FootnotesWethankAnmingZhangandKenSmallforhelpfulcomments.Anyshortcomi - PDF document

FootnotesWethankAnmingZhangandKenSmallforhelpfulcomments.Anyshortcomingsinthepaper,however,areourresponsibility.1SeeBrueckner(2008),BruecknerandVanDender(2008),Verhoef(2008),DanielandPahwa(2000),Daniel(2001),DanielandHarback(2008a,b,c),ZhangandZhang(2006),BassoandZhang(2007),JohnsonandSavage(2006),andMorrisonandWinston(2007).2Arelatedconcernaboutmanipulationarisesinmarketsforpollutionpermits,wherelargepolluterscouldmanipulatethepermitpricetotheiradvantage.SeeHahn(1984)foranearlystudyinvestigatingthisissue.Notethatasimilarphenomenoncouldemergeinthemarketforairportslots,whichrepresentanalternateinstrument(analogoustopollutionrights)fordealingwithairportcongestion(seeBrueckner(2008)andVerhoef(2008)foranalysesofslotmarkets).3Notethattheadjective\classical"isusedtodenoteaparametrictollthatissetunderthecorrectpredictionoftheoptimum.Thetermisthereforenotusedtodistinguishtollswithandwithoutpartialinternalizationofcongestion.4ItshouldbenotedthataclassicalPigouviantollisnotsubjecttothekindofmanipulationdescribedinthissection.SinceaclassicaltollcomesfromevaluatingthePigouvianruleatthesocialoptimum,theonlywaycarrierscouldaectthetolllevelisbysomehowchangingtheregulator'sperceptionoftheoptimum.Suchbehaviorisconceivable,forexample,inasettingwheretheregulatorreliesonthecarriersforinformationonthecostparameter,usedincomputingf.Carrierscouldmisrepresenttheircosts,thusin\ruencingtheregulator'scomputedfandultimatelythelevelofthetolltheypay.Explorationofsuchanalternativesetting,however,isbeyondthescopeofthepaper.5Notethattheregulator'sexpectationsarealsoconrmedintheclassicalPigouviancase.Theregulatorusesthesocially\rightvolumefincomputingthetoll,andthis\rightvolumeisindeedchosenbythecarriers.6In(25),equalsleft-handsideof(24)evaluatedatf1andf2,whileequalstheleft-handsideof(23)evaluatedattheoptimumminus(f1+f2).7Ifthermsproducethesamegood,thecommonmarketdemandisassumedtobeperfectlyelastic.21 Hahn,R.W.,1984.Marketpowerandtransferablepropertyrights.QuarterlyJournalofEconomics99,753-765.Johnson,T.,Savage,I.,2006.Departuredelays,thepricingofcongestion,andexpansionproposalsatChicago'sO'Hareairport.JournalofAirTransportManagement12,182-190.Mohring,H.,Harwitz,M.,1962.HighwayBenets:AnAnalyticalFramework.North-westernUniversityPress,Evanston,IL.Morrison,S.A.,Winston,C.,2007.Anotherlookatairportcongestionpricing.AmericanEconomicReview97,1970-1977.Pels,E.,Verhoef,E.,2004.Theeconomicsofairportcongestionpricing.JournalofUrbanEconomics55,257-277.Small,K.A.,Verhoef,E.T.,2007.TheEconomicsofUrbanTransportation.Routledge,London.Verhoef,E.T.,2008.Congestionpricing,slotsalesandslottradinginaviation.DiscussionpaperTI2008-030/3,TinbergenInstitute,Amsterdam-Rotterdam.Zhang,A.,Zhang,Y.,2006.Airportcapacityandcongestionwhencarriershavemarketpower.JournalofUrbanEconomics60,229-247.20 ReferencesBasso,L.J.,Zhang,A.,2007.Congestiblefacilityrivalryinverticalstructures,JournalofUrbanEconomics61,218-237.Brueckner,J.K.,2002.Airportcongestionwhencarriershavemarketpower,AmericanEconomicReview92,1357-1375.Brueckner,J.K.,2005.Internalizationofairportcongestion:Anetworkanalysis.Interna-tionalJournalofIndustrialOrganization23,599-614.Brueckner,J.K.,2008.Pricevs.quantity-basedapproachestoairportcongestionmanage-ment.Unpublishedpaper,UCIrvine.Brueckner,J.K.,VanDender,K.,2008.Atomisticcongestiontollsatconcentratedairports?Seekingauniedviewintheinternalizationdebate.JournalofUrbanEconomics64,288-295.Buchanan,J.M.,1969.Externaldiseconomies,externaltaxes,andmarketstructure.Amer-icanEconomicReview59,174-177.Daniel,J.I.,Pahwa,M.,2000.Comparisonofthreeempiricalmodelsofairportcongestionpricing.JournalofUrbanEconomics47,1-38.Daniel,J.I.,2001.Distributionalconsequencesofairportcongestionpricing.JournalofUrbanEconomics50,230-258.Daniel,J.I.,1995.Congestionpricingandcapacityoflargehubairports:Abottleneckmodelwithstochasticqueues,Econometrica63,327-370.Daniel,J.I.,Harback,K.T.,2008a.