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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.
2
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
3
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 :
4
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!
5
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
∗ ∗ ∗ ∗
6
∗
∗
≝ (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.
7
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.
8
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.
9
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.
10
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)
11
Andforthesecondterm,
∗
∗
, ∗,
(22)
BytheSecondFundamentalTheoremofCalculus,(21)becomes:
, ∗,
(23)
Nownotethatbecauseofthefactthatforevery theproductionfunctionisthesameinthetwocountries,itisprovenintheappendixthat ∗, , ,where
∗.
So,wehavethat:
, 1 ∗,
, 1 ∗
, 11
(24)
Then,
, 1 1
Sosocialwelfareismaximizedwhen
12
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)
14
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.
15
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:
17
Orintheforeigncountry:
∗ ∗ ∗ ∗ ∗So
∗∗ ∗ ∗ ∗
(28)
Let’sexaminethecostratio ∗ .Cancelingoutthe terms,weget:
∗ ∗ ∗ ∗ ∗
Dividingthenumeratoranddenominatorby weseethat
∗
1∗ ∗ ∗ ∗
1∗ ∗ ∗ ∗
∗ ∗
∗ ∗
1∗ ∗
1∗ ∗
∗ ∗
Sincetheproductiontechnologyisthesameforagivenindustryinthetwocountries,thefactorshareratiosareequal:
∗ ∗
∗ ∗
1∗ ∗
So,
∗ ∗ ∗ (29)
Sincecapitalismobile,capitalwillflocktothecountrythatpaysthehighestrent,
andthusitwillbethecasethateventuallyeitherallofthecapitalislocatedinonecountry,or:
∗ ∗with ∗
18
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)
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