Chukyo University Institute of Economics
Discussion Paper Series
July 2021

No. 2102
Environmental policies in a differentiated oligopoly: a note

Akira Yakita and Donglin Zhang

Jan14,2021: CORRECTION25Jan21:Added16Feb2021

Environmental policies in a differentiated oligopoly: a note
Akira Yakitaa* and Donglin Zhangba Faculty of Economics, Nanzan University
b Graduate School of Social Sciences, Nanzan University

Abstract
This paper shows that, irrespective of whether free entry and exit are allowed or not,there might be a range of moderate degree of pollution emission per unit of goods within
which the optimal environment tax rate is higher than the marginal environmental
damage, i.e., the optimal tax rate is higher than the “Pigouvian tax rate,” when thepollution function of firms and the social environmental damage function differ in their
responses to a pollutant per unit of goods. However, outside the range of pollution
emission, the tax rate is lower than the marginal environmental damage.
Keywords: optimal environmental policy, oligopoly, pollution emission function, productdifferentiation
JEL Classification: H21, H23, L13, Q58
Declaration of Conflicting Interests
The authors declare that they have no potential conflict of interest with respect to theresearch, authorship, or publication of this article.
Data Availability Statement
Data sharing is inapplicable to this paper because no new data were analyzed.
Acknowledgments
We thank Makoto Hirazawa, Tohru Naito, and the seminar participants in the Nagoya
Macroeconomics Workshop for their insightful comments and suggestions.

* Correspondence: A. Yakita, Faculty of Economics, Nanzan University, 18 Yamasato-cho,Showa-ku, Nagoya 466-8673, Japan
E-mail: yakita@nanzan-u.ac.jp
11. Introduction
This paper presents examination of the optimal environmental policy in oligopolistic
markets with differentiated goods. The optimal pollution tax in a competitive industry
is well known to be equal to the marginal damage inflicted by the pollution (Pigou 1932).
It has also been shown that the optimal pollution tax on a monopoly is less than themarginal damage (Buchanan 1969; Barnett 1980). Since the publishing of these works,
many authors have investigated the optimal environmental policy in imperfectlycompetitive markets. 1 In such market structures, the economy has two inefficiencies:
underproduction caused by imperfect competition of oligopoly and externalities causedby environmental pollution.
Simpson (1995) reports that, in a Cournot duopoly, if firms have different production
costs, then the optimal tax rate might exceed the marginal damage because the tax canbe an effective instrument for allocating production from the less to the more efficientfirm. Lahiri and Ono (2007) report that, in an oligopolistic market, an emission standard
is welfare-superior but emission-inferior to an emission tax when the number of firms isfixed. However, with free entry and exit, the results might be reversed. Fujiwara (2009)
reports that optimal environmental taxation differs in the short term, with a fixed
number of differentiated firms, from those in the long term, which have free entry andexit. Lian et al. (2018) demonstrate that, in both oligopoly market cases of no entry andof free entry of firms, the tax rates are smaller than the marginal environmental damage,irrespective of the degree of pollution in goods production, although assuming anonlinear polluting technology of firms.
However, most reports of the literature assume that the pollution emission technology
of firms is linear in the degree of pollution emission, e.g., a one-to-one correspondence
between output and pollutant. Therefore, our research question is whether theenvironmental tax rate is greater than, equal to, or smaller than the marginalenvironmental damage when it is not linear. We call the square of a pollutant per unit ofdifferentiated goods the degree of pollution emission. A salient feature of this paper isthat it distinguishes the pollution (nonlinear) function of firms from the social
environmental damage (linear) function in the assessment of pollution emission. 2 Lianet al. (2018) also introduce such a distinction in an oligopolistic market structure. 3
However, they do not fully explore the effects on optimal environmental taxation.
The optimal environmental policy in oligopolistic markets has been analyzed by many
researchers, e.g., Katsoulacos and Xepapadeas (1995), Damanis (1996) and Yin (2003).
2 We do not assume consumers’ environmental awareness for expositional simplicity.
3 In contrast, Fujiwara (2009) assumes that production of a unit of goods emits a unit ofpollution as social damage.
1

2

The main results are the following. Both in the short-term and the long-term
symmetric oligopolistic equilibrium, the optimal environmental policy depends on thepollution emission per unit of differentiated goods. Both in the short term and in the longterm, although the optimal environmental tax rate is lower than the marginalenvironmental damage at higher degrees of pollution emission, the tax rate might
become higher than the marginal social damage when the per-unit pollution emission
becomes lower. When the degree of pollution emission becomes even lower, the optimal
tax becomes lower. The effect of the difference between pollution emission and theenvironmental damage on the optimal environmental policy has not been analyzed
formally.
A model of differentiated oligopoly is introduced in the next section. Section 3 analyzes
the optimal environmental tax policy in the short term, i.e., with a fixed number of firms,
and the policy in the long term, i.e., with free entry and exit. Section 4 concludes thepaper.