(When)Dohubairlinesinternalizetheirself-imposedcongestiondelays?JournalofUrbanEconomics,63,583-612.Daniel,J.I.,Harback,K.T.,2008b.Pricingthemajorhubairports.JournalofUrbanEconomics,forthcoming.Daniel,J.I.,Harback,K.T.,2008c.PricingCanadianairports.Unpublishedpaper,UniversityofDelaware.Forbes,S.J.,2008.Theeectofairtracdelaysonairlineprices.InternationalJournalofIndustrialOrganization26,1218-1232.19 willnolongerguidethemarkettothesocialoptimum.Toaddressthisproblem,thepresentpaperhasderivedalternative,manipulabletollrules,whicharedesignedtoachievethesocialoptimumwhenagentsanticipatethefullimpactoftheiractionsontollliabilities.Theanalysisshowsthat,althoughthemarginaltollsdonotdierbetweentheconventionalPigouvianandmanipulablecases,theaverageandtotaltollliabilitiesgenerallywillbedierent.Inaddition,revenuesfrommanipulabletollingarelowerthanunderparametricPigouviantolling,furtherreducingthescopeforself-nancingofcapacity.Theapproachusedinthepapercanbeex-tendedtoderivemanipulablePigouviantaxesinothercontextswherelargeagentsgenerateexternalitiesandunderstandtheimpactoftheiractionsoncorrectivetaxliabilities.18 twormsgeneratepollutionasaby-productofproduction,withtheexternalitydamageafunctionofthesumofthesetwolevels(thisdamageissueredbyotheragents,notthermsthemselves).Arm'spollutionleveldependsonitsoutputalongwiththelevelofacostlyabatementactivity,andthedemandforeachrm'soutputisassumedtobeperfectlyelastic.7Inthiscase,thesocialoptimumrequiresequalitybetweenthemarginalprotfromanextraunitofoutputandthemarginalpollutiondamagefromthatunit(equaltotheadditionalpollutiontimesmarginalpollutiondamage).Inaddition,themarginalbenetfromabatement(equaltoabatementproductivitytimesmarginalpollutiondamage)mustequalthemarginalabatementcost.Together,thesetwoconditionsdeterminetherms'optimaloutputandabatementlevels.Providedthatthermsviewanypollutiontaxasparametric,theoptimumcanbegeneratedbyaPigouviantaxequalsetequaltomarginalpollutiondamage,evaluatedatthesociallyoptimalpollutionlevelundertheclassicalapproachorattheexpectedequilibriumlevelunderthealternateapproach.WhenrmsmanipulatethePigouviantax,theoptimumfailstoemerge,andamanipulabletaxisthenrequired.Theformofthistaxcanbepredictedgiventhepreviousanalysis.Inparticular,eachrmshouldpayataxliabilityequaltototalpollutiondamage,adjustedbysomeconstant.Whensuchatax,whosestructureisunderstoodbytherms,issubtractedfromprot,optimaloutputandabatementlevelsarethenchosen.Itshouldnotedthat,sincepollutiondamageissueredbyagentsotherthanthermsthemselves,themanipulabletaxstructuredoesnotshareakeyfeatureofthepreviouscongestiontolls:chargingfordamagetotheotherrm.Insteadeachrmischargedfordamagedoneto(outside)pollutionvictims,adjustedbyconstant.Amanipulabletaxwouldhaveasimilarforminmanyothercontexts.6.ConclusionTherecentliteratureoncongestionpricingwithlargeagentscontainsaremarkablein-consistency:thoughagentsarelargeenoughtorecognizeself-imposedcongestionandexertmarketpoweroverprices,theydonottakeintoaccounttheimpactoftheirownactionsonthemagnitudeofcongestiontolls.Whenlargeagentsareconfrontedwithtollsderivedunderthisparametricassumptionbutunderstandtheruleusedtogeneratethem,thetollsystem17 levelandderivativeofthedesiredaveragetollfunctiont(
).Theseequationscanbesolvedfortheseunknownquantities,yieldingt(f1+f2)=f12f1f2c01(f1;f2)f22f1f2c02(f1;f2)(23)t0(f1+f2)=f1f1f2c01(f1;f2)+f2f1f2c02(f1;f2)+D0(f1;f2):(24)Althougharestrictedtollfunctioncouldbederivedinthebasicmodelbyndingthesolutiontoadierentialequation,asimilaroutcomeisnotfeasiblehere.Toseethediculty,observethattheleft-handof(24)isnotequaltothederivativeofleft-handsideof(23),whichmeansthatafunctiont(
)satisfyingthesetwoconditionsforallvaluesoff1andf2doesnotexist.Whilethisconclusionmeansthatarestrictedmanipulabletollrulelikethatinsection3isinfeasible,itispossibletogeneratea\fake"manipulablerule,onethatiscomputedusingknowledgeofthesocialoptimum.Inthissense,thetollresemblestheclassicalPigouviantollin(19),whosecomputationrequiresalltheinformationneededtoderivethesociallyoptimalvaluesf1andf2.Inparticular,itcanbeveriedthatthelinearaveragetollfunctiongivenbyet(f1+f2)=+(f1+f2)(25)yieldsamanipulabletollrulethatgeneratesthesocialoptimum,wheretheslopeandintercepttermsandarederivedusing(23)and(24)andthemselvesdependonf1andf2.6Atollrulewithacommonaveragetollper\rightmaybeappealingonequitygrounds,butdespitethenegativeconclusionjustderived,equitycouldbeachievedinadierentfashion.Specically,sincetheairportauthoritycanadjusttollliabilitiesunderthegeneralmanipulabletollbyvaryingtheKitermsin(21),anydistributionalgoalcanbemeteventhoughthedesiredtypeofrestrictedtollfunctionisnotavailable.5.ManipulableTaxesinOtherContextsTheprinciplesoftheprecedinganalysiscanbeappliedinothercontextswheremanip-ulationofPigouviantaxesmayoccur.Consider,forexample,thepollutioncontext,where16 