J EconDOI 10.1007/s00712-012-0316-2

Optimal degree of privatization and the environmentalproblem

Kazuhiko Kato

Received: 8 February 2011 / Accepted: 14 October 2012© Springer-Verlag Wien 2012

Abstract We determine the optimal degree of privatization in a mixed duopoly whenthe environmental problem exists. With regard to the ownership of the private firms,we analyze two cases: (h) the private firm is owned by domestic private investors and(f) it is owned by foreign private investors. A comparison of the two cases presents thefollowing results. Partial privatization is always desirable in (h), and the optimal degreeof privatization is independent of the degree of environmental damage. However, in (f),whether partial privatization is desirable or not depends on the degree of environmentaldamage: there are cases where full privatization or full nationalization is optimal.

Keywords Environment · Mixed duopoly · Partial privatization

JEL Classification L13 · L33 · Q50

1 Introduction

Full privatization of the public firm has been one of the major issues in mixed oligopolytheory since De Fraja and Delbono (1989) have shown situations wherein full priva-tization of the welfare-maximizing public firm can enhance welfare. Along with fullprivatization, partial privatization also has been a major focus of interest in the mixedoligopoly theory from both real and theoretical aspects. In the real world, partial pri-vatization has been one of the key reform methods for public firms and is widely seenin both developed and developing countries.1 Furthermore, from theoretical aspects,

1 See Heywood and Ye (2010), who provide some excellent examples of this phenomenon and introducethe related previous works.

K. Kato (B)Faculty of Economics, Asia University, 5-24-10 Sakai, Musashino, Tokyo 180-8629, Japane-mail: kkato@asia-u.ac.jp

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Table 1 CO2 emissions fromfuel combustion, and SO2emissions in China from2004 to 2006

Source: ∗OECD (2011) (CO2)∗∗National Bureau of Statisticsof China (2010) (SO2)

Million tons

2004 2005 2006

CO2∗ 4,552 5,062 5,603

SO2∗∗ 22.5 25.5 25.9

Table 2 Number of state-owned and private industrial enterprises in China from 2004 to 2006

2004 2005 2006

Unit % Unit % Unit %

I State-owned 35,597 23 27,477 18 24,961 14

II Private 119,357 77 123,820 82 149,736 86

III Total (I + II) 154,954 100 151,297 100 174,697 100

Source: National Bureau of Statistics of China (2010)

Matsumura (1998) shows that partial privatization increases welfare more than fullprivatization: partial privatization is desirable in terms of welfare. In recent years,several analyses of the optimal degree of privatization from these aspects have beenconducted in various settings: product differentiation Fujiwara (2007), foreign own-erships of private firms or international competition (Cato and Matsumura 2012; Hanand Ogawa 2009; Lin 2007; Lin and Matsumura 2012; Wang and Chen 2010, 2011),conjectural valuations (Heywood and Ye 2010), and competition between two domes-tic public firms (Saha 2009). These previous works show that the optimal degree ofprivatization can be affected by several factors: the substitutability of the products, thenumber of firms, the share of foreign ownership, etc.

At the same time, recent years have also seen studies associated with the environ-mental problem in a mixed oligopoly (Bárcena-Ruiz and Garzón 2006; Beladi andChao 2006; Cato 2008; Chen and Wang 2010; Kato 2006, 2011; Naito and Ogawa2009; Ohori 2006a,b; Pal and Saha 2010; Wang and Wang 2009; Wang et al. 2009).The appearance of these works can be attributed to the escalation of environmentalproblems, such as acid rain, global warming, and air and water pollution, in the world.In countries that suffer from environmental problems or have high interest in envi-ronmental problems, regardless of whether they are developed or developing, mixedoligopolies can be widely observed.

