Les biens et services de dépollution sont actuellement le plus souvent livrés par une éco-industrie. Cette note reconsidère la taxe pigouvienne dans ce contexte. On montre qu'une taxe optimale sur les émissions polluantes se démarquera du coût social marginal de la pollution en fonction des pouvoirs de marché relatifs des pollueurs et des firmes environnementales.

Alain-Désiré Nimubona

Bernard Sinclair-Desgagné

English

Research Papers in Economics

 
 
The Pigouvian Tax Rule in the Presence of an Eco-Industry 
 
 
 
Alain-Désiré Nimubona 
Bernard Sinclair-Desgagné 
 
 
 
Décember 2005 
 
 
Cahier n° 2005-032 
 
 
 
ECOLE POLYTECHNIQUE 
CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE 
 
 
LABORATOIRE D'ECONOMETRIE 
1rue Descartes F-75005 Paris 
(33) 1 55558215 
 http://ceco.polytechnique.fr/  
mailto:labecox@poly.polytechnique.fr 
 
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The Pigouvian Tax Rule in the Presence of an Eco-Industry
* 
 
 
 
Alain-Désiré Nimubona
1 
Bernard Sinclair-Desgagné
2 
 
 
 
Décember 2005 
 
 
Cahier n° 2005-032 
 
Résumé:  Les biens et services de dépollution sont actuellement le plus souvent livrés 
par une éco-industrie. Cette note reconsidère la taxe pigouvienne dans ce 
contexte. On montre qu'une taxe optimale sur les émissions polluantes se 
démarquera du coût social marginal de la pollution en fonction des pouvoirs 
de marché relatifs des pollueurs et des firmes environnementales.   
 
Abstract:  Pollution abatement goods and services are now largely being delivered by a 
specialized "eco-industry.'' This note reconsiders Pigouvian taxes in this 
context. We show that the optimal emission tax must depart from the marginal 
social cost of pollution according to the polluters' and the environment firms' 
relative market power. 
 
Mots clés :  Taxe pigouvienne, Industrie de l'environnement 
 
Key Words :  Pigouvian taxes, Environment industry 
 
Classification JEL:  H23, L13 
 
 
                                                           