ingservicebetweenacommonsetofnuncongestedendpointsandthesamecongestedairport(whichmightbeahub).Sincetheyinvolvedierentendpointcities,suchmarketsmayhavedierentdemandfunctions,leadingtomarket-powerdistortionsofdierentmagnitudes.Thesedistortionscanthenbeaddressedbymarket-levelsubsidies,whilecongestionisaddressedbyseparateairport-leveltolls,eitherofPigouvianormanipulableform.Withtwoasymmetriccarriers,therequiredsubsidieswouldbebothcarrierandmarket-specic,yielding2ndierentsubsidiesinthevariousmarkets.Inaddition,twodierent,carrier-specictollswouldbeleviedatthecongestedairport.Thesubsidiesandtollswouldseparatelyrepresentthemarket-powerandcongestionterms,suitablygeneralizedtomultiplemarkets,fromeitherthePigouviantollformula(20)orthemanipulableformula(22).Analpointregardingthecorrectionofthemarket-powerdistortionconcernsrst-degreepricediscrimination.Undersuchbehavior,everyticketwouldbesoldatapriceequaltotheconsumer'smaximumwillingnesstopay,andthemarket-powerdistortionwouldvanish.Althoughpricediscriminationinfaresettingisstandardpracticeintheairlineindustry,itdoesnottakeonthisidealform,sothatsomedownwardadjustmentofcongestiontolls(orthepaymentofseparatemarket-levelsubsidies)willremainnecessarytocompensateforexcessivefares.4.2.FeasibilityofamanipulablerulewithacommonaveragetollAsseeninsection3.2,arestrictedmanipulabletoll,wherebothcarriersarechargedthesameaveragetoll,isfeasibleinthebasicmodel.Anaturalquestioniswhethersuchatollisfeasibleinthegeneralsettingunderconsideration.Onemightexpectthatcarrierasymmetrymakesarestrictedtolllikethatderivedaboveinfeasible,andtheensuinganalysisshowsthatthisconjectureisindeedcorrect.Eventhoughthisresultmaycomeasnosurprise,theanalysisfurtherilluminatestheissuesinvolvedinderivingmanipulabletollrules.Asbefore,whencarrierspayacommonaveragetollt(
),themarginalmanipulabletollisgivenby@Ti(f1;f2)=@fi=t(f1+f2)+fit0(f1+f2),sothatthemarginal-tollcondition(20)mustholdwiththelatterexpressionreplacingtheleft-handsideexpression.Inthepreviousanalysis,theanalogto(20)(namely,eq.(10))reducedtothesinglecondition(11)givensymmetryofthecarriers.However,withasymmetry,(20)remainsastwoseparateequationsinvolvingthe15 IntegratingthersttermsbypartsandimposingthepreviousrequirementKi=fici(fi)ontheconstantsofintegration,(21)reducestoTi(f1;f2)="Zfi0D(x+fi)dxfiD(f1+f2)#+fi(ci(f1+f2)ci(fi));i=1;2:(22)Notethat(22)equalstheincreaseintheothercarrier'scongestioncostduetocarrieri'soperationsminustheadditiontoconsumersurplusfromtheseoperations.Withtollliabilityrule(22)inplace,airlinei'sprotfunctionbecomeseectivelyequaltosocialsurplusabovethelevelachievedifonlyfiweresupplied.Thisfactcanbeveriedbysubstitutionof(22)intotheexpressiongivingprotnetofthetollandcomparingwith(16).Theintuitionissimple:ifatollmanipulatoristobeseducedtobehavesoastomaximizesocialsurplus,thetollliabilityshouldbedenedsothatprotinclusiveofthisliabilityshouldvaryperfectlyinparallelwithsocialsurplus.Inhindsight,itwouldhavebeensurprisingifthiswouldnothavebeentrue.Thedistributionalimpactofthetollschedule,rewardingthecarrierwiththefullincreaseinsocialsurplusitcreatesgiventhecompetitor'soutput,mayofcoursebeconsideredundesirable.TheKitermsmaythenbeusedtoadjustthisimpactbysettingthematvaluesdierentfromKi=fici(fi).Butwhattheanalysisshowsisthatpost-taxprotsshouldvaryperfectlyinparallelwithsocialsurplusforthemanipulabletolltobeoptimal,apropertythatisindependentofthechoiceofKi.ThisrequirementleavestheKitermsasinstrumentstoaddressdistributionalconcerns.AsinthecaseofthePigouviantollfrom(20),thetotaltollliabilityin(22)maybepositive(anettax)ornegative(anetsubsidy).Thelatteroutcomeemergeswhenconsumersurplusislargerelativetothecongestionexternality(requiringsucientlyinelasticdemand).Thisambiguityre\rectstheuseofasingletollinstrumenttoaddresstwoopposingdistortions:amarket-powerdistortionthatinecientlylimits\rightvolumes,andacongestionexternalitythatinecientlyin\ratesthem.AssuggestedbyBrueckner(2005),however,thesetwodistor-tionscouldbeaddressedbydierentinstrumentsinasituationwherethedistortionscanbeseparated.Suchasituationariseswhenthecarriersservemultiplemarkets,witheachprovid-14 Using(1),therst-orderconditionforprotmaximizationintheabsenceoftollsisD(f1+f2)+fiD0(f1+f2)ici(f1+f2)fici0(f1+f2)=0;i=1;2:(18)Notethatthesecondtermin(18)arisesfromtheexploitationofmarketpower,whichallowsthecarrierstoraisethefarebylimiting\rights.WhenaclassicalPigouviantollisusedtoeliminatethedierencebetween(17)and(18),carrieri'stollequalszi=fiD0(f1+f2)+fici0(f1+f2);(19)asseeninPelsandVerhoef(2004).Theasterisksagaindenotesociallyoptimalvalues,whicharenowasymmetric,asarethePigouviantollsthemselves.Notethatthersttermin(19)isnegativewhendemandisimperfectlyelastic,indicatingthatthetollisadjusteddownwardtomitigateover-pricingbythecarriers.Thiscorrectionvanishesascarriersbecomeinnitesimallysmallandfiapproacheszero.ThedownwardadjustmentofPigouviantaxesundermarketpowerwasoriginallyderivedbyBuchanan(1969),anditfurthererodesthescopeforself-nancingofoptimalairportcapacity.Observethatthemarket-poweradjustmentin(19)islargerforbiggercarriers,osettingtheirgreaterincentivetorestrictoutputinordertoraisethefare.Asbefore,themanipulabletollruleTi(f1;f2)forcarrierishouldbesetsothatthemarginaltollcoincideswiththePigouviantollrule.Using(19),thisrequirementyields@Ti(f1;f2)@fi=fiD0(f1+f2)+fici0(f1+f2);i=1;2:(20)Themanipulabletollruleisfoundbyintegrating(20),whichyieldsTi(f1;f2)=Zfi0xD0(x+fi)dx+fici(f1+f2)+Ki;i=1;2:(21)13 