Taxes versus quantities for a stock pollutant with endogenous abatement costs and asymmetric

39 Pages·2012·0.43 MB·English

EconTheory(2012)49:371–409
DOI10.1007/s00199-010-0561-y
SYMPOSIUM
Taxes versus quantities for a stock pollutant
with endogenous abatement costs and asymmetric
information
LarryKarp · JiangfengZhang
Received:12September2008/Accepted:10August2010/Publishedonline:1September2010
©TheAuthor(s)2010.ThisarticleispublishedwithopenaccessatSpringerlink.com
Abstract Wecompareemissionstaxesandquotaswhena(strategic)regulatorand
(non-strategic) firms have asymmetric information about abatement costs, and all
agentsuseMarkovperfectdecisionrules.Firmsmakeinvestmentdecisionsthataffect
theirfutureabatementcosts.Forgeneralfunctionalforms,firms’investmentpolicyis
information-constrained efficient when the regulator uses a quota, but not when the
regulator uses an emissions tax. This advantage of quotas over emissions taxes has
notpreviouslybeenrecognized.Foraspecialfunctionalform(linear–quadratic)both
policies are constrained efficient. Using numerical methods, we find that a tax has
someadvantagesinthiscase.
Keywords Pollution control · Investment · Asymmetric information · Rational
expectations·Choiceofinstruments
JELClassification C61·D8·H21·Q28·C72·H4·Q54
JiangfengZhang:TheopinionsexpressedinthispaperdonotnecessarilyreflecttheviewsoftheAsian
DevelopmentBank.
Webenefittedfromcommentsbytwoanonymousreferees.Theusualdisclaimerapplies.
B
L.Karp( )
DepartmentofAgriculturalandResourceEconomics,UniversityofCalifornia,Berkeley,USA
e-mail:[email protected]
J.Zhang
AsianDevelopmentBank,Manila,Philippines
e-mail:[email protected]
123
372 L.Karp,J.Zhang
1 Introduction
Thedangerthatgreenhousegas(GHG)stockscauseenvironmentaldamagehasledto
arenewedinterestintheproblemofcontrollingemissionswhenthereisasymmetric
informationaboutabatementcosts.Althoughhybridpolicies,e.g.,capandtradewitha
priceceiling,aremoreefficientthaneitherthetaxorquantityrestriction(Pizer1999),
thecomparisonoftaxesandquotasremainsanimportantpolicyquestion.SinceGHGs
areastockpollutant,theregulator’sproblemisdynamic.Mostofthecurrentlitera-
tureonthisdynamicproblemassumesthatnon-strategicfirmssolveasuccessionof
staticproblems.If,however,afirm’sabatementcostsdependonitsstockofabatement
capital,thefirmmakesadynamicinvestmentdecisionaswellasthestaticemissions
decision.Westudytheregulatoryproblemwithasymmetricinformationwhenfirms
investinabatementcapital.Wefindthatforgeneralfunctionalforms,quotashavean
advantageovertaxesthathadnotpreviouslybeenrecognized:quotas,butnottaxes,
are“information-constrained”optimalwithrespecttoinvestment.Wethenconsidera
specialfunctionalform(linear–quadratic)wherethatadvantagedisappears,andhere
wefindthattaxescanincreaseabatementandwelfare.
Earlierliterature,e.g.,ChichilniskyandHeal(1995),providedpolicymakerswith
the economic framework needed to compare different types of policies, thereby
contributingtothecreationofthecarbonmarketcurrentlyusedbysomeKyotosig-
natoriesintheEuropeanTradingScheme.Ellerman(2010)discussesthecurrentstate
ofthismarket.Ourpapercontributestopolicydiscussionsbyhelpingtoilluminatea
difference between taxes and quotas when forward-looking firms make investments
thatreduceabatementcosts.Inoursetting,investmentdecreasescostsbyincreasing
thestockofabatementcapital.Wecanalsothinkofthefirms’activityasinvestment
in(excludable)R&D.