Here, we pick up China as an example of the environmental problem in a mixedoligopoly, particularly focusing on the relationships between privatization and theenvironment. Table 1 shows the increase in CO2 and SO2 emissions in China from2004 to 2006, while Table 2 gives the decline in the number of public firms and theincrease in the number of private firm during the same period. It can thus be said thatthese exists the possibility that privatization worsened the environment in China.2

2 Of course, it is too early to conclude solely on the basis of these data that privatization in China doesaffect its environment. We can consider other factors: an increase in the total number of firms, an expansion

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As for the theoretical aspects, in the works on environmental problems in anoligopoly, all players have traditionally been assumed to be private profit-maximizers.However, when considering a mixed oligopoly, the public firm too can be regarded asan important player. In the mixed oligopoly theory, the objective of the public firm isdifferent from that of the private firm, and is often assumed to be welfare maximiza-tion. This difference in the objectives of the public and private firms can affect theequilibrium outcome and the effect of the environmental policy to a large extent. Assuch, we can see the growing interest for environmental problems in a mixed oligopoly.

Taking into consideration the above two streams of researches on mixed oligopoly,we pose the following question from the theoretical aspects. How does the degree ofenvironmental damage affect the optimal degree of privatization? Since the welfare-maximizing public firm takes into consideration environmental damage, the optimaldegree of privatization may be affected by the degree of environmental damage.

As for studies with motivation similar to ours, we have Naito and Ogawa (2009),Wang et al. (2009), and Pal and Saha (2010) who consider the optimal degree of privati-zation in an environmental problem under emission tax in the domestic market.3 Thesestudies provide deep insight about the relationships between the emission tax and theoptimal degree of privatization.4 However, in these studies, lots of effects determinethe optimal degree of privatization, and thus, it is somewhat difficult to understand thebasic property for the optimal degree of privatization in an environmental problem.In order to see the basic property in the simplest framework, we do not consider anyenvironmental policies and any abatement investment to reduce its emission in ourmodel. If the emission tax is considered, the optimal degree of privatization is influ-enced not only by the direct effect of the environmental damage but also the indirecteffect through the emission tax. Thus, the effects on the optimal degree of privatizationare more complicated. Moreover, if environmental policies other than the emission taxare considered, the optimal degree of privatization might differ (Kato 2010). As forthe abatement investment, this is a specific endogenous variable in the environmentalproblems. Comparing the optimal degree of privatization in the environmental prob-lem with that in other models, we can see the differences more clearly when the firmscannot invest any abatement effort. Furthermore, Naito and Ogawa (2009) show thatfull nationalization is optimal for the government when firms can invest abatementeffort under no environmental regulation. This result strongly depends on this setting:the private firm does not abate its emissions under no environmental regulation.5

Footnote 2 continuedin the output due to economic development, etc. However, from these facts, it cannot be denied that priva-tization affects the environment in China.3 Ohori (2006a) foremost shows that full nationalization of the public firm is not optimal for the govern-ment under emission tax. However, he considers the situation where one partially privatized domestic firmcompetes with one partially privatized foreign firm in the third country. This case is quite different fromother previous works. As such, we do not cite Ohori (2006a) here.4 Naito and Ogawa (2009) analyze not only the emission tax but also the emission standard in which thegovernment directly sets the abatement level of each firm.5 They consider the mixed duopoly where the public firm competes with a domestic private firm. By simplecalculation, we can derive that full nationalization is also optimal for the government in their model in amixed duopoly where the competitor of the public firm is a foreign private firm.

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Our analyses have another characteristic that is different from Naito and Ogawa(2009), Pal and Saha (2010), and Wang et al. (2009). We consider two types of own-erships of the private firm: (h) the private firm is perfectly owned by domestic privateinvestors and ( f ) it is perfectly owned by foreign private investors. The aforemen-tioned previous works limit their analyses to the case where the public firm competeswith a domestic private firm. In the real world, globalization is resulting in competitionbecoming more multifaceted: the public firm does not always compete with a domesticprivate firm (Matsumura et al. 2009; Wang and Chen 2010, 2011). Meanwhile, in theliterature on the mixed oligopoly theory, we often observe different results between(h) and ( f ).6 From both real and theoretical aspects, we explore the optimal degreeof privatization for the two cases: (h) and ( f ).