3.5.AirportcostrecoveryThewell-knownself-nancingtheoremfromroadpricingtheorysaysthat,whenroadsarebuiltwithneutralscaleeconomiesandanadditionalzero-degreehomogeneityassumptionholdsforthecongestionfunction(i.e.,adoublingoftrac\rowandcapacitywouldleavethetraveltimeconstant),thecostoftheoptimal-sizeroadisexactlycoveredbyPigouviancongestiontollrevenue(MohringandHarwitz,1962).Thisresultfailstohold,however,intheairportcontextduetointernalizationofcongestion,whichgeneratessmallertollsthanintheroadsetting(foraformalderivation,seeBrueckner(2002,2008)).Giventheconclusionsderivedabove,therevenueshortfallisevenlargerunderarestrictedmanipulabletollrule.Inotherwords,sincetheaveragemanipulabletollissmallerthanthePigouviantollinequilibrium,andsincethelattertollitselfalreadyfailstocoverairportcost,alargerdecitemergesinthemanipulablecase.SincethegeneralmanipulabletollalsogeneratesasmallerliabilitythanthePigouviantollwhenittakestheformin(11)andc(
)isconvex,costrecoveryagainfailsinthiscase.Thedecitunderbothtypesofmanipulabletollsmustbecoveredbyadditionallump-sumcharges.4.TheGeneralModel:AsymmetricCarriersandImperfectlyElasticDe-mand4.1.TheunrestrictedmanipulabletollConsidernowthemoregeneralcasewherecarriersneednotbesymmetricanddemandisnotnecessarilyperfectlyelastic.SocialsurplusisnowdenedbyW=Zf1+f20D(x)dx2Xj=1fj(j+cj(f1+f2));(16)andthesocialoptimumrequiresD(f1+f2)ici(f1+f2)fici0(f1+f2)fici0(f1+f2)=0;i=1;2;(17)wherethesubscriptionc(
)denotesthefunctionbelongingtocarrieri'scompetitor.Notethat(17)diersfrom(2)inthelackofsymmetryandtheappearanceofD(
)inplaceofp.12 Butrecognizingthatthe\rightvolumesaresymmetricinequilibrium,f1=f2=F2holds,whereFisthetotal\rightvolume.Makingthissubstitution,(12)canberewrittenast(F)+F2t0(F)=F2c0(F):(13)Thisconditionisalinear,rst-orderdierentialequationintheunknownfunctiont(F).Giventhepresenceofthe1=2factor,theleft-handsideexpressioncannotbeintegrated,ap-parentlypreventingderivationofageneralsolution.However,asolutioncanbederivedunderafairlygeneralfunctionalformforc(
).Inparticular,ifc(F)F,where1,thenat(F)functionwiththesameexponentbutadierentmultiplicativefactorcansatisfy(13).Lettingt(F)=F,(13)reduces,afterdierentiationandsubstitution,tothefollowingrequirement:F+F2F1=F2F1;(14)whichissatisedwhen==(2+).Thus,themanipulableaveragetollfunctionforthisspecialcaseisgivenbyt(F)=2+F(15)Whencarrierifacesatollliabilityoffit(f1+f2)=fi(2+)(f1+f2),theresultingprot-maximizing\rightvolumesaresociallyoptimal.TheequilibriumtollpaymentunderthisrestrictedmanipulablefunctionissmallerthaninthePigouviancase.Undertheassumedformofc(
),themarginalPigouviantollper\right(eitherintheclassicalcaseorthenon-classicalequilibrium)is(F=2)c0(F)=(=2)F,whichexceedstheaveragemanipulabletollinequilibrium,givenby(15)evaluatedatF(thesociallyoptimaltotal\rightvolume).Thisrelationshipalsoholdsingeneral,ascanbeseenfrominspectionof(13).Sincetheright-handsideevaluatedatFisthePigouviantollper\right,itfollowsthatt(F)mustbesmallerthanthismagnitudeaslongast0(
)ispositive.Intuitively,theaveragemanipulabletollislessthanthePigouviantollwhentheaveragetollisincreasingsincethiseectprovidesanadditionaldeterrentinlimitingacarrier's\rightvolume.11 plusaconstant.WhileKicouldbesetataxednumericalvalue,itneedonlybeindependentoffiandthuscouldbeafunctionoffi,theothercarrier's\rightvolume.Giventhispossibility,anaturalchoiceistosetKi=fic(fi),sothatTi(f1;f2)=fic(f1+f2)fic(fi):(11)Then,carrieri'smanipulabletolliszerowhenfi=0andequalstheincreaseintheothercarrier'scongestioncostduetothe\rightsoperatedbycarrieri.ItisinterestingtocomparethetollliabilitiesinthemanipulableandPigouviancases.Whenthemanipulabletolltakestheformin(11),thetollliabilitycanberewrittenasRfi0fic0(x+fi)dx,whileunderPigouviantollingitequalsfific0(fi+fi),withbothexpressionsevaluatedatthesocialoptimumtogenerateequilibriumvalues.Ifc(