For a variety of pollution problems, capital costs comprise a large part of total
abatement costs (Vogan 1991) and investment in abatement capital depends on the
regulatory environment. In these cases, the endogeneity of investment is an impor-
tant aspect of the regulatory problem. Several recent papers (Buonanno et al. 2001;
GoulderandSchneider1999;GoulderandMathai2000;Norhaus1999)assumethat
theregulatorcanchooseemissionsandalsoinducefirmstoprovidethefirst-bestlevel
ofinvestment,e.g.,bymeansofaninvestmenttax/subsidy.
Weconsiderthesituationwheretheregulatorhasasinglepolicyinstrument,either
a sequence of emissions taxes or a sequence of quotas. This assumption is consis-
tentwithmanyregulationsandproposalsthatinvolveanemissionspolicybutignore
endogenous investment (e.g., the Kyoto Protocol). In virtually any real-world prob-
lem, the regulator is likely to have fewer instruments than targets. Our model is an
exampleofthisgeneraldisparitybetweenthenumberofinstrumentsandtargets,and
therefore is empirically relevant. We identify a previously unrecognized difference
betweentaxesandquantityrestrictions,andweprovideasimplemeansofsolvingthe
regulatoryproblemwhenacertainconditionholds.1
1 Jaffe et al. (2003) and Requate (2005) survey the literature on pollution control and endogenous
investment.Manypapersinthisliterature,includingBiglaiseretal.(1995),GersbachandGlazer(1999),
Kennedy and Laplante (1999), Montero (2002), Fischer et al. (2003), Moledina et al. (2003), and
123
Taxesversusquantitiesforastockpollutant 373
We now describe the problem in more detail. In each period, the representative
firmobservesanabatementcostshockthatisprivateinformation.Ifthiscostshockis
seriallycorrelated,theregulatorlearnssomethingaboutitscurrentvaluebyobserving
past behavior. The firm knows the current value of the cost shock and therefore is
better informed than the regulator. Both the regulator and firms obtain information
over time. We examine a subgame perfect equilibrium in which the regulator and
firmsconditiontheirdecisionson“directlypayoff-relevant”information,asdistinct,
forexamplefromtheentirehistoryofactions.Thatis,weconsideraMarkovperfect
equilibrium(MPE).
For the regulator, the payoff-relevant information consists of the aggregate stock
ofabatementcapital(whichaffectstheindustry-widemarginalabatementcosts),the
stock of pollution (which determines marginal damages) and the regulator’s beliefs
about the current cost shock (which also affects the industry’s marginal abatement
costs).Forthefirms,thepayoff-relevantinformationconsistsofthecurrentpolicylevel
(thetaxorquota),thecurrentcostshock,theindividualfirm’slevelofabatementcap-
ital,theaggregateindustrycapitalandthepollutionstock.Thelasttwostatevariables
arepayoff-relevantforthefirmbecausethefirmunderstandsthatthesevariablesaffect
theevolutionofaggregatecapitalstockandpollutionstock,andthefirmunderstands
thattheregulatorconditionsfuturepolicylevelsonfuturevaluesofthosetwostocks.
TheassumptionofMarkovperfectionmeansthattheregulatorcannotmakebinding
commitmentsregardingfuturepolicies.Firmshaverationalexpectations;theytakethe
currentemissionspolicyasgivenandtheyunderstandhowtheregulatorchoosesfuture
policies. The non-atomic representative firm is not able to affect the economy-wide
variables that determine future policies. The representative firm therefore behaves
non-strategically,butnotmyopically,andalsousesMarkovpolicies.
Theregulatorunderstandsthatfutureemissionspoliciesaffectthecurrentshadow
value of abatement capital and thus affect current investment. For example, firms’
anticipation that future emissions policies will be strict would increase the shadow
value of abatement capital, thereby increasing the current level of investment. The
regulatoratthecurrenttime,t,mightwanttocommittoafuturepolicy,implemented
att(cid:2) >t,asameansofaffectingcurrentinvestmentinabatementcapital.Thisincen-
tive is the source of the familiar time-consistency problem. After time t, the time-t
(cid:2)
investmentispredetermined,sothemotivationforthetimet policychoicethatwas
optimalattimet haschanged.Oursettinghastheusualingredientsthatleadtothis
problem: the regulator with a second-best instrument (the emissions tax or quota)
wantstoinfluenceforward-lookingagents.