2. ModelWe consider an oligopolistic market with horizontally differentiated and homogeneous
goods. Both are produced from labor. The homogeneous goods are produced with aunitary input coefficient so that the wage rate is unity, i.e., numeraire goods. The varietyof the differentiated goods is designated as n > 1 . All firms in the differentiated goods
industry have the same production technology. Output of xi units of the differentiatedgood requires cxi + f units of labor ( i = 1,..., n ) , where c > 0 and f > 0 respectively

represent the marginal variable and fixed labor input. Following Lian et al. (2018), one
unit of differentiated goods emitsθ units of pollution, where θ ∈ (0,1] is called thedegree of pollution emission herein. 4 Total pollution of the industry is given asnei=
Z = ∑ i ei , whereθ xi [≡ P( xi ;θ )] is a pollution function. One unit of pollution islevied a pollution tax at a rate 0 < τ < 1 . The profit of each firm is given as

π i = ( pi − c) xi − τ ei − f .

(1)

The utility function of a representative consumer is assumed asn=u α ∑ i xi −

β −γ

n

γ

n

s

( xi )2 − (∑ i xi )2 + M − Z 2 .
∑
i
2
2
2

(2)

Fujiwara (2009) assumes that each unit of differentiated goods emits one unit ofpollution, i.e., θ = 1 .

4

3

Parameter α is the utility weight on total consumption of the differentiated goods. Also,β and γ respectively represent the preferences for product variety and for productdifferentiation. As commonly assumed in reports of the literature, we assume thatβ > γ . Variable M denotes consumption of homogeneous goods other than the
differentiated goods. The social damage of the total pollution emission Z is given in a2

quadratic form: − sZ / 2 , where s measures the strength of damages from pollution.
Utility maximization with respect to a differentiated good, subject to the budget
constraint, engenders the following inverse demand function:npi =α − ( β − γ ) xi − γ ∑ i xi ( i = 1,..., n ).

(3)

Now we assume a symmetric equilibrium in the oligopolistic market, i.e., x=
i x=
j x.
In

the

symmetricequilibrium,

profits

of

each

firm

can

be

writtenasn

π i = [α − c − τ θ − ( β − γ ) xi − γ ∑ j x j ]xi − f . The first-order condition for profit
maximization of firms is given asdπ i= α − c − τ θ − [2 β + γ (n − 1)]x = 0 ,
dxi

(4)

from which we obtain the output-consumption level of each differentiated goods asx=

α − c −τ θ.
2 β + γ (n − 1)

(5)

3. Optimal environmental policies in short-term and long-term equilibrium
This section presents an analysis of the effects of the degree of pollution in producing
differentiated goods on the optimal environmental policy, i.e., the optimal environmental
tax rate relative to the marginal environmental damage. We first consider the short-term
equilibrium with a fixed number of firms and then long-term equilibrium with free entry
and exit of firms.
3.1. Short term with a fixed number of firms
In the short term, the number of firms is constant at n > 1 . Following Lian et al.
(2018), the objective function of the government is assumed as

W = CS + nπ − GED + T ,where CS represents the consumer surplus and where GED designates the social
4environmental damage caused by pollution. The consumer surplus is defined as

CS =(α nx −

β −γ
2γnx 2 − n 2 x 2 ) − pnx . Under a symmetric equilibrium, because each2firm emits e = θ x units of pollution, the total pollution emitted is given as Z = ne .
Following Fujiwara (2009) and Lian et al. (2018), the social environmental damage is2
defined by a quadratic form of the total pollution, i.e., =GED sZ=/ 2 sθ (nx)2 / 2 . The

tax revenue is =T τ=
en τ θ xn . Therefore, the objective function can be rewritten as

