此为英文文档nber -Corporate Social Responsibility and Voting over Public Goods

NBERWORKINGPAPERSERIES

CORPORATESOCIALRESPONSIBILITYANDVOTINGOVERPUBLICGOODS

AndrewA.Samwick

SophieWang

WorkingPaper31633

/papers/w31633

NATIONALBUREAUOFECONOMICRESEARCH

1050MassachusettsAvenue

Cambridge,MA02138

August2023

WethankthePresidentialScholarsfundatDartmouthforfinancialsupportandRajeshAggarwal,JulianLamprecht,NafisaLohawala,ErzoLuttmer,XuejuanSu,CurtWelling,seminarparticipantsattheWEAI98thAnnualConferenceandtheIIPF79thAnnualCongress,OmraneGuedhami(theeditor)andananonymousrefereeforusefulcomments.Anyerrorsareourown.TheviewsexpressedhereinarethoseoftheauthorsanddonotnecessarilyreflecttheviewsoftheNationalBureauofEconomicResearch.

NBERworkingpapersarecirculatedfordiscussionandcommentpurposes.Theyhavenotbeenpeer-reviewedorbeensubjecttothereviewbytheNBERBoardofDirectorsthataccompaniesofficialNBERpublications.

©2023byAndrewA.SamwickandSophieWang.Allrightsreserved.Shortsectionsoftext,nottoexceedtwoparagraphs,maybequotedwithoutexplicitpermissionprovidedthatfullcredit,including©notice,isgiventothesource.

CorporateSocialResponsibilityandVotingoverPublicGoods

AndrewA.SamwickandSophieWang

NBERWorkingPaperNo.31633

August2023

JELNo.D72,H41,M14

ABSTRACT

Thispaperanalyzestheimpactofcorporatesocialresponsibility(CSR)onthetotalprovisionofpublicgoodsinaframeworkinwhichconsumerswhomaymakesuchvoluntarycontributionstopublicgoodsviaCSRarealsovoterswhodecideontheleveloftaxestofinancepubliclyprovidedpublicgoods.Themainresultindicatesthat,relativetoaneconomyinwhichallpublicgoodsarepubliclyfinanced,theintroductionofCSRlowersthetotalamountofpublicgoods,asvotersrationallyanticipatethathigherCSRwillpartiallyoffsettheconsequencesoflowerpublicfunding.TheresultsofferacautionarytaleaboutthepromotionofCSRinaneconomywithheterogeneouspreferencesforthepublicgood.

AndrewA.Samwick

6106RockefellerHall

DepartmentofEconomics

DartmouthCollege

Hanover,NH03755-3514

andNBER

andrew.samwick@

SophieWang

DartmouthCollege

sophie.wang32@

1

1Introduction

TheBusinessRoundtablemadeheadlinesin2019when,forthefirsttimeinovertwodecades,itissueda

StatementonthePurposeofaCorporation

thatacknowledgedspecificresponsibilitiestostakeholdersbeyondshareholders,includingcustomers,em-ployees,suppliers,andcommunities.SuchCorporateSocialResponsibility(CSR)hasbeengrowingmarkedlyinrecentyears.

Dattanietal.

(2015)establishabaselineof

$20billionspentonCSRin2013byGlobalFortune500companies.

Stobierski

(2021)

reportsthatbetween2011and2019,theshareofS&P500companiespublishingaCSRReportrosefrom20to90percent.

InareviewoftheearlyeconomicsliteratureonCSR,

KitzmuellerandShimshack

(2012)defineCSRas“thecorporateprovisionofpublicgoodsorreductionofnegative

externalitiesbeyondwhatisrequiredbylaw.”MuchofthescholarlydiscussionofCSRhasfocusedonthemotivesthatshareholderswouldhaveforCSR,startingwithandoftenrespondingto

Friedman

(1970)’sseminalessay.

Hisconclusionthattheonlysocialresponsibilityofbusinessistoincreaseprofitsisfacilitatedbyassumptionsthatshareholderscanachievetheirphilanthropicobjectivesequallywelloutsidethefirmandwouldprefertodoso.

