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FindingtheOptimalTaxwithPollutionHavenEffects

AleciaWaite1

Inthispaper,Iseekasolutiontothegovernment’sproblemofimposinganoptimalpollutiontaxtocontrolemissions.Iuseamodelwithtwocountriesandforeigndirectinvestment.Firmsleavethecountryiftheircostsofproductionbecomeprohibitivelyexpensive.Thismimicstherealworld,inwhichindustriesmayfleeandproduceabroadifregulationistoostrict.Inmymodel,thegovernmentderivesutilityfromconsumption,whichisincreasinginthenumberofindustriesthatstayathomeandadecreasingintemperature.IcombineLevinsonandTaylor’s(2008)theoreticalframeworkforpollutionabatementandtradewithWeitzman’s(2009)globalwarmingutilityfunctions.Then,Iseektoanswerthequestions,“Towhatextentshouldgovernmentsregulate?Howdoesthisdependoncountrysize?”I.Introduction

ThePollutionHavenHypothesis(PHH)positsthatenvironmentalregulationchangestradeflowsbecauseproductionof“dirty”goodsisoutsourcedtocountrieswithlessstringentenvironmentalregulation.(CopelandandTaylor,2004)Stricterenvironmentalregulationscauseacountry’simportdemandforagivendirtygoodtoincrease,soforanyworldprice,thecountryimportsmoreofthatgood,causingemissionsoftradepartnerstorise.

ThePHHhasimportantimplicationsforenvironmentalregulationandtradefortwo

reasons.First,whereitoccurs,itunderminesunilateralcarbondioxideemissionspolicy.Whenonecountrydecidestomorestrictlyregulate emissions,ithopestodecreasetotalworldemissions.Regrettably,ifthePHHholds,productionisoutsourcedandthusemissionsmerelyshifttoanothercountry.Hence,worldemissionsdonotchange.Second,thepollutionhaveneffectcreateseconomicdistortions.Considerforamomenttwocountries,AandB.AssumethatcountryAhasacomparativeadvantageinproductionofdirtygoods.WhencountryAimplementsenvironmentalregulations,productionofdirtygoodsshiftstocountryB,thecountrythatdidnothaveacomparativeadvantageinproductionofdirtygoods.Insum,thepollutionhaveneffectunderminesunilateralenvironmentalregulationanddistortsefficienttradepatterns.

SomepreviousempiricalstudiessupportthePHH.AicheleandFelbermayr(2010)findthattheKyotoprotocolledtosubstantialcarbonleakage.TheyalsofindthatKyotomembershadmoreincentivesandsubsidyprogramstargetedatemissionreduction.GretherandMathys(2008)notethatproductionhasshiftedfastereastwardsthan

                                                            1 AleciaWaitegraduatedfromDukeUniversitywithaMaster’sinEconomicsin2012.SheiscurrentlypursuingaPhDinEconomicsattheUniversityofMichigan.Shecanbereachedataawaite@umich.edu.ThankstoCharlieBecker,OzgurKibris,EdTower,andTravissCassidyforvaluablecomments.

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economicactivity,andthisfindingsupportsthecarbonleakagehypothesis.KellenbergandLevinson(2010)exploreinternationalwastetradeandfindthattheBanAmendmenttotheBaselConventionontheControlofTransboundaryMovementsofHazardousWasteandTheirDisposalwasinfluentialinchangingtradepatterns.Caves(1982)andBrainard(z)showthatallelseequal,foreigninvestorswilllocateincountrieswherefactorstheyuseinhighproportionsarerelativelyinexpensive.XingandKolstad(2002)undertakeanempiricalstudyofUSFDIindirtyindustriesandfindapositiverelationshipbetweenFDIandpollutionintensity.

Manyviewsexistonwhythepollutionhaveneffectisobserved.Oneimportantexplanationisfirmresponsetopolicyintheformofabatement.Cole,Elliot,andFrederickson(2007)andSwee(2003)explainthatchangesinfactorreturnsareduetoincreasedabatementactivity.Abatementactionthatfirmsundertakeinresponsetotaxesleadstoanambiguousoutcomewhenitcomestotheeffectofenvironmentaltaxesoninvestmentandoutputdecisions.Shouldanindustrychoosetoinvestmorecapitalinthehomecountryinordertomakeabatementrequirementslessexpensivetocomplywith,theindustrymayincreaseoutputathometotakeadvantageoflowermarginalcosts.

