POLICY R-ESEARCH WORKING PAPER 231 9
Extelirnalities and Production Environmental improvements
should be sought from
Efficieric different polluters (public or
private, producer or
consurner, rich or poor) at the
same cost, regardless of the
nature of the polluting
activity. Under a plausible
structL re of monitoring costs,
emissions standards play a
central role.
The World Bank
Development Research Group
Public Economics U
April 2(000
POLICY RESEARCH WORKING PAPER 2319
Summary findings
Eskeland brings together two of govemrnent's primary producer, or government, but they cannot differentiate
challenges: environmental protection and taxation to such instruments (or commodity taxes) by personal
generate revenues. characteristics or make them nonlinear in individual
If negative externalities can be reduced not only by emissions.
changes in consumption patterns but also by making each Among Eskeland's findings and conclusions:
activity cleaner (abatement efforts), how shall Abatement efforts and consumption adjustmerts at all
inducements to various approaches be combined? If stages are optimally stimulated by a uniform emission tax
negative externalities are caused by agents as different as levied simply where emissions occur.
consumers, producers, and government, how does It simplifies things that optimal abatement is
optimal policy combine inducements to reduce independent of whether the car is used by government,
pollution? firms, or households - for weddings or for work.
Intuitively it seems right to tax emissions neutrally, It also simplifies implementation that the stimulus to
based on marginal damages - no matter which activity abatement at one stage (say, the factory) is independent
pollutes or whether the polluter is rich or poor, of whether it yields emission reductions from the factory
consumer or producer, private or public. Eskeland or from others (say, from car owners who buy the
provides a theoretical basis for such simplicity. factory's products).
Three assumptions are critical to his analysis: Finally, ministers of finance and of the environment
* Returns to scale do not influence the traditional should coordinate efforts, but they need not engage in
problem of revenue generation. each other's business. The minister of environment need
3 Consumers have equal access to pollution abaternent not know which commodities are elastic in demand and
opportunities (but he also relaxes this assumption). thus would bear a low commodity tax. The finance
* Planners can differentiate policy instruments minister need not know which comnmodities or agents
(emission taxes or abatement standards) by polluting pollute or who pays emission taxes.
good, and by whether the polluter is a consumer,
This paper - a product of Public Economics, Development Research Group - is parr of a larger efforr in the group to
establish principles for public intervention. Copies of the paper are available free from the World Bank, 1818 H Street NW,
Washington, DC 20433. Please contact Hedy Sladovich, room MC2-609, telephone 202-473-769S, fax 202-22-11iS4,
email address hsladovich( aworldbank.org. Policy Research Working Papers are also posted on the Web atrwww. .vorldbanik.
org/research/workingpapers. The author may be contacted at geskelandcaworldbank.org. April 2000. (43 pages)
The Policy Research Working Paper Series dissemiinates the findings of work in progress to encourage the exchaonge of ideas about t
development issues. An objective of the series is to get the findings oot quickly, even if the presentatosnsare less than fullv polished. The I
papers cany the names of the authors and should be cited accordingly. The findings, interpretations, and corclusions expressed in this
paper are entirely those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors, or the
coPdntries they represeuct.ch e
Produced by the Policv Research D7issemination Center
Externalities and Productioin Efficiency
Gunnar S. Eskeland
The World Bank
The findings, interpretations, and conclusions expressed in this paper are entirely those of
the authors. They do not necessarily represent the views of the World Bank, its Executive
Directors, or the countries they represent. The paper shculd not be cited without the
permission of the authors.
Correspondence: Gunnar Eskeland, The World Bank, Development Research Group,
1818 H St. NM., Washington, DC 20433, USA. Phone/fax: 1 202 473 7938/:522 1154.
geskeland@worldbank.org.
I. Introduction and Summary
In this paper, we combine two challenges of government: taxation for revenue
generation and environmental protection. First-best intuition would say that emissions be
taxed neutrally according to their marginal damages, withcout reference neither to the type
of activity that pollutes nor to whether the polluter is rich or poor, consumer or producer,
private or public. However, there has yet been no theoretical basis for assuming such
simplicity if government has a revenue need that requires distortionary taxation.
Our study builds in particular on two important contributions. First, Diamond and
Mirrlees, 1971, demonstrated conditions under which optimal taxation involves
production efficiency. Their findings imply that the input-.output vector is at the
aggregate production frontier, a solution that can be implemented by confrontirg all
producers, privalte as well as public, with the same producer prices. Thus, a social planner
may want to insert distorting tax wedges between consumers, and between the set of
consumers and the set of all producers, but not between producers, whether private or
public. With an external effect from producers and government, a production efficiency
result follows directly from Diamond and Mirrlees' treatment. The environmental good is
an additional output (or' input) of productive sectors, and shall consequently be provided
at the same marginal rates of transformation for the set cf producers and government seen
as a whole. However, the literature including externalities in models of optimal taxation
has focused on cases with only one type of polluting activity or source, thus putting aside
the question of producltion efficiency in environmental protection.
The second contribution on which the present stady builds is Sandmo s seminal
study "Optimal taxation in the presence of externalities" (Sandmo, 1975). Apart from
providing the amalytical framework used subsequently -- and in this study - Sandmo made
two findings we shall highlight here. First, he concluded that "even in a world of
distortionary taxation'...there is scope for taxing exteniality-generating commodities
according to the Pigovian principle." Second, he noted that the optimal comrnodity tax
structLre is "characterized by what might be called an additivity property; the marginal
social damage of commodity m enters the formula for that commodity only..." (m is the
externality-creating good) and "the optimal tax rate on the externality creating
commodity is a weighted average of two terns, of which the second is the marginal
1
social damage of commodity m. The first term... .is composed of the efficiency terms
familiar from the theory of optimal taxation".
An important aim of the present study is to understand conditions under which
optimal taxation - and Sandmo's framework - requires production efficiency, including
in environmental protection, when such protection can take many avenues. Diamond and
Mirrlees considered briefly whether production efficiency would hold if there is an
external effect between consumers, but then without including environmental protection
in the concept of production efficiency. They concluded "it seems quite likely that
efficiency will be desired in realistic settings". The concept of production efficiency
tested here is a broader one, since we include the environmental good in the vector
proposed to be at the aggregate frontier'. For Sandmo, the proposition of production
efficiency in environmental protection was not at the table, since there was only one
polluting activity.
To examine the question of whether production efficiency can include efficiency
in the protection of the environment, a key assumption is to include pollution abatement
as an additional avenue for pollution reductions: a polluting consumer (or producer) may
spend resources - say on a filter - to reduce emissions per unit consumed (or produced).
This allows us to test a proposition of production efficiency more broadly defined: UJnder
what conditions will marginal costs of abatement - per unit of emission reductions
achieved - be equalized across activities and agents?
In our model, the set of polluters is not only producers (as in Cremer et al. 1998,
Cremer and Gahvari, 1999), or only consumers (as in Sandmo, 1975, Diamond and
Mirrlees, 1971), but comprise consumers, producers and government. Briefly put, we ask
whether the social planner would tax emissions from different activities (or from
producers, consumers, government) differently.
Our model (Section II of the paper) is simple - a structure with fixed coefficients
of transformation between private goods is expanded with an external effect, a public
good. The public good is in the outset provided by nature, but is reduced as a negative
external effect (we call it pollution) results from consumption and production activities.
The model involves five minor modifications to Sandmo's 1975 model. First,
2
governmrent, firms and consumers are all polluters. Second, in addition to substitation
towards non-polluting goods and services, the model allows the polluter to expend
resources on pollution abatement to reduce emissions. This term includes efforts such as
the consumer's maintenanace of her car, the producer's installation of a catalytic converter
in her product or a filter in her smokestack, modifications of practices or of compounds
such as fuels and detergents, and finally cleanup efforts. Third, we allow multiple
polluting goods, or activities (we use the word activity to comprise consumption and
production). Fourth, we allow nonunifornity across pollulers in how much they pollute
per unit of activity (more precisely, they differ in their costs of pollution abatement).
Finally, these modifications themselves invite expansions of the set of policy instruments
relative to those allowed by Sandmo and subsequent authors. One the one hand, emission
taxes no longer are mere extensions of commodity taxes when pollultion abatement is
possible (this distinction is also used by Cremer and Gahvari, 1999). Also, we show,
standards for abatement (or for emission per unit) can play a role under plausible
restrictions on the observability of emissions.
Several studies have provided approaches preparing the ground for this treatment.
Bovenberg and van der Ploeg, 1994, introduce abatement, but as public production rather
than related to own emissions (a good example might be a municipal wastewater
treatment plant). Their discussion is centered on how increased environmental concern
influences provision of public goods and consumption oF private goods. Gould,er et al.,
1998, allow abatement amongst producers, and focus on the interaLction between
envirornental instruments and pre-existing taxes. Both these studies employ assumptions
giving the labor/leisure choice, not only the environmental good, ;a particular r ole in
preferences. Cremer et al. 1998 analyze optimal taxation and focus on the interaction
between the environmental tax and other instruments, mnuch in the same way as did
Atkinson and Stiglitz, 1976, for the interaction between direct and indirect instruments in
the traditional i roblem without externalities. Cremer ard Gahvari, 1999, closest to the
questions asked here, allow several polluting industries with uniforrn technology.
Amongst their findings is a uniform emissions tax, implicitly showing how Diamond and
'Since they were proposing an external effect from consumers to consumers, it was quite natural in their
context not to exipand with the environmental good the vector of inputs and outputs proposed to be at the
aggregate frontier.
3
Mirrlees' production efficiency result must apply if the input or output of producers is
expanded with one element - the environmental good.
In section III, we characterize optimal policy assuming equal access to technology
for all consumers. We investigate when instruments such as emission taxes will be
applied 'neutrally' as expected according to Pigovian principles, including under
plausible restrictions on the monitoring of emissions2. These results can be viewed as
generally extending those of Sandmo, 1975. Also, they extend the results of Eskeland,
1994, on the combination of emission standards and presumptive Pigovian taxes (levied
on inputs and outputs), to a case in which taxation is costly.
For polluting producers, production efficiency applies as expected even when
firms differ in their access to abatement technology (section III). In section IV of the
paper, we introduce nonuniform emission fumctions for consumers. If the planner can
differentiate emission taxes across polluting activities, when are the emission taxes equal
across activities, and equal to the one applied to producers? We find that the 'one tax'
breaks down if the pattern of nonuniformity across consumers in emission functions lends
itself to nonenvironmental goals of the planner, such as to redistribute, or to minimize the
distortionary effects of taxation. Certain covariance formulas identify these cases. In the
case when emission taxes are not available, presumptive Pigovian taxes on goods and
emission standards are no longer 'first-best' when emission functions are nonuniform, so
results are modified for that reason. While this result is new in a setting of distortionary
taxation, it naturally extends results from a literature examining indirect Pigovian
instruments when the externality generating good is itself unavailable or imperfect as a
3
base for a corrective instrument3.
II. The Model
We introduce some variations to existing treatments of optimal taxation in the
presence of externalities (Sandmo, 1975, in particular, but also Cremer et al, 1998). The
importance of these variations lie in their practical relevance, and we will thus intersperse
the text with some examples for illustration.