Forexample,underlyingFriedman’scontentionistheassumptionthattherearenoadvantagestocontributionsbyshareholdersactinginconcertviathecorporationrelativetocontributingasindividualsontheirown.Contrarytothisassumption,

B´enabouandTirole

(2010)positthatthefirmmayfacelowertransactionscostsin

makingdonationsormitigatingexternalitiesasarationaleforshareholdersdelegatingphilanthropythroughCSR.

1

Inabriefessaycommemoratingthe50thanniversaryof

Friedman

(1970),

Hart

(2020)expandsthisargumentbynotingthatgivingthrough

thecorporationmayhelpovercomefree-riderproblemsinindividualgivingtopub-licgoods,aseachshareholder’s“donation”isconditionalonthedonationsofalltheothershareholders.Additionally,asseveralauthors,mostnotably

Baron

(2001),have

noted,CSRmaybe“strategic”inthesensethatitrespondstostakeholderprefer-encesforthisvoluntaryovercomplianceand,indoingso,mayenhanceprofitability.

2

1ElGhouletal.

(2017)showthatCSRismorepositivelyassociatedwithfirmvalue(asmea

-suredbyTobin’sq)incountrieswithweakermarketinstitutions,suggestingthatCSRishelpingtocounteractgreatertransactioncostsandlimitedaccesstoresources.

2Thatcompaniesarerespondingtostakeholderpreferencesissuggestedby

Rubin

(2008)’sfindings

thatcompanieswithhighCSRratingsaretypicallyin“blue”statesorcountieswhereascompanieswithlowCSRratingsaretypicallyin“red”statesorcounties.Similarly,

DiGiuliandKostovetsky

2

Friedmanacknowledgedthispossibilityinhisessayandfocusedinsteadonvalue-reducingCSRthatrespondstomanagerialpreferencesthatdifferfromthoseoftheshareholders.

However,CSRcanalsoemergeastheexpresseddesireofshareholdersratherthanasanagencyconflictbetweenshareholdersandmanagers.As

HartandZingales

(2017)argue,shareholdersoftenactinprosocialways,attheirownexpense,intheir

isolateddecisionsasindividuals.Itisnaturalthattheywouldalsoseektodosointheirrolesasshareholders,whentheycansharethecostsoftheirphilanthropicbehaviorwithothers.Inlightofthis,maximizingshareholderwelfareneednotbethesameasmaximizingmarketvalue.Further,

Baron

(2007)developsatheoryof

CSRinananalyticalmodelthatreflectstheeconomicenvironmentenvisionedby

Friedman

(1970)andshowsthatprosocialentrepreneurswillformfirms,evenata

financialloss,toopenupnewopportunitiesforotherstoengageinCSR

.3

LesswelldevelopedintheexistingliteraturearethedownstreamconsequencesoftheavailabilityofCSRonthebehaviorofthestakeholdersotherthanshareholders.Toanalyzesuchconsequences,weabstractfromtheshareholderconsiderationsnotedabove,assumeidenticalcostsforthepublicgoodacrossprivateCSRandtax-financedpublicprovision,andpositfreeentrybyfirmsthatdriveseconomicprofitstozeroforanydegreeofCSR.

4

Ourmodeltakesasitsstartingpointthatof

BesleyandGhatak

(2007),whoshowthatastylizedversionofCSRcanbemodeledintheframework

ofvoluntary,privatecontributionstopublicgoodsintroducedby

Bergstrometal.

(1986)

.Inamodelofpurepublicgoodswithtwotypesofconsumers–thosewhocareaboutthepublicgoodandthosewhodonot–theydemonstratethatCSRgeneratesaParetoimprovement,ascaringconsumerswillengageinCSRandneutralconsumerswillnotchangetheirbehavior.Theyfurthershowthatanexogenous

(2014)findthatfirmswithDemocraticfoundersorCEOsspendmoreonCSR.Whiletheyalsofind

thatincreasesinfirmCSRratingsareassociatedwithnegativefuturestockreturnsanddeclinesinthefirm’sreturnonassets,

Friedeetal.