OtherempiricalstudiesseemtocontradictthefindingsonthePHH.Eskelandaand

Harrison(2003)questionwhetherthepollutionhavenhypothesisisupheldintheUSandfindthattheevidenceisweak.Ederingtonetal(2004)undertakesastudyofUStradeliberalizationandfindthatthereisnoevidencethatpollutionintensiveindustriesaredisproportionatelyaffectedbytariffchanges.

TherearealsoplausibleexplanationsastowhywewouldnotobservethePHH.

Ederingtonetal(2005)showthatdirtyindustriesarelessmobilethanotherindustries,supportingtheassertionthatliberalizationitselfwouldnotcauseapollutionhaveneffect.Brainard(1997)explainsthattariffbarriersandtransportcostsmeanthatevenifconditionsarefavorable,industriesmaynotrelocate(ormoreplausibly,fewerwillrelocate).Economiesofscalealsoplayapartinarelocationdecision.

Interestingly,someauthorshavearguedthattherelationshipbetweengovernment

stringency(orlackthereof)andFDIcouldgobothways.ColeandElliot(2006)andCole,Elliot,andFrederickson(2007)commentona“reversePollutionHavenHypothesis,”thatis,aneffectofFDIonthestringencyofenvironmentalpolicy.Foreignfirmslobbyandbribehostcountrygovernmentsinordertoweakenenvironmentalregulation.

MoststudiesofthePHHdiscusshow,if,andwhyitoccurs.Iaminterestedin

calculatinganoptimaltaxforonegovernment,takingthepollutionhaveneffectasgiven.IdothisbecausethepreponderanceofevidenceisinfavorofthePHHholding,andwhereitdoesnothold,thereareoftenspecificandanomalousreasons,suchasimmobilityoffirms.

ArelatedpaperbyLight(1999)considerstheproblemofallcountriestaken

togetherandcalculatesanoptimaltaxgivenatargetreductioninglobalcarbonemissionusinganumericaltechniquecalledMathematicalProgramswithEquilibriumConstraints(MPEC).Hefindsthattheoptimaltaxwouldneedtobalancethebenefitsofequalmarginal

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abatementcostsagainstthecostsofadditionalcarbonabatementasaresultofcarbonleakagetoothercountries.Inthecasethatthesecondeffectisstronger,Lightparadoxicallyfindsthatitmaybebesttoexemptsomepollutersfromthetax.

OnerecentpaperbyWu(2004)asksasimilarquestiontooursbutgoesaboutthe

analysisinaverydifferentway.Wuestablishesagametheoreticframeworkfortheprobleminordertoexploretheinteractionsbetweengovernmentsandpollutingmultinationals.Wumakestheassumptionthatthefirmhasprivateinformationaboutthepollutionintensityofitsproductionprocessthusinformationalrentsexist.Inoneparticularlyinterestingcase,thiscausesthehomegovernment,whichisprivytotheprivateinformation,toencouragefirmstocheattheforeigngovernmentsothattheforeigngovernmentmustpayahigherinformationrenttofirms,sinceintheendthisbecomesthehomegovernment’staxrevenue.Wufindsthatcooperationbetweengovernmentsincreaseswelfareevenasthedirtyindustrymovestowardtheforeigncountrywithmorepollution.Moreover,Wuexplainsthatanefficiency‐seekinggovernmentneedstodesignaregulatoryregimethatinducesafirmtorevealitstechnology.

II.DefinitionofthePollutionHavenHypothesis

Forsimplicity,Idefinethepollutionhavenhypothesisasthehypothesisthathigherdomesticenvironmentalstandardswillcauseindustriestomovetoforeigncountries(whichhavelowerenvironmentalstandards).Wewillassumethatstandardstaketheformofataxonemissionsofcompositegasobjectz.III.TheModel

FollowingLevinsonandTaylor(2008),productioninarepresentativeindustryisgivenby

1 , (1)whereyisproduction, isthefractionofthefactorsusedforabatement,Kiscapital,andLislabor.Weassumeconstantreturnstoscale.Pollutionofarepresentativeindustryisafunctionofintensityofabatement:

, (2)

Wedefine 1 ,where isbetween0and1(andhence1/α>1),sincethisisadecreasingfunctionof ,sowhen ishigh,wehavelesspollution.Rearranging,weseethat 1

,

Hence,wecanrewriteyas:

, (3)Below,Icalculatethefirstandsecondderivativesof :

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11 0

(4)

1

1 0(5)

whichimpliesthat isdecreasingin atandecreasingrate.Thisassumesthatthemorepollutionwehaveabated,theharderitistofurtherabate,i.e.themarginalimpactoftheabatementonthepollutionissmaller.Thisisconsistentwithwhatwewouldexpect,becausewerunoutoflow‐hangingfruitinourproductionprocessaftersomepoint.

