupposebycontr

48 adictionthat0forsometype.Letdenotethehig
adictionthat0forsometype.Letdenotethehighestsuchtype.Since,asnoted,spendinglevelsstrictlybetweennotallowed,itfollowsfrom(63)andthattherulemustsetMoreover,sincebyconstructiontherulefeaturesincreasingveriÞcationatcondition(14)inlemma7impliesHowever,given(12)and(13),equation(68)violatesassumption2.Itfollows1forall.QEDWecannowprovetheproposition.WebeginbyrulingoutdecreasingveriÞca-tion.Supposebycontradictionthatanincentive-compatiblerulefeaturesdecreas-ingveriÞcationatsome.Bylemma12,theremustexistatypesatisfyingtheconditionsinthelemma.Weproceedintwosteps.Step1.Thenitfollowsfrom(14)and(63)thatHowever,(12)and(69)togetherwiththefactthat(by[13])implythatassumption2isviolated,acontradiction.Step2Bystep1,anyincentive-compatiblerulewithdecreasingveriÞcationmusthaveateachpointatwhichthereisdecreasingveriÞcation.Nowconsidertheprincipalsproblem(58)(60).Letbethehighestnon-veriÞedtype.SincethetypeswithdecreasingveriÞcationareatomisticandtheruleispiecewisecontinuous,followingadecisionofnoveriÞcationtheprincipal Byproposition1,thesolutionassignsfor andsome.However,inthiscase,conditions(13)and(54)(whichrequire4570journalofpoliticaleconomy and,respectively)cannotbesatisÞedatapoint atwhichthereisdecreasingveriÞcationandthuscontradiction.Theclaimsaboveshowthatunderlimitedcommitment,anyincentive-compatiblerulefeaturesweaklyincreasingveriÞcationeverywhere.Analogousargumentstothoseintheproofsoflemmas1and2canthenbeappliedtoshowthataTECruleisoptimalifarulewithveriÞcationthatisweaklyincreasingeverywhereisoptimal.Therefore,underlimitedcommitment,ifveriÞcationisoptimal,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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