Pigouvian tax, abatement policies and uncertainty on the environment

31 Pages·2011·0.4 MB·English

JEcon(2011)103:221–251
DOI10.1007/s00712-011-0199-7
Pigouvian tax, abatement policies and uncertainty
on the environment
DonatellaBaiardi · MarioMenegatti
Received:11August2010/Accepted:15February2011/Publishedonline:6March2011
©Springer-Verlag2011
Abstract ThepaperexaminestheeffectsofenvironmentaluncertaintyonPigouvian
tax and abatement policy used, either separately or contemporaneously, to counter-
actpollution.Wediscusstheseeffectsbyintroducingthreekindsofrisk:riskonthe
environmentalquality,riskontheimpactofpollutionandriskontheimpactofabate-
ment.Foreachcasewedeterminetheconditionsensuringthatriskincreasesthesize
ofpublicinterventionandprovideaneconomicinterpretationandsomeparallelisms
withotherriskproblems.Thelastpartofthepapergeneralizessomeofourresultsto
thecaseof N-thorderriskchanges.
Keywords Pigouvian tax · Abatement policies · Environment · Uncertainty ·
Bivariateutility
JELClassification H23·D81·Q5
1 Introduction
Nowadaysenvironmentalproblemsareacentralissueformanydevelopedanddevel-
oping countries. Pollution and climate change are important challenges for human
healthandeconomicgrowth.Definingappropriateeconomicpoliciesforcontaining
D.Baiardi
DipartimentodiEconomiaPoliticaeMetodiQuantitativi,
UniversitàdiPavia,Pavia,Italy
B
M.Menegatti( )
DipartimentodiEconomia,UniversitàdiParma,Parma,Italy
e-mail:[email protected]
123
222 D.Baiardi,M.Menegatti
environmentaldegradationandeventuallyimprovingenvironmentalqualityisnowa
priorityformanygovernments.1
Theeconomiceffectsofpollutionandenvironmentdegradationhavebeenwidely
analyzedintheliterature.Ononehand,thesepapersshowtherelevanceofenviron-
mentalchangefordifferenteconomicissues.Inthisfield,inparticular,theeffectson
economic growth and welfare have been both studied from a theoretical standpoint
andquantifiedbysimulationsandcalibrations.2
Ontheotherhand,manypapersexamineappropriatepoliciesinordertoreduceenvi-
ronmentaldegradation.Twopolicyinstrumentsaremostimportant.Thefirstisabate-
mentpolicy,whichisanintervention,financedbytaxation,whichdirectlyimproves
environmentalquality.Thesecondistheso-called“Pigouviantax”,firstproposedby
Pigou(1924),whoarguedthatnegativeexternalityduetopollutioncanbeinternal-
izedinacompetitivemarketbyintroducingataxequaltothesocialmarginaldamage
causedbyenvironmentaldegradation.3
Althoughthereisabroadconsensusontheideathatpollutionhasimportanteffects
forenvironmentalconditionsandhumanlife,wealsofindawidespreaduncertainty
on the features and the sizes of these effects. A good example of this is the well-
known issue of global warming. First, although the majority of scientists in the cli-
materesearchcommunityassertthattherehasbeenanincreaseintemperaturesdueto
humanactivities,othersdisagree.4Second,evenifwebelievethatglobalwarmingis
takingplace,thereiswidespreaduncertaintyonhowtemperatureswillrise5andhow
thiswillimpactonhumansystems.6
Theaimofthisworkistostudytheeffectsofuncertaintyontheenvironmentwith
specificreferencetoitsinfluenceontheimplementationofpoliciesusedtocounteract
environmental deterioration. For this purpose we introduce three types of risk, i.e.
1 ArecentimportantsignalofthisimportanceistheattentionpaidinalldevelopedcountriestotheUnited
NationSummitsonClimateChangeinCopenhagen(December2009)andCancun(December2010).
2 ForasurveyoftheseissuesseeBaumolandOates(1988),Nordhaus(1994),NordhausandBoyer(2000)
andXepapadeas(2005).
