The measurement of environmental economic inefficiency with pollution-generating technologies

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ContentslistsavailableatScienceDirect

Resource

and

Energy

Economics

j o ur na l h o me pa g e:w w w . e l s e v i e r . c o m / l o c a t e / r e e

The

measurement

of

environmental

economic

inefﬁciency

with

pollution-generating

technologies

Juan

Aparicio

a

_,

_Magdalena

_Kapelko

b

_,

_José

_L.

_Zofío

c,d,∗

a_Center_of_Operations_Research_(CIO)._Universidad_Miguel_Hernández,_Elche,_Spain b_Department_of_Logistics,_Wroclaw_University_of_Economics_and_Business,_Wrocław,_Poland c_Department_of_Economics._Universidad_Autónoma_de_Madrid,_Madrid,_Spain

d_Erasmus_Research_Institute_of_Management,_Erasmus_University,_Rotterdam,_The_Netherlands

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received1July2019

Receivedinrevisedform9April2020 Accepted8June2020

Availableonline20June2020

JELclassiﬁcation: D20 D24 N52 Q50 Keywords:

Environmentaleconomicinefﬁciency Pollution-generatingtechnologies Technicalandallocativeefﬁciency measurement

Dataenvelopmentanalysis USagriculture

a

b

s

t

r

a

c

t

Thisstudyintroducesthemeasurementofenvironmentalinefﬁciencyfromaneconomic

perspective.Wedevelopourproposalusingthelatestby-productionmodelsthatconsider

twoseparateandparalleltechnologies:astandardtechnologygeneratinggoodoutputs,

andapollutingtechnologyfortheby-productionofbadoutputs.Whileresearchinto

environmentalinefﬁciencyincorporatingundesirableorbadoutputsfromatechnological

perspectiveiswellestablished,nosigniﬁcantattemptshavebeenmadetoextendittothe

economicsphere.Basedonthedefinitionofnetprofits,wedevelopaneconomicinefficiency

measurethataccountsforsuboptimalbehaviorintheformofforegoneprivaterevenue

andenvironmentalcostexcess.Weshowthateconomicinefﬁciencycanbeconsistently

decomposedaccordingtotechnicalandallocativecriteria,consideringthetwoseparate

technologiesandmarketprices,respectively.Weillustratetheempiricalimplementation

ofourapproachusingadatasetonagricultureatthelevelofUSstates.

BYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Measuringtheenvironmentalinefficiencyofproductionunitsisanincreasinglyimportanttopicofrecenteconomic research.Environmentalinefficiencyassessmentintegratesmarketed(desirable,intended,orgood)outputswithnegative environmentalexternalitiesintoproductionmodeling(theproductionofso-calledundesirable,unintended,detrimental,or badoutputs).Suchanalysisisimportantfromtheperspectiveofsustainableproductionbecauseitprovidesvaluableinsights forfirmsandindustrystakeholdersonhowtoadoptenvironmentallyfriendlystrategies,andforpolicymakerstoimprove thedesignofpollutant-abatementinstruments,accountingforenvironmentalchallenges.

However,theexistingenvironmentalefficiencymodelslacktheeconomicinefficiencydimensionoftheanalysis;that is,acomprehensivemeasurethatalsoconsiderstheforegoneprofits,intheformoflowerrevenuesand/orhighercosts, thatarenotonlyrelatedtoatechnologicalinefficientbehavior,butalsotoallocativeinefficiency.Thisimpliesthatfirms shouldnotonlypursuebeingtechnicallyefficientbyexploitingthepotentialoftheproductionfrontier,buttheyshouldalso

∗ Correspondingauthorat:DepartmentofEconomics.UniversidadAutónomadeMadrid,Madrid,Spain. E-mailaddresses:jose.zoﬁo@uam.es,jzoﬁo@rsm.nl(J.L.Zofío).

https://doi.org/10.1016/j.reseneeco.2020.101185

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usetheoptimalamountsof(goodandbad)outputsandinputsconsistentwithproﬁtmaximization(extensibletorevenue maximizationand/orcostminimization);i.e.,theyshouldbeallocativeefﬁcientbysupplyinganddemandingtheoptimal bundles(ormixes)ofoutputandinputs.

Followingtheliteratureoneconomicefficiencymeasurementanditsdecompositionintotechnicalandallocative com-ponentsinitiatedbyFarrell(1957),webringthistheoreticalframeworktothefieldofenvironmentaleconomics.Inthis framework,economicefficiencyanalysisnotonlyconsiderstheproductionofgoodoutputs(associatedtoprivaterevenue), butmustalsoaccountfortheeconomiccostofproducingbadoutputs,whichisproxiedthroughtheso-calleddamagecost functions;andsincebadoutputsareexternalities,theyarealsorelatedintheliteraturetosocialcostfunctions,e.g.,see BretschgerandPattakou(2019).Fromtheperspectiveofthefirm,thepossibilityofsimultaneouslyincreasingmarketgoods whilereducingenvironmentallydamagingoutputsiseconomicallyappealing.Ononehand,thereisaclear(private) incen-tivetoincreaserevenue,butalsocurrentsustainabilityandcorporatesocialresponsibilityconcernsareincreasinglybrought intofirms’decisionmaking.Thismeansthatinefficientfirms,fallingshortfromthebestpracticefrontier,canimproveits environmentalefficiencyatnoprivatecost,andthereforeitispossibletoreducetheexternalitiescausedbybadoutputs bymatchingbestpracticestandardsofefficiency.Nowadaysfirmsroutinelyadvertiseenvironmentalachievementsintheir annualreports;e.g.,fortheairlineindustrythereductionofthecarbonfootprintisbecomingincreasinglyimportant.

Fromtheperspectiveofenvironmentaleconomics,theaboveframeworkcanberelatedtothenotionofnetprofits(ornet revenuesifweconsiderproductiveinputsasgiven,aswedoforsimplicityinourmodel).Thismeansthattheobjective func-tionofthefirmcorrespondstothemaximumofitsmarketrevenuesnetoftheenvironmentalcoststhatitcausesthroughthe inevitableby-productionofbadoutputs(i.e.,thematerialsbalanceprincipleassociatedtothefirstlawofthermodynamics). Ofcourse,thereductionofbadoutputsentailsabatementcostsifthefirmisefficient,therebyproducingatthetechnological frontier(andthisisreflectedbythesubstitutabilitycharacteristicsofthetechnologybetweenthegoodandthebadoutputs, seeMurtyetal.,2012).Butifthefirmistechnicallyinefficient,bothgoodandbadoutputscanbefreelyincreasedand reduced,respectively.Consequently,inefficientfirmscanincreaseprivaterevenuewhilereducingenvironmentaldamage. Insum,theenvironmentaleconomicefficiencymodelthatweproposedintheveinofFarrell(1957)extendstheexisting technologicalmodelsforenvironmentalefficiencymeasurementtoaccountfortheseeconomicdimensionsbypostulating anobjectivefunctionthataimsatmaximizingprivaterevenuenetoftheenvironmentaldamage.

Thedeterminationofeconomicefficiencyisimportantfromamanagerialstandpointfocusedonmarket-oriented perfor-mance,butalsoforotherstakeholdersanddecisionsmakerssuchas,forexample,politicians,localandstategovernments, orregulators(e.g.,environmentalprotectionagencies).Forexample,managersareinterestedinincreasingperformance notonlyinphysicaltermsbytakingadvantageofthebesttechnologyavailable,butalsobyrealizingtheeconomicgains associatedwithallocativeefficiencyimprovements;thatis,thechoiceofoptimaloutputandinputmixes,leadingtoeither maximumprofit,revenueorminimumcost.Theinformationonallocativeefficiencythatoureconomicmodelyieldsis alsorelevantfortheaboveeconomicagentsasimprovingallocativeinefficiencyisarguablycheaperandeasierforfirmsto achievethanimprovingtheirtechnicalinefficiency.Inthissense,beingawareofthelevelofthisinefficiencyandrelated potentialforrevenueincreasesorcostsavingsenablesfirmsto“reapalow-hangingfruit”.Forregulatorsminimizingthe environmentalcostofproductionfromanallocativeperspectiveisalsocritical,asthecostofcarbondioxideemissions (e.g.,relatedtorespiratoryillnesses)couldbelowerthanthoseassociatedtotheuseofpesticides(e.g.,relatedtocancer treatments).Howthepollutinginputscausingbothdamagesshouldberegulatedintermsoftheireconomiccostscannow beaddressedthankstoournewframework.