The remainder of the paper is organized as follows. Section 2 describes the basicmodel of a mixed duopoly in an environmental problem. Section 3 derives the equi-librium outcome in cases (h) and ( f ) and compares them. Section 4 compares ourresults with those in some related works, and concludes. Detailed calculations for theequilibrium outcome in each case and the proofs of the propositions are given in theAppendices.

2 Model

Consider an economy where there exist one public firm (firm 0) and one private firm(firm 1) producing a homogeneous product. Firms 0 and 1 compete in quantity. Theoutput of firm i is denoted by qi (i = 0, 1). Total output is denoted by Q = q0 + q1.We assume that the cost function of firm i is given by ci (qi ) = cq2

i /2 (c > 0).7 Giventhe inverse demand function of the products, p(Q) = a − Q (a > 0), the profit offirm i is

πi (q0, q1) = (a − Q)qi − cqi

2.

In our model, pollution is generated by production and producing one unit of aproduct generates one unit of pollution. Thus, total pollution is Q. The pollution isconverted into environmental damage that reduces welfare. The total environmentaldamage is denoted as D(Q) = d Q2/2 (d > 0).

Now, we define welfare. Welfare is defined as the sum of consumer surplus, producersurplus, and environmental damage. We consider the following two cases with respectto the ownership of the private firm—case (h), where firm 1 is perfectly owned bydomestic private investors, and case ( f ), where it is perfectly owned by foreign private

6 For example, Fjell and Heywood (2004), Fjell and Pal (1996), Lu (2006), and Matsumura (2003) demon-strate different results in the case wherein the competitors of the public firm are domestic private firms andthe case wherein they are foreign private firms.7 This setting is very popular in the field of mixed oligopoly: as examples, consider De Fraja and Delbono(1989), which is a pioneer work in this field, and Naito and Ogawa (2009), Wang and Wang (2009),Heywood and Ye (2010), Kato (2011), and Wang et al. (2012), which are some of the recent works. We canalso consider another popular mixed oligopoly model: constant marginal costs with that for the public firmbeing higher. We discuss this case in Sect. 4.

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investors. In what follows, we refer to firm 1 in cases (h) and ( f ) as a domestic privatefirm and a foreign private firm, respectively. We show the welfare in each case byillustrating its components in detail.

First, we consider case (h), wherein firm 1 is a domestic private firm. Welfare isgiven by

W =Q∫

0

p(s)ds − cq20

2− cq2

1

2− d Q2

2. (1)

Second, we consider case ( f ). In this case, firm 1 is a foreign private firm. Welfareis given by

W =Q∫

0

p(s)ds − p(Q)q1 − cq20

2− d Q2

2. (2)

Here, we define the objective function of each firm. The objective functions ofpublic firm U0 and private firm U1 are, respectively, given as

U0 = αW + (1 − α)π0, α ∈ [0, 1], (3)

U1 = π1. (4)

Note that firm 0 maximizes (3) where W is defined as (1) in case (h) and maximizes(3) where W is defined as (2) in case ( f ); on the other hand, firm 1 maximizes its ownprofit in both cases. When α = 0, firm 0 is a pure profit-maximizer, and when α = 1,it is a pure welfare-maximizer. α is understood as the shareholding of the public sectorand 1 − α is that of the private sector.8

Finally, we consider the following timing of the game. In the first stage, the gov-ernment chooses the degree of privatization α to maximize welfare. In the secondstage, the two firms simultaneously choose their quantities q0 and q1 to maximizetheir objectives. To solve this game, we use the backward induction method.

3 Optimal degree of privatization in cases (h) and (f)

In this section, we derive the optimal degree of privatization α in cases (h) and ( f ).

3.1 Case (h)

We first consider the case where firm 1 is a domestic private firm. Note that welfare isdefined as (1) in this case.