2As is the tradition in the literature, we use the words tax and taxation whether the rate is positive or
negative (in everyday use, the negative rates would be called subsidies).
3 Some noteable contributions are: Diamond, 1973; Sandmo, 1976; Balcer, 1980; Wijkander, 1985;
Greenwald and Stiglitz, 1986.
4
Preferences
As is the tradition in the public finance literature, we analyze a. setting in which
consumers have preferences over private goods as well as a public good (with several
public goods, results extend straightforwardly). As a matter of terminology, a term such
as 'a public bad' could be used for pollution, but we shall generally try to describe the
social planner as providing (or procuring) a public good when using taxes or regulation to
stimulate pollution reductions. It is sometimes convenient to speak of goods in general,
and then let a vector of quantities include pollution as a p-ublic good, even though
consumers prefer less pollution to more.
Let H denote the set of consumers, and let h be a consumer, A E H. h's ulility
depends on her consumption xh of a set N of market goods, j = 0,1,.., n, as well as on a
pollution indicator, e:
(1 ) =Uh( XIXh Xh e)
We assume that lhe utility function is continuous, twice differentiable, quasicorncave, and
that uh < 04. In addition, we shall assume that preferences are separable between the
basket of market goods, j E N, and pollution, so that the marginal rates of substitution
between market goods are independent of pollution levels. A sufficient condition for this
to hold is that individual preferences can be described by a separable utility function:
1h = U h (Vh (Xo, Xi ., xn), e). In the literature on taxation in the presence of external
effects, separability is typically assumed.5
In assuming that pollution is experienced at the samne level by all consurmers, we
combine two properties. The important one is that pollution (or its absence) is a pure
public good in the sense of Samuelson, 1954, that there is no rivalry in its consumption.
If one person enjoys the low level of pollution, this does not reduce another person's
4Whenever possible without risking confusion, we shall use subscripts to denote partial derivatives.
u h < 0 is necessary for us to use words such as 'negative external effects' and 'pollution', but the results
are equally applicable also to positive external effects. An equilibrium in whic'h Pigovian taxation makes
distortionary taxation un-necessary is less plausible with positive externalities.
5An assumption of separability between leisure and other private goods is often also included (Cremer et
al., 1998, Cremer and Gahv7ari, 1999, Bovenberg and Goulder, 1996, Bovenberg and van der Ploeg, 1994).
In the traditional model wilthout externalities, this additional separability assuraption renders (other)
5
enjoyment. The less significant implied property is homogenous dispersion (which James
Meade, 1952, termed atmospheric pollution). It simplifies notation, by ensuring that the
marginal damages from emissions (or the benefits from emission reductions) are
independent of who or where the polluter is. If damages per unit of emissions vary, say
by location or by stack hight, accommodation of this fact must be made, and results
extend (the question should be asked, however, of whether instruments can be
differentiated accordingly).
Emissions and pollution abatement
In the world we try to capture with our model, emissions of pollution are caused
by several activities, in production stages as well as consumption stages. Examples that
we all know of are that emissions are caused in the production of gasoline, cars, and
detergents, and also as households and firms use car services and do their laundry. Also,
pollution abatement, or efforts to make each activity less polluting, may be undertak-en by
producers or consumers. Reduced emissions from cars, for instance, can result as the
manufacturer changes his product by adding a catalytic converter, as the refinery changes
the gasoline characteristics, and as the driver drives more carefully, buys a 'cleaner'
gasoline, and improves her maintenance. As these examples illustrate, efforts to abate
emissions may well be exerted at a production stage even if emissions occur later, fcr
instance in consumption.
The traditional treatment of externalities in the theoretical literature has been to
view substitution in consumption (towards non-polluting goods and services) as the only
way to reduce pollution (Cremer and Gahvari, 1999, made advances beyond this). In such
a case, it is of no importance whether emissions result from production or consumption -
since the assumption of equilibrium ensures that production and consumption move in
parallel. Sandmo's important (1975) contribution described emissions as caused by
consumption, but with results directly applicable for emissions caused by producers.
In a context with pollution abatement in contrast, it could be material whether
emissions occur in consumption or in production. To illustrate, if in optimum producers
and households face different price vectors and abatement options (they do), are marginal
commodity taxes redundant in the presence of a tax on labor. In a model with a polluting commodity. like
Sandmo, 1975, it would then suffice with a tax on labor and an emission tax.
6
abatement costs in. optimtm different for car manufacturers and users? Similarly., if a car
owner can reduce emissions, should policy stimuli depend on whether she is a consumer,
an enterprise or government?
Commodities are potentially polluting in both production and consumption, so we
may think of a set of 2n polluting activities. We choose conlsumer abatement and
emissions as our rnain presentational vehicle, in part becau.se consurner abatement is
novel and poses more interesting questions in a welfare economic perspective. To save on
notation, we do not introduce emissions from producers before later in this section.
Individual emissions are caused in association with the consuimption of polluting
goods and. services, as reipresented by emission factors fjh = fj (bj):
(2) eh = fj (bh).x forallgoods j=1,..,n, heH,
and equivalently for government. (2) reflects that the consumer may expend resources on
pollution abatement, b , in order to reduce the emissions per unit consumed of good
j. We assume, until we generalize in section IV, that consumers have access to the same
abatement technology, so that emission fimctions fj (b>) are uniform across consumers.
The assumption that emi.ssions display proportionality with quantity (though conditional
on the good in question, and abatement) is restrictive, but allows us to place our results in
a literatture based on constant returns to scale. To simplifr, we assume that the niumeraire
good is not polluting: f, = 0, and we describe all other goods as polluting:
for j X 0: fj > 0 . Abatement bj is nonnegative and conitinuous and fj is assumed to be
continuous and differentiable. We assume that abatemeni. reduces e,missions at a
decreasing rate, so that the marginal cost of emission reductions - 1/fjb is positive and
increasing6.
The pollution level, the argument in each consumer's utility finction, is simply
emissions aggregated across polluters and polluting goods7:
6The assumption that all goods j = 1,.., n are polluting simplifies notation. It means that a rLonpolluting
good is approximated as one for which emissions and emission taxes are trivially low at abatement levels
that are trivially low
7We skip irnportant detail here, and some deserve mention: i) the pollution level, here a scalar, may be a
vector (concentrations of dust and of ground-level ozone). Extensicn of results to several 'public goods' (or
bads: dust, ozone), with. one set of Pigovian taxes for each, is straightforward; :i) whether or rnot the
pollution level is a scalar (say, parts per million of ozone), ernissions contributing to the pollution level may
7
(3) e = t e,+e,
jeN heH
where eP denotes emissions resulting if the government uses goodj (the generalization
with producer emissions is straightforward, and will follow).
The consumer's problem8
Let ti and r ej be linear taxes levied respectively on consumption and emissions of
goodj, j = 0,.., n. The consumer faces a price pj + tj for each good. p6 = 1, to = 0, so
the numeraire good is untaxed. Our model is general in its treatment of private goods, so
it is not important whether one thinks of the numeraire good as leisure. In this respect,
our model differs from a number of recent contributions on Pigovian taxation in which
results are based in part on preferences that are separable in leisure versus other private
goods (additional results following from making that assumption are rather obvious)9.
We model consumer h as maximizing her utility uh with respect to consumption
and abatement, subject to her budget constraint: 10
(4) Max h :U (X h ,. X h e) s.t. b [pj + t1 + b, + 1ei fj (b ), = 0.
In h 's maximization, we shall assume that she considers the level of pollution,, e
(the sum of what is generated by all polluters), and also public sector revenue to be
independent of her own actions. This will be either accurate descriptions or close
be a vector (as with the precursors of ozone: tons ernitted of nitrogen oxides and of volatile organic
compounds); iii) we abstract from how the pollution indicator translates into damages (health effects.,
species extinction, corrosion, etc.), when we describe willingness to pay simply as a function of the
pollution level.
8We use the tenrs 'consumer' and 'household' as synonyms, and are thus unable to handle distributional
issues within the household. Bergstrom (1989) provides an interesting discussion of when this is
appropriate, and of the shortcomings (more recently highlighted in the literature focusing on gender and
children).
9 In a model with constant transfonnation coefficients, like ours (see production technology, below), lhe
choice of untaxed commodity is imnmaterial: it is easily checked that the relative prices obtained here
(including those inducing abatement) can be replicated with another choice of untaxed good.
to Our budget constraint is consistent with the traditional: n I (pi + t )x =-I where I is
endowment and I is leisure. With x0 1 - I (x0 is a negative figure), we have E . = (pi + t1 J = O.
When we introduce consumer abatement and emission taxes, we obtain
n > (pj +tj ±bj +rej fj)x. = 0.
8
approximations if the number of individuals, H, is large. These assumptions are rather
natural extensions of the assumptions of competitive equilibrium, under which producers
and consumers take prices as given. In the present model, they consicLer two additional
variables as independent of their actions: total pollution and. public revenue
The first-order conditions for h's individual optimum are her budget constraint and,
for all goodsj =l,..,n:
(5) ^ _pj +f +b +-f (b)=qh and
(6) =*
fjb (bj )
The first equality in (5) shows how the consumer will set marginal rates of substitution
between private goods equal to the relative marginal costs of these goods. In the second
equation in (5), we have taken advantage of the fact that these marginal costs are
independent of consunmption levels, so that the marginal cost is also a unit cost, and
introduced the symbol qh to represent this 'all-inclusive consumer price'. In (6). the
consumer sets her marginal cost of emission reductions equal to the emission tax rate.
These marginal costs would be equal across consumers even if emission functions
differed across consurners. However, with homogenous emnission fumctions, abatement
b and emission JFactors f, will also be the same across consumers, ensuring th,at the all-
inclusive consumer prices are uniform across consumers. 'We shall use this property to
suppress individual superscripts for bj and qj until section IV, in which we adcpt
heterogeneous emission functions.
Finally we may sketch a generalization. If emissions occur at production stages as
well, and if producers in sectorj face emission taxes and abatement opportunities, then
the producer price in (5) will itself be a sum components, to include, the producer's
abatement and taxes on emissions in production. Then, self interestzd producers will join
the consumer in an ef fort to minimize the all-inclusive consumer pnice.
1 1 See Sandmo, 197:5, or Eskeland, 1994, for some further treatment. If there are H individuals who take
into account the effect of their own actions, then our approximation error is to set H/(H-1) equal to one.
9
Production technology
To describe the economy's technological constraint - its ability to transform one
bundle of consumption goods into another - let capitalized variables without superscripts
denote aggregate quantities: Xj = E x x,.', with xp denoting government
hEH
consumption. A rather general description of technology would be:
(7) F(X1 ,.., Xj = Yo
where Y. is work, or endowment less leisure and abatement:
(8) Yo =I-I-E= 1b1X =-X0 -yj=1b1X1.
One assumption embodied in (7) and (8) is that the damages from pollution do not
affect production possibilities. Thus, the motivation for pollution abatement is found
solely in the way pollution affects household utility (1).