Consideringonlyatechnologicalperspective,theliteratureonmodelingproductiontechnologiesthataccountforbad outputshasdevelopedfollowingtwoapproachesmainly:oneinvolvingparametricmethods(suchasstochasticfrontier analysis,SFA;Aigneretal.,1977),andonebasedonnonparametricmethods(suchasdataenvelopmentanalysis,DEA;

Charnesetal.,1978;Bankeretal.,1984).Commontobothmethods,manydifferentapproacheshavebeenproposedto assessenvironmentalefficiencyofproductionunits.Lauwers(2009)classifiedtheseapproachesintothreegroups.Thefirst groupconcernsenvironmentallyadjustedproductionefficiencymodels,inwhichundesirableoutputsareincorporatedinto theproductiontechnology.Ingeneral,twomainbranchesofstudieswithinthisgroupcanbedistinguished:(i)treatingbad outputsasstrong(free)disposableinputs(Haynesetal.,1993;HailuandVeeman,2001)or(ii)treatingbadoutputsasweekly disposableoutputsandassumingthenull-jointnessofbothbadandgoodoutputs(Färeetal.,1986,1989).Thesecondgroup ofstudiesconsistsoffrontiereco-efficiencymodels(KorhonenandLuptacik,2004;KuosmanenandKortelainen,2005), whichdonotfollowaxiomaticproductionefficiencyframeworks,butrelateaggregateecologicaloutcomeswitheconomic outcomesonly.Inotherwords,eco-efficiencyismeasuredeitherthroughminimizationofenvironmentaloutcomesgiven economicoutcomes(e.g.,valueadded)orthealternativemaximizationofeconomicoutcomesgiventheenvironmental outcomes.Thethirdgroupofstudiesisbasedontheintroductionofthematerialsbalanceprincipleintoproductionmodels (LauwersandVanHuylenbroeck,2003;Coellietal.,2007;Førsund,2009).Thematerialsbalanceprinciplestatesthatflows intoandoutoftheenvironmentareequal,linkingtherawmaterialsusedintheproductionsystemtooutputs,bothintended andresidualones.

Dakpoetal.’s(2016)recentsurveyofenvironmentalefficiencystudiesextendedtheLauwers(2009)classificationinto thefourth,mostrecent,categoryofby-productionmodels,whicharebasedontheideaofdefiningtwosubtechnologiesin parallel:onethatgeneratesgoodoutputsandasecondthatgeneratesbadoutputs.ThisapproachwasintroducedbyMurty etal.(2012)and,asaconsistentandrelativelynewapproach,itsempiricalapplicationsareflourishing(e.g.,Dakpoetal., 2017;Arjomandietal.,2018;Rayetal.,2018),asareitsextensions(e.g.,Serraetal.,2014;Lozano,2015;Dakpo,2016;

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Førsund,2018).Werelyonthenovelby-productionapproachtointroduceoureconomicenvironmentalefﬁciencymodel. Nevertheless,itcouldbeeasilyparticularizedforpreviousapproaches.1_We_also_consider_recent_{qualiﬁcations}_of_the_original

by-productionmodelbyDakpo(2016)andFørsund(2018).2

Asanticipated,regardlessthemodelingapproach—parametricornon-parametric—underthefourlistedcategories,a commonfeatureofallpreviousstudiesisthattheyareonlycapableofmeasuringtechnicalefficiencybyfocusingonthe technologicalsideoftheproductionprocess,therebyneglectingthemeasurementofenvironmentalefficiencyfroman economicperspective.Thisallowsustosummarizewhatourmodeldoes;i.e.,enhancingtheexistingapproachesthroughthe introductionofameasureofenvironmentaleconomicinefficiencythat,groundedonthetheoreticalframeworkproposedby

Farrell(1957),considersbothgoodandbadoutputs,andenablesitsdecompositionintotechnicalandallocativecomponents. Tofillinthegapintheliteraturewepostulateacomprehensiveframeworkthatisconsistentwiththeeconomicbehaviorof organizationsintheirattempttomaximizerevenue,butalsoaccountsfortheenvironmentalinefficiencythatresultsfrom thefailuretominimizetheeconomiccostsassociatedtoenvironmentaldamage.Aspreviouslyremarked,thisresultsinthe definitionofaneteconomicfunctionthatmaximizesthedifferencebetweenprivate(market)revenuelessenvironmental (social)cost,usingpricesofgoodandbadoutputs.3 _In_this_regard,_our_framework_is_capable_of_balancing_private_gains

(revenue)andenvironmentaldamage(cost)intoameasureofeconomicinefﬁciencythatcanbedecomposedaccordingto technicalandallocativecriteria.

From an appliedperspective we rely onDEA techniques becausemost existing empirical applicationsfollow this approach:theyareflexible,donotimposerestrictiveassumptionsontheparametricspecificationofthetechnology,noron thedistributionofenvironmentalinefficiency.4_{Nevertheless,}_the_drawbacks_of_DEA_should_be_also_highlighted_and_these

includeitsdeterministicnatureandthesensitivitytooutliers(foracomprehensiveexpositionofstrengthsandweaknesses ofDEAsee,forexample,Stolp(1990),Berg(2010)).Awareofthesecaveats,whichcanbeeventuallyaddressedthrough,e.g., bootstrappingandotherresamplingtechniques(seethemethodsintroducedbySimarandZelenyuk(2006)employedinthe empiricalsection),wedeﬁnetheDEAprogramsthatallowtheempiricalimplementationofournovelapproach.5_Our_point_of

departureistheby-productionmodelintroducedbyMurtyetal.(2012),asitrepresentsthemostrecentextensionof previ-ousapproachesandcanarguablybeseenasageneralizationthat,byconsideringtwoindependenttechnologiesfordesirable andundesirableoutputs,avoidssomeoftheirinconsistencies(namely,themultiplicityofoptimalcombinationsof desir-ableandundesirableoutputsforagivenlevelofinputs,anderroneouslysignedmarginalratesoftransformation−shadow prices−betweenoutputsandinputs).

Wedemonstratethepracticalusefulnessofournewlydevelopedmethodologythroughanapplicationtostate-leveldata oftheUnitedStatesagriculturalsector.Agricultureinvolvestheproductionofnotonlygoodoutputssuchasprimaryfood commodities,butalsoofbadoutputsrelatedwith,forexample,theneedforfuel,theusageofpesticides,fertilizersand otheragriculturechemicals,orthemanagementofmanure(Skinneretal.,1997;Reinhardetal.,1999).Examplesofbad outputsassociatedtothesepollutinginputsinagriculturearegreenhousegasemissions,pesticideandnitrogenleaching andrunoff,risktohumanhealthandﬁshfromexposuretopesticidesandfertilizers,etc.(seeBalletal.,2001;Kellogetal., 2002;Dakpoetal.,2017).Intheempiricalapplicationwearecapableofconsideringtwoofthesebadoutputs:CO2emissions

andpesticideexposures.

Theremainderofthispaperisstructuredasfollows.Thenextsectionreviewstheby-productionmodelsoftechnical inefficiencyandintroducestheirmathematicalunderpinnings.Thesubsequentsectiondevelopsourextensionallowingthe measurementofeconomic(profit)inefficiency.Wethendiscussourempiricalapplication,brieflycommentingthedataset andpresentingtheresults.Conclusionsaredrawninthefinalsection.

2. Theby-productionmodels

Pittman(1983)andFäreetal.(1986)initiatedtheasymmetricmodelingofoutputswhenmeasuringefﬁciencydepending ontheirnature,increasingthosethataremarket-orientedwhilereducingthosethataredetrimentaltotheenvironment.A keyquestionishowtoaxiomaticallymodeltheproductiontechnologywhencalculatingtechnicalefﬁciencythroughdistance functions.Mostparticularly,ascommentedintheintroductiontothispaper,shouldtheaxiomsunderlyingtheproduction

1_Details_on_the_{characteristics}_of_the_{by-production}_approach_are_presented_in_the_next_section.

2_Although_we_are_aware_of_other_{methodological}_developments_that_rely_on_the_{by-production}_model,_such_as_Serra_et_al.₍₂₀₁₄₎_or_Lozano₍₂₀₁₅₎_,_we

havenotconsideredthemsincetheirgeneralideaistomixtheby-productionapproachwithotherefﬁciencyframeworks,andnotthemodiﬁcationofthe modelperse.Hence,ifapplied,theirresultswouldnotbecomparabletothoseoftheoriginalby-productionmodel.

3_The_model_can_be_easily_enhanced_to_include_the_minimization_of_inputs_cost,_but_instead_we_keep_the_definition_of_{“environmental}_profit_{inefficiency”}

asatrade-offbetweenprivaterevenueandenvironmentalcost.

4_See_Zoﬁo_and_Prieto₍₂₀₀₁₎_for_an_exposition_of_early_models_within_the_{non-parametric}_approach_based_on_the_output,_input,_and_hyperbolic_distance

functions,whichweresubsequentlyimplementedinaparametricframeworkbyCuestaetal.(2009).Duetal.(2016)relyonthelatterapproachtoestimate carbonabatementscoststhroughshadowprices.

5_Brännlund_et_al.₍₁₉₉₅₎_measured_proﬁt_{inefﬁciency}_under_a_quota_system_and_the_production_of_undesirable_outputs_by_DEA_models._However,_they

didnotusepricesforweightingthenegativeexternalitiesanddonotdecomposeprofitinefficiencyintoitsdrivers,somethingthatwewilldointhispaper. Additionally,wenotethatPhamandZelenyuk(2018)definerevenueinefficiencyinthebankingindustryaccountingfornonperformingloans(NPLs), whicharemodeledasundesirableoutputsundertheapproachofweakdisposability.However,themodelisinternaltothefirm(thatis,privaterevenue), asitdoesnotincludeenvironmentalindicators,whiletheydonotimplementitempirically.