Subgameperfectionisstrongerthantime-consistency;theformerrequiresthatno
agent wants to deviate from the equilibrium strategy at any possible subgame, and
thelatterrequiresonlythatnoagentwantstodeviatefromtheequilibriumstrategyat
Footnote1continued
TaruiandPolasky(2005,2006)assumethatfirmsbehavestrategicallywithrespecttotheregulator:firms
believethattheirinvestmentdecisionswillaffectfutureregulation.Severalpaper,includingMalueg(1989),
MillimanandPrince(1989),Requate(1998),RequateandUnold(2003)andKarp(2008)treatfirmsas
non-strategic.Papersthatdiscusstime-inconsistencyarisingbecauseofthedisparitybetweenthenumber
oftargetsandthenumberofinstrumentsincludeAbregoandPerroni(2002)andMarsilianiandRenstrom
(2002).
123
374 L.Karp,J.Zhang
subgamesthatactuallyoccurinequilibrium.TheassumptionofMarkovperfection(a
refinementofsubgameperfection)thereforeimpliestime-consistency.
In order to understand an important difference between emissions taxes and
quotas,itisusefultodeterminewhethertherestrictiontotime-consistencyisabind-
ing constraint. To this end, we ask whether the solution to the regulator’s problem,
in the absence of a time-consistency constraint, would in any case be time-consis-
tent. If the private level of investment under the equilibrium emissions policy (the
tax or quota) is socially optimal, then the regulator wants to use the emissions pol-
icyexclusivelytoinfluenceemissions,withouthavingtoconsiderhowtheemissions
policyaffectsinvestment.Inthiscase,theregulatorhasnoincentivetoalteraprevi-
ouslyannouncedpolicyrule.Inthiscircumstancethereisnotime-consistencyprob-
lem.HerewecanobtaintheMPEbysolvinganoptimizationproblemthatcontains
elementsoftheregulator’sandthefirms’problems.If,incontrast,firms’investment
decisions are not socially optimal given beliefs about the emissions trajectory, then
the regulator would like to choose the emissions policy partly to influence invest-
ment. In this case, the Markov perfect restriction, which implies time-consistency,
actually constrains the regulator. Here, we need to solve an equilibrium problem
(a dynamic game between the regulator and non-strategic firms) rather than a rel-
atively simple dynamic optimization problem. In other words, the type of problem
that we need to solve an equilibrium problem or an optimization problem depends
onwhethertheequilibriumlevelofinvestmentissociallyoptimal,conditionalonthe
emissionstrajectory.
There is another way of thinking about the time-consistency problem. The only
marketfailureisthatfirmsdonottakeintoaccountthesocialdamagesarisingfrom
emissions. If there were no cost shock, or if the regulator and firms had symmetric
information,eithertheemissionstaxorthequotawouldbesufficienttoinducefirms
to emit at the optimal level. In that case, firms’ investment decisions would be first
best.Therefore,ifinadditiontotheemissionspolicytheregulatorwereabletouse
an investment tax, the optimal level of that tax would be identically zero. However,
when there is asymmetric information about abatement costs, there is no assurance
that either the emissions tax or the quota leads to the first best level of emissions.
Therefore, with asymmetric information, the equilibrium level of investment under
theemissionspolicymightnotbe(information-constrained)sociallyoptimal.Inthat
case,theoptimallevelofaninvestmenttaxwouldbenon-zero.
Wecanaskourbasicquestionintwoequivalentways.(1)Istheoptimalemissions
taxorquotapolicytime-consistent?(2)Wouldaregulatorwhouseseithertheemis-
sionstaxorthequotaincreasewelfarebyadditionallyusinganinvestmenttax/subsidy?
(Inotherwords:istheoptimalinvestmenttax/subsidyidentically0?).
Weprovideasimpleanswertothesequestions.Theoptimalquotapolicyistime-
consistent;equivalently,whentheregulatorcanusebothanemissionsquotaandan
investmenttax,thelatterisidentically0.However,theoptimalemissionstaxpolicyis
time-inconsistent,unlessaparticular“separabilitycondition”holds;ifthiscondition
does not hold, the optimal investment tax, when used with the emissions tax, is not
identically0.