3 ForananalysisofthedesignandtheeffectsofthesetwopoliciesseerespectivelyGradusandSmulders
(1993),SmuldersandGradus(1996),EconomidesandPhilippopoulos(2008)andGuptaandBarman(2009)
forabatementpolicyandPigou(1924),BaumolandOates(1988),Mohtadi(1996)andHelfandetal.(2003)
forPigouviantax.Notethatotherpossibleinstrumentsforpublicinterventioninenvironmentaleconomics,
notanalysedinthiswork,arepropertyrights,bindingquotarestrictionsandmarketsforpollutionpermits.
4 Onthissee,forinstance,thedebatebetweentheIntergovernmentalPanelonClimateChange(IPCC)and
theNongovernmentalInternationalPanelonClimateChange(NIPCC).
5 AsemphasizedbytheIPCC:‘Modelsdifferconsiderablyintheirestimatesofthestrengthofdifferent
feedbacksintheclimatesystem,particularlycloudfeedbacks,oceanicheatuptakeandcarboncyclefeed-
backs,althoughprogresshasbeenmadeintheseareas.Also,theconfidenceinprojectionsishigherforsome
variables(e.g.temperature)thanforothers(e.g.precipitation),anditishigherforlargerspatialscales
andlongertimeaveragingperiods’(IPCC2007,p.73)and‘Projectionsofclimatechangeanditsimpacts
beyondabout2050arestronglyscenario-andmodel-dependent,andimprovedprojectionswouldrequire
improvedunderstandingofsourcesofuncertaintyandenhancementsinsystematicobservationnetworks’
(IPCC2007,p.73).
6 OnthistheIPCCFourthAssessmentReportclaimsthat:‘Effectsofclimatechangesonhumanandsome
naturalsystemsaredifficulttodetectduetoadaptationandnon-climaticdrivers’(IPCC2007,p.72).On
uncertaintyontheeconomiceffectsofclimaticchangesseealsoHealandKriström(2002).
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Pigouviantax,abatementpoliciesanduncertaintyontheenvironment 223
riskontheenvironmentalqualitylevel,riskontheeffectofpollutionandriskonthe
impactoftheabatementpolicy.
Toourknowledgetheeffectsofuncertaintyontheenvironmentwasstudiedinthe
literatureonlyfromastandpointdifferentfromthatconsideredinthiswork.Inpar-
ticular,amongthepapersexaminingthisissue,Heal(1984)andKeeleretal.(2004)
studytheeffectofuncertaintyonthetimeofafuturechangeinproductivitycausedby
pollution,insertinganuncertainclimatethresholdinamodelofoptimaldepletionand
inaDICEgrowthmodel.7UlphandUlph(1997)andPindyck(2000,2002)analysethe
optimaltimingfortheimplementationofanabatementpolicyinarealoptionframe-
work.Gollieretal.(2000)investigateconditionsonutilityfunctionswhichensurethat
scientificprogressreducinguncertaintyonthedistributionofdamagerelatedtocon-
sumption(inparticularpollutiondamage)induceanearlierpreventioneffort.Finally,
Soretz (2007) examines optimal pollution taxation in an endogenous growth model
whereproductivityisaffectedbyuncertaintyonenvironmentalqualityonthebasisof
arelationshipevolvingasastochasticWienerprocess.
This paper presents three main differences with regard to the previous literature.
First,thepaperscitedaboveeitherdonotexplicitlyconsiderenvironmentalpolicies,
or,asinthemajorityofthecases,theystudytheeffectofuncertaintyonabatementtax.
In this work, we analyze the effect of environmental uncertainty on both Pigouvian
andabatementtax,consideredbothasseparateinstrumentsandjointinstruments.
Second,thepaperscitedabovemakespecificassumptionsaboutthedistributionof
therandomprocessdescribingriskwhileourpaperdoesnot.8Wethusobtaingeneral
conclusionswhichdonotdependonthespecificdistributionintroduced.Inthisfield,
wealsointroduceafurthergeneralization,extendingourresultstothecaseoftheso
called‘N-thorderriskchange’.9
Finally,theexistingliteraturestudiesaone-argumentutilityframeworkwhereutil-
itydependsonlyonconsumption.Thisimpliesthatenvironmentalriskaffectsagents’
preferences through its effect on consumption, because it impacts on productivity.
Environmentalrisk,however,hasmainlyadirecteffectonutilitysinceenvironmental
qualitydeterminesaspecificdimensionofagentsutility,differentfromconsumption.