W N= α nx −β −γ2

nx 2 −γ
2n 2 x 2 − pnx + n[ p − c − τ θ ]x − nf−

= nx[α − c −β − γ + ( sθ + γ )n2

The optimal environmental tax rate τNx] − nf .

sθ(nx)2 + τ n θ x
2(6)

is chosen to maximize the objective function(6). The first-order condition for the maximization is given as 5

1 dW Ndx= {α − c − [ β − γ + ( sθ + γ )n]x} = 0 ,
n dτ
dτ

(7)

from which we obtain the optimal environmental tax rate asτN =(α − c)(nsθ − β ).
θ [ β − γ + (γ + sθ )n](8)

Variables with superscript N denote the values in short-term equilibrium with noentry or exit of firms. When θ = 1 , the optimal tax rate coincides with the one obtainedin Fujiwara (2009), i.e., τN> 0 if ns − β > 0 . In this case in which the number of firms

is constant, the environmental tax reduces a firm’s profits. Because the tax revenue isadded to the social objective, these merely offset one another. Therefore, theenvironmental tax affects the social objective only through changes in the output levelsof differentiated goods. We assume here that ns − β > 0 , as a benchmark case. 6
The marginal environmental damage is defined as an increase in the environmental
damagebrought

about

by

an

additional

unit

of

differentiatedgoods,d 2W

θ n[ β − γ + (γ + sθ )n]
=
−
<0.
dτ 2
[2 β + γ (n − 1)]2

5

The second-order condition is satisfied, i.e.,

6

The condition ensures a positive optimal tax rate, as described by Fujiwara (2009).
5

dGED / dX = sθ (nx) , where

X = nx . Letting

MED

denotethemarginal

environmental damage, we obtain the following from (5)MED N =nsθ (α − c − τ θ ).
2 β + γ (n − 1)

(9)To analyze the properties of the optimal policy in the presence of inefficiencies causedby pollution externality and imperfect competition, we examine the sign of τ − MED . If
it is equal to zero, then the tax is a Pigouvian tax, merely internalizing the external
diseconomies. If it has a positive sign, then it is optimal for the tax to depress the
production of differentiated goods more than internalizing the external diseconomies. By
contrast, if it is negative, then the tax should internalize the externalities only partially,
encouraging more goods production and consumption.
From (8) and (9) we obtain(α − c)[nsθ (1 − θ ) − β ].
τ N − MED N =θ [ β − γ + (γ + sθ )n](10)Equation (10) shows that the sign of τ − MED depends on the degree of pollution
emission θ . Lian et al. (2018) report that the optimal tax rate is lower than the marginalenvironmental damage for all θ ∈ (0,1) when the tax rate is determined after firms
enter

the

oligopolistic

market.

We

also

have

τ N − MED N=− β (α − c) / [ β − γ + n(γ + s )] < 0 when θ = 1 , as in Fujiwara (2009). Therefore, thequestion is whether τN− MED N < 0 for any θ ∈ (0,1) . 7 From (8), we obtaindτ N( sn + β / θ )[ β − γ + (γ + sθ )n] / 2 + sn( β − snθ )= (α − c).
dθθ [ β − γ + (γ + sθ )n]2
The sign of dτNpositive, i.e., dτ(11)/ dθ is indecisive a priori. However, we can assume that it is plausiblyN/ dθ > 0 . From (9), we also have1 θ dτ N
dMED N MED N
nsτ N θ
( +
).
= −
dθ
2 β + γ (n − 1) 2 τ N dθθ

(12)

The sign of the right-hand side of (12) is ambiguous, depending on the relative7

We have τN− MED N → −∞ < 0 as θ → 0 .
6

magnitudes of the two terms. If the elasticity of the tax rate with respect to the degreeof pollution emission (dτ N / dθ ) / (θ / τ N ) is sufficiently great, then dMED
has a small value relative to dτ

N

N

/ dθ

/ dθ . An increase in pollutants per unit ofdifferentiated goods requires a higher optimal environmental tax rate because it involves
a greater pollution emission per differentiated-good due to the concavity of pollutionfunction with respect to the degree of pollution emission. The tax increase decreases themarginal environmental damage by depressing the marginal environmental damage.
The high tax rate internalizes the environmental costs rather than retaining outputd N(τ − MED N ) > 0 . This means that wedθlevels per firm. In this case, we might havemight have τN− MED N > 0 for some θ ∈ (0,1) even if τ N − MED N < 0 when