Abatementcostsandregulationdifferacrossindustries.AsLevinsonpointsout,α(pollutionintensityofaproductionprocess)isdifferentforeachindustry.Wecanexpressthisasα(η),whereη,thesectorindex,isavariableonacontinuumfrom0to1,whereη=0istheleastpollution‐intensivefirmandη=1isthemostpollution‐intensivefirm.2Then,η η η , η and η 0,sothatα(η)isincreasinginη.Inthe

aboveexpression,z( istheamountofgreenhousegasemittedinsector , η isproductioninsector ,and η and arefactoruseinsector .Tointerprettheequationfor ,notethatwehavewrittenoutputasaCobb‐Douglasfunctionofpollutionandotherfactorsofproduction.

IV.SimplifyingAssumptions:Iadoptthefollowingsimplifyingassumptions:

1. AssumethatthePollutionHavenHypothesisisonlyaforeigndirectinvestmenteffect.3Thismeansthatfirmsfromthehomecountrydecidewhethertopolluteathomeorabroad.Theydonotgooutofbusiness,butrathersimplymovetheirproductiontoaforeigncountry.Theystillselltheirgoodstothesamemarket.

2. Theonlypollutantisacompositegasgoodz.Wedefinethistobeagasthatcontributestoglobalwarmingandaffectsallcitizensoftheworldequally.Thus,thedetrimentaleffectsofaunitofzemittedinonecountryaresharedamongallcountriesuniformly.4Allcountrieshavethesamedisutilityof andnocountriesdisputethedetrimentaleffectsof 5;further, followsadeterministicprocess,sothereisnouncertaintyovertheeffectsof .

3. Intermediategoodsandre‐exportsdonotexist.Allgoodsarefinalgoods.4. Therearenoshipmentcosts.

                                                            2Fromnowon,IdepartfromLevinsonsignificantly.3Winchesterpointsoutthatsinceclimatepoliciesindevelopedcountriesthatraisefossilfuelpricessimultaneouslylowerfossilfuelpricesincountrieswithoutfossilfuelpricerestrictions.Thisresultsinincreasedenergyconsumptioninothercountries.4Thisisaverysimplisticassumption.Considertheislandnations,whichmaycompletelydisappearwhentheoceanlevelrises.ThisassumptionsaysthattheresidentsofthesenationscarethesameaboutglobalwarmingasdothefarmersoftheUSMidwest,wherecropyieldsmayactuallyincrease.5NotethatthisassumptionimpliesthattheUnitedStatesbelievesinglobalwarming!

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5. FollowingLevinson,themodelispartialequilibrium,inthatfactorpricesandenvironmentaltaxesaretakenasconstants.Consumersspendaconstantfractionoftheirincomeongoodsfromeachfirm,whichmeansthatdemandforeachgoodisperfectlyinelastic.

6. Productssellforthesamepriceineverycountry.FactorsofproductionKandLarehomogeneousandnotmobile.Listhetotalpopulationinthehomecountry.Thisassumptionmeansthatcomparativeadvantageinthef‐goodisdrivenbyinitialendowmentsinthecountries.ThisassumptionbuildsontheworkofHeckscher(1919)andOhlin(1933).WewillseeinsectionXIwhythisisanecessaryassumption.

V.ProductionCostsandfirmlocation

Tradeisdeterminedbyproductioncostsinacountry.Becausethemarginalcostofproductionmustequalthemarginalproduct,theunitcostofproducingyis:6

1 (6)

Where isapollutiontaxand isthecostofproducing .

Ihaveassumedabove,unlikeLevinson,thateachindustrycontainsonefirmandproducesonegood.Thisisnotaparticularlyrestrictiveassumptionsinceifwewantedtomodelanindustrythatproducedmorethanonegood,wecouldjustusetwodifferent ′sforthetwogoods(ofcourse,thisignoresconsiderationsofeconomiesofscope).

