Taxes Versus Quantities for a Stock Pollutant with Endogenous Abatement Costs and Asymmetric ...

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Taxes Versus Quantities for a Stock Pollutant
with Endogenous Abatement Costs
and Asymmetric Information
LarryKarpandJiangfengZhang
1 Introduction
Thedangerthatgreenhousegas(GHG)stockscauseenvironmentaldamagehasled
to a renewed interest in the problem of controlling emissions when there is asym-
metric information about abatement costs. Although hybrid policies, e.g., cap and
tradewithapriceceiling,aremoreefficientthaneitherthetaxorquantityrestriction
(Pizer1999),thecomparisonoftaxesandquotasremainsanimportantpolicyques-
tion. Since GHGs are a stock pollutant, the regulator’s problem is dynamic. Most
of the current literature on this dynamic problem assumes that non-strategic firms
solveasuccessionofstaticproblems.If,however,afirm’sabatementcostsdepend
onitsstockofabatementcapital,thefirmmakesadynamicinvestmentdecisionas
wellasthestaticemissionsdecision.Westudytheregulatoryproblemwithasym-
metricinformationwhenfirmsinvestinabatementcapital.Wefindthatforgeneral
functionalforms,quotashaveanadvantageovertaxesthathadnotpreviouslybeen
recognized:quotas,butnottaxes,are“information-constrained”optimalwithrespect
toinvestment.Wethenconsideraspecialfunctionalform(linear-quadratic)where
JiangfengZhang:Theopinionsexpressedinthispaperdonotnecessarilyreflecttheviewsof
theAsianDevelopmentBank.
Webenefittedfromcommentsbytwoanonymousreferees.Theusualdisclaimerapplies.
ReprintedwithkindpermissionfromtheAuthors:OriginallypublishedinEconomicTheory,
Volume49,Number2,February2012DOI10.1007/s00199-010-0561-y.
L.Karp(✉)
DepartmentofAgriculturalandResourceEconomics,UniversityofCalifornia,
Berkeley,USA
e-mail:[email protected]
J.Zhang
AsianDevelopmentBank,Manila,Philippines
e-mail:[email protected]
©TheAuthor(s)2016 493
G.ChichilniskyandA.Rezai(eds.),TheEconomicsoftheGlobalEnvironment,
StudiesinEconomicTheory29,DOI10.1007/978-3-319-31943-8_22
494 L.KarpandJ.Zhang
that advantage disappears, and here we find that taxes can increase abatement and
welfare.
Earlierliterature,e.g.,ChichilniskyandHeal(1995),providedpolicymakerswith
theeconomicframeworkneededtocomparedifferenttypesofpolicies,therebycon-
tributingtothecreationofthecarbonmarketcurrentlyusedbysomeKyotosignato-
riesintheEuropeanTradingScheme.Ellerman(2010)discussesthecurrentstateof
thismarket.Ourpapercontributestopolicydiscussionsbyhelpingtoilluminatea
differencebetweentaxesandquotaswhenforward-lookingfirmsmakeinvestments
thatreduceabatementcosts.Inoursetting,investmentdecreasescostsbyincreasing
thestockofabatementcapital.Wecanalsothinkofthefirms’activityasinvestment
in(excludable)R&D.
For a variety of pollution problems, capital costs comprise a large part of total
abatementcosts(Vogan1991)andinvestmentinabatementcapitaldependsonthe
regulatoryenvironment.Inthesecases,theendogeneityofinvestmentisanimpor-
tantaspectoftheregulatoryproblem.Severalrecentpapers(Buonannoetal.2001;
GoulderandSchneider1999;GoulderandMathai2000;Norhaus1999)assumethat
the regulator can choose emissions and also induce firms to provide the first-best
levelofinvestment,e.g.,bymeansofaninvestmenttax/subsidy.