)isstrictlyconvex,thesecondexpressionislargerforanyvaluesoffiandfi,implyingthatthetollliabilityunderPigouviantollingexceedsthatundermanipulabletolling.However,giventhefreedomtoadjustKi,adierentchoicecanmakethemanipulabletoll'sliabilityexceedthatofthePigouviantoll.3.4.ArestrictedmanipulabletollThemanipulabletollin(10)hasaformthatmightbeviewedasunappealinginapracticalsense,giventhatitembodiesachargethatdependsontheothercarrier'scongestioncost.Toeliminatethisdrawback,adierenttollrulecanbederived,subjecttoamorenaturalrestrictionontherule'sform.Inparticular,supposethatacarrier'stollliabilityisrequiredtoequalitsown\rightvolumetimesafunction,commontobothcarriers,thatdependsonthetotal\rightvolumeattheairport.Stateddierently,therequirementisthattheaveragetollpaidbyacarrier,equaltoitstollliabilitydividedbyitsown\rightvolume,begivenbyacommonfunctionthatdependsonthe\righttotal.Lettingthisaveragetollfunctionbewrittent(f1+f2),carrieri'stollliabilityisthengivenbyfit(f1+f2).Thus,therestrictionimpliesT1(f1;f2)f1t(f1+f2)andT2(f1;f2)f2t(f1+f2).Underthisrestriction,@Ti(f1;f2)=@fi=t(f1+f2)+fit0(f1+f2)holds,sothat(9)becomest(f1+f2)+fit0(f1+f2)=fic0(f1+f2):(12)10 volumechoices,whichcanbeexploitedindecisionmaking.Althoughthetollruleisgivenforthecarrier,thetolllevelhasthus,intentionally,become\manipulable."LetTi(f1;f2)denotethemanipulabletollruleforcarrieri,whichgivesitstotaltolllia-bilityasafunctionofboth\rightvolumes(carrieri'stollper\right,ti,isthereforeequaltoTi(f1;f2)=fi).Facedwiththisfunction,carrieri'srst-orderconditionforchoiceoffiispc(f1+f2)fic0(f1+f2)@Ti(f1;f2)@fi=0;i=1;2:(8)ThegoalistochoosetheTifunctionssothatthesolutionstothetwoconditionsin(8)coincidewiththesocialoptimum.Thisgoalcanbeachievedifthemanipulabletollruleischosensothat(8)isthesameasthesocialoptimalityconditionin(2).Inspectionofthetwoconditionsshowsthatthiscoincidencerequires@Ti(f1;f2)@fi=fic0(f1+f2):(9)Theleft-handsideof(9)givesthemarginaltollasperceivedbyatoll-manipulatingcarrier:theadditionaltollliabilityresultingfromaddinganextra\right.Notethattheright-handsideof(9),whenevaluatedatthesocialoptimum,issimplyequaltotheleveloftheclassicalPigouviantollz,whichitselfrepresentsamarginaltollgiventhetollliabilityofzfi.Thus,(9)impliesthatthemarginaltollfromthemanipulablecase(whenevaluatedatthesocialoptimum)isequaltothemarginalPigouviantoll.Despitethiscoincidenceofmarginaltollsinthetwocases,itwillbecomeclearbelowthattheaveragetolls(zintheclassicalPigouviancasevs.Ti(f1;f2)=fiinthemanipulablecase)willnotbeequal,norwillthetotaltollliabilities.Integratingbothsidesof(9)withrespecttofidirectlyyieldsthemanipulabletollrule,whichisgivenbyTi(f1;f2)=fic(f1+f2)+Ki:(10)whereKiisaconstantofintegration.ThusT1(f1;f2)=f2c(f1+f2)+K1andT2(f1;f2)=f1c(f1+f2)+K2,sothatacarrier'stollliabilityequalstheothercarrier'stotalcongestioncost9 elicitsthesesame\rightvolumesaschoices.Thus,theregulator'sexpectationsincomputingthetollareconrmed.Sincethemodiedversionof(5)coincideswiththeoptimalitycondition(2),theresultingequilibriumisecient,justliketheonebasedontheclassicalPigouviantoll.4Nowconsiderthecasewheretheairlinesknowthetollrule,perhapsbecausetheparametrictollhasbeenadjustedanumberoftimesinresponsetochangesintheequilibrium(duetovariationinpor,forexample)orbecausethetollruleispublic.Inaddition,supposethatthecarriersusethisknowledgetotheiradvantage.Specically,carrieriwouldviewprotasbeinggivenby(p)fic(f1+f2)fific0(f1+f2)fi;(6)butthetollper\rightexpression(fic0(f1+f2))appearingthelasttermwouldnolongerbeviewedasparametric.Carrierswouldthenoptimizetakingaccountoftheeectoftheirdecisionsonthetollpaidper\right.Computingtherelevantrst-orderconditionandevaluatingatthesymmetricequilibrium,theresultingcommon\rightvolume,denotedef,satisespc(2ef)2efc0(2ef)ef2c00(2ef)=0:(7)Notethatconrmationoftheregulator's\right-volumeexpectationsisagainensuredby(7).Theregulatorexpectssymmetric\rightvolumesofefandthuschargesatollper\rightofefc0(2ef).But(7)ensuresthatcarrierchoices,whichnowinvolvemanipulationofthetoll,generateacommon\rightvolumeofef,conrmingtheseexpectations.Thekeyobservationfrom(7),however,isthatsincec00�0,theequationissatisedatanefvaluesmallerthanf.Thus,manipulationofthe(non-classical)Pigouviancongestiontollleadstoaninecientlylow\rightvolume.53.3.AmanipulablecongestiontollTocircumventmanipulativebehavior,theplannercouldabandonthePigouviantollandinsteadannouncetothecarriersacompleterulethatdeterminestheirtollliabilitiesasafunctionof\rightvolumes.Thistollruleisdesignedtobemanipulatedinthesensethatcarriersaregivenfullinformationabouttheconnectionbetweentollliabilitiesandtheir\right-8 Whenzfiissubtractedfromcarrieri'sprot,thenewrst-orderconditionsarepc(f1+f2)fic0(f1+f2)fc0(2f)=0;i=1;2:(5)whichyieldthesolutionf1=f2=f.Intheroadcontext,whereinternalizationofcongestiondoesnotoccur,thePigouviantollwouldequalthefullmarginalcongestiondamageevaluatedatthesocialoptimum,givenherebyMCD=2fc0.Butinternalizationmeansthatatollthislargeisnotrequired,withzinsteadequaltoMCD=2.3.2.IncentivesformanipulationWhilethe\classical"Pigouvianapproachembodiedin(4)assumesthattheregulatorhassucientinformationtocomputetheoptimum,anactualapplicationofthePigouvianmethodmightbebasedonincompleteinformation.Theregulator,forexample,maylackinformationonthecostparameterorthefullpricep.Assuming,however,thattheregulatorknowsthecongestioncostfunctionc(