Thisframeworkisclearlyrelatedtoarecentstrandofliteratureanalyzingtheeffect
ofthebackgroundrisk.Backgroundriskisinfactasecondsourceofrisk,distinctfrom
financial risk, studied in a two-argument utility model, where the second argument
oftheutilityisrelatedtovariablessuchashealthstatusorenvironmentalquality.In
thisliterature,Pratt(1988),Finkelshtainetal.(1999)andCourbage(2001)studythe
effectofbackgroundrisk,andthusofenvironmentalrisk,onriskaversion.Courbage
and Rey (2007) and Menegatti (2009a,b) analyse the effects of background risk on
precautionarysavingrespectivelyundersomespecificassumptionsonbivariaterisk
distributionandinthecaseofsmallrisks.FinallyDenuitetal.(2011)generalizethe
7 Asimilarmodelisusedtostudyuncertaintyonfuturepreferencesabouttheenvironment(giventhe
certainleveloffutureenvironmentalquality)inBeltrattietal.(1998),AyongLeKama(2001)andAyong
LeKamaandSchubert(2004).
8 TheonlyexceptiontothisisGollieretal.(2000),whostudyaframeworkwherethedistributionofa
genericrandomvariableisrevisedbyconsumersaccordingtotheBayesrule.
9 AsdefinedbyEkern(1980)andEeckhoudtandSchlesinger(2008).
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224 D.Baiardi,M.Menegatti
analysis of optimal saving and investment in the presence of two correlated risks,
financialandbackground.
Thispaperisorganizedasfollow.Section2introducesthemodel.Section3analy-
sestheeffectofriskonoptimalPigouviantaxation.Section4studiestheimplication
ofriskinamodelwithabatementpolicies.Section5examinesamodelwhereboth
Pigouvian tax and abatement policies are used together. Section6 presents a gener-
alizationofourresultsinthecaseofstochastic N-thdegreeriskincrease.Section7
concludes.
2 Themodel
Inthissectionwesetupasimplemodelwhichcompareschoicemadeintheprivate
sectorwithsociallyoptimalchoice.Followinganapproachwidelyusedinthelitera-
ture(see,forinstance,Mohtadi1996;Xepapadeas2005),10choiceintheprivatesector
isdescribedbythebehaviourofarepresentativeconsumer,11whilethesociallyopti-
malchoiceisdescribedbythechoicesmadebyabenevolentsocialplanner.Different
policyinstrumentswillbethenintroducedinnextsections.
Weassumethatconsumerpreferencesarerepresentedbyabivariateutilityfunction
U(c,q).Theutilityfunctiondependsonconsumptioncandonenvironmentalquality
levelq,wherec andq areexpressedwithdifferentunitofmeasurement.U(c,q)is
increasing and concave with regard to each argument and it is three times continu-
ouslydifferentiablewithrespecttocandq respectively.LettingU =∂U/∂c,U =
c q
∂U/∂q,U = ∂2U/∂c2, U = ∂2U/∂c∂q and so on, these assumptions mean
cc cq
U > 0,U > 0,U < 0 and U < 0. We emphasize that this last assumption
c q cc qq
implies aversion toward environmental risk. We introduce this hypothesis since we
want to compare optimal choice with and without uncertainty for agents who are
affectedbyrisk.Forcompletenessweprovidesomeresultsforthecaseofriskneu-
tralityinAppendixA.
Consumptionciscomposedbyexogenousincome y andbyincomederivedfrom
production activity. Production activity is financed by investment s and generates
an output given by f(s). As usual for production functions, we assume that f(s)
is increasing and concave with regard to s, such that f(cid:2)(.) > 0 and f(cid:2)(cid:2)(.) < 0.
Consumptioncisthusgivenbyc= y−s+ f(s).12Thevalueofqdependsontheini-
tialenvironmentalqualitylevelq andonthedamagefunctiong(s),whichexpresses
0
10 Theuniquedifferencewithreferencetothesetwopapersisthattheyexamineamultiperiodframework
whilewearestudyingastaticmodel.OtherpaperswithaverysimilarapproachareSmuldersandGradus
(1996),EconomidesandPhilippopoulos(2008)andAngelopoulosetal.(2010).