8 For a rationalization of this objective function, see Bös (1991) and Matsumura (1998).

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In the second stage, each firm maximizes its objective by choosing its quantity. Thefirst-order conditions of the maximization problems of firms 0 and 1 are, respectively,given as

∂U0

∂q0= a − (2 − α + c + αd)q0 − (1 + αd)q1 = 0, (5)

∂U1

∂q1= a − q0 − (2 + c)q1 = 0. (6)

Solving for q0 and q1 by using the above first-order conditions, we obtain

qh0 = a(1 + c − αd)

(1 + c)(3 + c) − (2 + c)α + (1 + c)αd, (7)

qh1 = a(1 + c − α + αd)

(1 + c)(3 + c) − (2 + c)α + (1 + c)αd, (8)

W h = 2a2(1 + c){(1 + c)(4 + c − 2d) − (5 + 2c − 4d − 2cd)α}2{(1 + c)(3 + c) − (2 + c)α + (1 + c)αd}2 ,

+ a2α2{3 + c − d(3 + 2cd)}2{(1 + c)(3 + c) − (2 + c)α + (1 + c)αd}2 , (9)

where the superscript h denotes the equilibrium outcome in the second stage in case(h) except for when referring to α, wherein the superscript denotes the equilibriumoutcome in the first stage. In what follows, this superscript is also used to representthe above meaning.

With respect to the parameters, we assume the following two conditions: (p1)d < 1 + c and (p2) c ≥ 1. The former condition is a sufficient condition for theinterior solution, which implies the positiveness of the equilibrium output of each firm.We only focus on the case of the interior solution. The latter condition is assumed tosatisfy the second-order condition of the maximization problem of the governmentin the first stage, and is a sufficient condition. If we consider d ≥ 1 + c, we have toconsider the corner solution: there is the case that the optimal degree of privatizationis continuous and includes full nationalization of the public firm. To omit this trivialcase, we assume that d < 1 + c.

Here, we examine the comparative statics for the equilibrium output of each firmwith respect to α. We obtain the following lemma.

Lemma 1 (i) If d > (<) 1/2, qh0 is decreasing (increasing) in α, qh

1 is increasing(decreasing) in α, and Qh is decreasing (increasing) in α. (ii) If d = 1/2, qh

0 , qh1 , and

Qh are independent of α.

Proof Simple differentiation of qh0 , qh

1 , and Qh in α yields the results in Lemma 1.��

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In terms of welfare, there is a possibility that there exist two distortions in thecountry. One is caused by underproduction with regard to the duopolistic marketand the other is caused by excess production with regard to environmental damage.Suppose that the degree of environmental damage is high (low), that is, d is high(low). In this case, the latter distortion dominates (is dominated by) the former one,and therefore, the public firm decreases (increases) its output when it gives greaterweight to welfare.

In the first stage, the government chooses α in order to maximize welfare. We obtainthe following proposition.

Proposition 1 When d = 1/2, any α ∈ [0, 1] becomes the optimal degree of privati-zation. When d �= 1/2, the optimal degree of privatization is

αh = (1 + c)2

(1 + c)2 + c.

Proof See Appendix 1. ��The result yields that αh ∈ (0, 1); that is, partial privatization is desirable for the

government. We also find that αh does not depend on the degree of environmentaldamage.9

The reasons why αh does not depend on d are as follows. First, we consider thefirst-best outcome. Solving q0 and q1 so as to maximize W that is defined in (1), weobtain the first-best outcome (q F B

0 , q F B1 ). As the cost function is symmetric between

firms 0 and 1, q F B0 = q F B

1 . In the q0 − q1 plane, the first-best outcome lies on the45-degree line that starts from the origin. As both q F B

0 and q F B1 depend on d, the

first-best outcome shifts on the 45-degree line with a change in d. Note that q F B0 and

q F B1 decrease as d increases. Second, we consider the equilibrium in the case of a

pure duopoly: this case corresponds to the case of α = 0 in this model. Then, theequilibrium output of each firm in the second stage does not depend on d and alsolies on the 45-degree line. Therefore, some value of d makes the equilibrium outputin the case of a pure duopoly to be the same as the output in the case of the first best.In particular, the value is 1/2 and the first-best outcome can be realized regardless ofα in the mixed duopoly when d = 1/2.