This result is useful for two reasons. First, under plausible circumstances the
separability condition does not hold. In these circumstances, an emissions tax cre-
123
Taxesversusquantitiesforastockpollutant 375
atesasecondaryinvestmentdistortion,whereastheemissionsquotadoesnot.Thus,
we have identified a difference between taxes and quotas that has previously been
unnoticed.
Second,whentheseparabilityconditiondoeshold,wecansolvethedynamicgame
by solving a much simpler dynamic optimization problem that combines elements
of the regulator’s and the firms’ optimization problems. The separability condition
holds for an important special case, the linear–quadratic model, which has been
previouslyusedtostudytheproblemofregulatingbothaflowandastockpollutant
underasymmetricinformation.Ourseparabilityresultmeansthatwecangeneralize
thelinear–quadraticmodelbyincludingendogenousabatementcapital.Thisgeneral-
izationenablesustolearnhowtheinclusionofendogenousabatementcostsaffects
therankingofthetwopolicies.Ourprincipalnumericalfindingisthat,inthelinear–
quadraticframework,includingabatementcostsincreasestheadvantageoftaxesover
quotas.
Insummary,explicittreatmentoftheinvestmentdecisionrendersthefirms’deci-
sionproblemsdynamic.Ingeneral,thisfeaturefavorstheuseofquotas,becausethese,
unlike taxes, introduce no distortion into firms’ investment decisions. However, in
specialcasesthatincludethelinear–quadraticmodel,thisdistortiondisappearswith
taxes.Forourcalibration,theexplicittreatmentoftheinvestmentdecisionfavorstaxes
inthelinear–quadraticsetting.
Thenextsectiondiscussesastaticproblemthatprovidestheintuitionforthesep-
arability condition. Subsequent sections describe the dynamic model and show the
roleoftheseparabilitycondition.Wethendiscussthelinear–quadraticspecialization,
andexplainhowendogenousinvestmentaffectsthecomparisonoftaxesandquotas
inthatsetting.Inclosingthepaper,wediscussotheraspectsofthetaxversusquantity
debate,asitappliestoclimatechangepolicy.
Other papers in this symposium discuss related aspects of the climate change
problem. Asheim et al. (2011), Lauwers (2011), Figuieres and Tidball (2011) and
Chichilnisky(2011)considertheethicalfoundationsofcriteriaforsustainability,and
thepossibilityofimplementingsuchprograms.LecocqandHourcade(2011)examine
therelationbetweenincomelevelsandabatementexpenditureinanefficientsolution.
DuttaandRadner(2011)showthatcapitalaccumulationcanexacerbatethetragedy
of the commons, and that income transfers can alleviate this problem. Rezai et al.
(2011)explainwhymeaningfulclimatepolicymayrequiresmaller(orevenzero)sac-
rificebythecurrentgeneration,contrarytotheconclusionsofmainstreamintegrated
assessmentmodels.BurniauxandMartins(2011)useacomputablegeneralequilib-
riummodeltoevaluatethekeyparametersindeterminingthemagnitudeof“carbon
leakage”.Ostrom(2011)emphasizestheimportanceofclimatechangepolicyatsub-
globallevels.ChipmanandTian(2011)elaborateontheroleoftheCoaseTheorem
inclimatechangepolicy.
2 Theone-periodexample
Thissectionusesaone-periodmodelthatdemonstrates,inasimplesetting,thediffer-
encebetweentaxesandquotaswhenabatementcostsareendogenous.Weshowthat
123
376 L.Karp,J.Zhang
the emissions quota is always time-consistent; equivalently, if the regulator uses an
emissionsquotaandalsoisabletouseaninvestmenttax/subsidy,theoptimalinvest-
menttaxis0.Incontrast,theemissionstaxisnottime-consistent;equivalently,ifthe
regulatorusesanemissionstaxandalsoisabletouseaninvestmenttax/subsidy,the
investmenttaxisnot0ingeneral.However,theoptimalemissionstaxistime-consistent
ifandonlyiftheprimitivefunctionssatisfyaparticular“separabilitycondition”.