Now,tomakethingsconcrete,wewillposittwoplayers:homeandforeign.Iuse

theseplayersbecausesizeofcountrywillhaveabearingontheoptimalamountofabatementthatacountrychooses.Wewilldenotethecostsofproducingyinindustry athomeby

1

(7)

Similarly,thecostofproducingyabroadis:

∗ 1 ∗ ∗

(8)

Anindustryislocatedathomeif7

                                                            6ThisisastandardresultthatcanbefoundinVarian(1992)7Dividing by ∗givesustheratioofproductioncostsincludingthetaxes:

∗

11

∗∗

∗ ∗

Whichmeansthat

∗ ∗ ∗ ∗

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∗

∗

≝ (9)

Where istheunitcostofproducingoneunitoffusinglaborandcapital, ∗ istheunitcostofproducingoneunitoffintheforeigncountry,and and *arethecostofemittingoneunitofginthehomeandforeigncountries,respectively.Noticethatfirmscareonly

aboutcostdifferentials,andthat,asLevinsonpointsout, ∗

iscompletelyindependentof ,

sinceitisaratioofcostsofnon‐pollutingfactorsintheindustry.Wehaveassumedthatforaparticularindustry , doesnotvarybetweenthetwocountries.Thatis,whilethecostsofproductionoffmayvaryinthecountry(wageandrenttocapitalmayvary,becauseintheCobb‐Douglascasetheydependontheamountoflaborandcapital),theintensityofpollutionintheindustrydoesnot.

Wecansaythisbecauseweknowthatitemsareshippedeffortlesslyandsellforthesamepriceineverycountry,soproductioncostsaretheonlyrelevantvariable.ThisisconsistentwiththePHHinthatanincreasein makesasectormorepollutionintensiveanddecreases ,thusincreasingthelikelihoodthatthegoodwillbeproducedabroad.

Belowwegraph and :8

                                                                                                                                                                                                

Thisratio=1when ∗

∗≝ ,whichmeanstheratiois<1(andthusitmakessensetoproduceat

home)when: ∗

∗≝ .

8 Theγfunctionmeetsthex‐axiswhentheforeigntaxis0,sinceinthatcase,allbusinessflockstotheforeigncountry.Ittouchesthey‐axiswhentheforeignanddomestictaxesareequal,sinceinthatcase,theforeigncountrycannotattractanybusiness. 

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Wesolveforthelastfirmthatwillleaveandobtaintheefficiencyequilibriumcondition:

ln ∗

ln ∗ ln∗

(10)

Forsimplification,assumethat .Thisisplausiblesince η 0,i.e.thehigherthepollutionintensityofthefirm,thehighertheCobb‐Douglascoefficientforpollutionintheproductionprocess.

So,anexpressionfor isgivenby:

ln ∗

ln ∗ ln∗

(11)

VI.Weitzman’sUtilityfunction

MartinWeitzman(2009)definescountryutilityforglobalwarmingasfollows:

∗, (12)Where:C*:potentialconsumption(consumptionunderscenarioofnoglobalwarming,T=0)C:realizedconsumptionT:TemperatureThesignsofthevariablesandpartialderivativesare:

∗ 0, 0, 0, 0, 0, 0WeitzmanexplainsthatTinfluencesUforagivenC*bydiminishingthe

“effectiveness”ofC*inproducinga“welfareequivalent”C.Inotherwords,positiveTnegativelyaffectsthetransmissionofpotentialconsumptionintoconsumptionC.

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VII.MyUtilityfunction:Sincetheutilityfunctionisthegovernment’sutilityfunction,myutilityfunction

departssignificantlyfromWeitzman’s.9Thegovernment,wishingtobere‐electedandcontrolledbypowerfulpoliticallobbyists,caresonlyaboutdomesticoutput.Politicsputspressureonthegovernmenttokeepproductioninthecountry,andthusthegovernmentwantstominimize1‐j,whichisthenumberoffirms/industriesthatareoutsourced.

IdepartfromWeitzmanbyintroducingadisutilityoftemperaturefunction, .I

modelacountry’sutilityas, T (13)

Where:

0, 0, 0, 0, T >0Iexpress inutilityunits,andhencewritethisas:10

(14)

(NotethatIamnotspecifyingthederivativesof T sinceIwillexploretwocases.)

Assumethatgovernmentutilityislinearinoutput.Then,wecanexpressthisas:

(15)

Thedisutilityfunctionisnowinunitsofoutput.Itcouldbethecasethat,asinWeitzman’smodel,temperaturehasanegativeeffectonoutput,thusdampeningpotentialoutput.Ifweexpresssome“realizedoutput”valueby ,wecanwrite:

(16)

Andthus

(17)

Notethat,similartoWeitzman’sapproach,Y(j)givesoutputofindustriesifthereisnodisutilityfrompollution. 1 givesoutputifj=1,thatis,ifthereisnoindustryflightduetotaxation.