9 We consider a simpler extension of case (h): there exist one public firm and m domestic private firms. Werefer to this case as case (m). We analytically obtain the optimal degree of privatization in case (m). Theresults are as follows. When d = 1/(m + 1), any α ∈ [0, 1] becomes the optimal degree of privatization.Suppose that d �= 1/(m + 1). Then, the optimal degree of privatization is

αm = (c + 1)2

(c + 1)2 + mc.

Since we focus on the interior solution that implies the positiveness of the equilibrium output of each firm,instead of assuming (p1) d < 1 + c in Sect. 3, we set a new assumption: d < (1 + c)/m. The proofs of thisresult and other results in the subsequent footnotes are available from the author upon request.

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Now, we reconsider the maximization problem of the government in the first stage.Partially differentiating W h with respect to α, we obtain

∂W h

∂α= (p − c′

0 − D′)∂qh

0

∂α+ (p − c′

1 − D′)∂qh

1

∂α,

= −(1 − α)(p′qh0 + D′)

∂qh0

∂α− (p′qh

1 + D′)∂qh

1

∂α,

= −(1 − α){(d − 1)qh

0 + dqh1

} ∂qh0

∂α−

{dqh

0 + (d − 1)qh1

} ∂qh1

∂α. (10)

The equation in the second line is derived by substituting the first-order conditions ofthe maximization problem of firms 0 and 1 in the second stage into the equation in thefirst line. The equation in the third line is derived by the model setting in this paper.

From the above discussion and Lemma 1, we find that both p′qhi + D′ = 0 and

∂qhi /∂α = 0 when d = 1/2; they have a common term: (1 − 2d). Therefore, αh does

not depend on d. Rather, these results depend on the symmetry of the cost function ofeach firm. In Sect. 4, we discuss this point.

As for the reasoning that partial privatization is optimal, the underlying intuitionis as follows. First, we show that full nationalization, α = 1, is not optimal. From(10), we can see that ∂W h/∂α|α=1 = −(p′qh

1 + D′)(∂qh1 /∂α) = −{−qh

1 + d(qh0 +

qh1 )}(∂qh

1 /∂α). When we consider a marginal decrease in α at α = 1, the marginalchange in the output of the public firm does not affect welfare since the objective of thepublic firm is to maximize welfare. From Lemma 1, when d is low (high), a marginaldecrease in α at α = 1 increases (decreases) the output of the private firm. This causesan increase (decrease) in consumer surplus and environmental damage. The formereffect dominates (is dominated by) the latter effect, and then, ∂W h/∂α|α=1 <0. Next,we show that full privatization, α = 0, is not optimal. From (10), we can see that∂W h/∂α|α=0 =−(p′qh

0 + D′)(∂ Qh/∂α)=−(2d − 1)qh0 (∂ Qh/∂α) since qh

0 |α=0 =qh

1 |α=0. When d is low (high), the marginal increase in α increases (decreases) the totaloutput from Lemma 1. This causes an increase (decrease) in consumer surplus andenvironmental damage. The former effect dominates (is dominated by) the latter effect,and then, ∂W h/∂α|α=0 > 0. Therefore, partial privatization is optimal in case (h).

Finally, from Lemma 1, when d = 1/2, the equilibrium output of each firm in thesecond stage is independent from α. Therefore, any α is optimal. We note that thefirst-best outcome is realized in this case.

3.2 Case (f)

We consider the case where firm 1 is a foreign private firm. Note that welfare is definedas (2) in this case.

In the second stage, each firm maximizes its objective by choosing its quantity. Thefirst-order conditions of the maximization problems of firms 0 and 1 are, respectively,given as

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∂U0

∂q0= a − (2 − α + c + dα)q0 − (1 − α + dα)q1 = 0, (11)

∂U1

∂q1= a − q0 − (2 + c)q1 = 0. (12)

Solving for q0 and q1 by using the above first-order conditions, we obtain

q f0 = a(1 + c + α − αd)

(1 + c)(3 − α + c + αd), (13)

q f1 = a(1 + c − α + αd)

(1 + c)(3 − α + c + αd), (14)

W f = a2{(1 + c)2(6 + c − 4d) − 2(1 + c)cα(1 − d) − (2 + 3c)(1 − d)2α2}2(1 + c)2(3 − α + c + αd)2 .