There is a simple explanation for this difference between taxes and quotas. An
emissionspolicychosenbeforeinvestmenthasa“directwelfareeffect”viaitsdirect
effectonemissions,andan“indirectwelfareeffect”,viaitseffectoninvestment.When
thepolicylevelischosenafterinvestment,theregulatortakesintoaccountonlythe
directwelfareeffect,sinceinvestmentisfixed.Theoptimalityconditionforthepolicy
chosenbeforeinvestmentrequiresthatthesumofthedirectandtheindirectwelfare
effectsissetequalto0.Theoptimalityconditionforthepolicychosenafterinvestment
requiresthatthedirectwelfareeffectissetequalto0.Thesetwooptimalityconditions
areequivalentifandonlyiftheindirecteffectis0wheneverthedirecteffectis0.That
equivalencedoesnotholdingeneralundertaxes,butitdoesholdwhentheseparability
condition is satisfied. In contrast, the indirect welfare effect under quotas is always
0,sotheoptimalquotaisthesameregardlessofwhetheritischosenbeforeorafter
taxes.Allofthisbecomesobviousoncewedevelopsomenotationandwritedownthe
optimalityconditions.
We normalize the initial level of abatement capital to 0. The non-strategic but
forwardlookingrepresentativefirmcanbuyk unitsofabatementcapitalatcostc(k);
thefirmobtainsbenefits B(x,k,θ)byemittingx unitsofemissionswhenitsstockof
abatement capital is k and the cost shock isθ.We can think of the function B(·) as
arestrictedprofitfunctioninwhichinputandoutputpricesaresuppressed.Alterna-
tively,wecaninterpret B(·)astheamountofavoidedabatementcosts.Forthelatter
interpretation,define xb astheBusiness-as-Usual(BAU)levelofemissions,i.e.,the
level of emissions under the status quo. Define a = xb − x as the level of abate-
ment,i.e.,thereductioninemissionsduetoanewregulatorypolicy.Theabatement
costs associated with the new regulations are A = A(k,θ,a). If xb is a function of
(k,θ), we can rewrite the abatement cost function as A(k,θ,a) = B(x,k,θ), with
A (·)= B (·):marginalabatementcostsequalthemarginalbenefitofemissions.
a x
Thebenefitfunctionisincreasingandconcaveinxandkandincreasinginθ(B >
k
0,Bθ > 0,Bx > 0,Bkk < 0,Bxx < 0).Moreabatementcapitaldecreasesthemar-
ginal cost of abatement and, therefore, lowers the marginal benefit of pollution, so
B <0.Ahighercostshockincreasesthemarginalbenefitsofabatementcapitaland
xk
emissions: Bkθ ≥0,Bxθ ≥0.
Thedamagefromemissions(externaltothefirm)is D(x).Theregulatorchooses
eitheranemissionstax poranemissionsquotax¯.Throughoutthispaper,weassume
thattheemissionsquotaisbindingforallrealizationsofθ.Boththeregulatorandthe
firmhavethesameinformationaboutthedistributionofθ beforethefirmobservesits
value.
Eachfirmhasmeasure0,andbychoiceofunitsthemassoffirmshasmeasure1.
Withthisnormalization,inasymmetricequilibriumk and x representtheindustry-
wide capital stock and aggregate emissions, as well as the firm level values. The
non-strategic firm chooses its (possibly constrained) level of k and x but takes the
123
Taxesversusquantitiesforastockpollutant 377
industry-widelevelsasexogenous.Inthissectionitisclearfromthecontextwhether
wemeanfirmoraggregatelevelvariables,butinalatersectionwemodifythenotation
toavoidthepossibilityofmisunderstanding.
Weconsiderthefollowingtwotime-lines:
TimeLineA TimeLineB
1.Theregulatorchoosesthe 1.Thefirmchooses
policylevel(porx¯). investment(k)
2.Thefirmchoosesinvestment(k) 2.Theregulatorchooses
thepolicylevel(porx¯).
3.Naturerevealsthecostshock(θ) 3.Naturerevealsthecost
shock(θ).
