The purpose of this note is to re-examine whether privatization improves the environment or not in a mixed duopolistic model. In our model, both firms adopt pollution abatement technologies in response to the environmental tax imposed by the government. It is shown that privatization unambiguously reduces the pollution levels of firms. Namely, privatization does improve the environment. Moreover, by implementing partial-privatization policy, social welfare can be enhanced.Environmental Taxes; Mixed Oligopoly: Privatization

Leonard F. S. Wang

Lihong Zhao

Ya-chin Wang

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 4 
  
Privatization and the Environment in a Mixed Duopoly with Pollution 
Abatement 
  
 
 
Leonard F. S. Wang  
Department of Applied Economics, National University of Kaoshiung 
Ya-chin Wang  
Department of Finance and Banking, Kun Shan 
University 
Lihong Zhao  
China Huanqiu Contracting & Engineering Corporation
Abstract 
The purpose of this note is to re-examine whether privatization improves the environment or not in a mixed duopolistic 
model. In our model, both firms adopt pollution abatement technologies in response to the environmental tax imposed 
by the government. It is shown that privatization unambiguously reduces the pollution levels of firms. Namely, 
privatization does improve the environment. Moreover, by implementing partial-privatization policy, social welfare can 
be enhanced.
 
Citation: Leonard F. S. Wang and Ya-chin Wang and Lihong Zhao, (2009) ''Privatization and the Environment in a Mixed Duopoly with 
Pollution Abatement'', Economics Bulletin, Vol. 29 no.4 pp. 3112-3119. 
Submitted: Oct 01 2009.   Published: December 16, 2009. 
 
     1
1. Introduction
Since 1990s, concerns over the environmental quality have been prominent in public
domain and government imposes green taxes or/and sets pollution standards to control
pollution. We have concurrently witnessed that mixed markets exist in a broad range of
industries such as oil, telecommunications, electricity and the postal sector, which were
formerly dominated by public monopolies in many developing economies. Such mixed
market with competition between firms with different objectives deserves further study. This
paper examines whether privatization improves (or deteriorates) the environment and social
welfare in a mixed duopolistic framework.
In the literature of pollution issues under oligopolistic framework, Simpson (1995) derived
the optimal pollution for a Cournot duopoly and found that if firms have different production
costs, the optimal tax rate may exceed the marginal damage, whereas Yin (2003) showed that
when the externalities are substantial and/or the number of polluters is large, the effluent
levies on these firms do not necessarily result in a deadweight loss.
1 To explore the
relationship between privatization and environment, Bárcena-Ruiz and Garzón (2006)
showed how the decision on whether to privatize a public firm interacts with the
environmental policy.
2 Beladi and Chao (2006) proved that privatization paradoxically exerts
a negative effect on the environment. Their result is, however, restricted to the case of
monopoly in the absence of pollution abatement. This paradox may be attributed to the
failure of a monopolistic firm to control pollution, since public monopolies are believed to
reflect inefficient management (see Vickers and Yarrow 1991) and high labor costs (see
Bradburd 1995). Cato (2008) investigated the privatization policy of an industry where the
production process generates emissions, and showed that the high degree of negative
externality leads to production substitution from the public firm to private firms; if the degree
of negative externality is sufficiently high, then a mixed oligopoly is preferable to a pure
oligopoly for social welfare, even if the number of firms in the market is large.
The model presented in this paper reflects the real-world scenario in Taiwan steel industry.
China Steel Corporation (CSC), the state-owned corporation possessing the largest market
share, competes against the private steel firm Yueh United Steel Corporation (YUSCO) in the
domestic market. In 1995, CSC was privatized by the Ministry of Economic Affairs of
Taiwan government. These two Kaohsiung-located corporations have been intensively
criticized by the public for contaminating the environment of Kaohsiung metropolis. Recently,
Taiwan government proposes green taxes to mitigate the unceasing increase of pollution
damage resulting from production and consumption-generated negative externalities.
Recognizing the industrial scenario and government-proposed policy, in this paper we want
to examine the effects of environmental tax and see whether privatization improves (or
deteriorates) the environment in a mixed duopolistic framework with pollution abatement,
and show that privatization unambiguously reduces the pollution levels of firms. Moreover, it
is proved that by implementing partial-privatization policy the social welfare can be
enhanced.
1 Bárcena-Ruiz and Garzón (2002) analyzed the environmental policy in a pure duopolistic model with
incentive schemes, and showed that firm owners have to pay a higher environmental tax and both
environmental damage and social welfare increase compared with the profit-maximization case.
2 Kato (2006) investigated the effects of tradable emission permits in a mixed oligopoly, and found that
under certain conditions, social welfare is greater (resp. smaller) under Tradable Emission Permits (TEP)
than under Non-Tradable Emission Permits (NTEP) when the weight of social welfare in each public
f i r m ’ s   o b j e c t i v e   f u n c t i o n   a n d   t h e   d e g r e e   o f convexity of the production cost function and that of the
abatement cost function are small (resp. large).2
2. The model
We depict a single market made up of one partially privatized firm (indexed by 0) and one
private firm (indexed by 1) producing a homogeneous commodity. For simplicity, the inverse
demand function is linearly defined:
1 i j p q q    , , 0,1 i j  , i j  . (1)
Accordingly, total consumer surplus is CS
2
0 1 ( ) /2 q q   . We assume that both firms have
identical technologies, and the production cost function takes the quadratic form
2 ( ) i i C q F q   , where 0 F  . It is noteworthy that the less cost efficiency of a partially
privatized firm is not emphasized here for focusing on the issues of degree of privatization
and pollution abatement.
The production of this commodity in both firms leads to pollution i e . The environmental
damage is measured by a quadratic form ED
2 ( ) /2 i ie  . However, each firm can prevent
pollution by undertaking abatement measures. Suppose that firm i chooses pollution
abatement level i a , then the pollution level of each firm is i i i e q a   . The cost of pollution
abatement of firm i is
2 /2 i a . Each firm has to pay an environmental tax t per unit of
pollutant emitted and as a result, the tax revenues collected by the government are T i i t e  .
The environmental tax is imposed by the government and its objective is to maximize
social welfare, which comprises the consumer surplus CS, the producer surplus 0 1    , and
the tax revenues collected by the government T, less the environmental damage ED. Thus, the
social welfare can be expressed as:
0 1 W CS T ED        , (2)
where the profit of firm i is given by:
2 2 /2 i i i i i pq q te a     . (3)
In Matsumura (1998), Matsumura and Kanda (2005) and Fujiwara (2007), the objective
function of privatized firm is defined as the weighted sum of social welfare and profit. The
private firm seeks profit maximization, whereas the partially privatized firm takes into
account both social welfare and profit considerations in mixed oligopoly. In Beladi and Chao
(2006), the public firm’ s objective function is defined as the sum of consumer surplus and the
firm’ s profit. We formulate the objective function of the partially privatized firm as the
weighted sum of social welfare and the firm’ s profit:
0 (1 ) U k W k    , 0 1 k   . (4)
where k is the degree of private ownership: the larger the value of k, the more the private
ownership. When k 0  , it is completely state-owned, and when k 1  , it is fully
private-owned.
We propose a two-stage game with the following timing. In the first stage, the government
sets the environmental tax to maximize the social welfare, i.e. max
t
W . In the second stage,
the two firms compete in the market by choosing output and abatement levels, i.e.
0 0 , max
q a U3
and
1 1
1 ,
max
q a
. As usual in such models, a backward induction is applied to obtain the
Subgame Perfect Nash Equilibrium (SPNE).
In the second stage of the game, differentiating (4) with respect to 0 q and 0 a for public
firm, and differentiating (3) with respect to 1 q and 1 a for private firm, we have the
following first-order conditions:
0 1 0 1
0
1 (1 )( ) 4 (2 ) 0
U
kt k a a q k q
q