Our model of the production side of the economy shall involve additional
restrictions: fixed factors of transformation between market goods (see, for instance,
Sandmo, 1975, or Cremer et al., 1998): 12
n
(9) Lc xi= Y,
j=1
where the vector c consists of the constant transformation coefficients. Though we have
used aggregate quantities, we may think of (9) as describing a generally available
conversion technology, possessed and controlled by many independent producers. WIhen
these producers compete in input and output markets, each handling their share of the
aggregate quantities, profits will be zero and producer prices will be equal to marginal
costs:
(10) pj =cj,allj=l,..,n, and p0 =1.
III. Optimal taxation
A benevolent planner
12 The assumption of fixed coefficients of transformation - or of constant producer prices - is motivat:ed by
our desire to compare with standard results in the optimal taxation literature. Diamond and MirTlees, 1971,
showed that the results based on constant producer prices apply also to the more general case of constant
returns to scale.
10
Let a benevolent planner's objectives be represented by a welfare function
defined over individual utility levels, w = w(v', v2 ..,vh ) where vh = vh (qh, J, e) is
the indirect utility function corresponding to (1), and qh is given by p, t, r, as described
by (5). He maximizes welfare subject to a constraint that revenues are equal to a
predetermined mirnimum - enough to finance exogenous public sector expenditures13 (we
initially assume that government abatement is exogenously given). Apart from the
instruments used here, we assume that the planner does not have available other
instruments for redistribution or revenue mobilization. However, our analysis applies also
to the case when there are other instruments with redistributive and revenue implications.
14
Such other instruments could be uniform poll taxes as well as non-linear income taxes
Commodity taxes are in the literature typically restricted to be linear, with a brief
justification being that, the planner's information includes aggregate quantities E. ,x , but
not the H vector xh. Thus, the tax man may observe liters of gasoline- exiting the refinery
gate or the gas station, but not individual purchases to an extent attributable to inLdividual
consumers. We may add that nonlinear commodity taxes (or commodity taxes
differentiated by personal characteristics) would involve not only costly information and
administration, but also distortions, as consumers would enigage in costly exchange of
goods and services. Thus, the restriction that commodity taxes be linear can rest on a
broader set of considerations than only information availability.
This lattei, broader justification is more appropriate when we assume that
emission taxes may be differentiated by polluting commodity (gaso:line versus heating
oil), but are confined to be linear and uniform across cons umers. We may think of
emissions as in principle observable by the planner at the emitting source (a meter on
each car, for instance., displaying the car's cumulative emissions at year-end), but that the
attribution of emissions to households would be costly and lead to distortions under
13 Individual budget constraints add up to the technology constraint (9) if we include that of the planner:
Zh Z. (t1 + Teji f jX = p + bj )xj ), where the right hnd side is public expenditures.
14 The analysis applies by viewing the presented sufficient conditions as a subsel of conditions for optimal
policy. In the tradition of MirTlees (1971), nonlinear income taxes are introduced by assuming that
individuals differ in endowment of timne in productivity units, but that only income (work times wage) is
observed and taxable by the planner. One approach is to assume a di
(12) A/3h[xh +ah- -~-Ade]+,+ E (xgf) dxj 1
hti gEH jeN ti
15 Examples to illustrate such difficulties: Multiple caT households, (tempoTary) exchange of cars and car
services, multihousehold heating and municipal waste-water discharge. The impossibility of using potential
infornation on individual emissions for nonlinear taxation is particularly clear in the case of pollutina
producers in a model with constant returns to scale, as ours. Nonlinear taxation of emissions would be ruled
out by the costless replication (or merger) of firms. For households, on the other hand, non-linear taxation
of emissions could with plausibility be feasible and attractive, save for reasons of administrative difficulty
and distortions. We learn something relevant for non-linear taxation of household emissions when we treat
differences in emission factors in section IV.
16 Use the definition of the indirect utility function and the envelope theorem:
u (x(q, I, e), e) = v(q, e, I) ':> - --. Divide by minus the marginal utility of income to express
willingness to pay in terns of the numeraire: --/ E = /-
12 j aj 03 & a
12
M =o
drei
de dxjg
(1 3) - Jh(f, x h ai +de -) E[ff Xi +x*eifbbifXi +b i, x+Vejfj)tj i I 0
h~~H i dz-). dTrei*
heH jl d ei geH jeN
In (12), we have used the fact that producer prices are independent of commodity taxes:
dp j1 v i, h dq _v ,h _ h*
-__ = 0, = - = _ ---=- , and Roy's identity: -= - xh . Irn (13), we
dti C7 Acqj dt1 d6 d
have used q1 = and dq = O, which follow from differentiation of qi (see
dTei drei
equation 5) and the enve]lope theorem. To develop these e.xpression further, we may use:
de dex( dde
(14) d=,2,fj b1) di and d f6bbi,xig +E fj d
di i. g di drei g i d' ei
The 2n equations (121 and (13), with (14) describe an optimal tax structure for the n
commodity tax rates and the n emission tax rates.
The uncompensated demand functions are in general definedl over prices, income,
and the quantity of the pablic good, xg= xg (q, 1 9, e). Let us now employ the assumption
of separability between pollution and other goods: xj = (
(15) dxg/dtj=xj and dx91/drei =x.fj,allij,g.
When using (15) and (14), (12) and (13) simplify to, for all i = l,..,n:
(16) - E x + a' E fj xgi 1+,u Ax + ±Z(tj + rejfj )x l=0, and
heH geHKjeN geH jeN
(17) _ E 1 h i x + Z' E(fiifbbirxF+ X fj xJfi )]
+ful E f Xj +-7eItfbbrXi9+Y(tJ +Tejfj ),[fi 0 °
;,rH jeNI
In order to gain firther -insights into the tax structure implied by (16) and (17), we shall
go via simplifying assunmptions.
Assumption 1: ivo abatement available: emission factors are exogenously given
13
The case with an exogenously given emission factor was analyzed by Sandmo, 1975.
When polluting goods cannot be made less polluting per unit (in our model's
terminology, when fib = 0, all i E N for any b ), pollution reductions will rely solely on
changes in consumption patterns towards goods that are less polluting (say: from
motorcycles to bicycles, from cigars to cigarettes). The model with exogenously given
emission factors is - fortunately - unrealistic in most practically interesting cases, but
provides important insights even for the more general case7. We show it here as a basis
for comparison with existing literature, and also to generalize to several polluting
activities (or goods).
With no abatement available, the equations in (6) do not apply, and every equation
in (17) is simply fi times the corresponding equation in (16). Thus, the 2n by 2n
coefficient matrix is at most of rank n and at most n instruments are required to
implement the optimal allocation. Assuming that the n equations in (16) are linearly
independent, we use this redundancy to set emission taxes all equal to zero and
implement the optimal solution using commodity taxes only (as in Sandmo's treatment).
Substituting -ei = 0, all j E N into (16), we obtain
(18) Etj Yx,t xhE fLx -I A a ENh
(18) ~~j Y ~fj Yxjgi, all i EN.
jeN gezH heH LK U)i jrN g
Insights are gained by rearranging to have tax rates on the left hand side. We display the
solution for the case with four goods (0, 1,2,e), two tax rates:'8
t--l)(xl X22 -X 21 E h ha
(19) t, =__ + f_ and
H xl xx22 - x21Px2 /
17 It is not unrealistic in all interesting cases, and realism depends on the level of generality in the model. In
the example of CO2, there are virtually no abatement technologies available to users if we examine (fuel)
efficient combustion technologies for each fuel (say: coal fired power plant). Thus, for a model
disaggregating to the individual fuel, the assumption of fixed emission factors would be quite appropriate.
In contrast, for a model with an energy aggregate only, one could represent the flexibility within this
aggregate (towards fuels that are less CO2 intensive) as abatement options.
"8In the more general case, we have tk = Ih ((f h /8u-1)ZX, F1k /IE| + fk Eh h h a h where
E is the coefficient matrix in (18), and F,k is the cofactor of row i, column k.
14
(----10(X2x -XlX2_) E_h_ h
t2 2 _ _ _ _ _ _ _ _ _ _
t2=- =- --) +t2
II x2 - x21 x12 J
which is the solution given by Sandmo, 1975. In (19), we have used consumer averages
(xy = Ei XyI /H) to highlight which terms in these formulas we"ghted by fi. The optimal
tax formula 'cares' about individual consumption xh arid willingness to pay a', but
about demand responsiveness only in aggregate.
Sandmo describes the optimal tax structure (19) as giving commodity taxes in the
presence of externalities an 'additivity property' (page 92). Of the two terms, the first is
equal to the formula for optimal commodity taxes in the traditio nal problem with no
external effects (see below), and the second is motivated by the need to correct for
external effects. The term for the corrective tax is, as Sandmo noted, zero for
commodities that are not polluting, and it is zero for all commodities if there. is no
willingness to pay fo:r pollution reductions (Eh flhah = 0). The addition to Sandmo's
result given here is onaly that with several polluting goods, the corrective tax element in
each tax formula is uniform per unit of public good (the emission factor in each formula
ensures this). This result, it can be argued, follows so diirectly from Sandmo's analysis, it
is implicit.
We shall now use the redundancy in tax instruments to explore a specific alternative
way to impleiment this allocation. Let us examine a solution including the following tax
rate levied on emissions uniformly across polluting goods:
(20) r1ek =fe] -,e h allk,l = l,..,n .
Substituting (20) into (16), we have:
(21) Z hxX +vEg[xjg +Xjt x =O, all i = n.
(20) and (21) also solves (17), so this system of comnmodity taxes and a unijform emission
tax implements the optimal allocation. Solving for the commodity tax rates. and again
assuming two taxed goods, (21) is satisfied for'9:
19 The more genieral case corresponds to the formula in footnote 19, eliminating the Pigovian element.
15
E XA -1(XJ X2 XX21 )ECA-)(2 Xl I XX12)
(22) t= h and t2 = l- -x--
H (x,, x HFx 22 - X21 , Hxlx22 -x21 x12
The formulas for the commodity taxes in (21) and (22) (and also those for the non-
Pigovian terms in (18), (19)) are equivalent to the expressions for the solution to the
traditional problem of optimal commodity taxation without pollution (i.e. Samuelson,
1951), but the actual tax rates in models with and without pollution will in general not be
the same. To highlight the implied structure for the non-Pigovian part, let us follow
Samuelson and use the Slutsky equation and the symmetry of the compensated demand
derivatives Sh (q,uh) = Sh (q,uhu) to see that (21) =>
(23) X1tZ,sJ D h XiJ tj &j + ph -1j, all i = 1,..,n.
As Samuelson pointed out, if one assumes an arbitrarily small revenue requirement and
identical consumers, (23) gives the same proportionate reduction in compensated demanld
for all commodities. Sandmo (1976) highlighted that this feature of (23) extends to hold
for substantive revenue requirements if all taxed goods have equal income elasticities.
Simplifications often used to illustrate the implications of (21) (or 23) are to assume that
the displayed cross price responses are zero, implying that taxes are inversely
proportional to own-price elasticities.20 The structure is also equivalent to the one
analyzed by Corlett and Hague, 1953, who showed that with two taxed goods the good be
taxed at a higher rate which has a higher degree of complementarity with the untaxed
good.