We consider the situation where the regulator has a single policy instrument,
either a sequence of emissions taxes or a sequence of quotas. This assumption is
consistentwithmanyregulationsandproposalsthatinvolveanemissionspolicybut
ignoreendogenousinvestment(e.g.,theKyotoProtocol).Invirtuallyanyreal-world
problem, the regulator is likely to have fewer instruments than targets. Our model
isanexampleofthisgeneraldisparitybetweenthenumberofinstrumentsandtar-
gets, and therefore is empirically relevant. We identify a previously unrecognized
differencebetweentaxesandquantityrestrictions,andweprovideasimplemeans
ofsolvingtheregulatoryproblemwhenacertainconditionholds.1
We now describe the problem in more detail. In each period the representative
firmobservesanabatementcostshockthatisprivateinformation.Ifthiscostshockis
seriallycorrelated,theregulatorlearnssomethingaboutitscurrentvaluebyobserv-
ingpastbehavior.Thefirmknowsthecurrentvalueofthecostshockandthereforeis
betterinformedthantheregulator.Boththeregulatorandfirmsobtaininformation
over time. We examine a subgame perfect equilibrium in which the regulator and
firmsconditiontheirdecisionson“directlypayoff-relevant”information,asdistinct,
forexamplefromtheentirehistoryofactions.Thatis,weconsideraMarkovPerfect
Equilibrium(MPE).
1Jaffeetal.(2003)andRequate(2005)surveytheliteratureonpollutioncontrolandendogenous
investment.Manypapersinthisliterature,includingBiglaiseretal.(1995);GersbachandGlazer
(1999);KennedyandLaplante(1999);Montero(2002);Fischeretal.(2003);Moledinaetal.(2003);
TaruiandPolasky(2005, 2006)assumethatfirmsbehavestrategicallywithrespecttotheregulator:
firmsbelievethattheirinvestmentdecisionswillaffectfutureregulation.Severalpaper,including
Malueg(1989);MillimanandPrince(1989);Requate(1998);RequateandUnold(2003);Karp
(2008)treatfirmsasnon-strategic.Papersthatdiscusstime-inconsistencyarisingbecauseofthe
disparitybetweenthenumberoftargetsandthenumberofinstrumentsincludeAbregoandPerroni
(2002);MarsilianiandRenstrom(2002).
TaxesVersusQuantitiesforaStockPollutantwithEndogenous... 495
Fortheregulator,thepayoff-relevantinformationconsistsoftheaggregatestock
ofabatementcapital(whichaffectstheindustry-widemarginalabatementcosts),the
stockofpollution(whichdeterminesmarginaldamages)andtheregulator’sbeliefs
aboutthecurrentcostshock(whichalsoaffectstheindustry’smarginalabatement
costs).Forthefirms,thepayoff-relevantinformationconsistsofthecurrentpolicy
level(thetaxorquota),thecurrentcostshock,theindividualfirm’slevelofabate-
mentcapital,theaggregateindustrycapitalandthepollutionstock.Thelasttwostate
variables are payoff-relevant for the firm because the firm understands that these
variablesaffecttheevolutionofaggregatecapitalstockandpollutionstock,andthe
firmunderstandsthattheregulatorconditionsfuturepolicylevelsonfuturevalues
ofthosetwostocks.
TheassumptionofMarkovPerfectionmeansthattheregulatorcannotmakebind-
ing commitments regarding future policies. Firms have rational expectations; they
take the current emissions policy as given and they understand how the regulator
chooses future policies. The non-atomic representative firm is not able to affect
theeconomy-widevariablesthatdeterminefuturepolicies.Therepresentativefirm
thereforebehavesnon-strategically,butnotmyopically,andalsousesMarkovpoli-
cies.
Theregulatorunderstandsthatfutureemissionspoliciesaffectthecurrentshadow
value of abatement capital and thus affect current investment. For example, firms’
anticipationthatfutureemissionspolicieswillbestrictwouldincreasetheshadow
valueofabatementcapital,therebyincreasingthecurrentlevelofinvestment.The
regulatoratthecurrenttime,t,mightwanttocommittoafuturepolicy,implemented
att′ >t,asameansofaffectingcurrentinvestmentinabatementcapital.Thisincen-
tiveisthesourceofthefamiliartime-consistencyproblem.Aftertimet,thetime-t
investmentispredetermined,sothemotivationforthetimet′policychoicethatwas
optimalattimethaschanged.Oursettinghastheusualingredientsthatleadtothis
problem: the regulator with a second-best instrument (the emissions tax or quota)
wantstoinfluenceforward-lookingagents.
Subgameperfectionisstrongerthantime-consistency;theformerrequiresthatno
agentwantstodeviatefromtheequilibriumstrategyatanypossiblesubgame,and
thelatterrequiresonlythatnoagentwantstodeviatefromtheequilibriumstrategyat
subgamesthatactuallyoccurinequilibrium.TheassumptionofMarkovperfection
(arefinementofsubgameperfection)thereforeimpliestime-consistency.