                                                            9 Myutilityfunctionisregardedastherulingagent’smaximand,ratherthantheprincipal’ssocialwelfarefunction.10Ihaveimplicitlyusedastrongseparabilityassumption.Intuitively,Iameliminatingthepossibilitythatdisutilityfromglobalwarming interactswithoutput.Ihavedonethistosimplifytheproblem,butmanycasescouldariseinwhichthisassumptioninvalidatesmyanalysis.Oneimportantcasewouldbethat,insteadof“dampening”outputdirectly,achangeinzaffectscapitalonly,i.e.anincreaseinzcausesimmediatecapitaldepreciation.Thisisnotimplausibleintheleast:itiscommonlyacceptedthatcoastalcitieswillbephysicallydamagedbyglobalwarming.Thankfully,politiciansgenerallyoperatewithinashortpoliticalcareerandthusmostlytake asgiven.

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Anotherinterpretationofourdisutilityfunctionisthatthegovernmentissomehowmotivatedtoreduceemissionsthroughpoliticalpressures.Inthiscase, doesnothavetodecreaseoutputdirectly,andcanbeanyfunctionof .VIII.Temperaturefunction

Now,weanalyzeT,whichisafunctionofemissions.Therelationshipbetweentemperatureandpollutionemissionsiscrucialtoourmodel.Liffman(2010)plottedtheobservedlinearcorrelationbetweenatmospheric concentrationinpartspermillion(NAOOWeb)andaverageGlobalTemperatureindegreesCelsius(NASAWeb)between1959and2009whichIhavereproducedbelow.

Thismotivatesustomodeltemperature(T)andemissions(Z)inalinearfashion:

(18)

Wewillnormalize,simplifyingthisto:

(19)

OurcountryisconcernedwithglobalemissionsandthuscaresaboutemissionstakingplaceinbothHomeandForeign,sowhenfirmsmoveabroad,theiremissionsstillcontributetoTinthesamewaythattheemissionsofthefirmsthatstayathomedo.We

denote asthetotalpollutionofallfirmsthatcouldpossiblylocateinthe

homecountry.Then,wecanwrite where = , isthepollutionfrom

theindustriesthatstayathomeand = ∗, isthepollutionfromtheindustries

whoseproductsareproducedabroadbecauseoftaxpolicy.

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IX.Solvingfortheoptimaltax

Thegovernment’sproblemistomaximizeasocialwelfarefunction usingataxthatwilldecreaseemissions,Z.Aswewillsee,thisisacomplicatedendeavor,andtheproblemneedstoberecalculatedfrequentlybecausethemaximizationproblem

willbedifferentwhentheratios ∗and∗change.Thus,thegovernmentexhaustsallofits

taxrevenueinhiringeconomiststofigureouttheoptimaltax.Tosolvefortheoptimaltax,wedifferentiatetheutilityfunction:

(20)

Where isdomesticoutput, isadisutilityfunctionwithparameter .

Now,forthefirstterm,

ByLeibnitz’sRule,

∗ 0

Andsince doesnotdependonj,thisbecomes:

0 ∗ 1 0

(21)

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Andforthesecondterm,

∗

∗

, ∗,

(22)

BytheSecondFundamentalTheoremofCalculus,(21)becomes:

, ∗,

(23)

Nownotethatbecauseofthefactthatforevery theproductionfunctionisthesameinthetwocountries,itisprovenintheappendixthat ∗, , ,where

∗.

So,wehavethat:

, 1 ∗,

, 1 ∗

, 11

(24)

Then,

, 1 1

Sosocialwelfareismaximizedwhen

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0 , 11

(25)

Whichimpliesthat

0 1 1

0 11

(26)

Andnow,notethatforthetaxtobeoptimal,thegovernmentmusttakeintoaccountthefirm’sreactiontothetax.Thatis,thegovernmentmustrecognizethatthefirmwillsetthetaxequaltothemarginalproductofpollution.Thus,forfirmsunderperfectcompetition,wehavethat:

1

Then, = ,so

1

1

Therefore,

0 1 1

11

1

Thus,

11

1

13  

Andso,

11

1

(27)

Wecanseealreadythatouroptimaltaxisgoingtodependnegativelyonj,thenumberoffirmsthatwillleavethecountrywhenthetaxiserected.Thismakessensebecausethegovernmentwillwanttolowertaxifitknowsthatmanyfirmswillleavethecountry.Thetaxwilldependpositivelyonourdisutilitycoefficient,whichmakessensebecauseifthedisutilitycoefficientisveryhigh,thegovernmentwillprefertotaxatahigherratetopreventtheglobaltemperaturefromrising.