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technologyreﬂecttheirstrongorweakdisposability,andeventually,bemodeledasoutputsorasiftheywereinputs?Among theexistingapproachesfordealingwithundesirableoutputsandefﬁciency,theby-productionmodelintroducedbyMurty andRussell(2002)andMurtyetal.(2012)iscurrentlyconsideredapreferredoption.

Theby-productionapproachpositsthatcomplexproductionsystemsaremadeupofseveralindependentprocesses (Frisch,1965).Inthismodel,thetechnologycanbeseparatedintosetsofsub-technologies;onefortheproductionofgood outputsandoneforthegenerationofbadoutputs.The“global”technologyimpliesinteractionsbetweenseveralseparate sub-technologies.Førsund(2018)andMurtyandRussell(2018)recentlyclassifiedtheby-productionapproachamongthe multi-equationmodelingapproachesandarguedthatanimportantadvantageofthisapproachisthatitrepresents pollution-generatingtechnologiesbyaccountingfortheMaterialBalancePrinciple,therebysatisfyingthelawsofthermodynamics. Additionally,asMurtyetal.(2012)remarked,theby-productionmodelavoidstwoinconsistenciesofpreviousapproaches. Inparticular,severaltechnicalefficiencycombinationsofgoodandbadoutputs,withvaryinglevelsofbadoutput,couldbe possiblewhenholding(pollutingandnon-polluting)inputquantitiesfixed.However,intheabsenceofabatementactivities implementedbythefirm,thistypeofcombinationiscontrarytothephenomenonofby-production,sinceby-production impliesthat,atfixedlevelsofinputs,thereisonlyonelevelofpollutionatthefrontieroftheproductionpossibilityset. Moreover,itispossibletoobserveanegativetrade-offbetweentheinputsassociatedwithpollution,likefuel,andtheir associatedbadoutput,suchasCO2,whichrepresentsaclearinconsistency(morefuelbutlessCO2).Thesearethereasonswhy

theby-productionapproachisutilizedinthecurrentstudytointroducetheconceptofenvironmentaleconomicinefﬁciency takingmarketpricesintoaccount.

Inordertoreviewthestandardby-productionapproach,letusformallydeﬁnex∈Rn

+asavectorofinputs,y∈Rm+ as

avectorofgoodoutputs,z ∈Rm

+ asavectorofbadoutputs(e.g.,pollutants),andletusassumethatpDMUshavebeen

observed.Murtyetal.(2012)presentedtheirmodelbysplittingtheinputvectorintotwogroups:non-pollutinginputs, x1 ∈Rn+1andpollution-generatinginputs,x2 ∈Rn+2,withn1+n2=n.6,7Theﬁrstsetcouldcompriseland,labor,andsoon,

whilethesecondset,inthecontextofourempiricalapplicationonagriculture,consistsofinputslikefuel,fertilizers,and pesticides,whichproducecertainpollutantsasby-products,suchasCO2emissionsandpesticideexposures.Inthisway,

the‘global’technology,denotedbyT ,istheintersectionoftwosub-technologies,T1andT2.WhereasT1isthestandard

productiontechnologywithonlygoodoutputs,T2representstheproductionofbadoutputs.InthemodelbyMurtyetal. (2012),bothtechnologiesarelinkedthroughthelevelofthepollutinginputs.Inmoredetail,Murtyetal.(2012)deﬁnein generaltermsthetechnologyas:

T=T1∩T2, (1) where T1=

(x1,x2,y,z)≥0:f (x1,x2,y)≤0

, (2) T2=

(x1,x2,y,z)≥0:z≥g (x2)

(3) andf andgarecontinuouslydifferentiablefunctions.ThesetT1 isastandardtechnologyset,reﬂectingthewaysin

whichtheinputscanbetransformedintotheintendedoutputs.Thestandardfree-disposabilitypropertiesmaybeimposed byassumingthat ∂f (x1,x2,y)

∂x1 ≤0,

∂f (x1,x2,y)

∂x2 ≤0and

∂f (x1,x2,y)

∂y ≥0.NotealsothatT1 imposesnoconstraintonz;thatis,it

isimplicitlyassumedthattheby-productdoesnotaffecttheproductionofbadoutputs.Ontheotherhand,T2 reﬂects

aresidual-generationmechanism.Itisworthmentioningthat,intheformulationofT2,pollutionisreallytreatedasan

output.Inparticular,Murtyetal.(2012)assumethat ∂g(x2)

∂x2 >0.Thisexpressionand theformulationofT2 capturethe factthatpollutionisanoutputoftheproductionprocessforwhichdisposalisnotfree.Thispropertywascalled“costly disposability”ofresiduals.InwordsofMurtyetal.(2012):“Costlydisposabilityimpliesthepossibilityofinefﬁcienciesin thegenerationofpollution(e.g.,ifagivenlevelofcoalgeneratessomeminimallevelofsmoketheninefﬁciencyintheuse ofcoalmayimplythatthislevelofcoalcanalsogenerateagreateramountofsmoke)”.

Inthenon-parametricframeworkofDEA,thetwosub-technologiesmaybeexpressedmathematicallyundervariable returnstoscale(VRS)as:

T1=

(x1,x2,y,z)≥0: p

d=1 dx1d≤x1, p

d=1 dx2d≤x2, p

d=1 dyd≥y, p

d=1 d=1,d≥0

, (4) T2=

(x1,x2,y,z)≥0: p

d=1 dx2d≥x2, p

d=1 dzd≤z, p

d=1 d=1,d≥0

. (5)

6 _Ayres_and_Kneese₍₁₉₆₉₎_proposed_these_two_same_groups_when_introducing_the_materials_balance_principle_to_economists.

7 _As_Murty_et_al.₍₂₀₁₂₎_,_we_assume_that_Decision_Making_Units_apply_uniform_abatement_factors_and,_{consequently,}_these_factors_are_not_explicitly

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T1in(4)istherepresentationofthegeneralsetT1in(2)underseveralpostulatesasconvexityandminimalextrapolation

(seeBankeretal.,1984).ThesamecanbesaidforT2in(5)withrespecttothegeneralexpressionofT2in(3).Notethatthe

sub-technologiesaredeﬁned,inthisframework,withtwodifferentintensityvariables:and.Otherwise,wewillhavea confusionbetweenthesevariablesintheabovetwoproductionpossibilitysets.

Regardingthemeasurementoftechnicalefﬁciency,Murtyetal.(2012)showedthatsomeconventionalapproaches,like thehyperbolicanddirectionaldistancefunctiondeﬁnedonT=T1∩T2,areinadequateinthecontextofby-production.

Weusetheterm“output-oriented”inthiscontextbecausethesedistancefunctionsmeasureefficiencywithrespectto both goodandbadoutputs simultaneously.In thisway,theweaknessis duetothefactthat thetwo aforementioned measuresusethesamecoefficient(decisionvariable)fordeterminingefficiencybothinT1 forthegoodoutputsandT2

forthebadoutputs.Thisimpliesthatitispossibletoreachtheefficiencyfrontierforsomeofthesub-technologies,but theobservationcanfallshortofachievingthefrontieroftheotherone.Forconsistency,efficiencyintheby-production approachrequiresmodelsthatprojecttheassessedobservationsontoboththeefficientfrontierofT1 andtheefficient

frontierofT2.

TheabovementioneddrawbacksofstandardapproachesmotivatedMurtyetal.(2012)toproposeadifferentmeasurefor dealingwithgoodandbadoutputsunderby-production.ForDMU0,thismeasureisgood-output-speciﬁcand

bad-output-speciﬁc,andisbasedontheindexpreviouslydeﬁnedbyFäreetal.(1985):

min 1 2

⎡

⎢

⎣

1 m m

j=1 j

standard efﬁciency + _m1 m

k=1 k

environmental efﬁciency

⎤

⎥

⎦

s.t. p

d=1 dxid≤xi0, i=1,...,n p

d=1 dyjd≥yj0/j, j=1,...,m p

d=1 d=1, p

d=1 dxid≥xi0, i=n1+1,...,n p

d=1 dzkd≤kzk0, k=1,...,m p

d=1 d=1, j≤1, j=1,...,m k≤1, k=1,...,m d≥0,d≥0, d=1,...,p (6)

Model(6)projectstheassessedDMU0(x10,x20,y0,z0) ontotheefﬁcientfrontierofthetechnologyTbyincreasinggood

outputsandreducingbadoutputs.Thesechangesarevariable-speciﬁc,usingadifferentdecisionvariableforeachdimension: j,j=1,...,m,andk,k=1,...,m.Theconstraintsofmodel(6)coincidewiththerestrictionsthatdeﬁnetheDEAproduction

possibilitysetsT1 andT2 in(4)and(5).Additionally,theoptimalvalueof(6)coincideswiththemeanofthestandard

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separable.Inthiscase,thismeansthattheoptimalvaluecanbedeterminedasthemeanofamodelthatminimizes 1 m m

j=1 j

onT1andamodelthatminimizesm1

m

k=1 konT2: min 1 m m

j=1 j s.t. p

d=1 dxid≤xi0, i=1,...,n p

d=1 dyjd≥yj0/j, j=1,...,m p

d=1 d=1, j≤1, j=1,...,m d≥0, d=1,...,p min 1 m m

k=1 k s.t. p

d=1 dxid≥xi0, i=n1+1,...,n2 p

d=1 dzkd≤kzk0, k=1,...,m p

d=1 d=1, k≤1, k=1,...,m d≥0, d=1,...,p (7)