),avariantoftheclassicalapproachcanbeenvisioned.Itgeneratesthesameoptimaloutcomeastheclassicalapproachwhencarrierstreattollsasparametric,buttheapproachalsoinvitestollmanipulation.Underthisalternateapproach,theregulatorcomputesthecongestiontollper\rightusingthePigouvianrulefic0(f1+f2).Ratherthanevaluatingtheruleatthesocialoptimum,whichtheregulatorisunabletocompute,theruleisevaluatedatthe\rightvolumesthereg-ulatorexpectstoemergeinequilibrium.Theseexpectationsmustbeconrmed,however,bycarrierchoices.Giventheirperceptionofthetoll(asparametricor,alternatively,subjecttomanipulation),the\right-volumechoicesofthecarriersmustmatchtheregulator'sexpecta-tionsregardingthesevolumes.Ifnot,theregulatorwouldadjusttheparametrictollsuntilexpectationsareconrmed(achievingequilibrium).Inthecasewherethecarriersviewthetollasparametric,thisequilibriumrequirementisstraightforward.Inparticular,thePigouviantollper\rightfic0(f1+f2),whichisviewedasparametric,issubstitutedinplaceoffc0(2f)in(5).EvaluatingthePigouvianruleatthe\rightvolumesthatsatisfy(5)yieldsatollthat,whenperceivedasparametricbythecarriers,7 3.1.Socialoptimum,laissez-faireequilibrium,andPigouviantollsWithperfectlyelasticdemand,consumersurplusiszeroandsocialwelfareequalstotalprot,givenby[pc(f1+f2)](f1+f2).Maximizingthisexpressionbychoiceoff1andf2yieldstwoidenticalrst-orderconditions,givenbypc(f1+f2)(f1+f2)c0(f1+f2)=0:(2)Thisconditiondeterminesasymmetricoptimal\rightvolume,givenbyf1=f2=f.Notethat(2)saysthatextra\rightsshouldbeoperateduptothepointwherethefullpriceminusoperatingandcongestioncostsper\rightequalsthemarginalcongestiondamagegeneratedbyanextra\right,givenby(f1+f2)c0(f1+f2)MCD.Thisexpressionequalsmarginalcongestioncostper\right,givenbytheslopeofthecongestioncostfunction,timesthenumberof\rightsexperiencingtheadditionalcongestion.Thelaissez-faireequilibriumisgeneratedbyprotmaximizationunderCournotbehavior.Carrieri'sprotequals[pc(f1+f2)]fi,andtherst-orderconditionformaximizationofthisexpressionispc(f1+f2)fic0(f1+f2)=0;i=1;2:(3)In(3),firatherthanf1+f2multipliesc0,indicatingthatcarrieridoesnottakeintoaccountthecongestiondamageimposedontheothercarrierwhenitschedulesanextra\right.Therefore,thesymmetricequilibrium\rightvolumes,equaltobf,aretoolarge,satisfyingbf�f.Notethat,eventhoughitignorestheimpactontheothercarrier,carrieridoesinternalizethecongestionitimposesonitself,viewingitsowncongestiondamage(fic0)aspartofthecostofoperatinganextra\right.APigouviantoll,ifviewedasparametricbythecarriers,canremedythisineciency.Thetollchargeseachcarrierforthecongestiondamagethatitdoesnottakeintoaccount,equalingfic0(f1+f2)per\rightforcarrieri,whereidenotestheothercarrier.Evaluatingthisexpressionatthesymmetricsocialoptimum,theclassicalPigouviantollisthengivenby3z=fc0(2f):(4)6 givenbyD(f1+f2)h(f1+f2).Thus,congestiongeneratesafarediscount,asveriedintheempiricalstudyofForbes(2008).Carrieri'srevenueisthenequalto[D(f1+f2)h(f1+f2)]fi,i=1;2.Inadditiontoraisingpassengertimecosts,airportcongestionincreasesairlineoperatingcosts,withtheeectallowedtodieracrosscarriers.Congestioncostper\rightforcarrieriisgivenbygi(f1+f2),i=1;2,withthesefunctionsagainassumedtobeincreasingandconvexovertherelevantrange.Anairlinealsoincursoperatingcoststhatareunrelatedtoairportcongestion,coststhatagainmaydieracrosscarriers.Assumingconstantreturns,thesecostsaregivenbyifi,whereiisoperatingcostper\right,i=1;2.Combiningtheaboveelements,airlinei'sprotisgivenbyi=[D(f1+f2)h(f1+f2)]fiifigi(f1+f2)fi=D(f1+f2)fiifici(f1+f2)fi;i=1;2(1)whereci(f1+f2)h(f1+f2)+gi(f1+f2)givespassengerplusairlinecongestioncost.Notethatthetwotypesofcongestioncostsentertheprotfunctionsymmetricallybecauseanincreaseinpassengertimecostsimpliesanequallylargedecreaseinthefarethatcanbechargedforagivenoutputlevel.Firmprotsarethereforeequallysensitivetoh(f1+f2)andgi(f1+f2).Socialwelfareismeasuredbytotalprot,1+2,plusconsumersurplus,whichisgivenbyRf1+f20D(x)dx(f1+f2)D(f1+f2).Useofthiswelfarefunctionrequirestheabsenceofincomeeectsondemand,inwhichcaseconsumersurplusisanexactmeasure.3.BaseCase:SymmetricCarriersandPerfectlyElasticDemandItisusefultobeginbyconsideringabasecasewherefurthersimplicationsareimposedonthemodel.Accordingly,supposethatdemandisperfectlyelastic,sothattheDfunctionfromaboveisequaltoaconstant,denotedp.Inaddition,letcostsbesymmetricacrosscarriers,sothatoperatingandcongestioncostsin(1)losetheirisubscripts.Afterderivingresultsforthisbasecase,theanalysisreturnstothegeneralcaseinsection4.5 