We modify the vertically differentiated duopoly model by André et al. (2009) replacing Bertrand with Cournot behaviour to show that firms may spontaneously adopt a green technology even in the complete absence of any form of regulation.

A. Tampieri

L. Lambertini

English

Research Papers in Economics

 
 
 
 
   
 
 
 
   
Vertical Differentiation in a Cournot 
Industry: The Porter Hypothesis and Beyond 
 
   
Luca Lambertini 
Alessandro Tampieri 
 
 
 
Quaderni  - Working Paper DSE N° 730   
   
   
 Vertical Diﬀerentiation in a Cournot
Industry: The Porter Hypothesis and
Beyond∗
Luca Lambertini§,# and Alessandro Tampieri§
§ Department of Economics, University of Bologna
Strada Maggiore 45, 40125 Bologna, Italy
luca.lambertini@unibo.it; alessandro.tampieri@unibo.it
# ENCORE, University of Amsterdam
Roetersstraat 11, WB1018 Amsterdam, The Netherlands
February 11, 2011
Abstract
We modify the vertically diﬀerentiated duopoly model by Andr´ e
et al. (2009) replacing Bertrand with Cournot behaviour to show
that ﬁrms may spontaneously adopt a green technology even in the
complete absence of any form of regulation.
JEL codes: L13, L51, Q55, Q58
Keywords: environmental quality, quantity competition, green
technology
∗We would like to thank HERA Spa for sponsoring this project. The usual disclaimer
applies.
11I n t r o d u c t i o n
The Porter Hypothesis (Porter, 1990, 1991) challenges the view according to
which any form of environmental regulation necessarily hinders ﬁrms’ proﬁt
perspectives. Indeed, according to Porter, environmental regulation may
induce eﬃciency and encourage innovations.
Several contributions nested in the debate triggered by this claim have
brought to the fore a number of scenarios where the Porter Hypothesis may
apply (see Porter and van der Linde, 1995; Simpson and Bradford, 1996;
Xepapadeas and de Zeeuw, 1999; and Hart, 2004). In particular, Andr´ e et
al. (2009) support the Porter Hypothesis by arguing that, in a vertically
diﬀerentiated duopoly with Bertrand competition, where ﬁrms can decide
whether to invest in green R&D or not, it is possible to drive the industry
to an equilibrium where both ﬁrms choose to adopt the green standard in
presence of a streamlined regulation taking the form of a lump-sum tax levied
on the brown technology.
Here, we extend the analysis carried out by Andr´ e et al. (2009) assuming
Cournot competition between ﬁrms at the market stage. The issue of why
and when ﬁrms choose to be quantity- or price-setting agents has been lively
discussed in the theory of industrial organization, pointing out that price
(respectively, quantity) competition ﬁts best industries where ﬁrms’ capacity
constraints are absent or loose (resp., tight).1 This appears to be the case,
e.g., in the car industry, where a ﬁrm’s plant size matters. Additionally, this is
a prototypical market where the environmental implications of production are
relevant, and the existence of appropriate incentives for investments aimed
at the attainment of green product qualities is a hot issue. In fact, casual
1As is well known, the ﬁrst analysis of capacity-constrained price competition is in
Edgeworth (1897). In the modern literature, see Beckman (1967); Levitan and Shubik
(1972); Kreps and Scheinkman (1983); Singh and Vives (1984); Osborne and Pitchik
(1986), to mention only a few.
2observation suggests that several large ﬁrms like Honda, Toyota and VW-
Audi are investing (or are about to invest) massive resources in R&D for
clean technologies, without being subject to a speciﬁc form of taxation on
any brown technology currently in use, which are instead regulated through
minimum quality standards (like Euro V). Hence, one is led to think that
these eﬀorts must be driven by proﬁt incentives largely - if not altogether -
independent of the regulatory framework.