ItisworthmentioningthattherecentpaperbyFørsund(2018)arguedthatnon-pollutioncausinginputsshouldalsobe includedintechnologyT2giventhatsubstitutionbetweenthetwogroupsofinputscanhelpmitigatethepollution.Dakpo etal.(2017)indicatedthatsomeadditionalconstraintsmustbeaddedtotheby-productionapproachofMurtyetal.(2012)in ordertoguaranteethattheprojectionpointsfortheinputdimensionsarethesameinT1andT2.Inparticular,thecondition

thatshouldbeincorporatedtomodel(6)wouldbe:

p

d=1 dxid= p

d=1

dxid,

∀

i.Hereafter,weuseTMtodenotetheproduction

possibilitysetdeﬁnedastheintersectionofT1andT2in(1)and(2),respectively,asawayofhighlightingthatthedeﬁnition

ofthistechnologycorrespondstotheoriginalproposalofMurtyetal.(2012).Inthesameway,weuseTD_to_denote_the

productionpossibilitysetdeﬁnedfromtheoriginalby-productionapproachbutincorporatingtheconstraints

p

d=1 dxid= p

d=1

_dx_id,

∀

i,aspointedoutbyDakpoetal.(2017).Finally,wewillutilizeTMF _to_denote_the_production_possibility_set

deﬁnedbyMurtyetal.(2012)butincorporatingnon-pollutinginputsintechnologyT2.Likewise,TDFdenotestheproduction

possibilitysetàlaDakpoetal.(2017)butagainconsideringnon-pollutinginputsinthedeﬁnitionoftechnologyT2.

Tointroduceoureconomicinefﬁciencymodel,weextendthestate-of-the-artofby-productionapproach(Murtyetal., 2012;Dakpoetal.,2017andFørsund,2018)byincorporatinginformationonmarketprices.Todothat,weresorttoduality theoryfollowingChambersetal.(1998),and,morerecently,Aparicioetal.(2015),Aparicioetal.(2016a),andAparicioetal. (2016b).Inparticular,werecallrelevantdualityresultsconcerningthedirectionaldistancefunction8_._{Consequently,}_we

startoutbydeﬁningthistypeofmeasurefromanoutput-orientedperspectiveinthecontextofby-production.Underthe viewpointintroducedbyMurtyetal.(2012),weneedameasurethatallowsustoprojecttheassessedobservationsonto

8 _Although_the_directional_distance_function_is_well-known_due_to_its_ﬂexibility_and_because_it_encompasses_the_Shephard_distance_functions,_it_presents

somedrawbacks.Thismeasureneglectsslacksand,therefore,itdoesnottakeintoaccountallsourcesoftechnicalinefﬁciency(see,e.g.,Ray,2004). Additionally,theuseofthedirectionaldistancefunctioncouldresultininfeasibilitiesundercertainconditions,whichareanalysedindetailinBriecand Kerstens(2009).

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theefﬁcientfrontiersofT1 andT2simultaneously.Inthisway,the“by-production”directionaloutput-orienteddistance

functionfortheMurtyetal.(2012)approachwithdirectionalvectorg= (0,y0,z0) isdeﬁnedasfollows: → B

x0,y0,z0;TM

= max ıT1_ˇT1+_ıT2_ˇT2 s.t. p

j=1 j0xij≤xi0, i=1,...,n1 (8.1) p

j=1 j0xij≤xi0, i=n1+1,...,n2 (8.2) − p

j=1 j0yrj+ˇT1yr0≤−yr0, r=1,...,m (8.3) p

j=1 j0=1, (8.4) − p

j=1 j0xij≤−xi0, i=n1+1,...,n2 (8.5) p

j=1 j0zkj+ˇT2zk0≤zk0, k=1,...,m (8.6) p

j=1 j0=1, (8.7) ˇT1_,_ˇT2_,_j0_,_j0≥₀ _(8.8) (8)

Model(8)projectstheassessedDMU0(x10,x20,y0,z0) ontotheefﬁcientfrontierofthetechnologyTbyincreasinggood

outputsandreducingbadoutputs.Inthismodel,thechangesinoutputsarenotvariable-specific,i.e.,itdoesnotutilize adifferentdecisionvariableforeachdimension.Instead,itusesthesameexpansionfactorforthegoodoutputs,ˇT1_,_and thesamereductionfactorforthebadoutputs,ˇT2_._Moreover,_the_constraints_of_model₍₈₎_coincide_with_the_restrictions thatdefinetheDEAproductionpossibilitysetsT1andT2in(4)and(5).Additionally,theexogenouscoefficientsıT1 ≥0and

ıT2 ≥_0,_ıT1+_ıT2=_1,_which_appear_in_the_objective_function,_are_weights_that_are_ﬁxed_exogenously_by_the_{corresponding} decisionmaker(manager,politician,regulator,etc.)toreﬂecttherelativeimportanceofthestandard(traditional)wayof producingversusthenewandcleanparadigmforgeneratinggoodsandservices.Additionally,itslineardualis:

→ B

x0,y0,z0;TM

= min n1

i=1

v

1 i0xi0+ n2

i=n1+1

v

1 i0xi0− m

r=1 u1_r0yr0+˛10+ − n2

i=n1+1

v

2 i0xi0+ m

k=1 u2_k0zk0+˛20 s.t. n1

i=1

v

1 i0xij+ n2

i=n1+1

v

1 i0xij− m

r=1 u1_r0yrj+˛01≥0, j=1,...,p (9.1) m

r=1 u1_r0yr0≥ıT1, (9.2) − n2

i=n1+1

v

2 i0xij+ m

k=1 u2 k0zkj+˛20≥0, j=1,...,p (9.3) m

k=1 u2 k0zk0≥ı T2_, _(9.4)

v

1 i0,

v

2i0,u1r0,u2k0≥0, (9.5) ˛1 0,˛20free (9.6) (9)

Finally,tocompletethisopeningsection,werecallthefirstadditivemeasureanddecompositionofeconomicinefficiency proposedintheliterature.WerefertotheNerlovianprofitinefficiencymeasure,whichcanbedecomposedintotechnical

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inefﬁciency(thedirectionaldistancefunction)andaresidualterminterpretedasallocativeinefﬁciency(Chambersetal., 1998).9

In the standard production context, considering private revenue and cost only, and given a vector of input and output prices (ω,q) ∈Rn₊+m and technology T , the proﬁt function ˘ is deﬁned as ˘T(ω,q)=

max x,y

_m

r=1 qryr− n

i=1 ωixi: (x,y)∈T

.ProfitinefficiencyàlaNerloveforDMU0isdefinedasoptimalprofit(thatis,the

valueof theproﬁtfunctionatmarketprices)minusobservedproﬁt,both normalizedbythevalueofa reference

vec-torg_{= (g}x_,_gy₎_∈_Rn+m + : ˘T(ω,q)−

_m r=1 qryr0− n

i=1 ωixi0

m

r=1 qrgyr+ n

i=1 ωigix

.Additionally,Chambersetal.(1998)showedthatproﬁtinefﬁciency

maybedecomposedintotechnicalinefficiencyandallocativeinefficiency,wheretechnicalinefficiencycorrespondstothe directionaldistancefunction→DT(x0,y0;gx,gy)=max

ˇ:(x0−ˇgx,y0+ˇgy)∈T

: ˘T(ω,q)−

_m

r=1 qryr0− n

i=1 ωixi0

m

r=1 qrgry+ n

i=1 ωigix =→DT

x0,y0;gx,gy

+AIN T

x0,y0;ω,q;gx,gy

(10)

Inmodel(10),theleft-handsidecorrespondstoameasureofprofitinefficiency,definedasthenormalizeddifference betweenmaximumprofitandactualprofitatobservedmarketprices.Thismaybedecomposedintotechnicalinefficiency, i.e.,thevalueofthedirectionaldistancefunction→DT(x0,y0;gx,gy),andpriceorallocativeinefficiencyAI_TN(x0,y0;ω,q;gx,gy).

3. Measuringeconomicinefﬁciencywithby-productionmodelsinDEA 3.1. EconomicinefﬁciencymodelconsideringMurtyetal.’s(2012)technology

Wefirstintroducesomenotationanddefinitions.Givenafixedlevelofinputx0= (x10,...,xn0)∈Rn+andafixedlevel

ofbadoutputz0= (z10,...,zm0)∈Rm+,letusalsodeﬁneasr (x0,z0,q,T ) themaximumfeasiblerevenuegiventheoutput

pricevectorq= (q1,...,qm)∈Rm₊₊: r (x0,z0,q,T )=sup y

_m

r=1 qryr: (x0,y,z0)∈T

=sup y

_m

r=1 qryr: (x0,y,z0)∈ [T1∩T2]

. (11)

Eq.(11)representsagenericformulationforexpressinghowtodeterminethemaximumrevenue

m

r=1

qryr forgood

outputsthatcanbeobtainedgivenatechnologyT=T1∩Tandﬁxedquantitiesofinputsx0andbadoutputsz0.