        

,
0 1 0 1
0
(2 ) (1 ) (1 )( ) 0
U
kt k a k a k q q
a

        

,
1
0 1
1
1 4 0 t q q
q
 
    

,
1
1
1
0 t a
a
 
  

.
From the above first-order conditions, we obtain
0 2
5 7 2( 5 )
25 6 4
t k kt
q
k k
  

 
, 1 2
5 8 (4 1 )
25 6 4
t k t kt k
q
k k
    

 
,
0 2
5 13 (19 9 4 )
25 6 4
t k t kt k
a
k k
    

 
, 1 a t  .
In the first stage of the game, the government sets the environmental tax to maximize the
social welfare. We substitute i q and i a ( 0,1 i  ) into (2), and thus social welfare is given
by,
2 3 4 2 3 4
2 2
2 3 4 2
2 2
125 60 40 9 2 (125 54 35 84 8 )
(25 6 4 )
(595 348 331 273 58 )
.
(25 6 4 )
k k k k k k k k t
W
k k
k k k k t
k k
        

 
   

 
Differentiating it with respect to t, we obtain the optimal environmental tax,
2 3 4
* 125 54 35 84 8 k k k k
t
D
   
 . (5)
Substituting
* t back to i a and i q , we get the SPNE outcomes denoted by asterisk, as
follows:
2 3 4
* *
1
125 54 35 84 8 k k k k
t a
D
   
  ,4
2 3 4
*
0
173 165 170 9 11 k k k k
a
D
   
 ,
2 3
*
0
3(91 78 69 26 ) k k k
q
D
  
 ,
2 3 4
*
1
3(66 34 35 32 9 ) k k k k
q
D
   
 ,
2 3 4
* * *
0 1
3(157 112 104 58 9 ) k k k k
Q q q
D
   
   ,
2 3 4 2
*
2
(173 117 107 81 8 )
2
k k k k
ED
D
   
 ,
2 3 4
* 3(167 96 86 84 11 )
2
k k k k
W
D
   
 ,
where D
2 3 4 2(595 348 331 273 58 ) k k k k      .
Our results are consistent with Bárcena-Ruiz and Garzón (2006), who without considering
partial privatization, have pointed out that both abatement and pollution levels of the firms
mainly depend on their output levels.
Lemma 1. 0 1 q q  , 0 1 a a  and 0 1 e e  .
Proof.
2 3
0 1
3(1 )(25 19 15 9 )
0
k k k k
q q
D
   