11 Theuseofarepresentativeconsumerimpliesthatweassumeallconsumersintheprivatesectortobe
homogeneous.
12 Notethatsincethemodelisstatic,wehavethatinvestmentsandproductionactivityrelatedtoitare
contemporaneous.Thismeansthattheagenthasonlytochoosetheoptimalfractionofexogenousincome
ytobedestinedtoproductionactivity(whichwilldependonmarginalproductivity).Forsimplicitywecall
thisfraction‘investment’eventhoughthemodeldoesnotexhibitanykindofcapitalaccumulation.Note
thatthissettingisequivalenttoaframeworkwheretheproductioniscarriedoutbyafirmwhichmaximizes
itsprofit f(s)−sandthattheconsumeristheownerofthefirm,sothattheprofitisapartofherincome.
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Pigouviantax,abatementpoliciesanduncertaintyontheenvironment 225
thenegativeinfluenceofpollutionthroughinvestmentinproductiveactivities.Since
largerproductionimpliesmorepollution,weassumethatg(cid:2)(.)>0.
Asinmuchliteratureonpollutionandenvironment(e.g.MichelandRotillon1995;
Mohtadi1996;Helfandetal.2003;Xepapadeas2005;Angelopoulosetal.2010),we
assume that pollution is an externality, meaning that consumers do not take it into
account when they choose the optimal level of s. This assumption generates a dif-
ferencebetweenconsumerbehaviourandsociallyoptimalchoice,andjustifiesintro-
ducing taxation on polluting activity in the next sections.13 A partially alternative
approachwillbediscussedattheendofthissection.
Theconsumermaximisationproblem,whichconsidersenvironmentalqualitylevel
asgivenandequaltoq ,isthusthefollowing:
0
maxU(y−s+ f(s),q ) (1)
0
s
Optimal investment s(cid:3) is determined as the solution of the first-order condition,14
which,aftersomecomputations,isequivalentto
f(cid:2)(s(cid:3))−1=0. (2)
Notethatcondition(2)simplyimpliestheequalitybetweenmarginalcostandmarginal
benefitofinvestments.
The externality related to pollution is neglected by the consumer, but considered
bytheplanner.Thismeansthat,whenweconsidertheplanner’sproblem,weneedto
taketheeffectofs onq intoaccount.Sotheplanner’soptimalchoiceisobtainedby
maximising
maxU(y−s+ f (s),q −g(s)). (3)
0
s
Thefirst-orderconditionis
U (y−s+f(s),q −g(s))(f(cid:2)(s)−1)−U (y−s+f(s),q −g(s))g(cid:2)(s)=0 (4)
c 0 q 0
wheresistheplanneroptimalinvestmentlevel.Weassumethatthesecond-ordercon-
dition ensuring ∂2U < 0 is satisfied.15 This condition implies the equality between
∂s2
social marginal cost and marginal benefit of investment s. Note that both cost and
13 In fact, without the externality the consumer would limit emissions in order to avoid an excessive
pollutionandtaxationonpollutingactivitywouldbenotnecessary.
14 Itisclearthat,givenourassumptiononU(c,q)and f(s)thesecond-orderconditionoftheconsumer’s
problemissatisfied.
15 Thesecond-orderconditionfortheplannerproblemisUcc(y−s+ f(s),q0−g(s))[f(cid:2)(s)−1]2+
Uc(y−s+ f(s),q0−g(s))f(cid:2)(cid:2)(s)+Uqq(y−s+ f(s),q0−g(s))g(cid:2)(s)2−Uq(y−s+ f(s),q0−
g(s))g(cid:2)(cid:2)(s)−2Ucq(y−s+ f(s),q0−g(s))[f(cid:2)(s)−1]g(cid:2)(s)<0.Notethat,giventhepreviousassump-
tionsonU(c,q)and f(s),theconditionisautomaticallysatisfiedifg(cid:2)(cid:2)(s) ≥ 0andUcq ≥ 0.Thislast
assumptionwouldmeanthatconsumptionandenvironmentalqualitylevelarecomplements.Asimilar
hypothesiswasmade,forexample,byMohtadi(1996),whoshowsthatanimprovementinenvironmental
qualityincreaseseconomicgrowthifUcq ≥0.