This is precisely the bottom line of our analysis. By replacing Bertrand
with Cournot, the resulting softer competition at the market stage delivers
the appropriate incentives for ﬁrms to standardise the industry onto the green
product, in absence of any regulation whatsoever. This opens a perspective
that stretches even beyond the Porter Hypothesis, prompting for the inves-
tigation of scenarios where ﬁrms governed by unconstrained proﬁti n c e n t i v e s
may pursue projects ultimately leading to socially eﬃcient outcomes.
2T h e m o d e l
We adopt largely the same setup as in Andr´ e et al. (2009). We describe a
vertically diﬀerentiated duopoly under complete information, where single-
product ﬁrms, 1 and 2, decide whether to supply a high- or low-quality good,
identiﬁed by qH >q L. Quality has both hedonic and environmental features;
in particular, while qL involves some undesirable environmental externality,
i.e., it is brown, while qH is green. the individual ﬁrm’s proﬁt function is
πij =( pij − cxij)xij,i , j = H,L, (1)
where xij indicates the output level of ﬁrm i (hence, in the remainder the ﬁrst
subscript will consistently refer to ﬁrm i while the second will refer to the
rival). As it appears from (1), the production technology involves variable
3costs, which are quadratic in the output level:
Cij = cx
2
ij,c > 0. (2)
For the moment, we abstract from any ﬁxed costs related to installed capacity
or the R&D eﬀort required for the attainment of either quality level. We
will come back to this aspect of technology in the remainder. Similarly,
we suppose that ﬁrms are not subject to any regulatory measure, unlike
what appears in Andr´ e et al. (2009), where ﬁrms pay a lump-sum tax when
supplying the brown variety.
Using (1) we may describe three possible cases: either both ﬁrms sup-
ply the green (resp., brown) standard, so that consumers indeed purchase a
homogeneous good, or they opt for diﬀerent quality levels.
T h ed e m a n ds i d ei sa si nM u s s aa n dR o s e n( 1 9 7 8 )a n dG a b s z e w i c za n d
Thisse (1979): there exists a continuum of consumers indexed by their marginal
willingness to pay for quality θ ∈ [θ,θ], with θ = θ − 1. The distribution of
c o n s u m e r si su n i f o r m ,w i t hd e n s i t yf(θ) = 1, so that the total mass of con-
sumers is also equal to one. Each consumer buys at most one unit of variety
i = H,L, whereby his net utility is either U = θqi − pij ≥ 0, or nil if he
doesn’t buy. As Andr´ e et al. (2009), we focus on the case where the mar-
ket is partially covered. The construnction of the demand system is best
approached by considering ﬁrst the case where ﬁrms supply diﬀerent quality
levels. If so, there exists a consumer who is indiﬀerent between buying the
low-quality good and not buying at all. His location along the spectrum of
the marginal willingness to pay is given by the ratio e θ = pLH/qL.G i v e n
generic prices and qualities, the location of the consumer indiﬀerent between
the two varieties is b θ =( pHL − pLH)/(qH − qL), so that the system of direct
market demands is xHL = θ −b θ and xLH = b θ −e θ. This is what one needs to
4use in order to model Bertrand behaviour, while inverse demands
pHL =
¡
θ − xHL
¢
qH − qLxLH
pLH =
¡
θ − xHL − xLH
¢
qL
(3)
are to be used under Cournot competition. If instead both ﬁrms adopt the
same quality standard (either green or brown), demands (3) symmetrically
collapse onto:
pii =
¡
θ − 2xii
¢
qi,i = H,L. (4)
The interaction between ﬁrms is fully noncooperative and takes place in
two stages. In the ﬁrst, ﬁrms set their respective quality levels; then, in
the second, which is the proper market stage, they compete in outputs ` al a
Cournot. The solution concept applied is the subgame perfect equilibrium
by backward induction.
3T h e g a m e