(2015)find,inaggregatingdatafromover2,000studies,that

thereisgenerallyanonnegative(thoughnotnecessarilycausal)relationshipbetweenenvironmental,social,andgovernancecriteriaandcorporatefinancialperformance.

3See

SchmitzandSchrader

(2015)foramoreextensivediscussionofthemotivescorporations

mighthaveforCSR.Inarecentsurveyoftheliterature,

HongandShore

(2022)concludethat

shareholderinterestinCSRisdrivenprimarilybynon-pecuniarymotives–awillingnesstosacrificereturnstoaddressvariousfirmexternalities–ratherthanbypecuniarymotiveslikethepursuitofabnormalratesofreturn.

4WethusalsoabstractfromtheinteractionofCSRwitholigopolisticcompetition.See

Bagnoli

andWatts

(2003)forsuchamodel

.

increaseinthepublicprovisionofthepublicgoodwillcrowdoutvoluntaryprovisionbyCSR.

OurinvestigationofCSRismotivatedbytheobservationthattheconsumerswhodecidewhethertoengageinCSRviatheirpurchasesarealsothevoterswhodecidewhethertotaxthemselvestoprovidethepublicgoodthroughthepublicsector.Toconsidertheimplicationsofthisduality,weaugmentthemodelof

Besleyand

Ghatak

(2007)intwoways.First,weallowformultiplecaringtypesofconsumers,

whoengageinCSRtovaryingdegreesaccordingtotheirpreferences.Thischangetothemodelnecessitatesusingtheversionoftheirmodelwithanimpurepublicgood,incorporatingthe“warmglow”ofgivingdescribedby

Andreoni

(1989,

1990).

5

Inourmodel,individualsdifferonlyintheintensityoftheirpreferenceforthepublicgood,withseparateparametersindexingthedirectutilityfromthetotalstockofpublicgoodsandthewarmglowofgivingthroughCSR.Second,weintroduceafirst-stageofvotingontheamountoftax-financedpublicgoodtobeprovided.The“warmglow”isassumedtobegeneratedbythevoluntarycontributionstopublicgoodsprovidedthroughCSRbutnotbytaxes.

EndogenizingtheamountofthepublicgoodprovidedthroughtaxesallowsustoconsidertheextenttowhichtheprospectofsubsequentCSRbyconsumerscrowdsoutpublicprovisionbyvoters.Wecomparethelevelofpublicgoodsunderthreeregimesdistinguishedbyhowtheycanbefunded:PublicOnlythroughtaxes,VoluntaryOnlythroughCSR,andPublicandVoluntarythroughboth.WederivetheequilibriumundereachregimeandshowthatthetotalprovisionofpublicgoodsislowerwhenCSRandpublicprovisionarebothpossiblecomparedtowhenallpublicgoodsareprovidedpubliclythroughtaxes.VotersrationallyanticipatethatlowerCSRwillpartiallyoffsettheconsequencesofhigherpublicfunding,andthisoffsetlowersthemarginalbenefitofsupportinghighertaxestofinancepublicprovision.Foragivenmarginalcostofprovidingthepublicgood,thisreductioninthemarginalbenefitreducestheequilibriumamountofthepublicgood.Ingeneral,theadditionofCSRopportunitiesdoesnotgenerateaParetoimprovement.Weshowconditionsunderwhichamajoritycoalitionprefersthehybridregime,despiteitslowerlevelofpublicgoods.

Ourresultsaremostsimilartothepriorworkof

Calverasetal.

(2011)and

Epple

5SeeCornesandSandler(1994)foranearlyanalysisofthecomparativestaticpropertiesofsuch

amodel.

3

4

andRomano

(2003),whoexploreprivatecontributionstopublicgoodsinframeworks

withoutexplicitCSR.

Calverasetal.

(2011)defineamore“altruistic”societyasone

inwhichwarmglowpreferencesforprivatecontributionsarestronger.Theyshowthatmajorityvotinginamorealtruisticsocietyleadstoalowertaxrateandmayalsoleadtolowerequilibriumprovisionofthepublicgood.