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226 D.Baiardi,M.Menegatti
benefitin(4)aremeasuredinutils.Thesameistrueforconsumer’sfirst-ordercondi-
tionbeforethesimplificationdetermining(2),duetothesymmetriceffectofcostand
benefitonconsumer’sutility.
(cid:3)
Thecomparisonbetweens andsclearlyshowsthattheconsumerchoosesasubop-
timallylargerinvestmentlevelsduetopollutionexternality.16Thissuboptimalresult
canberemovedbytheplannerbyimposingaPigouviantaxequaltothemarginalpol-
lutiondamage.Furthermore,environmentalqualitylevelcanbeincreasedbymeans
of abatement policies. The following sections compare the effects of the Pigouvian
taxationandabatementpoliciesunderrisk.
Beforeexaminingtheseeffects,itisimportanttorememberthat,asexplainedabove,
theframeworkanalyzedinthispaperassumesthattheconsumercaresaboutenviron-
mentalquality(weconsideratwo-argumentutilityfunction)butwithouttakinginto
accounttheinfluenceofherinvestmentonit(generatinganexternality).17Apartially
differentapproachcouldassumethattheconsumerdoesnotcareabouttheenviron-
ment(shehasaone-argumentutility)andtheplannermaximizesaweightedsumof
consumerutilityandenvironmentalquality.Inthiscase,wehavethattheconsumer’s
maximizationproblembecomes
maxV (y−s+ f (s)) (5)
s
andtheplanner’sproblembecomes
maxαV(y−s+ f(s))+(1−α)G(q −g(s)) (6)
0
s
whereαand(1−α)aretheweightsofconsumerutilityandenvironmentalquality.
Inthefirstofthesetwoproblemsitisclearthattheconsumer’sfirst-ordercondition
isexactlyequalto(2).18Intheplanner’sproblemwegetthatsociallyoptimallevelof
investments isgivenby
αV(cid:2)(y−s+ f (s))(f(cid:2)(s)−1)=(1−α)G(cid:2)(q −g(s))g(cid:2)(s).
0
It is clear that this condition is a particular case of (4) where U (y − s + f(s),
c
q −g(s))=αV(cid:2)(y−s+f(s))andU (y−s+f(s),q −g(s))=(1−α)G(cid:2)(q −g(s)).
0 q 0 0
ThisoccurssincetheutilityfunctionαV(y−s+ f(s))+(1−α)G(q −g(s))can
0
be seen as a separable formulation (i.e. a formulation where the cross-derivate U
cq
isnull)offunctionU(y −s + f(s),q −g(s)).Giventhisequivalence,theresults
0
obtainedusingtheframeworkdescribedbyEqs.(1)and(3)alsoholdforthealternative
frameworkdescribedbyEqs.(5)and(6).
16 Theright-handsideof(4)ispositive,whiletheright-handsideof(2)isnull.Since f(cid:2)(s)isdecreasing
ins,thisclearlyimpliesthats(cid:3)≥s.
17 AsforinstanceinMichelandRotillon(1995),Mohtadi(1996),Helfandetal.(2003),Xepapadeas(2005)
andAngelopoulosetal.(2010).
18 Thisisobvioussinceinbothsettingstheconsumerconsidersonlytheeffectofinvestmentonconsump-
tion.
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Pigouviantax,abatementpoliciesanduncertaintyontheenvironment 227
3 Pigouviantaxanduncertainty
The excess of investment in the consumer’s optimal choice shown in the previous
sectioncanberemovedbymeansofaPigouviantax,i.e.ataxproportionaltoinvest-
mentequaltomarginalpollutiondamage.IfweintroducethePigouviantax T ,the
p
consumer’sproblembecomes
(cid:2) (cid:3)
maxU y−s+ f (s)−sT ,q .