θ = 1 and θ ≈ 0 .
It is apparently difficult to solve algebraically. Therefore, we consider a numericalexample. For our purposes, we assume an oligopoly market with 15 firms, i.e., n = 15 .
Other parameters are set as (α , c, β , γ , s, n) = (1, 0.8, 0.9, 0.3, 0.5,15) . The dependence ofthe optimal policy ( τN− MED N ) and output level ( x N ) on the degree of pollution

emission in production of differentiated goods is shown in Table 1 and Fig. 1. For large
degrees of pollution emission, the optimal environmental tax rate is lower than themarginal environmental damage. As the degree of pollution emission becomes smaller,i.e., for θ ∈ [0.3, 0.6] , the optimal tax rate becomes higher than the marginalenvironmental damage. However, if the degree of pollution emission becomes evensmaller, then the optimal tax rate becomes lower than the marginal environmental
damage.
Table 1 Short-term output level, environmental tax rate, and optimal policy

θ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xN

0.034

0.030

0.027

0.025

0.023

0.021

0.019

0.018

0.017

τN

-0.016

0.041

0.067

0.082

0.091

0.097

0.010

0.103

0.104

τ N − MED N-0.042-0.005

0.006

0.008

0.006

0.003

-0.001-0.005-0.010

These results can be interpreted intuitively as the following. The pollution emissionfunction is a concave function of the degree of pollution emission. In the intermediate7

degree of pollution emission, the pollution emission per unit of goods is high. Therefore,
it is optimal to set the tax rate higher, thereby greatly depressing the pollution emission
level. With a constant number of firms, changes in the tax rate only affect the profit
levels of firms. The social benefit from raising the tax revenue is offset by decreasing
profits of firms (i.e., the producers’ surplus).
Figure 1 Optimal policy and output level for degrees of pollution emissions in a duopoly

0.020
0.010
0.000
-0.010τ N − MED N

0.040

0.0080.006
0.006
0.003
-0.001θ-0.005-0.0050.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.010
0.9-0.040-0.050

0.034

0.030
0.027
0.025
0.023
0.021
0.019
0.0180.017

0.030

0.020

-0.020-0.030xN

0.010-0.0420.000(α , c, β , γ , s, n) = (1, 0.8, 0.9, 0.3, 0.5,15)

θ
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Therefore, we have the following proposition.8
Proposition 1. Assume that pollution emissions per unit of goods be a concave functionof the degree of pollution emission θ ∈ (0,1] and the social environmental damage be aquadratic function of the total pollution emissions. In an oligopolistic market with asufficient number of symmetric firms, the second-best optimal environmental tax rate
might be higher than the marginal environmental damage at an intermediate degree ofpollution emission per output of differentiated goods.
At intermediate degrees of pollution emission, the optimal environmental tax might be
set as more than internalizing the environmental externality. Although this result isobtained in a numerical example, it runs in contrast to that presented by Lian et al.
(2018).
It is noteworthy that the tax rate might always be lower than the marginal
From (10) we might obtain the relation between the number of firms and the degree ofpollution emission, which brings about the negative sign to τ − MED . It is sufficient forour purposes to report an example case of a positive sign.

88environmental damage for θ ∈ (0,1] when the number of firms is fixed as sufficiently
small. 9 When the firms are few, the inefficiency caused by imperfect competition is great.
Therefore, it might be optimal to induce firms to produce more by a lower tax rate.
3.2. Long-term policy with free entry and exit
In this subsection, firms choose whether to enter (or exit), and if entering, how many
goods to produce. Because firms enter the oligopolistic market to the extent that they
expect positive profits, the profit of each firm becomes zero in equilibrium. Government
chooses the tax rate to maximize the social objective, given these firms’ reactive
behaviors.
The first-order condition for the firm’s profit maximization is given as (4) as in thepreceding subsection. The level of output of each firm is given by (5). The maximized
profit is obtained by inserting the output level in (5) into the profit function

π = [α − c − τ θ − ( β − γ ) x − γ nx]x − f as=
π β[

α − c −τ θ 2] −f.
2 β + γ (n − 1)

(13)

From the zero-profit condition, the equilibrium number of firms is obtained asnE =(α − c − τ θ ) β / f − (2 β − γ )

γ

.