4.Thefirmmakesitsemissions 4.Thefirmmakesits
decision(x) emissionsdecision(x).
WithTimeLineA,theemissionspolicycaninfluenceboththelevelsofinvestment
andemissions.WithTimeLineB,theemissionspolicydependsonthelevelofinvest-
ment,andinfluencesonlytheemissionslevel.Intheone-periodgame,neitherofthe
twotimelineshasagreaterclaimtoplausibility,butthecomparisonofthetwohelps
tounderstandthetime-consistencyprobleminthedynamicsetting.
Iftheoptimalpolicyfortheregulatoristhesameunderbothtimelines,thenitis
obviousthattheregulatorusesthatpolicyonlytoaffecttheemissionsdecision,not
toinfluencetheinvestmentdecision.Inthiscase,theemissionspolicydoesnotcreate
a secondary distortion in the investment decision; if we were to add a period after
stage2andbeforestage3(“stage2.5”)toTimeLineA,atwhichtheregulatorwere
permittedtorevisethepolicyannouncedinstage1,theregulatorwouldnotwantto
makearevisionwhenpoliciesaretime-consistent.
Weshowthattheemissionsquotaisalwaystime-consistent,buttheemissionstax
istime-consistentifandonlyifaparticularseparabilityconditionholds.Equivalently,
if we were to add a “stage 0” to either time line, at which the regulator announces
aninvestmenttax,theoptimalinvestmenttaxisalways0whentheregulatorusesan
emissionsquota,butitis0whentheregulatorusesanemissionstaxifandonlyifthe
separability condition holds. To establish this claim, we examine each policy under
bothtimelines.
EmissionstaxesUnderTimeLineA, pispredeterminedwhentheindividualfirm
chooses k. Under Time Line B, p depends on aggregate capital, about which firms
have rational point expectations. Since firms take aggregate capital as given when
making their investment decision, and since they know the relation between p and
aggregatecapital,itis“asif”theytake pasgivenunderTimeLineBaswell.Inshort
Remark1 Underbothtimelines,theindividualfirmtakes pasgivenwhenchoosing
itslevelofcapital.
This remark simplifies comparison of the two time lines because we need only
considerhowthechoiceoftimelinesaffectstheregulator’sstrategicincentives;firms
donotbehavestrategically,sothechoiceoftimelinesobviouslydoesnotaffecttheir
strategicincentives.
123
378 L.Karp,J.Zhang
Consider Time Line A when the regulator uses a tax. The representative firm’s
payoffinstage2is
E[B(x,k,θ)− px −c(k)],
wheretheexpectationiswithrespecttoθ.Thefirmchoosesx inthelaststage,con-
ditionalonk,p,andθ.Itchooseskbeforeitlearnsθ.Thefirst-orderconditionsforx
andk andthecorrespondingdecisionrules(denotedusing∗)instages4and2are
B (x,k,θ)− p =0⇒ x = x∗(k,p,θ). (1)
(cid:2) x(cid:3) (cid:4) (cid:5)
E B x∗,k,θ −c(cid:2)(k) =0⇒k =k∗(p). (2)
k
Notethatk∗(p)isindependentofθ.Differentiatingthefirst-ordercondition(1)gives
thecomparativestaticsresult
∂x∗ −1 ∂x∗ −B (x,k,θ)
= and = xk . (3)
∂p B (x,k,θ) ∂k B (x,k,θ)
xx xx
Theregulator’sproblemunderTimeLineAis
(cid:2) (cid:3) (cid:4) (cid:3) (cid:4) (cid:3) (cid:4)(cid:5)
maxE B x∗,k∗,θ −c k∗ −D x∗ ,
p
leadingtothefirst-ordercondition
(cid:6) (cid:7) (cid:6)(cid:8)
(cid:2) (cid:3) (cid:4)(cid:5)∂x∗ (cid:2) (cid:3) (cid:4)(cid:5)∂x∗
E B (∗,θ)−D(cid:2) x∗ +E B (∗,θ)−D(cid:2) x∗
x ∂p x ∂k
(cid:9) (cid:7)
(cid:3) (cid:4) ∗
dk
+B (∗,θ)−c(cid:2) k∗ =0, (4)
k
dp
(usingthenotation∗=(x∗,k∗)).Becausek∗(p)isindependentofθ,wecantake dk∗
dp
outsidetheexpectationsoperatorandusethefirm’soptimalitycondition(2)towrite
theregulator’soptimalityconditionas
(cid:6) (cid:7) (cid:6) (cid:7)
(cid:2) (cid:3) (cid:4)(cid:5)∂x∗ dk∗ (cid:2) (cid:3) (cid:4)(cid:5)∂x∗
E B (∗,θ)−D(cid:2) x∗ + E B (∗,θ)−D(cid:2) x∗ =0, (5)
x ∂p dp x ∂k
Thefirsttermontheleftsideisthedirectwelfareeffectofthetaxandthesecondterm
istheindirectwelfareeffect,operatingthroughinvestment.