To begin with, we have to solve the three possible Cournot subgames. The
fastest way to do it is to focus on the case in which ﬁrms supply diﬀerent
standards. The ﬁrst order conditions for proﬁt maximization are:
∂πHL
∂xHL
=
¡
θ − 2xHL
¢
qH − 2cxHL − qLxLH =0
∂πLH
∂xLH
=
¡
θ − xHL − 2xLH
¢
qL − 2cxLH =0
(5)
yielding:
x∗
HL =
θ[2cqH +( 2 qH − qL)qL]
4c(qH + qL + c)+( 4 qH − qL)qL
x∗
LH =
θ(2c + qH)qL
4c(qH + qL + c)+( 4 qH − qL)qL
(6)
5The resulting Cournot-Nash proﬁts are:
π∗
HL =
θ
2
[2cqH +( 2 qH − qL)qL]
2 (c + qH)
[4c(qH + qL + c)+( 4 qH − qL)qL]
2
π∗
LH =
θ
2
(2c + qH)
2 q2
L (c + qL)
[4c(qH + qL + c)+( 4 qH − qL)qL]
2
(7)
The alternative cases where ﬁrms symmetrically choose the same standard
(either brown or green) obtain from (6-7) by imposing qi = qj.2 This yields:
xii =
θqi
3qi +2 c
,i = H,L,
π∗
ii =
θ
2
q2
i (qi +2 c)
2 (c + qi)
[3q2
i +4 c(2qi + c)]
2
(8)
Before proceeding any further, it is worth oserving that
∂xii
∂qi
=
2θc
(3qi +2 c)
2 > 0( 9 )
entailing that total industry output is strictly higher when both ﬁrms adopt
the green standard than when they adopt the brown one. The equilibrium
price in the symmetric cases writes as follows:
pii =
θqi (2c + qi)
3qi +2 c
,i = H,L, (10)
with
∂pii
∂qi
=
θ[4c2 + qi (4c +3 qi)]
(3qi +2 c)
2 > 0( 1 1 )
so that switching from brown to green qualities brings about an increase in
price. Therefore, ap r i o r i , if the whole industry increases the quality standard
2Under quantity competition, it is not necessary to build up the demand functions for
the homogeneous good case as it is instead under Bertrand competition (see Andr´ e et al.,
2009, p. 185)
6symmetrically, the net eﬀect on consumer surplus is ambiguous, depending
upon the balance between the increase in quality and output on one side and
the increase in price on the other.
In the general case where ﬁrms supply diﬀerentiated products, consumer
surplus is:
CSij =
Z b θ
e θ
(vqL − pLH)dv +
Z θ
b θ
(vqH − pHL)dv
=
qLx2
LH + xHL(qHxHL +2 qLxLH)
2
(12)
which, in the symmetric cases, becomes:
CSii =
2θ
2
q3
i
(3qi +2 c)
2 (13)
with
∂CSii
∂qi
=
6θ
2
q2
i (qi +2 c)
(3qi +2 c)
3 > 0( 1 4 )
On this basis, we can formulate:
Lemma 1 The symmetric switch from brown to green quality involves an
increase in consumer surplus.
This of course is driven by the fact that the balance between the increase
in all variables involved is such that consumers are happy to purchase larger
amounts of the green variety even if this happens at a higher price.
We are now in a position to investigate the symultaneous choice of the
quality level, either H or L, by the two ﬁrms. As in Andr´ e et al. (2009, p.
187), this is described by the reduced-form stage appearing in Matrix 1.
2
HL
1 H π∗
HH;π∗
HH π∗
HL;π∗
LH
L π∗
LH;π∗
HL π∗
LL;π∗
LL
Matrix 1
7The equilibrium analysis can be carried out by assessing the sign of three
key inequalities involving the payoﬀs apeparing in Matrix 1. To begin with,
consider the main diagonal, along which we have:
π
∗
HH − π
∗
LL ∝ 4c(qH + c)[cqH +( c +3 qH)qL]+( 2 c +3 qH)
2 q
2
L > 0. (15)
This establishes the superiority of the green standard in terms of the reuslting
proﬁti n c e n t i v e sf o rﬁrms, the explanation being that the higher production
costs associated with supplying the green standard rather than the brown one
are more than compensated by the corresponding increase in market price
generated by consumers’ hedonic inclinations.
Then, we move on to consider the inequalities by rows or columns:
π
∗
HH − π
∗
LH ∝ 16(qH + c)
3 (qL + c) − q
2
Hq
2
L > 0( 1 6 )
as q3
HqL >q 2
Hq2
L;m o r e o v e r ,
π
∗
HL − π
∗
LL ∝ [36qH + qL (qL − 16)] + 4c
£
3(10qH − qL)q
3
L+( 1 7 )
c(4c(c(qH + qL)+qL (5qH +3 qL))) + (37qH +9 qL)q
2
L
¤
> 0
since 36qH − 16qL > 0.
Therefore, on the basis of the foregoing analysis, we can formulate the
following claim:
Proposition 2 Pure proﬁt incentives yield (H,H) as the unique Nash equi-
librium at the intersection of strictly dominant strategies, in absence of any
form of regulation. The equilibrium outcome is also Pareto-eﬃcient for ﬁrms.
That is, the stage game depicted in Matrix 1 is not a prisoners’ dilemma.
T h es o u r c eo ft h es p o n taneous alignement of ﬁrms on the green quality in an
unregulated market is to be found in the less aggressive nature of Cournot
behaviour as opposed to Bertrand. This ultimately makes the lump-sum tax
appearing in Andr´ e et al. (2009) redundant.
8Combining Proposition 2 with the additional facts that (i) consumer sur-
plus is higher in (H,H)t h a ni n( L,L) - as we know from Lemma 1 - and (ii)
the negative environmental externality curtailing welfare in (L,L)d i s a p p e a r s
altogether in (H,H), it is also true that
Corollary 3 Social welfare is higher in (H,H) than in (L,L).
The foregoing analysis can be generalised as follows. The result outlined
in Proposition 2 clearly extends to the case where there exists a symmetric
ﬁxed cost F, and may indeed hold even in the more general case where
developing either quality standard requires some R&D eﬀort summarised in
a quality-speciﬁc ﬁxed cost FK,K= H,L, with FH >F L. This can be
easily appreciated by observing tha tt h ep r e s e n c eo fq u a l i t y - s p e c i ﬁc ﬁxed
costs modiﬁes the equilibrium values of objective functions as follows:
b π
∗
ij = π
∗
ij − F, (18)
whereby Proposition 2 applies provided that the inequalities
b π
∗
HH > b π
∗
LL ⇔ π∗
HH − π∗
LL >F H − FL
b π
∗
HH > b π
∗
LH ⇔ π∗
HH − π∗
LH >F H − FL
b π
∗
HL > b π
∗
LL ⇔ π∗
HL − π∗
LL >F H − FL
(19)
holds across the three admissible cases simultaneously. This can be sum-
marised in:
Corollary 4 If min{π∗
HH − π∗
LL,π∗
HH − π∗
LH,π ∗
HL − π∗
LL} >F H − FL, then
(H,H) is the unique Nash equilibrium in strictly dominant strategies of the
g a m e ,i np r e s e n c eo fa s y m m e t r i cﬁxed costs of quality improvement.
A straightforward addendum to the above Corollary is that, whenever
(H,H) is the equilibrium, then it remains also socially optimal.
94 Concluding remarks
We have revisited the vertically diﬀerentiated duopoly model by Andr´ e et al.
(2009) under quantity-setting behaviour, to highlight the striking fact that
softening market competition may have far-reaching consequences as far as
the environmental performance of the industry is concerned. Indeed, our
simple model is something more than a vindication of the Porter Hypothesis,
as it shows that private and social incentives towards the adoption of green
technologies may in fact be reciprocally aligned.
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11 

America’s green strategy. ScientiﬁcA m e r i c a n264: 96.

Edgeworth-Bertrand duopoly revisited,

Environmental policy and competitiveness: The Porter hypothesis and the composition of capital.

Growth, environment and innovation - a model with production vintages and environmentally oriented research,

La teoria pura del monopolio. Giornale degli Economisti 40: 13-31. Reprinted in English as: The pure theory of monopoly,

Monopoly and product quality.

Price and quantity competition in a diﬀerentiated duopoly.

Price competition in a capacityconstrained duopoly.

Price competition, quality, and income disparities.

Price duopoly and capacity constraints.

Strategic quality competition and the Porter hypothesis.

Taxing variable cost: Environmental regulation as industrial policy.

The Competitive Advantage of Nations,

Toward a new conception of the environment-competitiveness relationship.

Vertical Differentiation in a Cournot Industry: The Porter Hypothesis and Beyond

Vertical Differentiation in a Cournot Industry: The Porter Hypothesis and Beyond

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