(14)

Variables with superscript E denote the values in the long-term equilibrium with freeentry and exit. Inserting n

Efrom (14) into (4), one obtains the optimal output level ofthe differentiated goods per firm as=xE

α − c −τ θ=
2 β + γ (n E − 1)

f

β

.

(15)

The output level is independent of the tax rate and the number of firms.
As in the previous case, the objective function of the government is

W E = α nx −β −γ2

nx 2 −γsn 2 x 2 − pnx − θ n 2 x 2 + τ θ nx ,22where firm’s profits are zero, i.e., π

9

τ

E

= 0 . From (14) and (15), it can be rewritten as

In our numerical example, when the number of firms is less than 12, we have always

N− MED N < 0 for all θ ∈ (0,1) .

9

=W E nE xE [β − γ + n E (γ − sθ )2

x E + τ E θ )] .

(16)

By differentiating (16) with respect to τ , we obtain the optimal tax rate as 10τE

2[(α − c) / x E − (2 β − γ )]sθ − γ ( β − γ ) Ex .
=2 θ (γ + sθ )(17)

Setting θ = 1 engenders the optimal tax rate, which is obtained by Fujiwara (2009).11
Taking a case of the positive tax rate at θ = 1 as a benchmark, we analyze whether the
optimal policy makes the optimal tax rate lower than the marginal environmental
damage as the degree becomes lower.
The marginal environmental damage MED

MED E = sθE(α − c − τ θ ) − (2 β − γ ) x Eγ

is given from (14) as

.

(18)

Therefore, the optimal environmental policy is given asEτ E − MED
τE=

where τ

E

and x

E

γ + sθ θ
α − c − (2 β − γ ) x E
,
− sθ
γ
γ

(19)

are given respectively by (17) and (15). When θ = 1 , we haveτ E − MED E =−( β − γ ) / 2 < 0 . However, the sign of τ N − MED N also depends on the
degree of pollution emission in the long-term case. From (17) we obtain
E

dτ
x
=
dθ
2

2[(α − c) / x E − (2 β − γ )]s θ
γ + sθ
(γ − sθ ) + γ ( β − γ )[
+s θ]
2
2 θ

θ (γ + sθ )2.

(20)
If γ − s ≥ 0 , then we have dτE/ dθ > 0 . However, in this case, we also assume thatdτ E / dθ > 0 for all θ ∈ (0,1) . From (18) we also have102

The second-order condition is satisfied, d W / dτ =−θ (γ + sθ ) / γ

2

< 0.

A sufficient condition for a positive tax rate when θ = 1 is given by Fujiwara (2009,p. 244).

11

10

dMED E MED E sτ θ 1 θ dτ E
( +
).
=
−
dθ
θ
γ
2 τ E dθ(21)

If the elasticity of the tax rate with respect to the degree of pollution emission is
sufficiently great, then we have a small value of dMEDE

/ dθ , relative to the value ofdτ E / dθ . With a high pollution elasticity of tax rate, increases in the degree of pollution
emission engenders a higher environmental tax rate because of the concavity of pollutionfunction with respect to the degree of pollution emission. The high tax rate depresses thenumber of firms, mitigating the marginal environmental damage. The optimal
environmental policy mitigates environmental degradation rather than reserving theoutput levels. In the long term, because the output level per form is constant, the policy
depresses the number of firms. Therefore, in such a case, we might haveτ N − MED N > 0 for some θ ∈ (0,1) even if τ N − MED N < 0 when θ = 1 and θ ≈ 0 .
As in the previous case, we cannot derive their mutual relation algebraically.
Therefore, as in the previous case, we also consider a numerical example in which theparameter vector is set as (α , c, β , γ , s, f ) = (1, 0.8, 0.9, 0.3, 0.5, 0.003) . From (15), we
have xE= 0.058 . The results for optimal policy are presented in Table 2 and Figure 2.