Tosimplifythisexpression,weneedtorewrite(11)toget:

ln ∗

ln ∗ ln

(28)

Wewillreferto ∗as .Then,j=

Andnotethat1

11

1ln ∗

ln ∗ ln

ln ∗ ln

ln

ln ln

ln

(29)

And

1ln

ln lnln ln

lnlnln

(30)

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Hencethetaxthatmaximizesutilityisgivenby:

ln ln

1 1

(31)

Itistrivialtoshowthatthesecondorderconditionholds.11

1. Classiccase∗

1and ∗ 1

Thisisthetypicalcasewherethehomecountryisanadvancedeconomyandtheforeigneconomyisamorecompetitiveeconomywithlowerunitcosts;thus,undernotaxes,theforeigneconomyhasacomparativeadvantageinproducingthe good.

Weassumeallcostsandtaxesarepositive.12Weseethatln 0,1 <0, ,0<

1andsoallelseequal,wehavethat >0and <0.Thismeansthat

ifthe“disutility”thatthecountrygetsfromtemperature,B,increases,thenthegovernmentshouldincreasethetax.Also,ifthecostsofproducingthefgoodgoup,weshoulddecreasethetax.Thisisverynon‐intuitive:itissayingthatwemusttaxourmostefficientsectors!Anotherwaytoputthisisthatifcostsarerelativelyhigh,relativeinefficiencyinproductionisdoingthetaxingjobalready.Notethatcrucially,weareholding fixedinthisanalysis.So,ifthecostofproducingthefgoodgoesup,weneedtodecreasethetaxtomakesurethatdomesticproducersarestillcompetitiveonworldmarkets.

2. Caseofcountrywithlowtaxesandlowproductioncosts∗

1and ∗ 1

Inthiscaseweseethat

>0and >0.

Thatis,thegovernmentshouldtaxpollutionmorewhenthedisutilityparameterBgoesup,andshouldtaxpollutionmorewhencostsofproducingthefgoodincrease,solongascostsarestillbelowthatoftheforeigncountry.Thismakessensebecauseifthecostsarestillbelowthoseoftheforeigncountry,thehomecountryneedstotaxasmuchasitcan

                                                            11Justkidding!SeeAppendixforthederivation. 12 Thisisanunreasonableassumptionifgovernmentscareabouttheenvironmentverymuch,andthuswanttosubsidize(useanegativetaxfor)lowpollutionindustries.

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becauseofthedisutilityeffectfrompollution,butnotsomuchthatitlosesitscomparativeadvantage.

3. Caseofcountrywithlowtaxesandhighproductioncostsofthefgood

Assume∗

1and ∗ 1.Then,

<0, 1 >0.Thus,wehavetwo

subcases,bothwiththesameconclusion:

1. 0< 1if

Inthiscase, >0and <0

2. =1 0if

Inthiscase,

>0and <0

Theanalysisinthesetwocasesisthesameasthatoftheclassiccase.

4. Caseofcountrywithhightaxesandlowproductioncostsofthefgood

Assume∗

1and ∗ 1.Then,notethatln 0and1 <0.

Again,wehavetwosub‐cases:

1. 1 1if

Inthiscase,since

0and1 0,thetaxdependspositivelyonBand .This

tellsusthatwhenourcomparativeadvantageinproducingthefgoodisstrongenough,wecantaxwithoutworry,aslongaswearestillinthisrange.TheSOCarenotmetforthiscase;thisisacornersolution,aswewouldexpect.Aninterestingquestioniswhetherornottherearegoodsorcountriesintherealworldforwhichthiswouldhold.Apossiblecandidateisoilproduction:oilproductiontaxesaresometimesleviedonproducers,butdonotseemtocausefirmstorelocate,sincenoteverylocationisaviableproductionsite.

2. =1 1,0 if

Inthiscase,wegetanimaginarysolutionfor ,andagainwehaveacaseinwhich

theSOCarenotmet.

X.Doessizematter?

Nowwemovetothequestionofwhetherornotcountrysizewillhaveaneffecton

ourresults.Defineavariable ∗ where isthepopulationofthehomecountryand ∗

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isthepopulationoftheforeigncountry(Asexplainedintheassumptions,thesearebothexogenousandnotmobilebetweencountries).Then, isacountry’sfractionoftheworld.Ifthegovernmentcaresaboutthepeoplelivinginitscountry,andpresumablydoesnotcareaboutthepeoplelivingintheothercountry,itshouldbemoreconcernedwithitspollutionwhenthecountrysizeislargethanwhenitissmall,becausethedisutilityper

person willbehigh.Thus,weseetheneedforanewformulationforourdisutility

function:

, (25)

Because isexogenous,thiscanbewrittenas

(26)

where .