(15)

Here, we examine the comparative statics for the equilibrium output of each firmwith respect to α. We obtain the following lemma, which is similar to Lemma 1.

Lemma 2 (i) If d > (<) 1, q f0 is decreasing (increasing) in α, q f

1 is increasing

(decreasing) in α, and Q f is decreasing (increasing) in α. (ii) If d = 1, q f0 , q f

1 , andQ f are independent of α.

Proof Simple differentiation of q f0 , q f

1 , and Q f in α yields the results in Lemma 2.��

In the first stage, the government chooses α in order to maximize welfare. Solvingfor α, we obtain the following proposition.10

10 Results similar to those in case ( f ) hold even when the number of foreign private firms is more thanone. Here, we consider the mixed oligopoly market where one public firm and n foreign private firmscompete. We refer to this case as case (n). When d = 1, any α ∈ [0, 1] becomes the optimal degree ofprivatization. We consider the case where d �= 1. Then, the optimal degree of privatization is as follows.(1) Suppose that n > (1 + c){√c + √

4 + c}/(2√c). Then, αn = αn . (2) Suppose that 1 + c < n ≤

(1 + c){√c + √4 + c}/(2√

c). Then,

αn =

⎧⎪⎨⎪⎩

αn if 0 < d <cn

1 + c + cn,

1 ifcn

1 + c + cn≤ d <

1 + c

n.

(3) Suppose that (1 + c){c +√

8 + 4c + c2}/{2(2 + c)} < n < 1 + c. Then,

αn =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

αn if 0 < d <cn

1 + c + cn,

1 ifcn

1 + c + cn≤ d < 1,

0 if 1 < d <1 + c

n.

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Proposition 2 When d = 1, any α ∈ [0, 1] becomes the optimal degree of privatiza-tion. When d �= 1, the optimal degree of privatization is as follows.

α f =

⎧⎪⎨⎪⎩

α f i f 0 < d < c1+2c or 3+2c

2(1+c) < d,

1 i f c1+2c ≤ d < 1,

0 i f 1 < d ≤ 3+2c2(1+c) ,

where

α f = (1 + c){3 + 2c − 2d(1 + c)}(3 + 6c + 2c2)(1 − d)

. (16)

Proof See Appendix 2. ��The intuition underlying this result is as follows. Partially differentiating W f with

respect to α, we obtain

∂W f

∂α= (p − p′q f

1 − c′0 − D′)

∂q f0

∂α+ (−p′q1 − D′)

∂q f1

∂α,

= −(1 − α)(p′Q f + D′)∂q f

0

∂α− (p′Q f + D′ − p′q f

0 )∂q f

1

∂α,

= −(1 − α)(d − 1)Q f ∂q f0

∂α−

{(d − 1)Q f + q f

0

} ∂q f1

∂α. (17)

The development of the formula in (17) is as in (10).First, we consider the case where d is sufficiently low. We regard this case (to some

extent) as no environmental problem in a mixed duopoly. Suppose a marginal decreasein α at α = 1. As is seen in (17), the first term becomes 0; that is, a marginal decreasein the output of the public firm does not affect welfare. However, a marginal increase inthe output of the private firm increases consumer surplus to a large extent. Therefore,partial privatization enhances welfare.

Footnote 10 continued(4) Suppose that n < (1 + c){c +

√8 + 4c + c2}/{2(2 + c)}. Then,

αn =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

αn if 0 < d <cn

1 + c + cnor

1 + c + (2 + c)n

(1 + c)(1 + n)< d,

1 ifcn

1 + c + cn≤ d < 1,

0 if 1 < d ≤ 1 + c + (2 + c)n

(1 + c)(1 + n).