EppleandRomano

(2003)

useaframeworkinwhichhouseholdshavethesamepreferencesbutdifferintheirendowmentsofincome.Householdsvoteoveranincometaxbutdonotfacealabor-leisuretradeoff.Theyshowthatwhilepermittingprivatecontributionsmayleadtoareductionintotalprovisionofthepublicgood,amajorityalwaysfavorspermittingprivatecontributions.

Theremainderofthepaperisorganizedasfollows.InSection

2

,weexpandtheframeworkof

BesleyandGhatak

(2007)toincorporatemultiplecaringtypes

withawarmglowofCSR.WestartwiththesocialoptimumandthenderivetheequilibriumlevelsofpublicgoodswhenthereisafirststageofmajorityvotingbeforeCSRdecisionstakeplace.WecomparepublicgoodprovisionunderthethreeregimesinSection

3

,derivingthemainresultthatCSRreducesthetotalamountofpublicgoods.WeprovideconditionsforamajoritytopreferthehybridregimeinSection

4

andconsidertheimpactofchangesinpreferencestowardmorecaringforpublicgoodsinSection

5

.Section

6

discussestheimplicationsofourfindingsandconcludes.TheresultsofferacautionarytaleaboutallowingandpromotingCSRiftheobjectiveistoincreasethetotalamountofthepublicgood.

2Model

Ourmodelbuildsonthatof

BesleyandGhatak

(2007),whoanalyzeCSRinthe

contextofamodelwithtwotypesofconsumers,caringandneutraltowardCSR.Weadopttheirnotationwherepossible.Becauseweseektostudytheinterplaybe-tweenvotingforpubliclyprovidedpublicgoodsandCSR,weexpandtheirframeworktohavemultipletypesthatmightcare,todifferentextents,aboutthepublicgood,throughboththetotalamountofthepublicgoodandthe“warmglow”thatcon-sumersgetthroughtheirindividualamountsofCSR.Individualsdecide,asvoters,onanamountofthepublicgoodprovidedbythepublicsectorwhilethesameindi-viduals,asconsumers,decideonhowmuchCSRtoengageinviatheirconsumption.Inbothcases,ourassumptionsaremeanttobelessrestrictiveandmorerealistic,

5

acknowledgingthatvotersareultimatelyresponsibleforsettingtheleveloftaxesandthuspubliclyprovidedpublicgoodsandthatpreferencesforpublicgoodsaremorevariedthanapopulationthateithercaresornotaboutthelevelofpublicgoods.

Themodelof

BesleyandGhatak

(2007)issetinanenvironmentwherethereis

onetypeofpublicgoodandtwotypesofprivategoods.Thepublicgoodmustbeproducedwhileoneprivategoodisproducedandtheotherisendowedtoallproducersandconsumersintheamountb.ConsumerscanmakevoluntarycontributionstothepublicgoodthroughCSRbybuyingfromaproducerthatbundlestheprivategoodwithanamountθofthepublicgood.Eachunitofthepublicgoodcostsα,whetherproducedbythepublicsectorinamountGorcontributedviaCSR.TheyalsoassumethereisfreeentryfromproducerswhocompeteinBertrandfashionandthusearnzeroprofits.Theyproducetheprivategoodatcostcplusαθ,fortheamountoftheCSRassociatedwiththepurchase.Withzeroprofits,thepricefortheprivategoodbundledwithanamountθofthepublicgoodisp=c+αθ.Themodelthusabstractsfrombothentrepreneurs’motivationsforCSRandanyinteractionofCSRwithstrategiccompetitioninproductmarkets.

WeassumeapopulationofindividualsofsizeNwhoaredistinguishedbytheirpreferenceparametersforthepublicgood,γiandηi.Wespecifyautilityfunctionforindividualioftheform:

Ui=b.(c+αθi).+γif(G+Θ)+ηiv(θi)(1)

Publicgoodsprovidedbythepublicsectorarefinancedbyalumpsumtax,infirstisthroughthetotalamountofthepublicgood,G+Θ,whereΘ=!θi.Thefunctionf(G+Θ)isassumedtobeincreasingandstrictlyconcave,withf(>0andf((<0.ThesecondisthroughthewarmglowoftheCSR,v(θi),similarlywithv(>0andv((<0.