p 0
s
(cid:3)(cid:3)
Consumer’soptimalinvestments isthusdeterminedinthiscasebythefirst-order
condition19,20
(cid:2) (cid:3)
f(cid:2) s(cid:3)(cid:3) −1=T . (7)
p
ThismeansthatEq.(7)definesthedecreasingrelationship(f(cid:2)(s)isdecreasingins)
between optimal investment level and Pigouvian taxation. Given this condition, the
consumeroptimalchoicecoincideswiththesociallyoptimalinvestment(s =s(cid:3)(cid:3))if
theplannerchoosestheoptimallevelofT suchas f(cid:2)(s)−1=T .21Inthiscontext
p p
wecanintroducetwoalternativetypesofenvironmentalrisk:‘riskonenvironmental
quality level’ (q random) or ‘risk on the effect of pollution’ (g(s) random). Note
0
that the second case describes risk on the effect of human activities on the environ-
ment,whilethefirstconsidersriskonenvironmentfeatures.Forthisreason,thecase
of q random can represent a situation where we have risk on the present level of
0
environmental deterioration, but also a situation where we have risk on the possi-
bleoccurrenceofnaturaldisastersnotrelatedtohumanactivities(like,forinstance,
earthquakes,tsunamisandvolcaniceruptions).22
BeforestudyingoptimalPigouviantaxationunderdifferentkindsofrisk,ageneral
premise concerning consumer behaviour should be stated. Since all risks analyzed
involvetheenvironmentandsinceweassumethattheconsumertakesenvironmental
qualityasgiveninherchoice,consumeroptimalbehaviorunderenvironmentalriskis
19 Obviouslytheconditionimpliesequalitybetweenthenewmarginalcostofinvestment(incorporating
thetax)andinvestmentmarginalbenefit.
20 NotethatweintroducePigouviantaxasataxonpollutinginvestments.Alternativelyitcouldbeintro-
ducedasataxonoutput f(s).Itiseasytoseethat,inthiscase,thefirst-orderconditionofconsumer’s
problem(3)wouldbecome(1−T(cid:4)p)f(cid:2)(s(cid:3)(cid:3))−1 = 0whereT(cid:4)p isthelevelofPigouviantaxonoutput.
Consideringthistogetherwith(7)wegetT(cid:4)p=1−Tp1+1,implyingthatT(cid:4)pisanincreasingfunctionofTp.
Forthisreason,theresultsonthelargerorsmallersizeofTpunderrisk,derivedlateron,arealsocorrect
(cid:4)
forTp.
21 Notethat,using(4)and(7),wealsohavetheexplicitvalueofTpgivenby
Tp= UUqc((yy−−ss++ ff((ss)),,qq00−−gg((ss))))g(cid:2)(s).
22 Toformalizethisideaassume,forinstance,thatq˜0 =q0+(cid:5)˜where(cid:5)˜isarandomvariabledescribing
thepossibleoccurrenceofanaturaldisaster.
123
228 D.Baiardi,M.Menegatti
thesameaswehaveinthecertaintycase.Toclarifythispoint,considerforinstancethe
casewhereq israndom.Inthiscase,theconsumermaximizationproblembecomes
0
(cid:2) (cid:3)
maxU y−s+ f(s)−sT ,q˜ .
p 0
s
Computingtheconsumerfirst-orderconditionshowsthatitisequalto(7).
Itisimportanttonotethatthisoccursforallthekindsofenvironmentalriskstudied
in this paper, so that Eq. (7) describes consumer behavior in all the cases analyzed
below.
Unlikeconsumerbehavior,thesociallyoptimalleveloftaxationisdeterminedby
takingtheeffectonenvironmentintoaccount.Thislevelisthusaffectedbyenviron-
mental risk. For this reason, we will study the effect of each kind of environmental
riskbyexamininghowtheserisksaffectplanner’soptimalchoice.
(a)Riskonenvironmentalquality
Firstlywestudythecasewhereq israndom.Wedefineitasq˜ anditsexpectedvalue
0 0
asE(q˜)=q .Letusdenotes˜andT˜ theinvestmentandtaxlevelunderrisk,while
0 0 p
s andT arerespectivelyinvestmentandPigouviantaxinthecertaintycase.
p
Underrisk,theplanner’smaximisationproblembecomes
(cid:5) (cid:6)
maxE U(y−s+ f(s),q˜ −g(s)) .