Inordertounderstandanimportantdifferencebetweenemissionstaxesandquo-
tas,itisusefultodeterminewhethertherestrictiontotime-consistencyisabinding
constraint. To this end, we ask whether the solution to the regulator’s problem, in
theabsenceofatime-consistencyconstraint,wouldinanycasebetime-consistent.
Iftheprivatelevelofinvestmentundertheequilibriumemissionspolicy(thetaxor
quota)issociallyoptimal,thentheregulatorwantstousetheemissionspolicyexclu-
sivelytoinfluenceemissions,withouthavingtoconsiderhowtheemissionspolicy
affectsinvestment.Inthiscase,theregulatorhasnoincentivetoalterapreviously
announced policy rule. In this circumstance there is no time-consistency problem.
HerewecanobtaintheMPEbysolvinganoptimizationproblemthatcontainsele-
ments of the regulator’s and the firms’ problems. If, in contrast, firms’ investment
496 L.KarpandJ.Zhang
decisionsarenotsociallyoptimalgivenbeliefsabouttheemissionstrajectory,then
the regulator would like to choose the emissions policy partly to influence invest-
ment.Inthiscase,theMarkovPerfectrestriction,whichimpliestime-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-
ativelysimpledynamicoptimizationproblem.Inotherwords,thetypeofproblem
thatweneedtosolveanequilibriumproblemoranoptimizationproblemdepends
on whether the equilibrium level of investment is socially optimal, conditional on
theemissionstrajectory.
Thereisanotherwayofthinkingaboutthetime-consistencyproblem.Theonly
marketfailureisthatfirmsdonottakeintoaccountthesocialdamagesarisingfrom
emissions.Iftherewerenocostshock,oriftheregulatorandfirmshadsymmetric
information,eithertheemissionstaxorthequotawouldbesufficienttoinducefirms
toemitattheoptimallevel.Inthatcase,firms’investmentdecisionswouldbefirst
best.Therefore,ifinadditiontotheemissionspolicytheregulatorwereabletouse
aninvestmenttax,theoptimallevelofthattaxwouldbeidenticallyzero.However,
whenthereisasymmetricinformationaboutabatementcosts,thereisnoassurance
that either the emissions tax or the quota leads to the first best level of emissions.
Therefore,withasymmetricinformation,theequilibriumlevelofinvestmentunder
theemissionspolicymightnotbe(information-constrained)sociallyoptimal.Inthat
case,theoptimallevelofaninvestmenttaxwouldbenon-zero.
We can ask our basic question in two equivalent ways. (1) Is the optimal emis-
sions tax or quota policy time-consistent? (2) Would a regulator who uses either
theemissionstaxorthequotaincreasewelfarebyadditionallyusinganinvestment
tax/subsidy?(Inotherwords:Istheoptimalinvestmenttax/subsidyidentically0?).
Weprovideasimpleanswertothesequestions.Theoptimalquotapolicyistime-
consistent; equivalently, when the regulator can use both an emissions quota and
an investment tax, the latter is identically 0. However, the optimal emissions tax
policyistime-inconsistent,unlessaparticular“separabilitycondition”holds;ifthis
conditiondoesnothold,theoptimalinvestmenttax,whenusedwiththeemissions
tax,isnotidentically0.
Thisresultisusefulfortworeasons.First,underplausiblecircumstancesthesep-
arability condition does not hold. In these circumstances, an emissions tax creates
asecondaryinvestmentdistortion,whereastheemissionsquotadoesnot.Thus,we
haveidentifiedadifferencebetweentaxesandquotasthathaspreviouslybeenunno-
ticed.
Second, when the separability condition does hold, we can solve the dynamic
gamebysolvingamuchsimplerdynamicoptimizationproblemthatcombinesele-
mentsoftheregulator’sandthefirms’optimizationproblems.Theseparabilitycon-
dition holds for an important special case, the linear-quadratic model, which has
been previously used to study the problem of regulating both a flow and a stock
pollutantunderasymmetricinformation.Ourseparabilityresultmeansthatwecan
generalize the linear-quadratic model by including endogenous abatement capital.
Thisgeneralizationenablesustolearnhowtheinclusionofendogenousabatement
costsaffectstherankingofthetwopolicies.Ourprincipalnumericalfindingisthat,
TaxesVersusQuantitiesforaStockPollutantwithEndogenous... 497
inthelinear-quadraticframework,includingabatementcostsincreasestheadvantage
oftaxesoverquotas.