Now,ouroptimaltaxis:

lnln

11

(27)

whichislessthantheonewefoundbeforeforcases1,2,and3since 1.Forallthreeof

thesecases, >0.Thismeansthatasthepopulationofacountrygrowscompared

withthesizeoftheworldpopulation,theoptimaltaxgrowsbecausethegovernmentcaresmoreaboutemissions.ThisisconsistentwiththeCoaseTheorem,sincewhenthecountryislarger,itinternalizesmoreofthepollutionandthushasanincentivetopolluteless.XI.Relaxingtheimmobilityassumption

TheassumptionthatKandLareimmobile(Assumption6)isnecessaryforouranalysisbecauseitmeansthatthecostofproducingthefgooddiffersinthetwocountries.Ifthecostofproducingthefgoodinthetwocountriesdidnotdiffer,thennocountrywouldhaveacomparativeadvantageinproducingthefgood.Therefore,comparativeadvantagesinproductionwouldcomeonlyfromdifferencesinthepollutiontaxacrosscountries,andallindustrieswouldproduceineitheronecountryortheother(specifically,thecountrywiththelowerpollutiontax).

Importantly,theassumptionthatKismobileandLisimmobileisnotenough.Fora

concreteexample,supposeKismobileandLisimmobile.Then,sinceproductionofthefgoodisCRS,andthuscompetitive,weseethatprofitsare0forthefgood,andthustherevenuesequalcostsforthefgoodforanyindustryinbothcountries.Sinceaproducercouldproduceineithercountry,ifhewantstoproducefunitsthataresoldatacostof hecoulddosointhedomesticcountry:

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Orintheforeigncountry:

∗ ∗ ∗ ∗ ∗So

∗∗ ∗ ∗ ∗

(28)

Let’sexaminethecostratio ∗ .Cancelingoutthe terms,weget:

∗ ∗ ∗ ∗ ∗

Dividingthenumeratoranddenominatorby weseethat

∗

1∗ ∗ ∗ ∗

1∗ ∗ ∗ ∗

∗ ∗

∗ ∗

1∗ ∗

1∗ ∗

∗ ∗

Sincetheproductiontechnologyisthesameforagivenindustryinthetwocountries,thefactorshareratiosareequal:

∗ ∗

∗ ∗

1∗ ∗

So,

∗ ∗ ∗ (29)

Sincecapitalismobile,capitalwillflocktothecountrythatpaysthehighestrent,

andthusitwillbethecasethateventuallyeitherallofthecapitalislocatedinonecountry,or:

∗ ∗with ∗

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So,combining(28)and(29),

∗ ∗ ∗ 1

Then,wehaveshownthatnocountryhasacomparativeadvantageinproductionofthefgoodifcapitalismobile.Hence,afirmwillproduceathomeifandonlyif:

1∗

≝

(30)

Whichisequivalenttotheconditionthat∗

(31)

Noticethatthisisindependentofpollutionintensity ofthefirm.Thismeansthatallfirmsproduceinthecountrywiththelowesttax.Inthiscase,thegovernment’sproblemissimple:thegovernmentalwaysdoesbesttosetitstaxlessthantheforeigntax.Thisisthetypicalcaseofa“racetothebottom”inenvironmentalstandards,becausetheforeigngovernmentmustrespondbysettingitstaxevenlower.13

XII.Conclusion

Ihavefoundtheoptimaltaxthatacountryshouldsetgiventhevariouscostratiosandtaxratiosthatitfaces.Taxesshouldincreasewhendisutilityfrompollutionincreases.Apoorcountrywithlowtaxescanaffordtotaxmorewhenitsproductioncostsofthefgoodincrease,whereasarichcountrywherethecostsofproducingthefgoodarealreadyhighcannot.Sizedoesmatter:acountrywithalargerpopulationcomparedwiththeworldpopulationhasahighertax,otherthingsequal.ThisisconsistentwiththeCoaseTheorem.Lastly,onemobilefactorcouldcauseasituationinwhichitisalwaysoptimalforthedomesticgovernmenttosetitstaxlowerthanthatoftheforeigngovernment,whichwouldlikelygeneratetheclassic“racetothebottom”inenvironmentalstandards.

XIII.Appendix:Variable Interpretation

Pollutionintensityofaproductionprocess, Temperaturefunctioncoefficients, ∗ Costofproducingoneunitoffusinglabor

                                                            13 Onekeyassumptionbehindthisresultisthatfactorpricesarefixedinthismodel.Weretheynotfixed,bothfactorswouldneedtobemobileinordertoachievethisresult. 