Thus, it can be seen, the standard and recognized results for optimal commodity
taxes extend to the case with an environmental externality, as long as the externality is
20 Samuelson's (1951) rule should in part be attributed to F. P. Ramsey, 1927, who finds: "the production of
each commodity should be diminished in the same proportion". As noted in Munk, 1978, Ramsey's
changes in production are equal to changes in uncompensated demand, equal to those of compensated
demands if income elasticities are zero. Samuelson writes: "Aspects of the right answer have been hinted at
by Ramsey (1927)". Diamond, 1975, proposes to use the concept social marginal utility of income,
h nh h nh +P tj"Xi __
y ,rather than our h , with y j= + a t h . This is intuitively an equally attractive concept
and simplifies expression of certain results: "for each good the change in aggregate compensated demand is
proportional to the covariance between individual quantities demanded and social marginal-utility of
income" (page 338). A good orientation in this literature is provided by Auerbach, 1985.
16
taken care of by an appropriate tax levied on emissions. This is :in itself not an interesting
observation, first because there is redundancy in instrunents21, and second because the
fornal equivalence oi commodity tax formulas with an,d withoul presence of external
effects in no way wotuld imply equivalence in rates. Hcwever, there are two aspects of
this solution givinRg the emission tax an intuitive interpretation as a price. First,
unifonnity across polluting activities hint that emission reductions are elicited at the same
marginal cost wherever they can be found - alluding to a simple procurement rule.
Second, the expression itself consists of a weighted sum of the willingness to pay for
emission reductions, reminiscent of the Samuelson (1954) rule for optimal provision of
public goods.
Definition: When the purpose is to internalize external efficts such as emissions,
we shall use term Pigovian tax in the traditional way - to mean a corrective tax - if the
tax/subsidy is levied idirectly on emissions (or more generally on a measured contribution
to the public good). WVe shall use the term presumptive' Pigovian tax if the corrective tax
is levied on a commodity (such as an input or an outplt in the externality generating
activity) with the irate per unit of the commodity motivated by an emission factor, in
presurnption of emissions22.
Proposition 1: Fixed emission factors and presumptive Pigovian taxation
With fixed emission fcactors, two alternative tax structutres implementing the optimal
allocation are
a) as in Sandmo, 1975, a commodity tax structure in which the formula is the sum of
presumptive Pigovian taxes and the formula for cptimal commodity taxes in the
traditional problem without external effects (equcation 19, or more generally from
18).
b) a combination o a Pigovian tax (20) uniformly applied to emissions from all
polluting goods and services, and a commodity tax structure satisfying the formula
21 See Cremer et al., 1998, who operate with nonlinear instruments more general than ours.
22 Terms and definitions: We thus associate the term Pigovian tax with the cbjective of internalizing
externalities, but not a rule or a level (contrasting, for instance, Cremer et al., 1998). An indirect Pigovian
tax typically means z1 corTective tax levied not on the externality causing good itself, but on substitutes and
complements (Sandmo, 1976b). Outside the realm of Pigovian tixation, indirect taxes have a different
meaning (see, fir instance Atkinson and Stiglitz, 1976). The terz presumptive is for income taxes
17
for optimal commodity taxes in the traditional problem without external effects
(equation 22 or more generally from 21).
The proof is given above.
A historical note is worthwhile. Ramsey wrote in the introduction to his 1927
treatment of the traditional problem without externalities: "I shall suppose that, in
Professor Pigou's terminology, private and social net products are always equal, or has
been made so by State interference not included in the following." Sandmo, 1975
followed up to solve the twin tasks thus referred to by Ramsey. In his concluding
paragraphs on how to assess real-world taxes, Ramsey wrote: "In the case of motor taxes
we must separate off so much of the taxation as is offset by damage to the road. This part
should be so far as possible equal to the damage done. The remainder is a genuine tax and
should be distributed according to our theory; ". Thus, we may say Ramsey had in mind
something like Sandmo's 'additivity property'. Another aspect in Sandmo's forrnula was
that the damage component (reflecting benefits of public good provision) is adjusted by
the shadow price of public revenue. This adjustment points back to an important idea of
Pigou's, that when revenue generation has its own costs "expenditure ought not to be
carried so far as to make the real yield of the last unit of resources expended by the
government equal to the real yield of the last unit left in the hands of the representative
citizen."23 Pigou's conjecture later was found to require qualification, but the indicated
adjustment is assured if there is separability between the public good and taxed goods
(our model) and taxed goods are not predominantly inferior goods (See Atkinson and
Stem, 1973).
Assumption 2: Endogenous emission factors
When abatement technologies are available for a set M of polluting goods
(fib < 0, i E M)), the system (16) and (17) is at most of rank n+m. For simplicity, let us
assume that the rank is 2n so all n polluting goods have abatement technologies.
established, with a meaning parallel to ours for corrective taxes (See, for instance Musgrave and Musgrave,
1984, or articles in Newbery and Stem, 1984, and Gillis, 1989).
23 Pigou, 1947, 1949 reprint, page 34.
18
We may immediately substitute the Pigovian tax (20) into (16) and (17) to see that
this again reduces the set of equations to one of rank ^, and a formula equal to the one
defining the optimal commodity tax structure in the traditional problem without external
effects24. Thus, we may conclude with:
Proposition 2: Endogenous emission factors and Pigovian taxation
A combination of a Pigovan tax (20) and a commodity, tax structure satisfying the
formula for optimal commodity taxes in the traditional problem without external effects
(equation 22, or more generally 21) is optimal also in the context of endogenous emission
factors.
One way of this looking at this result is that one introduces a good-specified
emission tax to induce abatement. First, this tax is to be the same across polluting goods.
Second, in a context of commodity taxes satisfying the formula for optimal taxes in the
traditional problem without extemalities, this emissioni tax also induces optimal
substitution towards cleaner goods and services. Another way to communicate the result
is to suggest that the presumptive Pigovian part of Sanidmo's formula be replaced - when
possible - by a tax levied on emissions. Viewing Sandmo's fonnula as a sum of two
taxes, the presumptive Pigovian tax is transformed to a Pigovianr tax by moving the
emission factor from the tax rate to the base of a new tax. This scheme is strictly
preferred in a context in which abatement can be induced, and thus optimal in a wider
range of circumstances.
Assumption 3: Emissions are not observed (or not taxable) at the individual level
We here examine briefly the implications of two crudely defined constraints on the
monitoring of emissions. Let us first assume that the planner is not able to tax emissions,
but he can regulate abatement and levy commodity taxes. Standards for emission factors
(or for abatement) are often seen in the real world, and one interpretation of this is that it
is less costly to nmonitor emissionfactors or technology, than it is continuously to monitor
emissions (or to obtain a measure of cumulative emissions, say at year-end)25. Examples
24 In an earlier version of this paper (Eskeland, 1996), direct derivation of this tax structure was provided. It
can be made available upon request.
25 The existence of emission standards has been given several interpretations. The interpretation compatible
with our treatment here is that monitoring emissions at the source is prohibitively costly, but that abatement
19
are that vehicles and industries face emission standards, either mandating a particular
technology or defining a maximal rate for emissions per unit of output (say: grams per
mile or per gallon of fuel, for vehicles)26. Simultaneously, the vehicles may be subject to
odometer charges (by mile, or kilometer) or fuel taxes, and industries may be subject to
taxes on inputs and outputs.
With these assumptions, the planner has two instruments for each good (tj, bj), again
a total of 2n instruments at his disposal. Modifying the Lagrangian (11) with the
applicable budget constraint Zh E .tj x$ = R and instruments, the first order conditions
for optimum are the budget constraint and, for all i=1 ,..,n,
(24) - a fj (bj)xj]+8UXxi +tjXJ]=o) and
hf H geff j gEH j
(25) -, pL [xi h+(x E (fj (bj )X + fib (bj W )]+f[t ji .
where we have used aqj /8bj = l ,j=l,..,n (see equation 5).
It is easily checked that if the planner sets abatement or emission standards such
that marginal costs equal social benefits adjusted for the shadow price of public revenue:
(26) JEH =
(or technology, or emissionfactors) can be monitored cheaply ex ante (as a car model is approved by the
authorities) or periodically (at annual vehicle inspections) or even randomly. Such a structure is analyzed in
Eskeland, 1994, where it was shown that standards then should be combined with presumptive Pigovian
taxes, levied for instance on a car's odometer, or on a variable input, such as gasoline. A more
comprehensive practical discussion of these principles inthe light of monitoring and enforcement probleins
is given in Eskeland and Devarajan, 1996. Another important interpretation of standards and regulation in
general, as opposed to the tax treatment, is that they give the planner a way to distribute emission permits
(See Buchanan and Tullock, 1975, and Baumol and Oates, 1988).
26 Standards for emission factors and for abatement (or technology) have equivalent implications in our
model, but more generally instruments should be as open and flexible as possible. Thus, standards will be
more effective, ceteris paribus, if they define maximum emission factors than if they specify a technololyy
which meets that goal.
For reviews including discussion of monitoring costs and their consequences, see, for example,
Eskeland and Jimenez, 1992, and Cropper and Oates, 1996. Using a model with monitoring costs,
Sclmutzler and Goulder, 1997, conclude "Pure output taxes are optimal under sufficiently high monitoring
costs, sufficiently limited options for emission reductions by means other than output reduction, and
sufficiently high substitutability of the output".
For cars, a potentially important advantage of odometer charges (relative to fuel taxes) that is not
exploited in the literature nor in the real world is that it could implement a system where the corrective tax
is raised conditional on vehicle characteristics or emission factors. With fuel taxes, such differentiation will
be constrained.
20
then (25) reduces to (24) times the vectorf to be satisfied if (24) is satisfied. Then, the
formula for optimal commodity taxes including presumptive Pigovian taxes (19, or more
generally 18) satisfies (24) and (25).
Proposition 3: Emission standards and presumptive Pigovian taration
a) If the planner cannot tax emissions, but he can set standards for emissiol factors (or
abatement) and levy commodity taxes, then the optimal allocation is the same as
when emission taxes are available. The marginal cost of abatement per unit of
emissions is the same across activities, as if driven by optimal emission taxes (20).
Commodity taces will satisfy the formula for optimal commodity taxes including
presumptive Pigo-vian taxes (equation 19, or more generally 18, as in Sandmo, 1975).
b) If the planner cannot address abatement in any wav, then the optimal allocation is
one of commodity taxes including presumptive Pigovian taxes (19 or more generally
18) as in Sand,mo, 1975.
Part a) of Lemma 3 (the allocation is the same as with emission taxes) is seen by noticing
that abatement is identical, and that such abatement and the level of presumptive
Pigovian taxes result in the same all-inclusive consumer prices and the same public
reventLe. Part b) of Lemma 3 is seen by noticing that when the p tanner has only n
commodity tax rates as instruments, optimality is characterized only by the budget
constraint and the n equations in (24), equivalent to (13). Under the assumptions of b),
emission factors are higher and environmental costs (thle sum of abatement and Pigovian
taxes) weigh morc heavily in the 'all-inclusive consurrmer price' than in a).