The first-order condition for the firms’ emissions decision is the same under the
twotimelines,sinceinbothcasesthefirmconditionsitsemissionsdecisiononpre-
determinedvalues of p andk andtherealizedvalueofθ.InviewofRemark1,the
first-orderconditiontotheindividualfirm’sinvestmentproblemunderTimeLineBis
(apartfromaninessentialnotationaldifference)stillgivenbyEq.(2).Thisnotational
differenceisthatinsteadoftreating p asapredeterminedvariable,underTimeLine
Bthefirmtreats pasafunctionofaggregatecapital,whichthefirmtakesasgiven.
123
Taxesversusquantitiesforastockpollutant 379
Thefactthat(apartfromthenotationaldifference)thefirm’sfirst-orderconditions
arethesameunderthetwotimelinesmeansthatthefunctionsx∗(k,p,θ),andk∗(p)
arealsothesameunderthetwotimelines,althoughofcoursethevaluesof p inthe
twoscenarios(therefore,theequilibriumvaluesofkandx)mightdiffer.Thispossible
differenceisthekeytothetime-consistencyissue.UnderTimeLineB,theregulator
takes(aggregate)k asgiven,soitsfirst-orderconditionforthetaxis
(cid:6) (cid:7)
(cid:2) (cid:3) (cid:4)(cid:5)∂x∗
E B (∗,θ)−D(cid:2) x∗ =0. (6)
x ∂p
Notethatthisoptimalityconditionequatestheexpectedmarginalbenefitsandcosts
of the tax, not the expected marginal benefits and costs of emissions. The two need
notbethesame,becauseingeneral ∂x∗ dependsonθ.
∂p
Denote the optimal tax under Time Line B as pˆ. We assume that the regulator’s
problemisconcaveunderbothtimelines,sothatthesolutiontotherespectivefirst-
order condition is unique. Comparison of Eqs. (5) and (6) shows that the optimal
emissiontaxisthesameunderthetwotimelinesifandonlyiftheindirecteffectof
thetax,evaluatedat p = pˆ,iszero,i.e.,if
(cid:6) (cid:7)
(cid:2) (cid:3) (cid:4)(cid:5)∂x∗
E B (∗,θ)−D(cid:2) x∗ =0. (7)
x ∂k |p=pˆ
Werefertothefollowingasthe“separabilitycondition”:
∗
Condition1 (Separability)B andB ,evaluatedattheoptimalx ,areindependent
xx xk
ofθ.
Remark2 Equation(7)holdsforallfunctionsB(x,k,θ)ifandonlyiftheseparability
conditionholds.
Proof InordertoestablishthesufficiencyofCondition1,notethatitimplies(using
Eq.3)thatboth ∂x∗ and ∂x∗ areindependentofθ.Thisindependence,togetherwith
∂p ∂k
Eq. (6) and the fact that ∂x∗ (cid:6)= 0 imply that pˆ (the optimal tax under Time line B)
∂p
satisfies
(cid:2) (cid:3) (cid:3) (cid:4) (cid:4) (cid:3) (cid:3) (cid:4)(cid:4)(cid:5)
E B x∗ k∗,p,θ ,k∗,θ −D(cid:2) x∗ k∗,p,θ =0. (8)
x
That is, under Condition 1, the tax equates expected marginal benefits of emissions
withmarginaldamages.Theindependenceof ∂x∗ andθ,andthefactthat ∂x∗ (cid:6)= 0,
∂k ∂k
meanthatEq.(8)impliesEq.(7).Therefore,theoptimaltaxunderthetwolinesisthe
same.