UnderMurtyetal.’s(2012)approach, thisoptimizationproblemcanbealwayssolvedindependentlyonT1 andT2.

Therefore,asforT1,maximumfeasiblerevenuegiventheoutputpricevectorq= (q1,...,qm)∈Rm₊₊maybedeterminedby:

r

x0,z0,q,TM

=sup y

_m

r=1 qryr: (x0,y,z0)∈TM

=sup y

_m

r=1 qryr: (x0,y,z0)∈T1

. (12)

Again,(12)representsageneralformulationforstatinghowtodeterminethemaximumrevenue

m

r=1

qryrforgoodoutputs

thatcanbeobtainedgiventhesub-technologyT1andﬁxedquantitiesofinputsx0andbadoutputsz0.

9 _See_also_Koop_and_Diewert₍₁₉₈₂₎_and_Zieschang₍₁₉₈₃₎_for_earlier_{decompositions}_of_economic_(cost)_efﬁciency_into_technical_and_allocative

com-ponents.TheseauthorsimplementFarrell’s(1957)decompositionbasedontheradialinputmeasurewithinaparametric(Cobb-Douglas)deterministic approach.

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Next,weexplicitlyshowhowthevalueofr

x0,z0,q,TM

canbecalculatedinDEAundertheby-productionframework (seeRay,2004): r

x0,z0,q,TM

= max ,y s

r=1 qryr s.t. p

j=1 jxij≤xi0, i=1,...,n1 (13.1) p

j=1 jxij≤xi0, i=n1+1,...,n2 (13.2) − p

j=1 jyrj+yr≤0, r=1,...,m (13.3) p

j=1 j=1, (13.4) j≥0, j=1,...,p (13.5) yr≥0, r=1,...,m (13.6) (13)

Model(13)istheDEAimplementationofthegeneralexpressionin(12)fordeterminingmaximumrevenue.Theobjective functionisthesame,whiletheconstraintscoincidewiththosethatdeﬁneT1in(4).

Thedualprogramof(13)is(14):10

min c,d, = n1

i=1 c_i0xi0₊ n2

i=n1+1 c_i0xi0₊ s.t. n1

i=1 ci0xij+ n2

i=n1+1 ci0xij− m

r=1 dr0yjr+ 0≥0, j=1,...,p (14.1) dr0≥qr, r=1,...,m (14.2) ci0≥0 i=1,...,n (14.3) (14)

Being(14)thedualproblemof(13),thedecisionvariablesci0,i=1,...,n,anddr0,r=1,...,m,canbeinterpretedas

shadowpriceswhilethedecisionvariable 0maybeinterpretedasshadowproﬁt(seeAparicioetal.,2015).

Ifrevenuemaximizationisassumed,asisthecasehere,thefirmfacesexogenouslydeterminedmarketoutputprices. Followingthisline,wemaysupposethattheobjectiveoftheDMUistochoosetheoutputscombinationthatyieldthe maximumrevenueattheapplicableprices.Inthissense,revenueinefficiencymeasureshowcloseistheobservedrevenue oftheDMUunderevaluationtothemaximumfeasiblerevenue.Toevaluateeconomiclossduetorevenueinefficiency, inthecontextofthedirectionaloutputdistancefunctions,FäreandPrimont(2006)provedthatanormalizedmeasureof

revenueinefﬁciency,inparticulartheratio

r(x0,q,T )− m

r=1 qryr0 m

r=1 qrgr

,maybedecomposedintotechnicalinefﬁciency,→Do

x0_,_y0_;_g

_,

plusaresidualterminterpretedasallocativeinefﬁciencyintheFarrelltradition,wherer (x0,q,T ) and →

Do

x0_,_y0_;_g

_denote

the‘standard’revenuefunctionanddirectionaloutputdistancefunction,respectively,andgisthecorrespondingreference directionalvector.

Likewise,wecanintroducecostefﬁciencyfollowingthesamerationale,andbasedonthecostfunction.However,in ourcontextweareinterestedinenvironmentalcostfunctionsratherthanprivatecosts,representingameasureofthe (monetary)minimaldamagecausedbytheproductionofundesirableoutputs.Theenvironmentalcostfunctionrepresents a“monetizedmetric”oftheecologicalfootprintofthebadoutputs;see,forexample,Pearceetal.(1996)whorelatethe damagepertonofCO2withthesocialcostofcarbon(SCC).Correspondingly,anobservationiseconomicallyinefﬁcientin

environmentaltermsif,giventheamountofundesirableoutputsproduced,itcauseslargerdamagethanthatrepresented

10_Actually,_the_dual_program_of_model₍₁₃₎_has_an_additional_set_of_{non-negativity}_constraints_for_the_decision_variables_d

r0,r=1,...,m.However,this

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bytheminimumenvironmentalcostfunction(eitherasaresultoftechnicalorallocativeinefﬁciencies).Letusassumethat itispossibletoobserveorestimatepricesfortheundesirableoutputs:w= (w1,...,wm)∈Rm₊₊.UnderMurtyetal.’s(2012)

approach,theeco-damagefunctionwillbenon-parametricallydetermineddirectlyfromT2asfollows.

D

x0,y0,w,TM

= min ,z m

k=1 wkzr s.t. p

j=1 jxij≥xi0, i=n1+1,...,n2 (15.1) − p

j=1 jzkj+zk≥0, k=1,...,m (15.2) p

j=1 j=1, (15.3) j≥0, j=1,...,p (15.4) zk≥0, k=1,...,m (15.5) (15)

Model(15)minimizestheenvironmentalcost,

m

k=1

wkzr,associatedwiththeproduction/emissionofz(badoutputs)

givenaﬁxedquantityofpollution-generatinginputs.Theconstraintsin(15)coincidewiththerestrictionsthatdeﬁnethe sub-technologyT2in(5).

Thedualprogramof(15)is(16): max e,f, n2

i=n1+1 ei0xi0−0 s.t. n2

i=n1+1 ei0xij− m

k=1 fk0zkj−0≤0, j=1,...,p (16.1) f_k0_≤w_k, r₌1,...,m (16.2) ei0,fk0≥0 (16.3) (16)

Being(16)thelineardualofmodel(15),itiswell-knownthattheoptimalvaluesofbothmodelsarerelated.

Wenowderive,byduality,anormalizedmeasureofeconomicinefficiencyandshowhowitcanbedecomposedinto (desirable)revenueinefficiencyandeco-damageinefficiency.Inordertodothat,wefirstprovethefollowingtechnical proposition. Proposition1. LetıT1_,_ıT2 _>_0. Then, inf t,h

⎧

⎪

⎨

⎪

⎩

r

x0,z0,t,TM

− m

r=1 tryr0+ m

k=1 hkzr0−DT2

x0,y0,h,TM

:min

⎧

⎪

⎨

⎪

⎩

m

r=1 tryr0 ıT1 , m

k=1 hkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

≥1

⎫

⎪

⎬

⎪

⎭

≥→B

x0,y0,z0;TM

.

Proof. Let x0 ∈Rn₊, y0 ∈Rm₊, z0 ∈R₊m and let t ∈Rm₊, h∈Rm₊ such that min

⎧

⎪

⎨

⎪

⎩

m

r=1 tryr0 ıT1 , m

k=1 hkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

≥1. Let

c₀∗,d∗₀, ∗₀

be an optimal solution of (14) and let

e∗₀,f₀∗,∗₀

be an optimal solution of (16) when x0 ∈Rn+,

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v

1

0,u10,˛10,

v

20,u20,˛20

=

c₀∗,t, ₀∗,e∗₀,h,∗₀

is a feasible solution of (9). Constraints (9.5) and (9.6) are trivially satisﬁed. Regarding (9.1), n1

i=1 c∗_i0xij+ n2

i=n1+1 c∗_i0xij− m

r=1 tryrj+ 0∗

≥ by(14.2) n1

i=1 c∗_i0xij+ n2

i=n1+1 c∗_i0xij− m

r=1 d∗_r0yrj+ 0∗

≥ by(14.1) 0. As for (9.2), m

r=1 tryr0 ıT1 ≥1 since m

r=1 tryr0 ıT1 ≥min

⎧

⎪

⎨

⎪

⎩

m

r=1 tryr0 ıT1 , m

k=1 hkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

≥1. Therefore, m

r=1

tryr0≥ıT1. In the same way,

it is possible to prove that (9.3) and (9.4) are also satisﬁed. In particular, constraint (9.3) holds by (16.1) and (16.2). Consequently,

c∗₀,t, ∗₀,e∗₀,h,∗₀

is a feasible solution of (9). Regarding the objective function of (9) evaluated at this point, →B

x0,y0,z0;TM

≤ n1

i=1 c∗_i0xi0+ n2

i=n1+1 c∗_i0xi0− m

r=1 tryr0 + ∗₀− n2

i=n1+1 e∗_i0xi0+ m

k=1 hkzk0+ ∗₀ = r

x0,z0,t,TM

− m

r=1 tryr0+ m

k=1 hkzr0−D

x0,y0,h,TM

, since models (13) and (14) have the same optimal value and models (15) and (16) also have the same optimal value. In this way, →B

x0,y0,z0;TM

is a lower boundoftheset

⎧

⎨

⎩

n1

i=1 c_i0∗(t) xi0+ n2

i=n1+1 c_i0∗(t) xi0− m

r=1 tryr0+ ∗₀(t)− n2

i=n1+1 e∗_i0(h) xi0 + m

k=1 hkzk0+∗0(h) :