6

Ourapproachistoimposeminimalassumptionsonthejointdistributionofthepreferenceparameters,{γi,ηi},beyondtheirnon-negativityandatechnicalassumptionthatifγiisnon-zero,thenηiisalsonon-zero.Thus,ourmodeldiffersfrom

BesleyandGhatak

(2007)inthreemainways.First,weallowfor

anon-exogenouslevelofGwhichisfinancedbyalumpsumtax.Second,weallow

6Theseassumptionsareconsistentwiththefindingsof

Carpenter

(2021),whousesafieldexper

-imenttoshowthatfordonorsmotivatedbythewarmglowofgiving,itsshapeisincreasingandconcave.

6

formultipletypesbeyondγ=0orγ=1.Third,theexistenceofmultipletypesengaginginCSRrequirestheassumptionof“warmglow”utility,which,as

Besley

andGhatak

(2007)note,leavestheirmainresultsessentiallyunchanged.Insodoing,

weintroduceaseparatepreferenceparametertoindexthestrengthofthewarmglow,asdistinctfromthestrengthofthepreferenceforthetotalamountofthepublicgood.

2.1ASocialPlanner’sProblem

Beforederivingequilibriaindecentralizedmodels,weconsiderasocialplannerwhomaximizesaggregatewelfare,basedonEquation(

1

):

W=[b.(c+αθi).+γif(G+Θ)+ηiv(θi)](2)

Wenotethatindividualsdifferonlywithrespecttotheparameters,γiandηi,andnotwithrespecttotheirendowment,b.Thus,thereisnoredistributivemotiveforthetaxbasedonunequalendowmentsandnoparticularreasontoweightdifferentindividuals’utilitiesdifferently.Thelumpsumtaxisakintoamembershipfeeinagroupofindividualswhodifferonlyintheirpreferenceparameters.

SummingovertheNconsumers,wecanwrite:

W=N(b.c).α(G+Θ)+(γi)f(G+Θ)+(ηiv(θi))(3)

ThesocialplannerwouldchooseGand{θ1,...,θN}tomaximizethisexpression,subjecttotheconstraintsthatGc0andθic0,Ⅴi.IntheabsenceofCSR,thesocialplannerwouldchooseGaccordingtothefirst-ordercondition:

α=(γi)f((FB)(4)

Equation(

4

)isthefamiliaroptimalityconditionthatthemarginalcostofthepublicgoodshouldequalthesumofthemarginalutilitiesofthegoodacrossallconsumers.

7

Herewemaketheimplicitassumptionthatthemarginalutilityatzero

publicgoodsissufficientlyhigh,f((0)>,sothatFB>0.WithCSRavailable

atthesameresourcecostaspublicprovision,however,itisclearfromEquation(

3

)

7WedenotethischoiceofGasFB,withthetildeindicatingthatthisisnotavaluechosen

simultaneouslywiththevaluesof{θ1!!!!!θN}whenCSRisavailable.

7

thatasocialplannerwouldsetG=0giventheassumptionthatv(>0.AnypositiveamountofGcouldbereallocatedtosomeindividual’sθi,generatingawarmglowandthusanincreaseinaggregatewelfare,whilekeepingthetotalcostofthepublicgood,.α(G+Θ),andthedirectutilityfromthetotalamountofthepublicgood,f(G+Θ),unchanged.

WithGFB=0,thesocialplannerwouldchooseeachvalueofθjaccordingtothefirst-ordercondition:

α=!γi(f((ΘFB)+ηjv((θ)(5)

Inthiscase,themarginalcostofproducingthepublicgoodmustbeequaltothesumofthemarginalutilitiesofthepublicgoodacrossallconsumersplusthemarginalutilityofthewarmglowforindividualj.Theassumptionthatv((<0ensuresthat

θisincreasinginηjamongthoseengaginginCSR.Comparingthetwofirst-order

conditions,wecanseethatbecausev(>0,f((ΘFB)<f((FB),andthatbecause

f((<0,itmustbethatΘFB>FB.Thus,undertheseconditions,thepresence

ofCSRwithawarmglowincreasesthesociallyoptimalamountofthepublicgoodprovided.