The contribution of Lemma 3 is a modest one, since it is well known that the
efficiency properties of a quota for emissions can be the same as those for an emission
tax (See, for instance, Baumol and Oates, 1988, or Tietenberg, 1992). What we do here is
to introduce a generalizing and a restrictive feature. We generaliCze by looking at the use
of quotas (or standards) in a context with distortionary revenue generation. One of the
lessons thus learned is that such a system of optimal slandards and presumplive taxes in
our model has the same allocative and distributive impacts as a system with emission
charges. On the restrictive side, as we generalize to iniroduce constraints on monitoring
and enforcement, we assume that these allow a separate policy instrument to make
21
activities cleaner per unit of activity. 'Emission quotas' often come in the form of a
standard for emissions per unit of output, a fact that has formerly been afforded scant
notice and interpretation in the public finance literature27. A contribution of Eskeland,
1994, was to show that emission quotas of this kind make activities cleaner, but fail to
give appropriate incentives to reduced consumption, and thus should be accompanied by
a presumptive Pigovian tax 28.
Finally, we should emphasize that Sandmo's 1975 result should be read as holding
for any given level of abatement. Building on this, Lemma 3 contributes with optimal
abatement.
Assumption 4: Producers and government abate and pollute
Proposition 4: Production efficiency
Optimal abatement is efficient in the sense that the marginal cost of abatement per unit oqf
emissions is the same not only across activities but also across agents: government,
households andfirms.
Consider first the case of firms, and the simple case in which production of goodj
involves firms with the same costs, abatement opportunities a3 and consequences
fj (aj ) . Let us first assume that abatement in production influences production-stage
emissions (i.e. at the car-maker's smoke-stack, rather than at his customer's tail-pipe).
Then the producer price for goodj (see equation 10) will include not only the producer's
costs of abatement but also his emission taxes, aijfi:
(27) pj = cj + aj + i-. fj (aj).
27 Important textbooks such as Baumol and Oates, 1988, and Tietenberg, 1992, do not mention monitoring
costs as possibly favoring (or explaining) emission standards. Uncertainty in estimates of benefits and costs
is an accepted consideration in quantity instruments versus prices (Weitzman, 1974), but that argument
does not rely on costly monitoring of emissions.
28 One can argue that such emission standards implicitly award emission quotas to operators of polluting
processes: You may emit more, but the same amount per unit, if you drive more, or if you produce more
steel. That perspective is even more important when existing facilities are 'grandfathered' (given more
lenient treatment) in regulations. For analysis of such differential treatment, see Crandall et al., 1986, and
Harrington, 1997, on automobiles, and Nelson et al., 1993, on EPA's new source emission standards.
Grandfathering has positive and negative connotations: 'Grandfather clauses allow rents to be shifted to
those grandfathered without distorting supply responses' (Wittman, 1989).
22
It is easily checked that an emission tax Taj =e as in (20) is optimal (substitute (27) into
(5), simplify by setting consumer emissions and abatement to zero, and modify (11)
accordingly).
By the same argument, if two producers of j are active but have different
technologies and emission factors, their emissions are ltaxed at the same rate in optimum.
Note that two (or more) firms with different emission functions and different emission
factors can be active at only one level of the emission lax, since if the emission tax is
raised slightly, the producer with the higher emission factor shuts down (by ihe envelope
theorem). However, if there are latent technologies, then at any emission tax level
technologies wvith diffCerent emission factors can be active, and we have shown they shall
be taxed at the same rate per unit of emissions. Thus, tle marginal cost of abatement per
unit of emissions reduiced will be the same in productive sectors as amongst consumers:
(28) -=a= h
fija fa fib
For government, the Lagrangian (11) assumed that the government consumption
vector xP as vvell as the government abatement vector bP was given. Modify (11) to
reflect a choice of abatement, and partially differentiate with respect to b in addition to
the previously appliedi instruments tej ,e. No changes in expressions are imLplied for the
previously established set of first order conditions. For the additional first order
conditions reflecting optimal abatement for government, we have for all j=i',..,n:
(29) ,h ,h df e - p VCP = O.
Using - = . , from (2) and (3) we can see that
i i
(30)
Thus, in optirnum, the marginal costs of abatement per unit of emissions reduced will be
equalized across firms, government and households.
As a matter of implementation, if government agencies are geared to pursue their
respective goals vvhile maximizing some appropriate 'profit' function, then these agencies
23
should be exposed to the same emission tax (or abatement requirements) as the one levied
on consumers and firms.
We have now shown that abatement should be stimulated by the same emission tax
when abatement reduces own emissions. The generalization remaining is to allow
abatement at any stage to influence emissions or abatement opportunities at other stages
as well (as when the car's emissions can be reduced by abatement efforts in the car-
factory, at the service station, in the oil refinery and by the driver). It is intuitive, now,
that the emission tax (20) provides optimum stimulus in this more general setting. As
producers and consumers join forces to minimize private costs - including emission
taxes, emission reductions are provided effectively. Showing this involves additional
notation, and is left to the reader29.
We are now ready to summarize our findings:
Summary of central findings: Pigovian principles and production efficiency
Assume constant returns to scale, that the environmental good is separable from
other goods, that within each activity consumers have uniform emission functions, that
linear taxes on inputs, outputs and emissions can be differentiated by commodity (or thait
emission standards can be differentiated by commodity), and that different regimes can
apply for consumers, producers and government.
i) Welfare optimum is characterized by the marginal rates of transformation
between abatement and emission reductions a) equal across polluting activities (i.e.
goods, sectors, j(-N), b) equalfor emissions from consumers, producers and government,
and c) equal to the welfare weighted sum of willingness to pay across consumers
29 It may be worthwhile to revisit with a practical perspective the issue of the untaxed good. Our
formulation states that consumer abatement is through application of the untaxed good. It is this feature
which allows equal rates of transformation between abatement and emission reductions to be implemented
by one emission tax faced by producers and consumers (since producers face pretax prices). Assume now
that leisure has to be the untaxed good and that consumers may abate emissions with leisure (using time to
drive more carefully, or to perform more laborious laundry with less polluting detergents) and by changing
filters, and that producers may abate by installing filters and through many other actions. If leisure is the
untaxed good, and filters can be taxed at zero rates when used in abatement, then the efficient solution can
be implemented by confronting consumers and producers with the same emission tax. If abatement cannot
be taxed at zero rates (when used in abatement by consumers), then the optimal allocation - still equalizing
the marginal costs of emission reductions - is implemented by a separate emission tax for consumers.
24
adjusted by the shadow price ofpublic revenue: - - -- _ -p = ah
fjb fja jb
ii) When abatement is untaxed and emissions are observable, such abatement can
be implemented by a tax levied uniformly on all emissions (20):
Te=
'U
iii) An emission tax satisfying this formula combined with commodity taxes
satisfying the formulafor optimal commodity taxes in ihe traditionalproblem without
externalities (21) implements the optimal allocation.
iv) When emissions are not taxable, but emission standards or abatement standards
can be used, the same allocation can be implemented by a combination of emission
standards (as in i), above) and commodity taxes that include presumptive Pigovian taxes,
as in Sandmo, 1975.
v) When abatement cannot be induced by the planner, the optimal allocation is
implemented by commodity taxes which include presumptive Pigovian taxes, as in
Sandmo, 1975.
With pollution just from producers and government, the equality of marginal rates
of transformation between abatement and emission reductions is a predictable
consequence of Diamiond and Mirrlees' (1973) result. They showed that the set of
producers and governnent should be treated as one, to all have equal marginal rates of
transformation between goods. This clearly should apply even when an additional input
valued by consumers, the environment, is included in the model. Amongst the findings of
Cremer and Gahvari (1999) is that an emission tax should apply uniformly across
industries. We adid that this holds also for pollution from government, from firms with
heterogeneous emission functions, and from consumers - only the latter one of which
does not follow almost directly from Diamond and Mfrrlees' treatment. Also, we show
how standards can take the place of emission taxes under some plausible restrictions on
monitoring.
The result least to be expected, that production e,fficiency shall include pollution
abatement amongst consumers, must be understood irL a context of assumptions about
available instruments. Also, that result depends on the assumption that consumers face
25
the same abatement opportunities, implying that they have the same emission factors
when exposed to emission taxes. We relax this assumption in the section to follow.
IV. Non-Uniform Emission Functions
In the previous section, we established that producers shall be taxed uniformly on
emission irrespective of whether they have uniform emission functions. For consumers,
the analysis till now has assumed uniform emission functions. We now investigate the
consequences of heterogeneity across consumers in emission functions, reintroducing
individual superscripts for emission functions, and thus abatement: fjg (bg).
We should note that the additivity in the relationship between emissions and the
environmental good (equation 3) is retained. What we now allow implies only that
consumers may differ in terms of emissions per unit consumed of the polluting good. An
alternative formulation - also important in practice - could be that polluters differ in the
relationship between emissions and the enviromnental good (so the damages could differ
per unit emitted, rather than per unit consumed, which is our formulation). Results are
very similar in nature, though with qualifications regarding instrument availability. The
reason is that an emission tax is still 'first best' with respect to environmental protection
when emission functions differ. If damages per unit emitted are different, in contrast, the
emission tax is first best only if each polluter can be taxed at the same rate per unit of
damages. With presumptive Pigovian taxes levied on each unit of the polluting good, the
parallel is more direct, since that instrument loses its first best properties in both
formulations.
Uniformity of emission functions for a given commodity is more plausible the
more narrowly one can define each polluting commodity. Examining the model, this is a
question of whether consumption with different emission functions can be differentiated
in the commodity tax structure. If consumption can be differentiated in the commodity
tax structure, so that within each "commodity" uniform emission functions result, then
the results of the previous section apply.
To give a practical example, assume first that emissions are taxable, and that car
travel is more polluting when using leaded gasoline than when using unleaded, but with
emission functions that are uniform amongst users of leaded gasoline, and amongst users
of unleaded gasoline. If the two fuels can be taxed separately in the commodity tax
26
structure, then the results of the previous section apply. Assume in contrast, that emission
functions differ by car or user characteristics (old versus new/young, male versus
female). To the extent that commodity taxes cannot be: conditioned on these (perhaps they
could, if odoineter charges were used), one has set the scene for the topic of this
30
section
We assume polluters are exposed to non-individualized linear instruments:
commodity taxes ancl emission taxes or uniform abatement requirements, b' = . Under
regulation, then, abaltement is uniform by assumption, and emission factors may differ if
emission fuinctions are heterogeneous. Under an emission tax, consumers equalize
marginal abatement costs -1/ fjhh (bjh (Terj)) /fJgb (b (r)) = rj (re equalion (6)), and
abatement as well as emission factors may differ if emission functions are heterogenous.
As an important background, in a setting with costless redistribution and revenue
generation, an emrission tax is a first-best instrument even when emission functions are
heterogenous. In contrast, if the planner cannot tax emtissions, then presumptive taxation
of goods would be an imperfect instrument under heterogeneous emission fanctions. This
difference should be on our mind as we set out to ana].yze the cases with and without
emission taxes separately.
Taxation of emissioins
Corresponding to (16) and (17), our first order conditions for optimum are, for all
(31)-E / x i +a fg Xji ]+ l [Xig + (tj + Tej fJg})k,i°'
hEH gEH geH j
(32) -jhxh +a h f.b9MxF + , fjgx,fj '
hell geH jeN
30 Empirical aspects of vehicle characteristics and emission factors are well known amongst practitioners,
and were recently docurnented and discussed in Harrington, 1997. In Eskeland and Kong, 1998, the
distributional consideration is examined in detail. One stylized fact found is that the expansion path in
household energy use is toward energy carriers with lower emission factors (say, from coal and wood to
electricity and natural g;as). Another is that emission factors are lower for newer equipmerlt, both because
designs improve with vintage and because of deteriorating funcions (emission control, combustion).