∀

(t,h)∈S0

, where

c∗₀(t) ,d∗₀(t) , ∗₀(t)

is any optimal solution of (14) when q=t and

e∗₀(h) ,f₀∗(h) ,∗₀(h)

is any opti-mal solution of (16) when w=h. Note that

⎧

⎨

⎩

n1

i=1 c∗_i0(t) xi0+ n2

i=n1+1 c∗_i0(t) xi0− m

r=1 tryr0+ ∗₀(t)− n2

i=n1+1 e∗_i0(h) xi0 + m

k=1 hkzk0+∗0(h) :

∀

(t,h)∈S0

=

r

x0,z0,t,TM

− m

r=1 tryr0+ m

k=1 hkzr0−D

x0,y0,h,TM

:

∀

(t,h)∈S0

,withS0=

⎧

⎪

⎨

⎪

⎩

(q,w) ∈Rm₊+m:min

⎧

⎪

⎨

⎪

⎩

m

r=1 qryr0 ıT1 , m

k=1 wkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

≥1

⎫

⎪

⎬

⎪

⎭

.Now,giventhat theinﬁmumofa setisthegreatestlowerbound

ofthatset,weseethatinf

t,h

r

x0,z0,t,TM

− m

r=1 tryr0+ m

k=1 hkzr0−D

x0,y0,h,TM

: (t,h)∈S0

≥→B

x0,y0,z0;TM

, whichistheinequalitythatwewereseeking.

Let (q,w) ∈Rm₊₊+mbemarketpricesforgoodandbadoutputs,respectively.Then,(˜q, ˜w)= (q,w)

min{ m

r=1 qr yr0 ıT1 , m

k=1 wkzk0 ıT2 } ∈S0= {(q,w)∈Rm+m₊ :min{ m

r=1 qryr0 ıT1 , m

k=1 wkzk0 ıT2 }≥1}. Consequently,applyingProposition1,weget

r

x0,z0, ˜q,TM

− m

r=1 ˜qryr0+ m

k=1 ˜ wkzr0−D

x0,y0, ˜w,TM

≥ inf t,h

r

x0,z0,t,TM

− m

r=1 tryr0+ m

k=1 hkzr0−D

x0,y0,h,TM

: (q,h)∈S0

≥→B

x0,y0,z0;TM

. (17)

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Thisinequalitywillbeusefulfor statingthedesiredrelationshipbetweeneconomicenvironmentalinefﬁciencyand

→

B

x0,y0,z0;TM

.

Finally,giventhatr

x0,z0,t,TM

isafunctionhomogeneousofdegree+1intandD

x0,y0,h,TM

isafunction homo-geneousofdegree+1inh,then

r

x0,z0,q,TM

− m

r=1 qryr0

+

_m k=1 wkzr0−D

x0,y0,w,TM

min

⎧

⎪

⎨

⎪

⎩

m

r=1 qryr0 ıT1 , m

k=1 wkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

≥→B

x0,y0,z0;TM

. (18)

Notethattheleft-handsideof(18)maybeinterpretedasa(normalized)measureofeconomicenvironmentalinefficiency. Additionally,followingFarrell’stradition,theright-handsidecanbeinterpretedas(environmental)technicalinefficiency andtheresidualtermassociatedwithclosingtheinequalitycouldbeinterpretedasallocativeinefficiency.Moreover,itis possibletodecomposetheleft-handsideof(18)into

r

x0,z0,q,TM

− m

r=1 qryr0+ m

k=1 wkzr0−D

x0,y0,w,TM

min

⎧

⎪

⎨

⎪

⎩

m

r=1 qryr0 ıT1 , m

k=1 wkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

OverallInefﬁciency = = r

x0,z0,q,TM

− m

r=1 qryr0 min

⎧

⎪

⎨

⎪

⎩

m

r=1 qryr0 ıT1 , m

k=1 wkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

(Good)RevenueInefﬁciency + m

k=1 wkzr0−D

x0,y0,w,TM

min

⎧

⎪

⎨

⎪

⎩

m

r=1 qryr0 ıT1 , m

k=1 wkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

Eco-DamageInefﬁciency . (19)

However,notethatthenormalizationtermusedin(18)and(19)−thatis,min

⎧

⎪

⎨

⎪

⎩

m

r=1 qryr0 ıT1 , m

k=1 wkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

−dependsontwo

differentterms,incontrasttowhathappenswithrespecttotheNerlovianproﬁtinefﬁciencymeasurein(10).Thismeansthat,

dependingontheobserveddataforeachDMU,overallinefﬁciencyisnormalizedbyeither

m

r=1 qryr0 ıT1 or m

k=1 wkzk0 ıT2 .Inother words,inthesamesample,theDMUscouldusedifferentnormalizationfactorsfortheirmeasureofoverallinefﬁciency. Somethingthatmakesdifﬁcultthecomparisonofresults.Hence,byanalogywiththestandardapproachbasedonthe directionaldistancefunction,wesuggestresortingtoanendogenousvalueforıT1_and,_therefore,_also_for_ıT2=₁−_ıT1_,_such

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that m

r=1 qryr0 ıT1 = m

k=1 wkzk0

ıT2 .Inotherwords,thevalueofthisendogenousıT1makesthetwocomponentsbeequal.Itiseasyto checkthatthisvalueisıT1∗=

m

r=1 qryr0/

_m

r=1 qryr0+ m

k=1 wkzk0

.

3.2. EconomicinefﬁciencymodelconsideringDakpoetal.’s(2017)approach

WenowturntoDakpoetal.’s(2017)approach.Inthiscase,theprojectionpointsinthetwosubtechnologiesfortheinput dimensionsmustcoincide.The“by-production”directionaloutputdistancefunctionunderthisapproachisasfollows:11

→ B

x0,y0,z0;TD

=max ıT1_ˇT1+_ıT2_ˇT2 s.t. p

j=1 j0xij≤xi0, i=1,...,n1 (20.1) p

j=1 j0xij≤xi0, i=n1+1,...,n2 (20.2) − p

j=1 j0yrj+ˇT1yr0≤−yr0, r=1,...,m (20.3) p

j=1 j0=1, (20.4) − p

j=1 j0xij≤−xi0, i=n1+1,...,n2 (20.5) p

j=1 j0zkj+ˇT2zk0≤zk0, k=1,...,m (20.6) p

j=1 j0=1, (20.7) − p

j=1 j0xij+ p

j=1 j0xij≤0, i=n1+1,...,n2 (20.8) ˇT1_,_ˇT2_,_j0_,_j0≥₀ _(20.9) (20)

Model(20)islikemodel(8),wheretheDakpoetal.’s(2017)approachhasbeenconsideredthroughconstraint(20.8), whichinterconnectstheprojectionsofthepollution-generatinginputsinT1andT2.

11_Constraints_(20.2)_and_(20.5)_imply_that₋ p

j=1 j0xij+ p

j=1

j0xij≥0,foralli=n1+1,...,n2.Thisinequality,togetherwith(20.8),implies p

j=1 j0xij= p

j=1

j0xijforalli=n1+1,...,n2,whichcoincideswiththeconstraintrelatedtoDakpoetal.’s(2017)approach.Weprefertoinclude(20.8)insteadof p

j=1 j0xij= p

j=1

(14)

Itslineardualis: → B (x0,y0,z0;TD)=min n1

i=1

v

1 i0xi0+ n2

i=n1+1

v

1 i0xi0− m

r=1 u1_r0yr0+˛10+ − n2

i=n1+1

v

2_i0xi0+ m

k=1 u2_k0zk0+˛20 s.t. n1

i=1

v

1 i0xij+ n2

i=n1+1

v

1 i0xij− m

r=1 u1 r0yrj+˛10+ (21.1) − n2

i=n1+1 i0_x ij≥0,j=1,...,p m

r=1 u1_r0yr0≥ıT1, (21.2) − n2

i=n1+1

v

2_i0xij+ m

k=1 u2_k0zkj+˛20+ n2

i=n1+1 i0xij≥0,j=1,...,p (21.3) m

k=1 u2 k0zk0≥ıT2, (21.4)

v

1 i0,

v

2 i0,u 1 r0,u2k0, i0_≥_0, _(21.5) ˛1 0,˛20free (21.6) (21)

Models(20)and(21)arerelatedbythetheoryofLinearProgramming.

