Recreation Demand Models with Taste Differences ov

RecreationDemandModelswithTasteDifferencesOverPeoplebyKennethE.TrainDepartmentofEconomicsUniversityofCalifornia,BerkeleyJuly1997Forthcoming,LANDECONOMICS,Vol.74,No.2,May1998.ABSTRACT:Weestimaterandom-parameterlogitmodelsofanglers'choiceoffishingsite.Themodelsgeneralizelogitbyallowingcoefficientstovaryrandomlyoveranglersratherthanbeingfixed.Themodelsdonotexhibittherestrictiveindependencefromirrelevantalternativespropertyoflogitandcanrepresentanysubstitutionpattern.Estimationexplicitlyaccountsforthefactthatthevariationincoefficientsoveranglersinducescorrelationinunobservedutilityovertripsbythesameangler.Willingness-to-payforimprovedfishstockandthevaluetoanglersofspecificsitesarecalculatedfromthemodelsandcomparedwiththeestimatesobtainedfromastandardlogitmodel.1I.IntroductionRecreationdemandmodelsareusedtoforecastdemandforrecreationalactivitiesaswellastodeterminethevaluethatrecreatorsplaceonthevariousfactorsthataffecttheirchoices.Aprominentexampleisfishingmodels,whichdescribeanglers'choicesofwhethertotakeafishingtripduring1agivenperiod(e.g.,week),whichspeciesoffishtotarget,and/orwheretogofishing(i.e.,thesite).Themodelsrelateanangler'schoicestothecharacteristicsoftheavailableoptions(suchasthetimeandcostoftravelingtoeachsite,theavailabilityoffishateachsite,theavailabilityofcampgrounds,andsoon)andtothecharacteristicsoftheangler(suchasage,gender,andincome.)Themodelsprovideestimatesofanglers'willingnesstopayforchangesinsiteattributes,suchasincreasedfishstocksorreducedcontaminantsatspecifiedsites,aswellasthevalueofindividualsitesthemselves.2Recreationaldemandmodelshaveusuallybeenspecifiedaslogitornestedlogitmodels.These3specificationshaveseveraladvantages,includingsimplicityofestimation.However,theyimposeseveralwell-knownrestrictions(McFadden1973,1978;Train1986.)First,thecoefficientsofvariablesthatenterthemodelareassumedtobethesameforallpeople.Thisassumptionimpliesthatdifferentpeoplewiththesameobservedcharacteristicshavethesamevalue(i.e.,tastes)foreachfactorenteringthemodel.Second,logitandnestedlogitexhibittheindependencefromirrelevantalternatives(iia)property.(Logitexhibitsthispropertyforallalternatives,andnestedlogitexhibitsitwithineachnest.)Becauseofthisproperty,themodelsnecessarilypredictthatachangeintheattributesofonealternativechangestheprobabilitiesoftheotheralternativesproportionately.Thissubstitutionpatterncanbeunrealisticinmanysettings.Third,insituationswithrepeatedchoicesovertime,logitandnestedlogitassumethatunobservedfactorsareindependentovertimeforeachdecision-maker.Inreality,however,onewouldexpectunobservedfactorsthataffectadecision-makertopersist,atleastsomewhat,overtime.Inthecurrentpaperweestimaterecreationdemandmodelswithaspecificationthatisageneralizationoflogitandavoidstheselimitations.Inparticular,weestimatearandom-parameterslogit(RPL)modeloffishingsitechoice.RPLgeneralizeslogitbyallowingthatthecoefficientsof42observedvariablestovaryrandomlyoverpeopleratherthanbeingfixed.Withthisgeneralization,themodeldoesnotexhibittheiiapropertywithitsrestrictivesubstitutionpatterns.Infact,anypatternofsubstitutioncanberepresentedarbitrarilycloselybyanRPL(McFaddenandTrain1997).Thevariationincoefficientsoverpeopleimpliesthattheunobservedutilityassociatedwithanyalternativeisnecessarilycorrelatedovertimeforeachdecision-maker.Thiscorrelationisincorporatedintotheestimationwhenthereareobservationsonmorethanonechoicesituationforeachperson.ThespecificationandestimationofRPLaredescribedinthefollowingsection.TheapplicationtofishingsitechoiceisdescribedinsectionIII.II.Random-ParametersLogitRPLmodelshavetakendifferentformsindifferentapplications;theircommonalityarisesintheintegrationofthelogitformulaoverthedistributionofunobservedrandomparameters.Theearlyapplications(BoydandMellman1980,andCardellandDunbar1980)wererestrictedtosituationsinwhichexplanatoryvariablesdonotvaryoverdecisionmakers,suchthattheintegration,whichiscomputationallyintensive,isrequiredforonlyonedecisionmakerusingaggregatesharedataratherthanforeachdecisionmakerinasample.Advancesincomputerspeed,aswellasgreaterunderstandingofsimulationmethodstoapproximateintegration,haveallowedestimationofmodelswithexplanatoryvariablesvaryingoverdecisionmakers.ExamplesincludeErdem(1995),Ben-AkivaandBolduc(1996),Bhat(1996a,b),BrownstoneandTrain(1996),Mehndiratta(1996),andReveltandTrain(1996).TheformoftheRPLthatweutilizeinourinvestigationisdescribedasfollows.AnanglerchoosesamongJpossiblesiteseachtimehe/shetakesafishingtrip.TheutilitythatanglernwouldobtainfromsitejintriptisU=1x+wherexisavectorofobservedvariables,njtnnjtnjtnjtnisavectorofcoefficientsthatisunobservedforeachnandvariesrandomlyoveranglersrepresentingeachangler’stastes,andisanunobservedrandomtermthatisdistributediidextremevalue,njtindependentofandx.nnjtThisspecificationisthesameasforlogit,exceptthatnowthecoefficientsvaryinthepopulationn3ratherthanbeingfixed.Thevarianceininducescorrelationinutilityoversitesandtrips.Innparticular,thecoefficientvectorforeachangler,,canbeexpressedasthesumofthepopulationnmean,b,andin