27
+g[ igXg + ,ei (fifbiri )+ ,(t1 + Terfjg )xiig 1=0 -
gEH j
The reader may verify that straightforward application of Pigovian principles
(Z*ei = ~,hh /,u, all i) leaves (31) solved by commodity taxes satisfying the formula
for optimal commodity taxes in the traditional problem without pollution, but that this
emission tax is inconsistent with solving the set as a whole with only n remaining
instruments. Thus, emission taxes cannot in general comply with Pigovian principles
when emission functions are heterogeneous. We proceed to qualify and interpret these
deviations from Pigovian principles.
Without loss of generality, let us split the taxes levied on emissions in (31) and (32)
in two parts: one 'environmental tax' Te which we set according to Pigovian principles
(zTe = E 'a h h/u ), and a supplementary emission tax (or subsidy) -r, which we leave
for further investigation:
(33) rei=e - + *i ,
(34) Te =Eh a ha/hu.
Also, to simplify exposition, let us introduce the following expression (it is the derivative
of revenue from consumer g with respect to ti, except the part rvE figj,i):
g j
(35) g + ji
Substituting (33), (34) and (35) into (31) and (32), we have, for all i=l,..,n,
(36) hXh aRh =\
(36) ~h jXi - ) =at 0, and
BhfhXh [ aR h
(37) hfhXh A fih + i i
(37) can be rewritten using averages and covariances across consumers as follows:
(38) f ZrflX I P § H[/ifibbiXi COv (3xi,f1)+,uCov (-R f)]=0
heHK at, j L at2
28
We can see that if the covariances in (38) are zero, then we can set ri =0, all i, and each
equation in (38) is simply fi times (36). Thus, when l;hose covariances are zero,
emissions are taxed according to Pigovian principles (34) only, and a commodity tax
structure which solves (36) is optimal. When rT is zero for all polluting goods, (36) is
also the solution to the traditional optimal commodity tax problem (i.e. without
pollution).
More generally, let us observe that (38) is a sum of two terms, where l:he first is
simplyf times (36), so we may think of the optimal tax structure as follows. (36) is at
most of rank n , so the n commodity tax rates can be reserved to solve (36), conditional
on a set of supplementary emission tax rates. Thus, the system (,36) and (38) has a
solution for wvhich the supplementary emission tax rates render zero the bracket term in
(38). Assuming that ffibbirXi 0, and using
Cov(fi, aR/3tj) == Cov(fi, xi) + E jtjCov(fi, xji) + E1r Cov(fi, fjxji), the term in
brackets of (38) is ze,ro for
CoV(fi, i/)-P cov(fi xi) + Etjcov(fi ,xji) + z -Cov(f1Jjjixj)1
(39) Ti =- - J j
wufibbirxi
all i=1,..,n). This is no explicit solution: not only are there tax rates on the right hand
hand side, but all the expressions may be functions of the tax rates. Nevertheless, from
(39) we learn that it is a specific set of covariances that gives a potential role to "non-
Pigovian" supplementary taxation of emissions. To gain some additional insight, let us
make the assumption that only one activity is polluting: fj = 0, j X i, and assume that
1+ u (fifixii) 0:
fib bir Xi
Cov(ffiX)1i) - U{Cov(f xj) + EtjCov(>j xji)
(40) T=- -
Aftibbi,xi + Cov(fi, fixii))
(40) is still a complicated combination of effects, but all intuitively play a role given that
the social plamner compares the effects of supplementary emission taxes to the effects of
commodity taxes. The two terms in the denominator represent the responsiveness of
29
abatement and emissions to the emission tax. These responses are, from a first-best
perspective, wasteful when emission taxes differ from Pigovian principles, so the
absolute value of their sum ceteris paribus reduces the value of supplementary emission
taxes. In the numerator, the first covariance represents the planner's evaluation of the
distributive pattern of the emission tax (as compared to the commodity tax). As an
example, assume that the denominator is negative (fibbiTxi < 0 by assumption) and that
the bracket term is zero. If the emission factor falls with income (as when wealthier have
newer cars and these are less polluting) and fixi falls (rises) with income, the
supplementary non-Pigovian emission tax will be negative (positive).31
The combined term in brackets distinguishes between the emission tax and the
commodity tax in terms of the marginal effect on revenue. If the bracket term is negative,
then it means that increasing the emission tax on good i raises revenue less than fi times
a change in ti, an effect which ceteris paribus reduces the emission tax (assuming the
denominator < 0).
To focus on revenue and redistributive considerations, assume that the denominator
is negative and the tax weighted term with demand responsiveness in (40),
, j tj Cov(fi, xji) is zero: For ri to have a determined sign a priori, Cov(fi, /xi ) and
Cov(fi, Xi) must be of opposite sign. For a normal good, a "steep" ,8 is sufficient to
ensure a sign, and the sign is given by whetherf is increasing or declining with income.
Giving a practical illustration, emission factors will often be declining in income.
Assuming /J steep enough that Cov(fi,,ixi) is positive even though Cov(fi, xi) is
negative, these effects lead to a downward adjustment in emission taxes from Pigovian
levels.
Let us finally focus on the possible covariance between the emission factor and
demand responsiveness. To illustrate simply, assume that in (40) all cross price
elasticities are zero, and that Cov(xii, fi ) >0, so that the more polluting consumers are
less responsive in their demand. Assume further that the denominator is negative and thal:
3 lntuition: In case ,Ai and fi fall with income, the poor are hurt more by the emission tax than by the
commodity tax. A slight change in taxation from emissions to the commodity redistributes from rich to
poor.
30
the other covariances are zero. If ti is positive (negative), then non-Pigovian taxation of
emissions is positive (negative). In this case, the planner takes the opportunity for 'price
discrimination' simply to reduce distortions (this effect does not depend on the vector
,/ ): More polluting consumers are less price responsive (for good i) and should therefore
face a higher effeictive price for reasons well known in the literature (Ramsey-pricing). A
distortionary effect of this is that emissions are taxed 'too heavily', so 'too much
abatement' is executed.
We should highlight again that these perspectives often will point us back to ask
for a more different:iated commodity tax structure, ralher than to actually modifying
emission taxes with non-Pigovian objectives. When emission taxes are brolught to differ
from Pigovian principles, here, it is because they take on roles in redistribution and
revenue generation that are left unsolved by other instruments. Using emission taxes for
these purposes have separate, identifiable costs, and can be att-active only if other
available instruments entail costs as well.
Presumptive Pigovian taxes
In the case of non-uniform emission functions, taxation of commodities in
presumptiort of emissions has a weaknesses in additicn to the potential weakness of not
inducing abatement: consumption by dirtier consumers and cleaner consuraers is
discouraged with 'equal pressure'. This may present a probleni of fairness and
distribution., but also of efficiency, since the emission factor determines the emission
reductions 'bought' when consumption is reduced.
Let us initiate this analysis by assuming that emission factors are given. This
problem is similar to the problem of imperfect corrective pricing analyzed by Diamond,
197332. Our resujlts also compare with several studies on taxation of substitutes and
32 In the literature, 'corrective taxation' is used synonymously with 'Pigovian taxation', and corrective
pricing refers to prices that include corrective elements, equivalent to prices that include our 'presumptive
Pigovian taxes'. Diamond's problem is more general than ours in the sense that he makes no separability
assumption, sa his 'public good' (absence of congestion) may influence demand. On the other hand, our
problem is more general in including distortionary revenue generation, and in including effects across
markets in the external effects as well. These differences are illustrated in one of his concluding passages:
"..the optimal surcharge will be small relative to the average externality when individuals who contribute
greatly to congestion per unit demanded.. tend to have demands which are congestion sensitive .. and price
insensitive ..". In our maodel, the analogue to congestion sensitivity is zero due to separability (the case for
31
complements to externality-creating goods when the ideal corrective instrument is not
available 3. Analogously to (16), first order conditions for optimal commodity taxes are:
(41) - ,Lx +a X yfjgx i+,{ xxi + I(tj + teji= 0, i=L*..-n.
In (41), we have followed steps in previous sections to 'artificially' split the commodity
tax rates in two parts. The system is then indeterminate, and we can choose one part
arbitrarily. Let us choose tj, j = 1,.., n such as to solve the traditional problem of optimal
commodity taxes when there is no pollution:
(42) -h fiX, + ±i [x +E tj xJ] =0, all i=1,..,n .
Then, for (41) to be solved, we must have
(43) h ia IfJjgi=8tjEXygi = l.,
j g j g
Assuming that the coefficient matrix in (43) is nonsingular, we may use Cramer's rule to
develop more explicit expressions. For the two-good case the presumptive Pigovian tax
on good one is
19 Ag (XigiX22 X1g2X21) Egf2g (42 X22 - X2g2X21)E aha
(44) tel = ! _ )
H(x 11x22 - x12x21) H(x11x22 - X12 ) X
where the denominator is positive.
Corresponding formulas, for te2 or for systems with more goods are straightforward
to derive. We may consider a system consisting of traditional commodity taxes (42) and
presumptive Pigovian taxes (44) as a generalized version of commodity taxes which
include presumptive Pigovian taxes (19). The large parenthesis in (44) then plays the
role of the emission factor fl, and the first fraction in this parenthesis is indeed a
weighted average for f1, which equals f1 if the covariances Cov(fi, x 1) and
Cov(f1, x,2) are zero (as when f1 is uniform). The second fraction in the parenthesis
represents emission spillovers via cross-price elasticities to good 2, and is zero if the
such sensitivity is more compelling for congestion), and our results with respect to price sensitivity will be
less clear cut, due to cross price effects both in revenue generation and in extemal effects.
33 Noteable contributions are: Green and Sheshinski, 1976; Sandmo, 1976, Balcer, 1980; Wijkander, 1985,
Greenwald and Stiglitz, 1986.
32
covariances Cov f2 , X21) and Cov(f2, x,2 ) are zero. Interestingly, good I may be taxed
with reference to the Pigovian objective even if only good 2 is polluting, a result
observed in the literalture on indirect instruments (see below). To focus on covariances,
let us rearrange. (44) =>
(45)
tl 4 X22 [COV(J,Xll) + CoV(f2,X21)]-x2j[Cov(.f1,X12) + CoV(f2,x221 E
xi1 X22 -X12 X21
The four covariances all are between an emission factor and the price responsiveness of
the good to which it applies. If we assume that own price elasticities are negative and
cross price elasticities are positive, then positive covariances result in a tax rate lower
than under unweighted average emission factors. The reason for this is that positive
covariances raise the marginal costs of emission reductions relative to that indicated by
the average coefficient (illustration: Cov( fl, xl) > O == Cov(f1, jxl |) < 0, so individuals
with high emission factors adjust consumption less tham average).