0
s
Computingtheplanner’sfirst-orderconditionweget
(cid:5) (cid:6)(cid:2) (cid:3)
E U (y−s˜+ f (s˜),q˜ −g(s˜)) f(cid:2)(s˜)−1
c(cid:5) 0 (cid:6)
−E U (y−s˜+ f (s˜),q˜ −g(s˜)) g(cid:2)(s˜)=0. (8)
q 0
In this case too, once s˜ is determined by (8), the planner can push the consumer
tochoosethesociallyoptimallevelofinvestmentbyconsideringEq.(7),describing
consumer’soptimalbehaviour.TheplannershouldthusimposeaPigouviantaxsuch
as f(cid:2)(s˜)−1=T˜ .Considering(4),(7)and(8)together,wehavethat
p
Proposition1 Under risk on environmental quality level, conditions U ≤ (≥)0
cqq
andU ≥(≤)0aresufficienttohaveT˜ ≥(≤)T .
qqq p p
Proof SeeAppendixB. (cid:5)(cid:6)
Proposition 1 shows that the comparison between the optimal level of Pigouvian
taxwithandwithoutriskdependsonthesignofthird-orderderivativesoftheutility
function.Thisconclusionisnotunusualintheanalysisofriskeffects.Themostimpor-
tantresultinthisfieldisthewell-knownconditionensuringprecautionarysavingin
aunivariateutilityandtwo-periodframework.AsfirstshownbyLeland(1968)and
clearly explained by Kimball (1990),23 positive precautionary saving occurs in this
23 Inthisfield,seealsoSandmo(1970),DrèzeandModigliani(1972)andMenegatti(2001).
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Pigouviantax,abatementpoliciesanduncertaintyontheenvironment 229
caseifthethirdderivativeoftheutilityfunction(withregardtoitsuniqueargument)
ispositive.
Moving from a univariate to a bivariate utility framework, however, the role of
U andU (called‘crossprudence’byEeckhoudtetal.2007)isshowninmany
qqq cqq
works.Inabivariateutilitytwo-periodmodel,inparticular,thesignsofU andU
qqq cqq
areshowntodetermineprecautionarysavingindifferentcontextstogetherwithU
ccc
(seeCourbageandRey2007;Menegatti2009a,b).Similarlythesamederivativesare
relevantforthedeterminationofsavinginthepresenceofcorrelatedrisks(seeDenuit
et al. 2011). Finally, similar conditions affect the optimal choice of labor supply in
aoneperiodbivariateutilitymodelofchoicebetweenconsumptionandleisure(see
ChiuandEeckhoudt2010).
AninterpretationofconditionsU ≥ 0andU ≤ 0canbeobtainedstarting
qqq cqq
fromthatofprecautionarysavingconditionprovidedbyMenegatti(2007).24 Infact,
followingMenegatti(2007),thechangesinthesizeof−U canbeseenasamea-
qq
sureofthechangesinthedisutilityduetoriskonenvironmentalquality.25Condition
U ≥ 0ensuresthatthisdisutilityisreducedifweincreaseenvironmentalquality
qqq
and thus if we reduce pollution. Condition U ≤ 0 ensures that this disutility is
cqq
reducedifwedecreaseconsumptionandthusifwereduceinvestment(s˜ ≤s)below
itsoptimalvaluedeterminedbycondition(4).Inconclusion,boththeeffectsdescribed
abovearerelatedtoprudence.Thefirstisakindof‘prudenceabouttheenvironment’
(greaterriskontheenvironmentimpliesthatabetterenvironmentallevelisdesirable),
whilethesecondisakindofnegativecross-prudence(greaterriskontheenvironment
impliesthatlessconsumptionisdesirable).Thesetwofactstogetherimplythat,under
risk,wepreferlessinvestmentandlesspollution.
(b)Riskontheimpactofpollution
Wenowturntoriskonthepollutioneffect.Forsimplicity,weassumethatthedamage
function g(s)islineardefining g(s) = ks.Notethatunder thisassumption,Eq.(4)
becomes
(cid:2) (cid:3)
U (y−s+ f (s),q −ks) f(cid:2)(s)−1 −kU (y−s+ f (s),q −ks)=0 (9)
c 0 q 0
˜
Underthisassumptionweintroduceriskonparameterk.Wedenotekasrandomvalue
ofkanditsexpectedvalueasE(k˜)=k.Asbefore,wedenoteass˜andT˜ investment
p
andPigouviantaxunderrisk.Theplanner’smaximisationproblemnowbecomes
(cid:7) (cid:8) (cid:9)(cid:10)