* We are grateful to Hassan Benchekroun, Robert Cairns, Maia David, Andrew Leach, and seminar audiences at 
HEC Montréal, McGill University and the 2005 EAERE Conference in Bremen for helpful comments. This 
research currently benefits from the financial support of the French Ministry of Research through the Concerted 
Action (ACI) grant ''Economic Modeling of Sustainable Development". 
1 Institute of Applied Economics, HEC Montréal, Canada H3T 2A7. 
2 Laboratoire d'Econométrie, CNRS et École polytechnique, Paris; CIRANO, CIRAIG, and HEC Montréal, 
Canada H3T 2A7. 
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81. Introduction
In his classical analysis of market failure, Arthur Pigou (1920) showed that the negative
externalities caused by pollution would be internalized by the market if polluters paid a
tax equal to the marginal social cost of polluting emissions. This proposition, derived
under the assumption of perfect competition, was later amended by Buchanan (1969)
and Barnett (1980): when the polluting industry is imperfectly competitive, an emission
tax should be set lower than the marginal social cost of pollution, because it trades oﬀ
the desire to provide incentives for abatement and the necessity to prevent a greater
contraction of output. Several authors (see, for example, Katsoulacos and Xepapadeas,
1995; Long and Soubeyran, 1999; Morgenstern, 1995; and Smith, 1992) have now explored,
qualiﬁed and reﬁned the latter conclusion under more speciﬁc industry structures.
All these studies, however, (in fact, all environmental economics so far) postulate that
a polluting ﬁrm possesses its own internal abatement technology. But nowadays, abate-
ment goods and services are largely procured from specialized environment ﬁrms, and
this so-called “eco-industry” looks rather concentrated (see Barton, 1997; Davies, 2002;
the European Commission, 1999; Karliner, 1999; the OECD, 1996; and the World Trade
Organization, 1998). In a recent paper that ﬁrst acknowledges this situation, David and
Sinclair-Desgagné (2005) amended Pigou’s basic framework and found that an optimal
emission tax should now be set higher than the marginal social cost of pollution. In-
tuitively, imperfect competition between environment ﬁrms results in abatement prices
larger than the marginal cost of abatement; emission taxes must then be raised in order
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8to make polluters reduce their emissions suﬃciently. This note examines whether and to
what extent a similar conclusion holds in the Buchanan-Barnett context. The upshot is
that the optimal emission tax needs in general to adjust to the polluters’ and the abate-
ment suppliers’ relative market power. It may happen, for instance, that the distortions
respectively present on the product and the abatement markets oﬀset each other to the
p o i n tt h a tt h eo p t i m a le m i s s i o nt a xt u r n so u tt ob et h eo n ep r e s c r i b e di n i t i a l l yb yP i g o u .
The paper unfolds as follows. Section 2 presents the model. Section 3 derives and
discusses our results. Section 4 contains concluding remarks.
2. The model
First consider n identical ﬁrms competing àl aCournot. Each ﬁrm i produces a
quantity xi of a ﬁnal good at a cost C(xi), where the latter function is twice diﬀerentiable,
strictly increasing and convex. Consumers’ preferences for this ﬁnal good are captured
by the inverse demand function p(X), X =
Pn
i=1 xi , p0(X) < 0.E a c hﬁrm also generates
polluting emissions according to an emission function e(xi,a i),w h e r eai represents ﬁrm
i’s abatement eﬀort. We assume that e(xi,a i)=w(xi) − ai , w0(xi) > 0 and w00(xi) ≥ 0.1
Now, let the abatement goods and services be supplied by a Cournot oligopoly com-
prising m similar environment ﬁrms. Firm j’s cost of delivering an amount aj of such
1Our qualitative results remain valid without assuming an additively separable emission function. This
assumption greatly simpliﬁes the upcoming computations, however. It has been used in several previous
works (Barnett, 1980; Katsoulacos and Xepapadeas, 1995; Farzin and Kort, 2001; David and Sinclair-
Desgagné, 2005). It also ﬁts end-of-pipe abatement activities, such as solid waste management, waste
water treatment, air pollution control, contaminated soil and groundwater remediation, which currently
draw (by far) the largest share of abatement expenses (European Commission, 1999).
3
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8g o o d sa n ds e r v i c e si sg i v e nb yG(aj),w i t hG0 and G00 strictly positive. When the mar-
ket for abatement is characterized by the inverse demand function q(A), A =
Pm
j=1 aj,
q0(A) < 0, each environment ﬁrm oﬀers a quantity aj that maximizes the proﬁtf u n c t i o n
Πj(aj)=q(A)aj − G(aj) .
Satisfying simultaneously the (necessary and suﬃcient) ﬁrst-order conditions
q(A)=G
0(aj) − q
0(A)aj , j =1 ,...,m (1)
then yields a Cournot-Nash equilibrium for the eco-industry.
From now on, we shall focus on symmetric equilibria, so a representative polluter i’s
production and abatement are respectively given by x = X
n and ai = A
n, and a represen-
tative environment ﬁrm j’s delivered quantity of abatement is aj = A
m.
3. Optimal emission taxes
Suppose that each unit of pollution bears a positive social cost ν.Ab e n e v o l e n ta n d