Inthiscontextwenowdeﬁneanewsupportfunction,representingproﬁtinDakpoetal.’smodel,as

x0,q,w,TD

:

x0,q,w,TD

=max m

r=1 qryr− m

k=1 wkzk s.t. p

j=1 _j0x_ij_≤x_i0, i₌1,...,n1 (22.1) p

j=1 j0xij≤xi0, i=n1+1,...,n2 (22.2) − p

j=1 j0yrj+yr≤0, r=1,...,m (22.3) p

j=1 j0=1, (22.4) − p

j=1 _j0x_ij_≤_−x_i0, i₌n1+1,...,n2 (22.5) p

j=1 j0zkj−zk≤0, k=1,...,m (22.6) p

j=1 j0=1, (22.7) − p

j=1 j0xij+ p

j=1 j0xij≤0, i=n1+1,...,n2 (22.8) yr,zk,j0,j0≥0, (22.9) (22)

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whichmaximizesthedifferencebetweenprivaterevenueandeco-damagecostsinourby-productioncontext.Notethat theDakpoetal.’s(2017)approachhasbeenconsideredinmodel(22)throughconstraint(22.8).

Thelineardualof(22)is:

x0,q,w,TD

=min n1

i=1 ci0xi0+ n2

i=n1+1 ci0xi0+ 0+ − n2

i=n1+1 ei0xi0+0 s.t. n1

i=1 ci0_x ij+ n2

i=n1+1 ci0_x ij− m

r=1 dr0_y rj+ 0− n2

i=n1+1 ai0_x ij≥0, (23.1) j=1,...,p, dr0≥qr, (23.2) − n2

i=n1+1 ei0_x ij+ m

k=1 fk0_z kj+0+ n2

i=n1+1 ai0_x ij≥0, (23.3) j=1,...,p, fk0≤wk, (23.4) ci0_,_d r0,ei0,fk0,ai0≥0 (23.5) 0,0free. (23.6) (23)

ByLinearProgramming,theoptimalvaluesofmodel(22)and(23)arerelated. Next,weshowarelationshipbetween

x0,q,w,TD

and→B

x0,y0,z0;TD

. Proposition2. LetıT1_,_ıT2 _>_0. Then, inf t,h

⎧

⎪

⎨

⎪

⎩

x0,t,h,TD

− m

r=1 tryr0+ m

k=1 hkzr0:min

⎧

⎪

⎨

⎪

⎩

m

r=1 tryr0 ıT1 , m

k=1 hkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

≥1

⎫

⎪

⎬

⎪

⎭

≥→B

x0,y0,z0;TD

.

Proof.FollowingthesamestepsthaninProposition1,wegetthedesiredresult. ApplyingProposition2,withmarketprices (q,w),wegetthefollowinginequality.

x0,q,w,TD

−

_m

r=1 qryr0− m

k=1 wkzk0

min

⎧

⎪

⎨

⎪

⎩

m

r=1 qryr0 ıT1 , m

k=1 wkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

≥→B

x0,y0,z0;TD

. (24)

Theleft-handsidein(24)maybeinterpretedasameasureofeconomicenvironmentalinefficiency,whichcouldbe decomposedintotechnicalinefficiency(theright-handsidein(24))andaresidualterm,interpretedasallocativeinefficiency.

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3.3. EconomicinefﬁciencymodelconsideringFørsund’s(2018)proposal

Finally,itispossibletoincorporateFørsund’s(2018)proposal,adaptingMurtyetal.(2012)andDakpoetal.(2017).To dothis,itissufﬁcienttoincludethenon-pollutinginputsinthesub-technologyT2.TheresultsofProposition1and2are

validfor→B

x0,y0,z0;TMF

and→B

x0,y0,z0;TDF

.Hence,wehavemodel(25).

→ B

x0,y0,z0;TMF

= max ıT1_ˇT1+_ıT2_ˇT2 s.t. p

j=1 j0xij≤xi0, i=1,...,n1 (25.1) p

j=1 j0xij≤xi0, i=n1+1,...,n2 (25.2) − p

j=1 j0yrj+ˇT1yr0≤−yr0, r=1,...,m (25.3) p

j=1 j0=1, (25.4) − p

j=1 j0xij≤−xi0, i=1,...,n (25.5) p

j=1 j0zkj+ˇT2zk0≤zk0, k=1,...,m (25.6) p

j=1 j0=1, (25.7) ˇT1_,_ˇT2_,_j0_,_j0≥_0, _(25.8) (25)

Model(25)islikemodel(8),wheretheFørsund’s(2018)approachhasbeenconsideredbychangingi=n1+1,...,n2by

i=1,...,ninconstraint(25.5).And D

x0,y0,w,TMF

= min ,z m

k=1 wkzr s.t. p

j=1 jxij≥xi0, i=1,...,n (26.1) − p

j=1 jzkj+zk≥0, k=1,...,m (26.2) p

j=1 _j₌1, (26.3) j≥0, j=1,...,p (26.4) zk≥0, k=1,...,m (26.5) (26) with r

x0,z0,q,TM

− m

r=1 qryr0+ m

k=1 wkzr0−D

x0,y0,w,TMF

min

⎧

⎪

⎨

⎪

⎩

m

r=1 qryr0 ıT1 , m

k=1 wkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

≥→B

x0,y0,z0;TMF

. (27)

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Model(26)allowstodeterminethedamagefunctionwhentheMurtyetal.(2012)approachisadaptedthroughFørsund’s (2018)proposal.Additionally, theleft-handsidein (27)maybeinterpreted asa measureofeconomic environmental inefﬁciency.Inparticular,itispossibletodecomposeitinto

r

x0,z0,q,TM

− m

r=1 qryr0+ m

k=1 wkzr0−D

x0,y0,w,TMF

min

⎧

⎪

⎨

⎪

⎩

m

r=1 qryr0 ıT1 , m

k=1 wkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

OverallInefﬁciency = = r

x0,z0,q,TM

− m

r=1 qryr0 min

⎧

⎪

⎨

⎪

⎩

m

r=1 qryr0 ıT1 , m

k=1 wkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

(Good)RevenueInefﬁciency

+ m

k=1 wkzr0−D

x0,y0,w,TMF

min

⎧

⎪

⎨

⎪

⎩

m

r=1 qryr0 ıT1 , m

k=1 wkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

Eco-DamageInefﬁciency . (28)

RegardingDakpoetal.’s(2017)model,includingFørsund’s(2018)extension,wehave: B(x0,y0,z0;TDF)=max ıT1ˇT1+ıT2ˇT2 s.t. p

j=1 j0xij≤xi0, i=1,...,n1 (29.1) p

j=1 j0xij≤xi0, i=n1+1,...,n2 (29.2) − p

j=1 j0yrj+ˇT1yr0≤−yr0, r=1,...,m (29.3) p

j=1 _j0₌1, (29.4) − p

j=1 j0xij≤−xi0, i=1,...,n (29.5) p

j=1 j0zkj+ˇT2zk0≤zk0, k=1,...,m (29.6) p

j=1 j0=1, (29.7) − p

j=1 j0xij+ p

j=1 j0xij≤0, i=n1+1,...,n2 (29.8) ˇT1_,_ˇT2_, j0,j0≥0 (29.9) (29)

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And

x0,q,w,TDF

=max m

r=1 qryr− m

k=1 wkzk s.t. p

j=1 j0xij≤xi0, i=1,...,n1 (30.1) p

j=1 j0xij≤xi0, i=n1+1,...,n2 (30.2) − p

j=1 j0yrj+yr≤0, r=1,...,m (30.3) p

j=1 j0=1, (30.4) − p

j=1 j0xij≤−xi0, i=1,...,n (30.5) p

j=1 j0zkj−zk≤0, k=1,...,m (30.6) p

j=1 j0=1, (30.7) − p

j=1 j0xij+ p

j=1 j0xij≤0, i=n1+1,...,n2 (30.8) yr,zk,j0,j0≥0, (30.9) (30)

whichresultsinthefollowinginequality:

x0,q,w,TDF

−

_m

r=1 qryr0− m

k=1 wkzk0

min

⎧

⎪

⎨

⎪

⎩

m

r=1 qryr0 ıT1 , m

k=1 wkzk0 ıT2

⎫

⎪

⎬

⎪

⎭

≥→B

x0,y0,z0;TDF

. (31)

Inequalities(27)and(31)makeitpossibletodeﬁnetechnicalandallocativetermsasdriversofthecorrespondingmeasure ofeconomicenvironmentalinefﬁciency.IntheempiricalapplicationwesolvethemodelscorrespondingtoMurtyetal.(2012)

andDakpoetal.(2017),enhancedwithFørsund’s(2018)proposal.Thisrepresentsatotaloffourmodels. 4. Empiricalapplication

4.1. Datasetandvariables

Theempiricalillustrationreliesonstate-leveldataintheUnitedStatesthatcomesfrommultipleagencies.Thedataset consistsofaggregatedﬁrmdataforeachstateas,unfortunately,andduetostatisticalconﬁdentialityreasons,wedonot haveaccesstotheindividualmicrodata.12_The_main_source_of_data_is_the_U.S._Department_of_Agriculture_(USDA)_Economic

12 _We_perform_a_macro-level_analysis_in_our_empirical_application,_assuming_that_the_DMUs_(the_states)_can_be_compared._A_more_suitable_analysis_would

consistinestimatingameta-frontier(O’Donnelletal.,2008;Batteseetal.,2004)usingthedataforallthefirmsinallthestatesandthen,decomposing inefficiencyintowithin-stateinefficiencyandagapbetweenthetechnologyofeachstateandtheglobalfrontier.Thislinecouldbeagoodavenuefor furtherresearch.