demandisperfectlyelasticandcostsaresymmetric.Section3.1derivesthesocialoptimumforthiscase,andthencomputesthePigouviancongestiontollsrequiredtosupportit,assumingthatthetollsareviewedasparametricbythecarriers.Then,section3.2explorestheincentivesfortollmanipulation,relyingonthenon-classicalversionofthePigouvianapproachsketchedabove.Manipulabletolls,requiredinresponsetomanipulativebehavior,areanalyzednext.Theanalysisinsection3.3derivesthemostgeneralformforsuchtolls,andsection3.4considerstheirformunderaplausiblerestriction,whichrequiresacarrier'stollliabilitytoequalitsown\rightvolumetimesatollper\rightthatiscommonacrosscarriers.Section3.5considerstheissueofairportcostrecoveryunderthedierenttollregimes.Section4focusesonthegeneralmodel,inwhichcarriersareasymmetricanddemandisnolongerperfectlyelastic,leadingtoinecientpricingmark-upsascarriersexercisemarketpower.Sections4.1and4.2respectivelyconsidergeneralandrestrictedtollschedulesforthegeneralcase,withthelatterinvolvingacommontollper\rightacrossthenow-asymmetriccarriers.Section5considersapplicationofthelessonsoftheanalysistootherexternalitycontexts,andsection6oersconclusions.2.TheModelTheanalysisfocusesonasingletravelmarketwithacongestedairportatoneendpoint.Incontrasttosomeearlierpapers,themodeldoesnotdistinguishbetweenpeakando-peakperiods,sothatcongestionisalwayspresent.Themarketisservedbytwoairlines,denoted1and2,whichinteractinCournotfashion.Letfidenotethenumberof\rightsoperatedbycarrieri,andletthenumberofpassengersper\rightbeconstantandnormalizedtounity,sothatf1+f2representsboththetotal\rightvolumeandthetotalnumberofpassengers.Thedemandfor\rightsisgivenbytheinversedemandfunctionD(f1+f2),whichgivesthemarginalwillingnesstopayfortravel.Passengervolumeisdeterminedbyequatingthiswillingnesstopaytothe\fullprice"oftravel,whichincludestheairfareandthevalueoflostpassengertimeduetoairportcongestion.Withcongestiondependingontotal\rightsattheairport,timecostperpassengerisgivenbyh(f1+f2),afunctionthatisassumedtobeincreasingandconvexovertherelevantrangeof\rightvolumes(itmaybezeroatlowvolumes).Sincetheairfareplustimecostequalsthefullprice,itfollowsthatthefareis4 atomistic,raisingconcernsaboutmanipulation.Thisisthecaseofairportcongestion,whicharisesfromusageofacapacity-constrainedairportbyarelativelysmallnumberofairlines,someofwhichmayaccountforanappreciableshareofthetotal\rights.Forconcreteness,thepaper'sanalysisisdevelopedintheairportcontext.Departingfrompreviousstudies,arecentliteratureonairportcongestionrecognizesthenon-atomisticnatureofairlineswhileexploringaparticularconsequenceofthisalterationofthestandardroad-orientedmodel:internalizationofcongestion.Internalizationoccursbecauseanon-atomisticcarrier,inschedulinganextra\right,takesintoaccounttheadditionalcongestioncostsimposedontheother\rightsitoperates.Asaresult,aPigouviancongestiontollneedonlychargeanairlineforthecongestionimposedonothercarriers,excludingthecongestiontheairlineimposesonitself.Oneimplicationofthisruleisthat,whencarriersareasymmetric,theyshouldpaydierenttolls.Acarrierwithalarge\rightshareshouldpayalowtollgiventhatitinternalizesmostofthecongestionfromitsoperationofanextra\right,whileasmallcarrier,whichinternalizeslittleofthecongestionitcreates,shouldpayahightoll.Usingasimulationmodel,Daniel(2005)wasthersttorecognizethepotentialforinternalizationofcongestion,whileBrueckner(2002,2005)andPelsandVerhoef(2004)exploredtheimplicationsofinternalizationusingsimpleanalyticalmodels.Aburgeoningliteraturehasfollowedtheseinitialstudies.1Asthisdiscussionindicates,therecentanalysisofairportcongestionpresumestheuseofPigouviancongestiontolls,eventhoughthenon-atomisticnatureofairlinessuggeststhepotentialformanipulationofsuchtolls.Thepresentpaperismeanttoredressthisomission.Byanalyzingthenatureofthemanipulablecongestiontollsinanairportcontext,itaddsamissingcomponenttothenewtheoryofcongestionpricinginthepresenceofnon-atomisticagents.Theresults,however,applymoregenerallytothetheoryofcorrectivetaxation,showingthatmanipulabletaxesmayneedtoreplacePigouvianchargesinothercontexts.2Theplanofthepaperisasfollows.Section2presentsthemodel,whichtakesthesimplestpossibleform.Twoairlinesserveasingletravelmarket,withacongestedairportatoneendpoint.Whilethegeneralmodelhasanelasticdemandfortravelandcostfunctionsthatpotentiallydieracrosscarriers,section3beginsthediscussionbyconsideringabasecasewhere3 