Insummary,explicittreatmentoftheinvestmentdecisionrendersthefirms’deci-
sion problems dynamic. In general, this feature favors the use of quotas, because
these, unlike taxes, introduce no distortion into firms’ investment decisions. How-
ever,inspecialcasesthatincludethelinear-quadraticmodel,thisdistortiondisap-
pearswithtaxes.Forourcalibration,theexplicittreatmentoftheinvestmentdecision
favorstaxesinthelinear-quadraticsetting.
Thenextsectiondiscussesastaticproblemthatprovidestheintuitionforthesep-
arabilitycondition.Subsequentsectionsdescribethedynamicmodelandshowthe
role of the separability condition. We then discuss the linear-quadratic specializa-
tion, and explain how endogenous investment affects the comparison of taxes and
quotasinthatsetting.Inclosingthepaper,wediscussotheraspectsofthetaxversus
quantitydebate,asitappliestoclimatechangepolicy.
Other papers in this Symposium discuss related aspects of the climate change
problem. Asheim et al. (2016); Lauwers (2016); Figuieres and Tidball (2016);
Chichilnisky (2016) consider the ethical foundations of criteria for sustainability,
and the possibility of implementing such programs. Lecocq and Hourcade (2016)
examinetherelationbetweenincomelevelsandabatementexpenditureinanefficient
solution.DuttaandRadner(2016)showthatcapitalaccumulationcanexacerbatethe
tragedyofthecommons,andthatincometransferscanalleviatethisproblem.Rezai
et al. (2016) explain why meaningful climate policy may require smaller (or even
zero)sacrificebythecurrentgeneration,contrarytotheconclusionsofmainstream
integratedassessmentmodels.BurniauxandMartins(2016)useacomputablegen-
eralequilibriummodeltoevaluatethekeyparametersindeterminingthemagnitude
of“carbonleakage”.Ostrom(2016)emphasizestheimportanceofclimatechange
policy at sub-global levels. Chipman and Tian (2016) elaborate on the role of the
CoaseTheoreminclimatechangepolicy.
2 TheOne-PeriodExample
Thissectionusesaone-periodmodelthatdemonstrates,inasimplesetting,thedif-
ferencebetweentaxesandquotaswhenabatementcostsareendogenous.Weshow
thattheemissionsquotaisalwaystime-consistent;equivalently,iftheregulatoruses
an emissions quota and also is able to use an investment tax/subsidy, the optimal
investment tax is 0. In contrast, the emissions tax is not time-consistent; equiva-
lently, if the regulator uses an emissions tax and also is able to use an investment
tax/subsidy,theinvestmenttaxisnot0ingeneral.However,theoptimalemissions
taxistime-consistentifandonlyiftheprimitivefunctionssatisfyaparticular“sep-
arabilitycondition”.
There is a simple explanation for this difference between taxes and quotas. An
emissionspolicychosenbeforeinvestmenthasa“directwelfareeffect”viaitsdirect
effect on emissions, and an “indirect welfare effect”, via its effect on investment.
498 L.KarpandJ.Zhang
When the policy level is chosen after investment, the regulator takes into account
only the direct welfare effect, since investment is fixed. The optimality condition
forthepolicy chosen beforeinvestment requiresthatthesumofthedirectand the
indirectwelfareeffectsissetequalto0.Theoptimalityconditionforthepolicycho-
sen after investment requires that the direct welfare effect is set equal to 0. These
twooptimalityconditionsareequivalentifandonlyiftheindirecteffectis0when-
ever the direct effect is 0. That equivalence does not hold in general under taxes,
butitdoesholdwhentheseparabilityconditionissatisfied.Incontrast,theindirect
welfareeffectunderquotasisalways0,sotheoptimalquotaisthesameregardless
of whether itis chosen before or after taxes. Allof this becomes obvious once we
developsomenotationandwritedowntheoptimalityconditions.