To enhance intuition further and to compare with1 the literature on indirect Pigovian
instruments (see above), let us consider again the case with two taxed goods and assume
that good 1 is not polluting. The Pigovian parts of the commodity tax rates in this context
are
_22_V( 2 X21)XCoV(f2,X22) Xfihah
(46) tel ( X2 for the nonpolluting good, and
x11X22 -x12X21
(47) t -- x11'CoV(f2 ,X22) -X12C'oV(f2 ,X21) Y ,flhah'i
(42) te, = X ) ] , for the polluting good.
Xi I X22- X12 X21
Our model has greatest similarity with that of Balcer, 1980. The literature we refer
to does not include distortionary revenue generation, but compares with our formula for
the presumplive Pigovian tax which bares evidence of distortionary taxation only through
E fia/,u. Balcer focuses on the dimensions of "large offenders" (consumers g, for
whom f2/ > f2 , in our terminology) and "large offender complementarity"
(Cov(f2,X21) <0)
33
If we make the assumption of 'aggregate independence' in demand ( x2 = x12 = 0),
we can tabulate results analogous to some of those in Balcer's table 1:
Table 1: Presumptive Pigovian Taxation
under Aggregate Independence (x12 = x21 =0)
Assumption for Own -Price Own-price Own-Price
"large offenders": Responsive Neutral Non-responsive
Cov(f2, X22) < 0 Cov(2, X22) = 0 Cov(f2, X22 ) > 0
Result for A fi a -t_E
Direct hinstrument te2 > A fate 2 = fY,ate2< f2Y-ia
Assumption for Complementarity Neutrality Substitutability
"large offenders" Cov(f2,X21 ) <0 Cov(f2 , X21) = 0 Cov(f2 X21) > 0
Result for te > 0 t = 0 t < 0
Indirect Instrument '
The results under aggregate independence are quite intuitive. As examples, for the
direct instrument te the tax is raised by own - price responsiveness for large offenders,
since this reduces the costs of emission reductions when the price is raised equally for all.
For the indirect instrument, tel, the level will be negative if there is large offender
substitutability, since a subsidy then reduces consumption of good 2 amongst large
offenders but not for average offenders, thus providing emission reductions at low costs.
As an illustration, assume one group (say, the young) would substitute metro- for car
travel if metro fares were lower, but that for the old the two are complements, so that
aggregate demand for car travel is independent of metro-fares. If young people pollute
more than old when traveling by car, the commodity tax rate for metro travel would be
adjusted downwards for Pigovian reasons35.
34 Note: Additional results in the absence of aggregate independence are found by examining (46) and (47).
Examples: Average complementarity (X12 < 0, X21 < 0): the direct instrument te2 will be raised
(reduced), if there is large offender substitutability (complementarity). The indirect instrument te will be
raised (reduced) if large offenders are own-price nonresponsive.
35 Little systematic knowledge exists about demand responsiveness for polluting goods, let alone for
disaggregate groups. In practical discussions of the air pollution control program for Mexico City
(Eskeland, 1994), the responsiveness of demand for travel, including how it might vary by groups of
vehicles, was one of the issues on the agenda. Eskeland and Feyzioglu (1997b) estimated responsiveness in
demand for gasoline and vehicles in Mexico. Eskeland and Feyzioglu (1997) found very unfortunate
consequences of a scheme to ration trips in Mexico City: Households bought additional, used cars at
34
Our results ifor the indirect instrument, t,, are similar to those of Balcer, though his
results are sharper due to more restrictive assumptions36. For the direct instrument, t,, ,
we report howv the tax level compares to f2 E , hA hbi, whereas Balcer cornpares to the
tax level without any taxation of the associated good (i.e. t, =0). For this reason, his
results are not qualified by own-price responsiveness (x22 and Covf2,x22)).
Emission stanidards a7nd commodity taxes including presumptive Pigovian taxes
Equation (41} defines optimal commodity taxes including presumptive Pigovian taxes for
any given abatement levels. If we assume that the plainer can regulate abatement but
must do this uniformly for all consumers (though emiSsion functions differ), then first
order condition for optimal abatement are, for all i=l,..,n:
(4 8 ) -f Li +i ± fiXi + pi" ± ] (t[ J + tej )Xii ]o -
Commnodity taxes are optimal, so we may subtract (43), to obtain
(49) -2 I a E fgxg -,j xF = 0
n g g
This results in (J'gxg < 0 by assumption)
L 3hah Egi -xi
(50) =- =
Efibgxf Xifib+CoV(fib,Xi)
g
which simplifies to (26) E ,fiha /# = -l/fib if the covariance between consumption of
the polluting good aind the marginal cost of abatement is zero. (50) reflects a rather
intuitive relationship between the instrument at hand and the costs of emission
reductions. The planner has to ask high consumption individuals to abate proportionally
more than average consumption individuals (they must abate equally per unit). If the
covaiiance Covffib Xi) is negative (positive), so that high - consumption individuals
have low (high) marginal costs, then optimum is found in a point with - ceteris paribus -
unexpected rates to circumvent the regulation. Initially, the program rationing car use was seen as
politically attractive because of its distributional implications.
36 Balcer has zero incorme effects, so x12 = x21. His results on the indirect instrument are not qualified by
aggregate independence.
35
higher (lower) abatement standards and lower (higher) emissions than if the covariance
were zero.
Concluding, when presumptive taxes and standards are used, covariances with
demand patterns influence instruments, but only for reasons related to efficiency in
environmental protection. The demand patterns are important for efficiency because - for
both instruments - they determine the marginal costs of reducing emissions when, these
instruments are unable to equalize costs across consumers. The reason why distributional
considerations do not directly affect these instruments under nonuniform emission
functions (though they do for emission taxes) is simply that in this model neither
instrument can do anything towards redistribution that commodity taxes cannot.
V. Discussion
Our aim with this study was to examine whether intuition about 'pricing' the
environment applies in more general contexts than explored earlier. Does Sandmo's
'additivity property' (1975) apply in such a way that different polluting activities be
treated in the same fashion? If negative externalities can be reduced not only by changes
in consumption patterns, but also by making each activity cleaner (abatement efforts),
how shall optimal policy combine inducements to these various approaches? Finally, if
negative externalities are caused by agents as different as consumers, producers and
government, how does optimal policy combine efforts from these to reduce pollution?
Three assumptions are critical when we show that the marginal costs of emission
reductions, per unit of emissions, shall be the same across activities (goods, sectors), and
across polluters. The assumption of constant returns to scale is widely applied in the
literature, and is required in the present context since we want to see how established
results on production efficiency extend. Second, we assume that consumers have equal
access to pollution abatement opportunities (but also examine results of relaxing this
assumption). Third, we assume that the planner can differentiate his policy instruments
(emission taxes or abatement standards) by polluting good, and by whether the polluter is
a consumer, a producer or government, but he cannot differentiate such instruments - or
the commodity taxes - by personal characteristics, or make them non-linear in individual
emissions.
36
Comparing our results with Sandmo's results, they represent generalizations that
are very simple at a f^ormal level: One may replace the presumptive Pigovian part of his
commodity tax rates with an emission tax applied uniformly across agents and goods:
The emission. factors that are part of the expression for Sandmo's tax rates will now form
the base of an emission tax. Such a tax, combined wit: commodity taxes that satisfy the
fornula for optimal taxation in the traditional problenil without external effects, induces
optimum substitution towards less polluting activities as well as optimal abatement
everywhere.
The paper adds that the applicability of these principles is not limited to contexts in
which emissions are monitored at the source. Emission standards (or abatement
standards) may be implemented with more limited monitoring capabilities (car model
certifications, for instance), and a combination of emission standards and commodity
taxes that include presumptive Pigovian taxes can uncler the applied assumptions
implement th.e same allocation as the one implementable by commodity taxes and
emission taxes.
The results also extend the production efficiency result of Diamond arnd Mirrlees,
to include efficiency in environmental protection. For polluters that are producers and in
government, production efficiency (in a sense including equal mnarginal costs of emission
reductions) is to be expected. As an additional public good - the environment - is
included in the relevant input-output vector, the resullt that the optimal vector is at the
aggregate productioni frontier prevails.
When production efficiency applies also for polluting consumers - in the sense that
they too shall abate pollution at the same marginal rale of transformation as firms and
government -- it is rnore surprising. One might expect that the planner - in his desire to
redistribute or collect revenue at minimal distortionary costs -would choose to apply
different pressures to abate pollution in different activities in order to pursue these goals.
When consumners have equal access to abatement technology, however, emission taxes
differentiated according to polluting activity (i.e. goods) are redundant instruments for
redistributioni and revenue generation: they have the same dimensionality as commodity
taxes, and commodity taxes dominate since they do not induce additional resource costs
by making abatement deviate from efficient patterns.
37
When emission functions differ across producers, no deviation from Pigovian
principles result. The consequences when emissions functions are heterogeneous across
consumers are of two kinds, both related to covariances between emission factors and
consumption patterns and demand responses. First, when emission taxes are available, the
planner has a Pigovian instrument that is first best, and deviations from Pigovian
principles come if the differentiated pattern of emission taxes and abatement costs lend
themselves to his goals of redistribution and revenue generation. Second, when emission
taxes are not available but emission standards are used, non-unifornity of emission
functions influence policy because the instruments are no longer first best from a
Pigovian perspective. These resulting adjustments are related merely to the objective of
correcting externalities at least costs - not to revenue generation or redistribution. As an
example, if for a goodj marginal costs of emission reductions covary negatively with
consumed quantities, then this enhances the cost effectiveness of the emission standard,
relative to the case with no covariance.
Finally, simplicity in principles in this case also seems to simplify
implementation. Think about how to stimulate pollution reductions from those making
cars, roads, tires and fuels, and those using cars. First, it simplifies implementation that
the stimulus given to abatement at one stage (say at the factory) is independent of
whether the abatement yields i) emission reductions at that stage (the factory), ii)
emission reductions at some other stage (in the refinery, in the commute), or iii) enhanced
abatement opportunities at some other stage (the refinery, the commute). This allows
abatement efforts at all stages optimally to be stimulated by a uniform emission tax levied
where emissions occur. Second, it simplifies things that optimal abatement is independent
of whether the car is used by government, firms or households, for weddings or for work.
Finally, principles could be helpful also in simplifying the organization of
intervention for revenue and environmental protection, and perhaps in reducing the scope,
for wasteful political battles in environmental policy making. As an illustration, notice
that the emission tax that induces optimal abatement in its formula refers only to benefits
of environmental protection, not to price elasticities for polluting goods. Nevertheless, it
also induces optimal substitution towards less polluting goods, in the sense that this
emission tax should be combined with commodity tax rates satisfying the formula for
optimal taxation in the traditional problem without external effects. Thus, at a very
38
intuitive level, the environmental minister is conceme(d about pricing the environment -
and the finance minister may think about him as such. The revenues will contribute to the
general treasury aind thereby influence the shadow price of public revenue. T7hereby, the
environmental minister's agenda influences the optimal commodity tax prob lem of the
finance minister, but the finance minister need not think about the environmental costs or
opportunities in each activity. Similarly, the environmental minister need not think about
whether he - when taxing polluting sectors - tax sectors that are important for other
reasons, such as revenue or redistribution.
39
REFERENCES
Atkinson, Anthony B. and Joseph E. Stiglitz. 1976. "The Design of Tax Structure: Direct
Versus Indirect Taxation", Journal of Public Economics, Vol. 6, 55-75.