Inthisfirst-bestallocation,wecandefinethesetofcontributors,CFB,asthepairsof{(γi,ηi)}suchthattheright-handsideofthefirst-orderconditioninEquation(

5

)withθ=0exceedstheleft-handside:

CFB={(γi,ηi)|α<γif((ΘFB)+ηiv((0)}(6)

Thefirst-bestwillbesuchthatallindividualswith(γi,ηi)∈CFBwillengageinCSRasgivenbyEquation(

5

)andthosewith(γi,ηi)CFBwillnotengageinCSR

.8

2.2VotingasaFirstStage

Wenextconsiderhowequilibriumamountsofthepublicgoodaredeterminedinadecentralizedmodelwithtwostages.Inthefirststage,individualsvoteontheamount

8Inhisanalysisoftheoptimaltaxtreatmentofprivatecontributionsforpublicgoods,

Diamond

(2006)discussescircumstancesunderwhichitwouldnotbeappropriatetoincludethewarmglowin

thesocialplanner’sobjectivefunction.Inthiscase,thetotalamountofG+ΘisgivenbyEquation

(4),butthedistributionofthatamountacrosspublicandprivatecontributionsisnotdetermined

.

8

ofthepublicgood,G

.9

Inthesecondstage,individualsmaketheirconsumptiondecisionsoverhowmuchCSRtoengagewiththeirprivategoodpurchase,θi,atanadditionalcostofαθi.Whenindividualsvoteinthefirststage,theyanticipatehowtheamountofthepublicgoodprovidedthroughtaxeswillaffecttheCSRchoicesofallpotentialconsumersinthesecondstage.

ReturningtotheindividualutilityfunctioninEquation(

1

),andtemporarilysuppressingtheisubscriptsforreadability,wecanconsidertheconsumer’schoiceinthesecondstagebyholdingGfixedin:

b.(c+αθ).+γf(G+Θ)+ηv(θ)(7)

ForagivenamountofG,eachindividual(inthesecondstage)willmaximizethisutilitywithrespecttoθ,yieldingthefirst-ordercondition:

γf((G+Θ)+ηv((θ)=α(8)

ThisequationimplicitlydefinesarelationshipbetweenthechoicesofθandthevalueofG

.10

Asabove,thesetofcontributors,C,includesallcombinationsof{γ,η}suchthat:

C={(γ,η)|α<γf((G+Θ)+ηv((0)}(9)

Forsuchconsumers,themarginalbenefitoftheadditionalCSR,throughboththedirectutilityofthepublicgoodandthewarmglow,isgreaterthantheresourcecostofproducingit.Fortheseconsumerswith(γ,η)∈C,thefirst-orderconditioninEquation(

8

)determinestheirchoiceofθ.

Movingbacktothefirststage,takingthederivativeofEquation(

7

)withrespecttoGyieldsthefirst-orderconditionforavotercontemplatingwhethertosupportan

increaseinG:

9Individualsinthemodelarenotmakingalabor-leisuretradeoff,andincomeisequalacrossindividuals,soanincometaxisidenticaltoauniformlumpsumtaxasmodeledhere.

10Equation(

8

)showswhythewarmglowassumptionisnecessaryinthismodelinwhichmorethanonecaringconsumerengagesinCSR.Withf()thesameacrossconsumers,absentthev≥()term,therewouldbenowayforthisfirst-orderconditiontoholdsimultaneouslyfortwodifferentvaluesof.Similarly,thisisthereasonforthetechnicalassumptionthatmustbepositiveforanyconsumerwithpositive.