19  

andcapital∗ Potentialconsumption CRSfunctionofcapitalandlabor Nonlinearfunctionoftaxratio Industry,firm,orsectorindex Themostpollution‐intensiveindustrythat

staysathome, Factorsusedinsectorη∗, ∗ Foreigncapitalandlabor, Homecapitalandlabor Productioncostratio Thetotalnumberofindustriesthatwould

existinacountryinthecaseofnoabatement.Alsodenotestheindexforthemostpollution‐intensiveindustryorfirm.Inourmodel,wenormalizethisto1.

Functionalformofintensityofabatement Pollutiontaxratio

, ∗ Renttocapitalinhomeandforeigncountry,respectively

Sizeofcountryasafractionoftheworld Temperature Pollutiontax Fractionofthefactorsusedforabatement Utilityofacountry Disutilityfromtemperatureincreasesin

consumptionunitsY Output

Outputinindustry

Realizedoutput Greenhousegaspollutionemittedinsector

Greenhousegaspollutioninallsectors

SecondOrderconditions:

First,wecalculate:

lnln ln

1 ∗

ln ln1

20  

ln ln

(32)

Notethatsincejand arealwayspositive,intheclassiccase(∗

1and ∗ 1 ,

wehavethatthisispositive.Keepinmindthatthisisevaluatedattheoptimal ,whichiswhyifjgoesup,weshouldincrease tomaximizeutility.Inessence,whenthegovernmentimplementstheoptimaltaxtheyareindirectlychoosinganoptimalj;whenjincreases,thegovernmenthasthelibertytotaxmorewhilestillabovetheoptimalj.

Now,

1

ln ln ln ln

1ln ln

1

1 1ln ln

1

(33)

Also,notethat , <0,sinceforanyfirmzwithapollutionintensityj,whenthetaxrises,

inordertomaximizeprofitsthefirmwillset

,

(34)

Thismeansthat:

,

(35)

Sinceweholdjconstantwhentakingthederivative,

, ,,

(36)

Andweknowthat , 0,so , 0

21  

Now,ourtaskistodifferentiate:

, 11

(37)

Whichwerewriteas:

, 1 1

So

, 1 1

, 1 1

(38)

Wepluginourexpressionfor toget

, 1 1

1 1ln ln

1 , 1 1

, 1 1

1 1ln ln

1 , 1 1

, 1 1

1 1ln ln

1 , 1 1

1 1

1 1ln ln

1 , 1 1

22  

Andnoticethat issimplyapartialderivativeofourproductionfunctionwithrespecttooneofitsinputs,andhencebydefinitionispositive.

Rewriting,wehavethat

11

1 1ln ln

1 , 1 1

(39)

ClassicCase:∗

1and ∗ 1:

0, , <0, 1 0, 1 0=>thefirstterminthe

bracketsisnegative

Noticethat 1 <0, 0, 0, 1 0,and

thereforethesecondterminbracketsisnegative.Hence,wehaveamaximum.

Caseofcountrywithlowtaxesandlowproductioncosts:∗

1and ∗ 1

0, , <0, 1 0, 1 0=>thefirstterminthe

bracketsinequation(39)ispositiveif 1 andthesecondtermin

bracketsispositiveif 1 .Noticethatthisallisequivalenttothe

condition .If forall ,thenwehave:

(40)

And

Also,takingthederivativeofbothsidesof(40),weget:

23  

Clearlythisisproblematic:wecanrewritetheSOCas:

0 1

1 1ln ln

1 , 1

(41)

Then,If forall ,thentheoptimalz=0andtheoptimalj=0,sothisneedstoberuledout.

Ifweassume for ,thenweseethattheSOCaresatisfied.

Case3canbeanalyzedinasimilarwaytoCase1.

Proofthattheratioofamountsofpollutionusedinthetwocountriesistheinverseoftheratiooftaxesinthetwocountries

Ourassumptionthatconsumersspendafractionoftheirincomeongoodsfromeachfirmmeansthatagivenfirm willproduceexactly unitsofgood tosatisfydemand.Then,iftheindustryislocatedathome,

(42)

Andiftheindustryislocatedabroad,

∗ ∗

(43)

Then,nomatterwheretheindustryislocated,itwillsetthemarginalproductofpollutionequaltotheamountitmustpayforemittingaunitofpollution:

(44)

And

∗ ∗

(45)

Thenweseethatifindustry needstoproduce units,itwillbeindifferentbetweenproducinginthetwocountriesif

∗ ∗ (46)

24  

Whichmeansthat

∗ ∗

Andsowehaveshownthat:

∗, , .

(47)

XIV.References

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