maxE U y−s+ f(s),q −k˜s
0
s
24 OnthisseealsoEeckhoudtandSchlesinger(2006).
25 FollowingMenegatti(2007)E[U(y−s+ f(s),q˜0−g(s))]canbewrittenasU(y−s+ f(s),q0−
g(s))+(cid:6)(y−s+f(s),q˜0−g(s)),where(cid:6)(y−s+f(s),q˜0−g(s))=E[U(y−s+f(s),q˜0−g(s))]−
U(y−s+f(s),q0−g(s)).Theindex(cid:6)(.,.)iscalledutilitypremium(orgeneralizedriskmeasure)andit
isameasureofthedisutilitycausedbyuncertainty.FollowingStone(1970)andMenegatti(2007)wehave
that,forsmallrisks,(cid:6)(.,.)∼=−21Uqq(.,.)σ2.
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230 D.Baiardi,M.Menegatti
implyingafirstorderconditiongivenby
(cid:7) (cid:8) (cid:9)(cid:10)(cid:2) (cid:3) (cid:7) (cid:8) (cid:9)(cid:10)
E U y−s˜+f(s˜),q −k˜s˜ f(cid:2)(s˜)−1 −E k˜U y−s˜+f(s˜),q −k˜s˜ =0.
c 0 q 0
(10)
Asinthepreviouscase,condition(9)and(10)determinethesociallyoptimallevels
of investment respectively without risk (s) and under risk (s˜). Given these optimal
˜
levels,T andT aredeterminedbyEq.(7).Bycomparing(9)and(10)andusing(7),
p p
weget
Proposition2 Under risk on the effect of pollution, conditions U ≤ (≥)0 and
cqq
−ksUqqq ≥(≤)−2aresufficienttohaveT˜ ≥(≤)T .
Uqq p p
Proof SeeAppendixB. (cid:5)(cid:6)
Inthecaseofriskonthepollutioneffect,wehaveahigherPigouviantaxifU ≤0
cqq
and−ksUqqq ≥−2.Thefirstconditionisthesameasobtainedinthecaseofriskon
Uqq
environmentalquality,anditsinterpretationisthesametoo.Thesecondconditionis
differentfromthepreviouscase.Inordertoexplainit,itmustfirstbeemphasisedthat
theindex−ksUqqq isarelativeprudenceindex.26 Thelevelofrelativeprudencehas
Uqq
beenprovedtoberelevantincaseswhereareturnisuncertain.Inparticular,Rothschild
andStiglitz(1971)showedthatriskontheinterestrate(thereturnofsaving)increases
savingiftheindexofrelativeprudenceislargerthan2.SimilarlyChiuandEeckhoudt
(2010) showed that the same conditions in Proposition 2 ensure that risk on wages
raisesthelaborsupply.27Thesetworesultstogetherindicatethataconditionofrelative
prudenceisrelatedtoriskonreturns(eitherthereturnsfromsavingorthereturnfrom
labor).
Theinterpretationofcondition−ksUqqq inProposition2isconsistentwiththese
Uqq
conclusions.Indeed,inourframework,parameterk measurestheeffectofaunitof
investment on pollution, and it is thus a kind of ‘negative return’ of investment in
termsofpollution.Forthisreason,asintheproblemscitedabove,theeffectofrisk
onitdepends ontheindexofabsoluteprudence.Furthermore,thedifferenceinour
conditioncomparedtothatinRothschildandStiglitz(1971)(athresholdlevelof−2
insteadof2)occurssincetheeffectishereanegativereturninsteadofapositivereturn
(welikeinterestandwedislikepollution).28
26 FollowingPratt(1964)andKimball(1990)foranunivariateutilityfunctionU(x)wecandefinethefol-
lowingindexes:absoluteriskaversion−Uxx(x)/Ux(x),relativeriskaversion−xUxx(x)/Ux(x),absolute
prudence−Uxxx(x)/Uxx(x)andrelativeprudence−xUxxx(x)/Uxx(x).
27 ChiuandEeckhoudt(2010)showedthattheseconditionsaresufficientforanincreaseinlaborsupply,in
caseofamean-preservingspreadinwages.Thecomparisonbetweenaknownlevelofwageandarandom
onecanbeseenasaspecificcaseofmean-preservingspread.
28 Onthislastconclusionseealsotheresultsinnextsection.
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