9

.α.+γf((G+Θ)╱1++ηv((θ)=0(10)

whichcanberewrittenas:

=γf((G+Θ)╱1++(ηv((θ).α)(11)

IfthisvoterengagesinCSR,withθ>0,thensubstitutinginthefirst-orderconditionforθfromEquation(

8

)intothelastterminEquation(

11

)yields:

=γf((G+Θ)╱1+.(12)

IfinsteadthevoterdoesnotengageinCSR,withθ=0,then=0,andEquation(

11

)reducesto:

=γf((G+Θ)╱1+(13)

Inbothequations,theleft-handsideistheindividualvoter’smarginalcostofanadditionalunitofG,andtheright-handsideisthevoter’smarginalbenefitofthatadditionalunit.Thefirst-orderconditionincorporatesthepossibledependenceofeachθ,andthereforeΘ,onG.Inbothequations,themarginalbenefitofthepublicgoodfinancedthroughthelumpsumtaxisequaltotherelativeweightonthepublicgoodintheutilityfunction,γ,multipliedbythemarginalutilityofanadditionalunitofthepublicgood,f((G+Θ),multipliedbytheneteffectoftheincreaseinGon

thetotalstockofthepublicgood,╱1+..Aswewilldiscussbelow,because

f((<0,thetotalamountofthepublicgood,G+Θ,ispositivelyrelatedtothisneteffect.

Toobtainananalyticalexpressionforthisneteffect,wetakethetotalderivativeofthefirst-orderconditionforθinEquation(

8

):

γf(((G+Θ)>dG+γf(((G+Θ)>dΘ+ηv(((θ)>dθ=0

Dividingthroughbyγf(((G+Θ)>dGyields:

1++>>=0

dΘηv(((θ)dθ

dGγf(((G+Θ)dG

(14)

(15)

10

Equation(

15

)showsthatwithv()andf()strictlyconcave,crowdingoutwillbe

imperfect,1+>0,aslongascrowdoutexistsforsomeindividual,

<0.(Note

thatcrowdingoutwouldbeperfectforv((=0.)Thisexpressionfor1+canbesubstitutedintoEquation(

12

)toobtain:

=.γf((G+Θ)(1+>

(16)

Thelaststepistofindananalyticalexpressionfor.Reintroducingsubscriptsiandjtodistinguishdifferentindividuals,Equation(

15

)impliesthat:

(17)

>>=>>

foranyijsuchthatθi,θj>0.Thus,

Bydefinition,

dθidG

=ΣiEC

=

>>

dG

>dθj

(18)

wherethesummationonlyappliestothesubsetof

consumerswhoengageinCSR.Nowthatwehaveanexpressionforintermsof,wecansubstitutethisintoEquation(

15

)andrearrangetoget

1+(>>+>=0

andsubstitutingbackintoEquation(

16

)gives:

α

N

=γjf((G+Θ)

(1+>

(ΣiEC>>+

Factoringout>,givesafinalexpression:

α((1+>

N=γjf(G+Θ)(1+ΣiEC>

(19)

(20)

(21)

Whenthisvoterdoesnotengageitcanbeshownthatthenewexpressionforthisvoter’schoiceofGis:

11

11Thestepsinthederivationarethesame,recognizingthatthe1inthenumeratorinEquation

(20)isnotpresentbecausethevoterisnotengaginginCSR.

11

αN

=γjf((G+Θ)

1

╱1+!iEC>

(22)

IfvoterpreferencesforGaresingle-peaked,thenthemedianvotertheoremwillapply,andwecanexpressEquations(

21

)and(

22

)as:

αN

=γmedf((G+Θ)Kmed

(23)

Here,Kmedisthefinaltermineitherofthetwopriorequationswithjchosentobethevoterwiththemedianvalueofγj.AsinEquations(

12

)and(

13

),itcorrespondstotheneteffectofanincreaseinGontheamountofCSRcontributedbyconsumersotherthanthemedianvoter.Kmedispositivebecausef()andv()arestrictlyincreasingandconcave.ItislessthanonebecausethedenominatorislargerthanthenumeratoraslongasatleastoneconsumerengagesinCSR

.12

Thus,publicprovisionofthepublicgoodimperfectlycrowdsoutvoluntarypro-visionthroughCSR.Thedegreeofcrowdingoutdependsontherelativemagnitudesof|f((|and|v((|.Kmedislower,andthuscrowdingoutishigher,when|f((|ishighrelativeto|v((|.Insuchcases,agivenchangeinGgeneratesalargeoffsettingchangeinθithrougheachconsumer’sfirst-ordercondition.

Equation(

23

),combinedwiththefirst-orderconditioninEquation(

8

)foreachconsumerwhoengagesinCSRo