   ;
2 3
0 1
3(1 )(16 21 24 )
0
k k k k
a a
D
   
  
3;
2 3
0 1
3(1 )(9 2 9 10 )
0
k k k k
e e
D
   
   .
The effects of increasing privatization (i.e. increasing k) on relevant variables can be
studied. The following proposition summarizes our main comparative-statics results.
Proposition 1. In equilibrium, an increase in the degree of privatization of the partially
privatized firm:
(a) Decreases the output level of the partially privatized firm but increases the output level of
the private firm.
(b) Decreases the total output level, and thus increases the market price.
(c) Decreases the abatement level of the partially privatized firm.
(d) Improves the environment.
Proof. Since the exposition of the equilibrium value of each variable is rather complicated,
we adopt numerical simulation for proof. The figures below illustrate the relationship
between increasing privatization and the equilibrium value of relevant variables.
As in Figures 1 and 2, 0 / dq dk and 1 / dq dk are always negative and positive,
respectively. Therefore, part (a) of proposition 1 holds. In Figures 3, / dQ dk is always
negative, showing that part (b) of proposition 1 holds. As in Figures 4, 5, and 6, both
0 / da dk and / dED dk are always negative even though the sign of 1 / da dk depends on
3 Numerical simulation shows that the numerator of this equation is positive.5
the degree of privatization. Consequently, we can argue that the abatement level of the
partially privatized firm and the environmental damage decrease with privatization, i.e. part
(c) and (d) of proposition 1 holds.
0.2 0.4 0.6 0.8 1.0
k
 0.07
 0.06
 0.05
 0.04
 0.03
 0.02
 0.01
dq0
dk
0.0 0.2 0.4 0.6 0.8 1.0
k
0.005
0.010
0.015
0.020
dq1
dk
Figure 1. Privatization Effect on 0 q Figure 2. Privatization Effect on 1 q
0.2 0.4 0.6 0.8 1.0
k
 0.05
 0.04
 0.03
 0.02
 0.01
dQ
dk
0.2 0.4 0.6 0.8 1.0
k
 0.07
 0.06
 0.05
 0.04
 0.03
 0.02
 0.01
da0
dk
Figure 3. Privatization Effect on Q Figure 4. Privatization Effect on 0 a
0.2 0.4 0.6 0.8 1.0
k
 0.005
0.005
0.010
0.015
da1
dk
0.2 0.4 0.6 0.8 1.0
k
 0.0020
 0.0015
 0.0010
 0.0005
0.0005
dED
dk
Figure 5. Privatization Effect on 1 a Figure 6. Privatization Effect on ED6
The above results are interpreted with the following economic intuition. Due to an
obligation towards social welfare, the public firm tends to produce more than under profit
maximization. However, along with the increasing degree of privatization, the privatized firm
attaches higher weight to profit and thus produces less, and the profit-maximizing private
firm then produces more. Nevertheless, the higher output level of the private firm is not able
to outweigh the lower output level of the privatized firm. Therefore, the total output decreases,
the market price increases and consumers are worse off. Moreover, because the privatized
firm is less concerned about social welfare and decreases its output, the abatement effort is
reduced. However, even though the pollution abatement of the privatized firm is decreased,
the less total output along with an increasing degree of privatization decreases total
environmental damage, which may then improve the environment. Our result on
privatization-environment is different from the result of Beladi and Chao (2006) in the case
of monopoly without considering pollution abatement.
3. The Welfare Implication of Partial Privatization
The fact that
*
0
( )
| 0 k
dW k
dk
 implies that privatization policy can improve the social
welfare. From the equilibrium value of social welfare, we can derive the critical value of
* 0.16 k  such that *
*
0.16
( )
| 0
k
dW k
dk

 and *
2 *
2 0.16
( )
| 0
k
d W k
dk

 . The importance of this
critical value is that it explicitly depicts the degree of privatization that maximizes social
welfare. Therefore, we obtain the following Corollary.
Corollary 1. The government maximizes the social welfare by implementing partial-
privatization policy for an optimal environmental tax.
Proof. See Figures 7 and 8.
0.2 0.4 0.6 0.8 1.0
k
 0.005
 0.004
 0.003
 0.002
 0.001
0.001
0.002
dW
dk
0.0 0.2 0.4 0.6 0.8 1.0
k
0.2080
0.2085
0.2090
0.2095
0.2100
0.2105
0.2110
W
Figure 7. Privatization Effect on W Figure 8. Welfare Improvement
4. Concluding Remarks
Proposition 1 paves the way for us to analyze the environmental effect of privatization.
Taking the output effect of privatization into account, the above results indicate that the
abatement and the pollution level of the privatized firm decrease with privatization and,
although the private firm reacts by increasing its output and pollution level, this7
pollution-increasing effect does not offset the first one. In sum, we can argue that the
environment is improved, since the total pollution level decreases with privatization.
Moreover, by implementing partial-privatization policy, the social welfare can be enhanced.
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Privatization and the Environment in a Mixed Duopoly with Pollution Abatement

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