Inordertoestablishnecessity,notethatifeither B orB arenotindependentof
xx xk
θ,itisstraightforwardtoconstructexamplesunderwhichtheregulator’sfirst-order
conditionsfor pdifferunderthetwotimelines. (cid:7)(cid:8)
123
380 L.Karp,J.Zhang
Itisalsoeasytoshow:
Remark3 Supposethattheregulatorusesanemissionstax.Ifwemodifyeithertime
linesbyaddingastage0atwhichtheregulatorisabletochooseaninvestmenttax/sub-
sidy,theoptimallevelofthispolicyisidentically0forallfunctions B(x,k,θ)ifand
onlyiftheseparabilityconditionholds.
Weomittheproof,whichparallelstheproofofRemark2.
We see that time-consistency requires that Eq. (6) implies Eq. (7). Equation (6)
statesthatthefirst-orderchangeinwelfareduetoachangeinthetax,(holdinginvest-
mentfixed),iszero.Equation(7)statesthatthefirst-orderchangeinwelfaredueto
a change in investment (holding the tax fixed) is zero. In general, of course, there
isnoreasonthatoneequationimpliestheother,soingeneralthetaxchosenbefore
investment is not time-consistent. However, the separability condition implies two
thingsabouttheproblem:(1) ∂x∗ isindependentofθ,sothatsettingtheexpectednet
∂p
marginalbenefitofthetax(holdinginvestmentfixed)equalto0isequivalenttosetting
theexpectednetmarginalbenefitofemissionsequalto0;and(2) ∂x∗ isindependent
∂k
ofθ,sothatsettingtheexpectednetmarginalbenefitofinvestment(holdingthetax
fixed) equal to 0 is equivalent to setting the expected marginal benefit of emissions
equalto0.WhenbothEqs.(6)and(7)areequivalenttosettingexpectednetmarginal
benefitofemissionsequalto0,thetwoareequivalenttoeachother.
EmissionsquotasBasedonthesamereasoningthatledtoRemark1,wehave
Remark4 Under both time lines, the individual firm takes the emissions quota as
givenwhenchoosingitslevelofcapital.
If the regulator uses quotas (that by assumption are binding for all θ) the firm’s
emissionsdecisionequalsx¯,and ∂x∗ =1.InviewofRemark4,thefirm’sfirst-order
∂x¯
conditionforthechoiceofk (forbothofthetwotimelines)is
(cid:2) (cid:5)
E B (x¯,k,θ)−c(cid:2)(k) =0⇒k =k∗(x¯). (9)
k
Aswasthecasewithtaxes,thereisanunimportantnotationalissue:withTimeLine
A,x¯ isliterallypredetermined,whilewithTimeLineB,thefirmtreatsx¯ asaknown
functionofaggregateinvestment,andthefirmtakesaggregateinvestmentasgiven.
UnderTimeLineA,theregulator’sfirst-orderconditionforx¯ is
(cid:6) (cid:7)
(cid:2) (cid:3) (cid:4) (cid:5) (cid:2) (cid:3) (cid:4) (cid:3) (cid:4)(cid:5) ∗
dk
E B x¯,k∗,θ −D(cid:2)(x¯) + B x¯,k∗,θ −c(cid:2) k∗
x k dx¯
(cid:10)(cid:2) (cid:3) (cid:4) (cid:5)(cid:11)
= E B x¯,k∗,θ −D(cid:2)(x¯) =0, (10)
x
wherethefirstequalityusesEq.(9),thefactthatk∗isindependentofθ,and dk∗ (cid:6)=0.
dx¯
The first-order condition under Time Line B is identical to the second equality in
Eq.(10).Thus,whennon-strategicfirmshaverationalexpectations,theoptimalquota
isthesameunderthetwotimelines.Theregulatorusesthequotatotargetonlyemis-
sions,andthefirm’sinvestmentdecisionisinformation-constrainedsociallyoptimal.
123