informed regulator may now want to impose a tax t on polluting emissions. Ignoring
redistribution and income transfer issues, this tax would maximize the social welfare
objective
W(t)=
Z Xt
0
p(z)dz − nC(x
t) − mG(a
t
j) − nve(x
t,a
t
i),
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8where by symmetry xt = Xt
n , at
i = At
n ,a n dat
j = At
m. After we drop the superscript t
(remembering throughout that abatement and output levels depend on the tax t), the
necessary and suﬃcient ﬁrst-order condition for an optimal emission tax is given by
W
0(t)=n[p(X) − C
0(x)]
dx
dt
− nG
0
³nai
m
´ dai
dt
− nυ
·
w
0(x)
dx
dt
−
dai
dt
¸
=0 .( 2 )
A representative polluter will react to this tax by maximizing the proﬁtf u n c t i o n
Πi(x,ai)=p(X)x − C(x) − q(A)ai − te(x,ai) .
This is achieved if the following ﬁrst-order (necessary and suﬃcient) conditions hold:
p(X) − C
0(x)=−p
0(X)x + tw
0(x),( 3 )
q(A)=−q
0(A)ai + t.( 4 )
Standard comparative statics on equations (3) and (4) conﬁrms that dx
dt and
dai
dt are respec-
tively negative and positive, meaning that a larger tax on polluting emissions generates
less production of the ﬁnal good and greater abatement eﬀorts.
Substituting (1), (3) and (4) into expression (2) now yields the general formula for the
Pigouvian tax rule:
t = υ +
X
np0(X)dx
dt +
¡
A
m − A
n
¢
q0(A)
dai
dt
w0(x)dx
dt −
dai
dt
(5)
N o t et h a tt h ed e n o m i n a t o ro nt h er i g h t - h a n ds i d ei sn e g a t i v e .
5
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8Were the eco-industry perfectly competitive, which amounts here to letting m →∞ ,
the regulator should set t = υ if n also tends to inﬁnity (as Pigou (1920) ﬁrst proposed)
or t<υ if n is ﬁnite (in agreement with Buchanan (1969) and Barnett (1980), but with
an additional downward adjustment due to the term −A
nq0(A)
dai
dt which expresses the
polluting ﬁrms’ oligopsony in the abatement market).
Let now m<∞, so the eco-industry is imperfectly competitive. By (1), the price
of abatement in this case is equal to the marginal cost G0 ¡
A
m
¢
plus am a r k u p A
mq0(A).
This markup further dissuades polluters from investing in abatement. To counter this,
the regulator has to play tougher on polluting emissions.2 Indeed, when n is inﬁnite
(so polluters are price-takers), formula (5) coincides with the one obtained by David
and Sinclair-Desgagné (2005) and we have that t>v .W i t h n also ﬁnite, however, the
regulator must deal with the exercise of market power by polluting ﬁrms in the ﬁnal
good market, which lowers consumer surplus, together with the simultaneous presence of
an oligopsony and an oligopoly in the abatement market, which both hinder pollution
reduction. After some algebra, the numerator on the right-hand-side of (5) indicates that
>>
t = υ if and only if
µ
n − m
n
¶µ
q(A)
dai
dt
¶
LECO =
µ
−p(X)
dx
dt
¶
LPROD (6)
<<
2In practice, environmental policy has to also cope with the possibility of exit from the polluting
sector and entry in the eco-industry, and take into account the redistribution of revenues from polluting
to environment ﬁrms. A careful treatment of these issues is obviously beyond the scope of this note.
6
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8where LECO and LPROD are the Lerner indices associated with the eco-industry and
the ﬁnal good market respectively.3 As the environment ﬁrms’ market power matters
relatively more (less), so n−m
n or
LECO
LPROD increases (decreases), the regulator would now
impose a higher (lower) tax on emissions. This tax will approximate Pigou (1920)’s
recommendation when the upstream and downstream industry structures are such that
LPROD
LECO
≈
m − n
n
q(A)
p(X)
dai
dt
dx
dt
.( 7 )
4. Concluding remarks
This paper showed that taxes targetting polluting emissions must adjust to the relative
market power of environment ﬁrms (on the abatement market) and polluters (on both the
ﬁnal good and the abatement markets). All things equal, a relatively more concentrated
and powerful eco-industry warrants higher emission taxes.
The impact of more complex and realistic industry structures - with endogenous entry
and exit, or privately informed and heterogeneous environment ﬁrms, for example - on
Pigouvian taxes and environmental regulation in general remains to be explored. Pursuing
this path, henceforth investigating the vertical relationships and actual division of labor
between polluting ﬁrms and their abatement suppliers, will certainly shed light on the
3Deﬁned as the diﬀerence between the ﬁrm’s price and its marginal cost, divided by the ﬁrm’s price, the
Lerner index is a well-known measure of a ﬁrm’s market power. Here we have that LECO =
q(A)−G(aj)
q(A) =
1
m|η| and LPROD =
p(X)−C
0(x)
p(X) = 1
n|ε|,w h e r eε and η are the price-elasticities of demand for the ﬁnal
good and for the abatement goods and services respectively.
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8current black box of abatement costs, for the greater eﬀectiveness of environmental policy.
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The Pigouvian tax rule under monopoly.

Waste management multinationals

The Pigouvian Tax Rule in the Presence of an Eco-Industry

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The Pigouvian Tax Rule in the Presence of an Eco-Industry

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