We normalize the initial level of abatement capital to 0. The non-strategic but
forwardlookingrepresentativefirmcanbuykunitsofabatementcapitalatcostc(k);
the firm obtains benefits B(x,k,𝜃) by emitting x units of emissions when its stock
ofabatementcapitaliskandthecostshockis𝜃.WecanthinkofthefunctionB(⋅)
asarestrictedprofitfunctioninwhichinputandoutputpricesaresuppressed.Alter-
natively, we can interpret B(⋅) as the amount of avoided abatement costs. For the
latter interpretation, define xb as the Business-as-Usual (BAU) level of emissions,
i.e., the level of emissions under the status quo. Define a=xb−x as the level of
abatement,i.e.,thereductioninemissionsduetoanewregulatorypolicy.Theabate-
mentcostsassociatedwiththenewregulationsareA=A(k,𝜃,a).Ifxbisafunction
of(k,𝜃),wecanrewritetheabatementcostfunctionasA(k,𝜃,a)=B(x,k,𝜃),with
A (⋅)=B (⋅):marginalabatementcostsequalthemarginalbenefitofemissions.
a x
The benefit function is increasing and concave in x and k and increasing in 𝜃
(B >0, B >0, B >0, B <0, B <0). More abatement capital decreases the
k 𝜃 x kk xx
marginal cost of abatement and therefore lowers the marginal benefit of pollution,
soB <0.Ahighercostshockincreasesthemarginalbenefitsofabatementcapital
xk
andemissions:B ≥0,B ≥0.
k𝜃 x𝜃
Thedamagefromemissions(externaltothefirm)isD(x).Theregulatorchooses
eitheranemissionstaxporanemissionsquotax̄.Throughoutthispaper,weassume
thattheemissionsquotaisbindingforallrealizationsof𝜃.Boththeregulatorandthe
firmhavethesameinformationaboutthedistributionof𝜃 beforethefirmobserves
itsvalue.
Eachfirmhasmeasure0,andbychoiceofunitsthemassoffirmshasmeasure1.
Withthisnormalization,inasymmetricequilibriumkandxrepresenttheindustry-
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
industry-widelevelsasexogenous.Inthissectionitisclearfromthecontextwhether
wemeanfirmoraggregatelevelvariables,butinalatersectionwemodifythenota-
tiontoavoidthepossibilityofmisunderstanding.
TaxesVersusQuantitiesforaStockPollutantwithEndogenous... 499
Weconsiderthefollowingtwotime-lines:
TimeLineA TimeLineB
1.Theregulatorchoosesthepolicylevel(por 1.Thefirmchoosesinvestment(k)
x̄)
2.Thefirmchoosesinvestment(k) 2.Theregulatorchoosesthepolicylevel(por
x̄)
3.Naturerevealsthecostshock(𝜃) 3.Naturerevealsthecostshock(𝜃)
4.Thefirmmakesitsemissionsdecision(x) 4.Thefirmmakesitsemissionsdecision(x)
WithTimeLineA,theemissionspolicycaninfluenceboththelevelsofinvest-
mentandemissions.WithTimeLineB,theemissionspolicydependsonthelevelof
investment,andinfluencesonlytheemissionslevel.Intheone-periodgame,neither
of the two time lines has a greater claim to plausibility, but the comparison of the
twohelpstounderstandthetime-consistencyprobleminthedynamicsetting.
Iftheoptimalpolicyfortheregulatoristhesameunderbothtimelines,thenit
isobvious thattheregulatorusesthatpolicyonlytoaffecttheemissionsdecision,
nottoinfluencetheinvestmentdecision.Inthiscase,theemissionspolicydoesnot
createasecondarydistortionintheinvestmentdecision;ifweweretoaddaperiod
afterstage2andbeforestage3(“stage2.5”)toTimeLineA,atwhichtheregulator
were permitted to revise the policy announced in stage 1, the regulator would not
wanttomakearevisionwhenpoliciesaretime-consistent.
We show that the emissions quota is always time-consistent, but the emissions
taxistime-consistentifandonlyifaparticularseparabilityconditionholds.Equiv-
alently, if we were to add a “stage 0” to either time line, at which the regulator
announcesaninvestmenttax,theoptimalinvestmenttaxisalways0whenthereg-
ulatorusesanemissionsquota,butitis0whentheregulatorusesanemissionstax
ifandonlyiftheseparabilityconditionholds.Toestablishthisclaim,weexamine
eachpolicyunderbothtimelines.
EmissionstaxesUnderTimeLineA,pispredeterminedwhentheindividualfirm
choosesk.UnderTimeLineB,pdependsonaggregatecapital,aboutwhichfirms
have rational point expectations. Since firms take aggregate capital as given when
makingtheirinvestmentdecision,andsincetheyknowtherelationbetween pand
aggregate capital, it is “as if” they take p as given under Time Line B as well. In
short
Remark1 Underbothtimelines,theindividualfirmtakespasgivenwhenchoosing
itslevelofcapital.