forexample).Then,ifthemarginaldamagefunctionisincreasing,thePigouviantaxwillvaryaswell,exhibitingpositivecorrelationwithboththeoverallactivitylevelandindividualactivitylevelswithinthesmallsetofagents.ThiscorrelationmayrevealtotheagentsthenatureoftheruleusedtocomputethePigouviantax,showingthatthetaxperunitdependsontheoverallactivitylevel.AnotherreasonwhyagentscouldbeawareoftheruleunderlyingthecalculationofPigouviantaxesisthattheprinciplesaccordingtowhichthetaxlevelisdeterminedmayhavetobemadetransparentandpubliclyaccessibleforlegalreasons.Agentscanthenobservetheruledirectly.Ineithercase,withthisknowledge,agentsmaythenattempttomanipulatethetaxregime,furtherrestrictingtheiractivitylevelsinordertodepressthemagnitudeoftheper-unittaxthattheypay.Suchbehavior,whichisshownbelowtoberelevantunderthealternate,non-classicalapproach,underminesthePigouviantaxregime,makingitincapableofachievingtheoptimum.Thepurposeofthepresentpaperistoexploreanalternativeapproachtocorrectivetax-ationthatrecognizesthepotentialexistenceofsuchmanipulativebehavior.Ourapproachconfrontsmanipulationhead-onbyreplacingtheconventionalPigouviantax,designedtobetreatedparametricallybyindividuals,withataxrulethatisdesignedtobemanipulated.Inotherwords,theplannerannouncesarulethatgiveseachagent'staxliabilityasafunctionofhisorherownactivitylevelandthelevelsofotheragents.Eachagentthenoptimizeswithfullknowledgeoftheruleusedtocomputethetaxliability,withtheoptimizationbeingcarriedoutconditionalonthechoicesofotheragents.Thus,insteadofusingaPigouvianrulethatoperatesbehindthescenestogenerateataxvaluemeanttobeviewedasparametric,ourapproachpresentsatransparent,manipulablerulethatisdirectlyexploitedbytheagentsinchoosingtheiractivitylevels.Thegoaloftheanalysisistoderivetheformofsuchmanipulabletaxrules,providingacomparisontothePigouviancase.Theanalysisfocusesoncongestionexternalities,wherePigouviantaxestaketheformofcongestiontolls.Traditionalanalysesofroadcongestionpricing(seeSmallandVerhoef(2007))aremostlyimmunetothemanipulationcritiquefromabove,aconsequenceofthefactthatroadusersaretypicallyatomisticandthusunabletomanipulatePigouviancongestiontolls.However,arecentliteraturefocusesonacasewheretheagentsgeneratingcongestionarenon-2 ManipulableCongestionTollsbyJanK.BruecknerandErikT.Verhoef*Pigouviantaxesarestandardinstrumentsfordealingwithnegativeexternalities.Theproblem,ofcourse,isthattheactivitygeneratingthenegativeexternaleectoccursattoohighalevel.ThePigouviantaxisdesignedtoraisethepricefortheactivity,reducingitsoverallleveltoonethatissociallyoptimal.Tocomputethetax,themarginalexternalitydamagefromtheactivityisevaluatedatthesocialoptimum,withthetaxoneachunitoftheactivitythensetattheresultingvalue.Facedwiththetax,theoendingagentsrestricttheirindividualactivitylevels,leadingtoanoptimaloveralllevel.Analternativetothis\classical"Pigouvianapproach,whichisusefulwhentheregulatorknowsthedamagefunctionbutlacksalltheinformationneededtocomputetheoptimum,istochargeatollequaltomarginaldamageevaluatedattheexpectedequilibriumactivitylevel.Inequilibrium,thisexpectationisconrmed,andthesociallyoptimalactivitylevelagainemerges.Inatypicalapplication,themarginalexternalitydamageisitselfafunctionoftheoverallactivitylevel,usuallyanincreasingone.However,thecomputedtaxisjustascalarvalue,generatedbyevaluatingthemarginaldamagefunctionatthesociallyoptimalactivitylevel(ortheexpectedequilibriumlevel,underthealternateapproach).Thus,whilethePigouviantaxruleinvolvestheleveloftheactivity,theper-unittaxultimatelychargedisjustanumbergeneratedbyevaluatingtheruleatparticularvalue.ThePigouvianapproachassumesthatagentstreattheresultingtaxasparametric,inde-pendentoftheirchosenactivitylevels.Thisviewisappropriatewhentheexternalityisjointlygeneratedbymanyagents,eachofwhommakesasmallcontributiontotheoverallleveloftheoendingactivity.However,whentheexternalityisgeneratedbyjustafewlargeagents,aparametricviewofthePigouviantaxmaybelessplausible.Tounderstandthispoint,imaginethatthesociallyoptimalactivitylevelvariesfromperiodtoperiodasaresultofchangesintheeconomy'sparameters(shiftsindemandorcostcurves,1 ManipulableCongestionTollsbyJanK.BruecknerDepartmentofEconomicsUniversityofCalifornia,Irvine3151SocialSciencePlazaIrvine,CA92697e-mail:jkbrueck@uci.eduandErikT.VerhoefDepartmentofSpatialEconomicsVUUniversityDeBoelelaan11051081HVAmsterdamTheNetherlandse-mail:everhoef@feweb.vu.nlJanuary2009AbstractTherecentliteratureoncongestionpricingwithlargeagentscontainsaremarkableinconsis-tency:thoughagentsarelargeenoughtorecognizeself-imposedcongestionandexertmarketpoweroverprices,theydonottakeintoaccounttheimpactoftheirownactionsonthemag-nitudeofcongestiontolls.Whenlargeagentsareconfrontedwithtollsderivedunderthisparametricassumptionbutunderstandtheruleusedtogeneratethem,thetollsystemwillnolongerguidethemarkettothesocialoptimum.Toaddressthisproblem,thepresentpaperderivesalternate,manipulabletollrules,whicharedesignedtoachievethesocialoptimumwhenagentsanticipatethefullimpactoftheiractionsontollliabilities.