1980. "Lectures on Public Economics", McGraw-Hill Book
Company, New York.
Atkinson, Anthony B. and N.H. Stern. 1974. "Pigou, Taxation and Public Goods",
Review of Economic Studies, Vol. 41, No. 1, 119-128.
Auerbach, Alan, J. 1985. "The Theory of Excess Burden and Optimal Taxation", In
Handbook of Public Economics, edited by Auerbach and Martin Feldstein, Vol. 1,
Chap. 2, North-Holland, New York.
Balcer, Yves. 1980. "Taxation of Externalities: Direct Versus Indirect", Journal of Pubi'ic
Economics, Vol. 13, 121-129.
Baumol, William J., and Wallace E. Oates. 1988. "The Theory of Environmental Policy",
Cambridge University Press, Cambridge, UK, 2nd Edition.
Bergstrom, Theodore C. 1989. "A Fresh look at the Rotten Kid Theorem- and Other
Household Mysteries." Journal of Political Economy, Vol. 97, No. 5, 1138-59.
Bovenberg, A. Lans and Lawrence H. Goulder. 1996. "Optimal Environmental Taxation
in the Presence of Other Taxes: General-Equilibrium Analyses", American
Economic Review, Vol. 86, No. 4, 985-1000.
Bovenberg, A. Lans and F. van der Ploeg. 1994. "Environmental Policy, Public Finance
and the Labour Market in a Second-Best World"; Journal of Public Economics,
Vol. 55, No. 3, 349-390.
Buchanan, James M. and Gordon Tullock. 1975. "Polluters' Profits and Political
Response: Direct Controls versus Taxes", American Economic Review, Vol. 65,
No. 1, 139-147.
Corlett, W. J. and D. C. Hague. 1953. "Complementarity and the Excess Burden of
Taxation", The Review of Economic Studies, Vol. 21, 21-30.
Crandall, R.W., H. Gruenspecht, T. Keller and L. Lave. ] 986. "Regulating the
Automobile", The Brookings Institution, Washington, D.C.
Cremer, Helmuth, Firouz Gahvari and Norbert Ladoux. 1998. "Externalities and Optimal
Taxation", Journal of Public Economics, Vol. 70, 343-364.
40
Cremer, Helmuth and Firouz Gahvari. 1999. "What to Tcax: Emissions or Polluting
Goods?", Mimeographed, University of Toulouse, Toulouse, France.
Cropper, Maureen L. and Wallace E. Oates. 1992. "Environmental Economics: A
Survey", Journal of Economic Literature, Vol. 30, No. 2, 675-740.
Diamond, Peter A. and James A. Mirrlees. 1971. "Optimal Taxation and Public
Production 1: Production Efficiency", American Economic Review, Vol. 61, 8-
27.
Diamond, Peter A. 1973. "Consumption Externalities and Imperfect Corrective Pricing",
The Bell Journal of Economics and Management Science, Vol. 4, No. 2, 526-538.
_1]975. "A Many-Person Ramsey Tax Rule", Journal of Public
Economics, Vol. 4, 335-342.
Eskeland, Gunnar S. and Emmanuel Jimenez. 1992. "Policy Instruments for Pollution
Control in I)eveloping Countries", World Bank Research Observer, Vol. 7, No.
2, 145-169.
Eskeland, Crunnar S. 1994. "A Presumptive Pigovian Tax: Complementing Regulation
to Mimic ani Emissions Fee", The World Bank Economic Review, Vol. 8, No. 3,
373-394.
Eskeland, Gunnar S. and Shanta Devarajan. 1995. "Taxing Bads by Taxing Goods:
Pollution Control with Presumptive Charges", The World Bank, Washington,
D.C
Eskeland, Crunnar S. and Tarhan Feyzioglu. 1997. "Rationing Can Backfire: The 'Day
without a Car' in Mexico City", World Bank Economic Review, Vol. 11, No. 3,
383-408.
Eskeland, Gunnar S. and Tarhan Feyzioglu. 1997b. "Is Demand for Polluting Goods
Manageable? An Econometric Study of Car Ownership and Use in Mexico",
Journal of Development Economics, Vol. 53, No.2, 42 3-445.
Eskeland, Gunnar S. and Chingying Kong. 1998. "Protecting the Environment and the
Poor: A Public Goods Framework Applied to Indonesia", Policy Research
Working Paper Series, No. 1961. The World Bank, Washington, D.C.
Gillis, Malcolm. 1989. "Tax Reform in Developing Countries" (editor), Duke University
Press, Durham, N.C.
Goulder, Lawrence H., 1. Parry, R. Williams III and D. Burtraw. 1999. "The Cost-
Effectiveness of Alternative Instruments for Environmental Protection in a Second-
Best Setting", Journal of Public Economics, Vol. 72, No. 3, 329-360.
41
Green, Jerry and Eytan Sheshinski. 1976. "Direct Versus Indirect Remedies for
Externalities", Journal of Political Economy, Vol. 84, No. 4, 797-808.
Greenwald, Bruce C. and Joseph E. Stiglitz. 1986. "Externalities in Economies wit:h
Imperfect Information and Incomplete Markets", Quarterly Journal cf
Economics, Vol. 101, No. 2, 229-264.
Harrington, Winston. 1997. "Fuel Economy and Motor Vehicle Emissions", Journal of
Environmental Economics and Management, Vol. 33, 240-252.
Meade, J. E. 1952. "External Economies and Diseconomies in a Competitive Situatioin",
The Economic Journal, Vol. 62, No. 245, 54-67.
Mirrlees, J. A. 1971. "An Exploration in the Theory of Optimum Income Taxatior",
Review of Economic Studies, Vol. 38, No. 2, 175-208.
Munk, Knud Joergen. 1978. "Optimal Taxation and Pure Profit", Scandinavian Journal
of Economics", Vol. 80, No. 1, 1- 19.
Musgrave, Richard A. and Peggy B. Musgrave. 1984. "Public Finance in Theory ald
Practice", McGraw-Hill, New York, 4th edition.
Nelson, R. A., T. Tietenberg and M. R. Donihue. 1993. "Differential Environmenlal
Regulation: Effects on Electric Utility Capital Turnover and Emissions", The
Review of Economics and Statistics, Vol. 75, No. 2, 368-373.
Newbery, David and Nicholas Stern. 1987. "The Theory of Taxation for Developing
Countries" (editors), Oxford University Press, New York.
Pigou, Arthur Cecil. 1947. "A Study in Public Finance", MacMillan & Co. LTD, London.
3rd edition, 1949 reprint.
Ramsey, F. P. 1927. "A Contribution to the Theory of Taxation", The Economic
Journal, Vol. 37, No. 147, 47-61.
Samuelson, Paul A. 1951. "Theory of Optimal Taxation", Memorandum to the U.S.
Treasury. Reprinted in Journal of Public Economics, 1986, Vol. 30, 137-143.
1954. "The Pure Theory of Public Expenditure", Review of
Economics and Statistics, Vol. 36, 387-389.
Sandmo, Agnar. 1975. "Optimal Taxation in the Presence of Externalities", Swedilsh
Journal of Economics, Vol. 77, 86-98 (the journal is now published under tile
name Scandinavian Journal of Economics).
1976. "Optimal Taxation - An Introduction to the Literature",
Journal of Public Economics, Vol. 6, 37-54.
42
_1976b. "Direct Versus Indirect Pigovian Taxation", European
Economic Review, Vol. 7, 337-349.
Schmutzler, Armin and Lawrence H. Goulder. 1997. "The Choice between Emission
Taxes and Output Taxes under Imperfect Monitoring", Journal of Environmental
Economics and Management, Vol. 32, 51-64.
Tietenberg, Tom. 1]992. "Environmental and Nattural Resource Economics", Harper
Collins Publishers, New York, 3rd edition.
Weitzman, M. L. 1974. "Prices vs. Quantities", Review of Economic Stucdies, Vol. 41,
477-491.
Wijkander, Hans. 1985. "Correcting Externalities through Taxes on Subsidies to Related
Goods", Journal of Public Economics, Vol. 28, 111-125.
Wittman, Donald. [989. "Why Democracies Produce Efficient Results.", Journal of
Political EcConomy, Vol. 97, No. 6, 1395-1424.
43
Policy Research Working Paper Series
Contact
Title Author Date for paper
WPS2302 Why Liberalization Alone Has Not Klaus Deininger March 2000 M. Fernandez
Improved Agricultural Productivity Pedro Olinto 33766
in Zambia: The Role of Asset
Ownership and Working Capital
Constraints
WPS2303 Malaria and Growth F. Desmond McCarthy March 2000 H. Sladovich
Holger Wolf 37698
Yi Wu
WPS23(04 Disinflation ano the Supply Side Pierre-Richard Agenor March 2000 T. Loftus
Lodovico Pizzati 36317
WPS23(05 The Impact of Banking Crises on Maria Soledad MAartinez March 2000 A. Yaptemco
Money Demand and Price Stability Peria 31823
WPS2306 International Contagion: Implications Roberto Chang March 2000 E. Mekhova
for Policy Giovanni Majnoni 85984
WPS2307 Surveying Surveys and Questioning Francesca Racanatini March 2000 P. Sintim-Aboagye
Questions: Learning from World Bank Scott J. Wallsten 37644
Experience! Lixin Colin Xu
WPS2308 How Small Should an Economy's Paul Beckermai March 2000 H. Vargas
Fiscal Deficit Be? A Monetary 38546
Programming Approach
WPS2309 What Drives Private Saving around Norman Loayza. March 2000 E. Khine
the World? Klaus Schmidt-Hebbel 37471
Luis Serven
'VVPS231 0 How Politics and Institutions Affect Mitchell A. Orenstein March 2000 M. Leenaerts
Pension Reform in Three 84264
PostcDmmunist Countries
WPS2311 The Buenos Aires Water Concession Lorena Alcazar April 2000 P. Sintim-Aboagye
Manuel A. Abdala 38526
Mary M. Shirley
WPS2312 Measuring Governance, Corruption, Joel S. Hellman April 2000 D. Bouvet
and State Capture: How Firms and Geraint Jones 35818
Bureaucrats Shape the Business Daniel Kaufmarin
Environment in Transition Economies Mark Schanker-nan
WPS2313 Hovv Interest Rates Changed under Patrick Honohan April 2000 A. Yaptenco
Financial Liberalization: A Cross- 31823
Country Review
Policy Research Working Paper Series
Contact
Title Author Date for paper
WPS2314 Technological Leadership and Beata K. Smarzynska April 2000 L.Tabada
Foreign Investors' Choice of 36896
Entry Mode
WPS2315 Investment in Natural Gas Pipelines Alejandro Jadresic April 2000 M. Salehi
in the Southern Cone of Latin America 37157
WPS2316 Distrubutional Outcomes of a Emanuela Galasso April 2000 P. Sader
Decentralized Welfare Program Martin Ravallion 33902
WPS2317 Trade Negotiations in the Presence of Keiko Kubota April 2000 L. Tabada
Network Externalities 36896
WPS2318 Regulatory Reform, Competition, Mark A. Dutz April 2000 H. Sladovich
and Innovation: A Case Study of the Aydin Hayri 37698
Mexican Road Freign Industry Pablo Ibarra