Thisremarksimplifiescomparisonofthetwotimelinesbecauseweneedonlycon-
siderhowthechoiceoftimelinesaffectstheregulator’sstrategicincentives;firms
donotbehavestrategically,sothechoiceoftimelinesobviouslydoesnotaffecttheir
strategicincentives.
500 L.KarpandJ.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.Thefirmchoosesxinthelaststage,con-
ditionalonk,p,and𝜃.Itchooseskbeforeitlearns𝜃.Thefirst-orderconditionsfor
xandkandthecorrespondingdecisionrules(denotedusing∗)instages4and2are
B (x,k,𝜃)−p=0⇒x=x∗(k,p,𝜃). (1)
x
[ ]
E B (x∗,k,𝜃)−c′(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
[ ]
maxE B(x∗,k∗,𝜃)−c(k∗)−D(x∗) ,
p
leadingtothefirst-ordercondition
{ }
{[
E [B (∗,𝜃)−D′(x∗)]𝜕x∗ +E [B (∗,𝜃)−D′(x∗)]𝜕x∗
x 𝜕p x 𝜕k
}
+B (∗,𝜃)−c′(k∗)]dk∗ =0, (4)
k dp
(usingthenotation∗= (x∗,k∗)).Becausek∗(p)isindependentof𝜃,wecantake dk∗
dp
outsidetheexpectationsoperatorandusethefirm’soptimalitycondition(2)towrite
theregulator’soptimalityconditionas
{ }
{ }
E [B (∗,𝜃)−D′(x∗)]𝜕x∗ + dk∗E [B (∗,𝜃)−D′(x∗)]𝜕x∗ =0, (5)
x 𝜕p dp x 𝜕k
Thefirsttermontheleftsideisthedirectwelfareeffectofthetaxandthesecond
termistheindirectwelfareeffect,operatingthroughinvestment.
Thefirst-orderconditionforthefirms’emissionsdecisionisthesameunderthe
two time lines, since in both cases the firm conditions its emissions decision on
predeterminedvaluesofpandkandtherealizedvalueof𝜃.InviewofRemark1,the
firstorderconditiontotheindividualfirm’sinvestmentproblemunderTimeLineBis
(apartfromaninessentialnotationaldifference)stillgivenbyEq.(2).Thisnotational
TaxesVersusQuantitiesforaStockPollutantwithEndogenous... 501
differenceisthatinsteadoftreatingpasapredeterminedvariable,underTimeLine
Bthefirmtreatspasafunctionofaggregatecapital,whichthefirmtakesasgiven.
Thefactthat(apartfromthenotationaldifference)thefirm’sfirstorderconditions
arethesameunderthetwotimelinesmeansthatthefunctionsx∗(k,p,𝜃),andk∗(p)
arealsothesameunderthetwotimelines,althoughofcoursethevaluesofpinthe
two scenarios (and therefore the equilibrium values of k and x) might differ. This
possibledifferenceisthekeytothetime-consistencyissue. UnderTimeLineBthe
regulatortakes(aggregate)kasgiven,soitsfirst-orderconditionforthetaxis
{ }
E [B (∗,𝜃)−D′(x∗)]𝜕x∗ =0. (6)
x 𝜕p
Notethatthisoptimalityconditionequatestheexpectedmarginalbenefitsandcosts
ofthetax,nottheexpectedmarginalbenefitsandcostsofemissions.Thetwoneed
notbethesame,becauseingeneral 𝜕x∗ dependson𝜃.
𝜕p
DenotetheoptimaltaxunderTimeLineBasp̂.Weassumethattheregulator’s
problemisconcaveunderbothtimelines,sothatthesolutiontotherespectivefirst
order condition is unique. Comparison of Eqs. (5) and (6) shows that the optimal
emissiontaxisthesameunderthetwotimelinesifandonlyiftheindirecteffectof
thetax,evaluatedatp=p̂,iszero,i.e.,if
{ }
E [B (∗,𝜃)−D′(x∗)]𝜕x∗ =0. (7)
x 𝜕k ∣p=p̂
Werefertothefollowingasthe“separabilitycondition”:
Condition1 (Separability)B andB ,evaluatedattheoptimalx∗,areindepen-
xx xk
dentof𝜃.