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Table1

Descriptivestatisticsofinput-outputdata(implicitrealquantities),2004.

Variable Mean SD Coefﬁcientofvariation

Non-pollution-generatinginputs

Capitalservices(million$) 541.068 450.804 0.833 Landserviceﬂows(million$) 650.678 738.002 1.134 Laborservices(million$) 1,292.827 1,186.855 0.918 Pollution-generatinginputs

Energy(million$) 166.647 144.484 0.867

Pesticides(million$) 164.921 163.858 0.994

Goodoutputs

Livestockandproducts(million$) 2,103.076 1,997.443 0.950

Crops(million$) 2,819.480 3,419.130 1.213

Badoutputs

CO2emissions(tonsofCO2equivalents) 996,394.520 914,820.595 0.918

Pesticideexposures(number) 2,458.708 2,256.445 0.918 Notes:SD=Standarddeviation.

ResearchService(ERS),whichcompiledthedatanecessarytocalculateagriculturalproductivityintheUS,and,inparticular, thepriceindicesandimplicitquantitiesoffarmoutputsandinputsforeachofthe48continentalstatesfor1960−2004.The datasethasbeenvalidatedandusedextensivelyinpreviousresearch(forexample,inBalletal.,1999;ZoﬁoandKnoxLovell, 2001;HuffmanandEvenson,2006;SabasiandShumway,2018).Acriticalreviewofthedatainlightofrecentdevelopments canbefoundinShumwayetal.(2015,2016).Toillustrateourmodels,weconsiderthemostrecentyearavailableinthe dataset(2004)andassumethattheproductionprocessischaracterizedbythefollowingthreenon-pollutinginputs(capital servicesexcludingland,landserviceﬂows,andlaborservices),twopollutinginputs(energyandpesticides),andtwogood outputs(livestockandcrops).ThepricesforthesevariablesweredirectlyobtainedfromtheERSdataset.Pricesindicesin theERSdatasetareconstructedusingtheTörnqvistformulation.13

Asfortheundesirableoutputproductiongeneratedbyenergyconsumption,weconsidercarbondioxide(CO2)emissions

fromtheagriculturalsectorassociatedwithfuelcombustion,alsofor2004(expressedintonsofCO2equivalents),obtained

fromtheU.S.EnvironmentalProtectionAgency(EPA).14_The_price_of_CO

2emissionsisproxiedbythemarketclearingprice

setinthestateofCalifornia(priceofcarbonemissionsexpressedinthousandsofdollarspertonofCO2equivalents),since

ageneralmarketforCO2forthewholeUSdoesnotexist.Inparticular,thisisthepriceofcarbonfortradableallowances

withafuturescontractthatoriginatesfromtheCaliforniangreenhousegasestradingmarketundertheCaliforniaCapand TradeProgram,2019.Weconsidertheaverage2012priceanddeﬂateitto2004usingtheconsumerpriceindexinabsence ofasuitabledeﬂator(USBureauofLaborStatistics).15_The_measure_of_bad_output_related_to_pesticides_is_the_number_of

pesticideexposuresperstatefor2004obtainedfromtheCentersforDiseaseControlandPreventionattheUSDepartment ofHealth&HumanServices.Astheapproximationofthepriceofthisbadoutputweusethecostofhospitalizedtreatment ofpesticide-relatedpoisonings(inthousandsofdollars)asestimatedinPimentel(2005).Becausethiscostisprovidedfor 1995,wefurtherexpressitto2004pricesusingthepriceindexformedicalservicesasobtainedfromtheU.S.Bureauof LaborStatistics.16

Tables1and2summarizethedescriptivestatisticsofinput-outputquantitiesandtheircorrespondingprices,respectively, fortheUSstatesin2004.AppendixA.1.oftheonlinesupplementalmaterialaccompanyingthepaperpresentsthisdatafor eachstate.

4.2. Results

4.2.1. Technical,allocativeandproﬁtfrontiers

Whensolvingourfourreferenceeconomicmodels–thatis,Murtyetal.(2012)andDakpoetal.(2017),each com-plementedwithFørsund’s(2018)proposal−itisrelevanttodetermine,fromatechnologicalperspective,thenumberof

13_The_details_on_the_method_of_construction_of_all_variables_are_contained_in_the_following_webpage_of_the_USDA-ERS:_{https://www.ers.usda.gov/}

data-products/agricultural-productivity-in-the-us/methods/.

14_Since_these_data_are_given_in_overall_terms_for_the_whole_country,_we_further_disaggregate_it_by_state,_using_for_that_purpose_the_share_that_each_state

hasinfarmproductionexpensesforgasoline,fuels,andoils,asreportedbytheU.S.DepartmentofAgriculture,expressedinthousandsofdollars.

15_{California’s}_GHG_emissions_program_is_the_fourth_largest_in_the_world_after_the_European_Union’s_Emissions_Trading_System,_South_Korean_Emissions

TradingSchemeandtheEmissionTradingSystemintheChineseprovinceofGuangdong.Fromitsbeginningin2012itcoveredthepowerandindustrial facilities,anditexpandedtonaturalgasandtransportationfuelsin2015,allowingittocoverapproximately85percentofCalifornia’sGHGemissions.

16_The_{consideration}_of_these_market_prices_for_the_bad_outputs,_which_can_be_considered_as_the_opportunity_costs_of_{environmental}_damage,_enables_the

analysisofallocativeinefficiencywithrespecttotheenvironmentallyefficientfrontier.Marketpricescanbecomparedtotheshadowpricescorresponding tothemarginalrateoftransformationofbadoutputsforgoodoutputs,whichareobservedattheoptimalprojectionofthefirmonthetechnological frontier.Comparingbothsetsofpricescanhelpregulatorstoadoptspecificpollutingabatementinstrumentssuchastaxes,permits,oremissionstandards. Onthistopic,arecentandcomprehensivestudycomparingshadowpricecalculationsforcarbondioxideemissionsinChina,usingbothparametricand non-parametrictechniques,canbefoundinMaetal.(2019).

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Table2

Descriptivestatisticsofpricesforinput-outputdata,2004.

Variable Mean SD Coefﬁcientofvariation

Non-pollution-generatinginputs

Capitalservices(indicesrelativetoAlabama) 1.105 0.028 0.026 Landserviceﬂows(indicesrelativetoAlabama) 1.134 0.636 0.560 Laborservices(indicesrelativetoAlabama) 1.158 0.325 0.280 Pollution-generatinginputs

Energy(indicesrelativetoAlabama) 1.441 0.166 0.115 Pesticides(indicesrelativetoAlabama) 1.142 0.252 0.220 Goodoutputs

Livestockandproducts 1.245 0.197 0.158

Crops(indicesrelativetoAlabama) 1.067 0.156 0.146 Badoutputs

CO2emissions(thousand$pertonofCO2equivalents) 0.011 0 0

Pesticideexposures(thousand$) 6.222 0 0

Notes:SD=Standarddeviation.Pricesofinputsandgoodoutputsvaryacrossstatesandareexpressedinrelativetermswithrespecttotheﬁrststate, Alabama,considering1996asthebaseyear.Pricesofbadoutputsareuniqueforallstatesandaredeﬂatedtothe2004referenceyear.

Table3

NumberofefﬁcientUSstates.

Model Technical Allocative Proﬁt

T T1 T2

Murtyetal. 6 19 11 4 4

Dakpoetal. 13 30 16 6 6

Murty&Førsund 10 19 24 4 4

Dakpo&Førsund 19 30 28 15 15

Fig.1. Averagetechnicalinefﬁcienciesbymodels(absolutevalues).

observationsthatareefficient,therebydefiningthefrontieroftheglobalby-productiontechnologyT,consistingofboth theintendedproductionT1 technology,(4)(hereafter,conventionalorstandardtechnology)andthepollution-generating technologyT2,(5)(hereafter,pollutingtechnology).Table3showsthatthenumberofobservationsdefiningtheproduction frontierisgreaterintheconventionaltechnologyT1thaninthepollutingtechnologyT2,exceptinthecaseoftheMurtyetal. (2012)modelincorporatingFørsund’sproposal.Clearly,notalltechnicallyefficientstatesareallocative-efficientandthereby achieveprofitefficiency.California,Delaware,IowaandVermontareefficientinallmodels.Approximately10percentof USstates(fourorsixoutof48)arefullyefficient,exceptinDakpoetal.’sapproachenhancedwithFørsund’sassumption, wherethenumberofprofit-efficientstatesincreasesto31.25percent17_.

4.2.2. Technicalinefﬁciency:resultswithinandbetweenmodels

Departingfromthisgeneralportraitofinefficiencyfrequenciesatthetechnical,allocative,andoverallprofitinefficiency levels,wenowfocusonthetechnologicalside,withFig.1portrayingtheaverageabsolutetechnicalefficiencyvaluesinT1,

T2andtheirglobalby-productionaggregateT,acrossthefourmodels.

The measurement of environmental economic inefficiency with pollution-generating technologies

Resource

and

Energy

Economics

The

measurement

of

environmental

economic

inefﬁciency

with

pollution-generating

technologies

Juan

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,

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Kapelko

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L.

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⎡

⎢

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⎢

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⎤

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