As the extent of pollution emission θ diminishes, the optimal tax rate becomes lower.
A lower tax affects firms’ expected profits positively, inducing entry of firms.
Consequently, the number of firms increases. Pollution emission function of firms are aconcave function of the degree of pollution emissions. The welfare damage is a linear
function of total pollution emissions. Therefore, these differences make τ
positive: aboutE− MED E

θ = 0.2 to θ = 0.5 . Outside this range of the degree of pollution

emission, setting the environment tax rate above the marginal environmental damageis optimal. In other words, it is optimal to set the environmental tax rate higher thanfull internalization of pollution externalities because a negative effect of increased
pollution damage brought about by firm entries overwhelms the positive effect ofincreased output and consumption.
Therefore, we obtain the following proposition.
Proposition 2. Presume that pollution emissions per unit of goods be a concave function11

of the degree of pollution emission θ ∈ (0,1] and the social environmental damage be aquadratic function of the total pollution emissions. In a symmetric oligopolistic market
equilibrium with free entry and exit of firms, the second-best optimal environmental tax
rate might be higher than the marginal environmental damage when the degree ofpollution emission in the differentiated good production is intermediate.
Table ２ Long-term number of firms, tax rate, and degree of pollution emissions

θ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

nE

6.469

5.660

5.031

4.528

4.117

3.774

3.483

3.234

3.019

τE

0.004

0.034

0.048

0.055

0.060

0.062

0.063

0.064

0.064

τ E − MED E-0.014

0.002

0.004

0.003

0.000

-0.003-0.007-0.011-0.014

Fig. 2 Long-term optimal policy and internalization of external diseconomies.
0.010
0.005
0.000
-0.005-0.010τ E − MED E
0.004
0.003
0.002
0.000

θ

-0.0030.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-0.007-0.011-0.014-0.014-0.015-0.020

E

n
6.469
5.660
6.000
5.031
4.528
5.000
4.117
3.774
3.4833.234
4.000
3.019
7.000

3.000
2.000
1.000
0.000

(α , c, β , γ , s, f ) = (1, 0.8, 0.9, 0.3, 0.5, 0.003)

θ
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

These results differ from those obtained by Lian et al. (2018), who analyze this freeentry-exit case as the ex-ante taxation case. They do not analyze the dependence of the
optimal policy on the degree of pollution emission. No report in the literature describes
such a study.

4. Conclusion
Results show that, irrespective of whether entry and exit of firms is allowed or not,the optimal environmental policy depends on the degree of pollution emission or
pollution emission per unit of differentiated goods. In both short-term and long-term
symmetric oligopolistic equilibrium, there might be a range of moderate degree ofpollution emission per output of differentiated goods in which it is optimal for the optimal
12

environmental tax rate to be set higher than the marginal environmental damage, i.e.,the optimal tax rate is higher than Pigouvian tax rate, which merely internalizes thecost of external diseconomies.
The results are robust. The pollution function of firms and the social environmental
damage function generally differ in their responses to the degree of pollution emission.
For a moderate degree of environmental pollution, environmental taxation works toalleviate the efficiency caused by imperfect competition rather than internalizing thecost of external diseconomies. By contrast, when the degree of pollution emission is either
sufficiently small or when it is sufficiently high, the optimal tax rate is lower than themarginal environment damage. The environmental taxation internalizes the cost of
external diseconomies only partially, inducing differentiated goods production. Although
the total output is adjusted in terms of the per-firm level of production in the short term,
the number of firms adjusts the total output by entry and exit in the long term.
The simple model in this note has some weak points that must be enhanced andextended. The results are provided only in numerical examples, but not algebraically.
Consumers’ environmental awareness is not considered. Environmental awareness is
expected to exert a positive effect on environmental policy (e.g., Yakita and Yamauchi
2011; Lian et al. 2018). These are interesting and important subjects to be analyzed infuture research.

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Environmental

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14

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Supplementary note [NOT TO BE PUBLISHED]The environmental tax rate is set in response to the degree of pollution emission
whereas the marginal environmental damage is proportional to the degree of pollution
emission (although the out put level changes). If the concavity of pollution function issevere, then that of the corresponding tax rate is also severe. The relation between theseare presented in the Fig. S1.

Fig. S1 Tax rate and marginal environmental damage(long-term case)(α , c, β , γ , s, f ) = (1, 0.8, 0.9, 0.3, 0.5, 0.003)

0.100
0.080
0.060
0.040
0.020
0.000
-0.020

0.033
0.019

0.044
0.048

0.052
0.055

0.059
0.060

0.065
0.062

0.070
0.063

0.075

0.064

0.078

0.064

0.034

θ
0.004
0.1

0.2

0.3
τ

0.4

0.5
MED

15

0.6

0.7
τーMED

0.8

0.9