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∗ ≠0 imply that p̂ (the optimal tax under Time line B)
𝜕p
satisfies [ ]
E B (x∗(k∗,p,𝜃),k∗,𝜃)−D′(x∗(k∗,p,𝜃)) =0. (8)
x
Thatis,underCondition1thetaxequatesexpectedmarginalbenefitsofemissions
with marginal damages. The independence of 𝜕x∗ and 𝜃, and the fact that 𝜕x∗ ≠0,
𝜕k 𝜕k
meanthatEq.(8)impliesEq.(7).Therefore,theoptimaltaxunderthetwolinesis
thesame.
Inordertoestablishnecessity,notethatifeitherB orB arenotindependentof
xx xk
𝜃,itisstraightforwardtoconstructexamplesunderwhichtheregulator’sfirstorder
conditionsforpdifferunderthetwotimelines. □
Itisalsoeasytoshow:
502 L.KarpandJ.Zhang
Remark3 Suppose that the regulator uses an emissions tax. If we modify either
timelinesbyaddingastage0atwhichtheregulatorisabletochooseaninvestment
tax/subsidy,theoptimallevelofthispolicyisidentically0forallfunctionsB(x,k,𝜃)
ifandonlyiftheseparabilityconditionholds.
Weomittheproof,whichparallelstheproofofRemark2.
Weseethattime-consistencyrequiresthatEq.(6)impliesEq.(7).Equation(6)
statesthatthefirst-orderchangeinwelfareduetoachangeinthetax,(holdinginvest-
mentfixed),iszero.Equation(7)statesthatthefirst-orderchangeinwelfareduetoa
changeininvestment(holdingthetaxfixed)iszero.Ingeneral,ofcourse,thereisno
reasonthatoneequationimpliestheother,soingeneralthetaxchosenbeforeinvest-
mentisnottime-consistent.However,theseparabilityconditionimpliestwothings
abouttheproblem:(1) 𝜕x∗ isindependentof𝜃,sothatsettingtheexpectednetmar-
𝜕p
ginalbenefitofthetax(holdinginvestmentfixed)equalto0isequivalenttosetting
theexpectednetmarginalbenefitofemissionsequalto0;and(2) 𝜕x∗ isindependent
of𝜃,sothatsettingtheexpectednetmarginalbenefitofinvestmen𝜕tk(holdingthetax
fixed)equalto0isequivalenttosettingtheexpectedmarginalbenefitofemissions
equalto0.WhenbothEqs.(6)and(7)areequivalenttosettingexpectednetmarginal
benefitofemissionsequalto0,thetwoareequivalenttoeachother.
EmissionsquotasBasedonthesamereasoningthatledtoRemark1,wehave
Remark4 Under both time lines, the individual firm takes the emissions quota as
givenwhenchoosingitslevelofcapital.
Iftheregulatorusesquotas(thatbyassumptionarebindingforall𝜃)thefirm’s
emissionsdecisionequalsx̄,and 𝜕x∗ =1.InviewofRemark4,thefirm’sfirst-order
𝜕x̄
conditionforthechoiceofk(forbothofthetwotimelines)is
[ ]
E B (x̄,k,𝜃)−c′(k) =0⇒k=k∗(x̄). (9)
k
Aswasthecasewithtaxes,thereisanunimportantnotationalissue:withTimeLine
A,x̄isliterallypredetermined,whilewithTimeLineB,thefirmtreatsx̄asaknown
functionofaggregateinvestment,andthefirmtakesaggregateinvestmentasgiven.
UnderTimeLineA,theregulator’sfirstorderconditionforx̄ is
{ }
[ ] [ ]
E B (x̄,k∗,𝜃)−D′(x̄) + B (x̄,k∗,𝜃)−c′(k∗) dk∗
x {[ k ]} dx̄ (10)
=E B (x̄,k∗,𝜃)−D′(x̄) =0,
x
wherethefirstequalityusesEq.(9),thefactthatk∗isindependentof𝜃,and dk∗ ≠0.
dx̄
The first order condition under Time Line B is identical to the second equality in
Eq. (10). Thus, when non-strategic firms have rational expectations, the optimal
quota is the same under the two time lines. The regulator uses the quota to tar-
get only emissions, and the firm’s investment decision is information-constrained
sociallyoptimal.Thereisnosocialvalueinusinganinvestmenttaxwhentheregu-
latorusesanemissionsquota.