Here,

αn = (1 + c){1 + c + 2n + cn − (1 + c)(1 + n)d}(1 + 2c + c2 + 2n + 3cn + c2n + cn2)(1 − d)

.

Note that we set new assumptions d < (1+ c)/n in this case, instead of assuming (p1) d < 1+ c in Sect. 3.

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Second, we consider the case where d is near the center of d = 1. In this case, wecan consider that the effects of a marginal change in α on consumer surplus (−p′Q f )and that on environmental damage (−D′) are balanced: −p′Q f − D′ = −(d −1)Q f .Consequently, we can regard ∂W f /∂α as −q f

0 (∂q f1 /∂α). If d < (>) 1, an increase

in α leads to a decrease (an increase) in the output of the foreign private firm, andleads to an increase (a decrease) in the profit of the public firm—∂W f /∂α|α=1 > 0(∂W f /∂α|α=0 < 0). Therefore, full nationalization (privatization) of the public firmis the best policy for the government.

Finally, we consider the case where d is sufficiently high. In this case, environmentaldamage is severe, and thus, the public firm produces less when α = 1 than when α = α.Suppose a marginal decrease in α at α = 1. A marginal increase in the output of thepublic firm does not affect welfare. However, a marginal decrease in the output of theprivate firm improves welfare because it reduces the environmental damage to a largeextent. Therefore, partial privatization enhances welfare.

We note that when d = 1, from Lemma 2, the equilibrium output of each firm inthe second stage is independent from α. Therefore, any α is optimal in this case.

Comparing the optimal degree of privatization in cases (h) and ( f ), we obtainthe following results. The optimal degree of privatization in case (h) always existsin the range (0, 1), whereas in case ( f ), it can be 0 or 1. Furthermore, in case (h),whether an environmental problem exists or not does not affect the optimal degree ofprivatization. From these results, the ownership pattern of the private firm has a largeimpact on the optimal degree of privatization. This result is partly in contrast to thatobtained by Heywood and Ye (2010) who compare the optimal degree of privatizationin case (h) with that in case ( f ) without an environmental problem in standard Cournotcompetition. They show that the optimal degree of privatization α is always lower whenthe private firm is domestic than when it is foreign: the public firm should place a highervalue on its profits when it competes with the domestic private firm. In our paper, thisresult also holds when the degree of environmental damage d is low.11 However, whenthe degree of environmental damage is high, the opposite result holds.

4 Concluding remarks

This paper examines the optimal degree of privatization of a public firm when thereexists an environmental problem. We analyze this problem by considering two own-ership patterns of the private firm.

Now, we make two short remarks on the above results. First, we consider the resultsby picking up Cato (2008). Cato (2008) shows that whether or not the public firm shouldbe privatized depends on d when the public firm competes with private firms. In hismodel, partial privatization is not allowed, and therefore, if it is allowed, we need not

11 Calculating α f − αh , we find that

α f − αh = (1 + c)(2c + c2 + d + cd)

(1 + 3c + c2)(3 + 6c + 2c2)(1 − d).

From the above result and Propositions 1 and 2, we obtain that α f − αh > 0 when d �= 1/2 and d < 1.

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worry about d in our results. However, we pay attention to his settings: the asymmetryof the cost function in output between the public and private firms emerges even if thecost function is a priori symmetric between them; the public firm abates its pollutionbut the private firms do not, and this leads to c′

0(q) > c′i (q) at the same output level

q0 = qi = q. If we consider the asymmetry of the cost function, the optimal degreeof privatization depends on d.12

Next, we consider the following extension: the private firm is jointly owned bydomestic and foreign private investors (case (J )). We use the parameter γ ∈ [0, 1]that represents the ratio of domestic private investors to all investors.13 The profit ofeach firm and welfare are defined as

W = aQ − Q2

2− (1 − γ )(a − Q)q1 − cq2

0

2− γ cq2

1

2− d Q2

2,

πi = (a − Q)qi − cq2i

2. (i = 0, 1)

With respect to the optimal degree of privatization in case (J ), we cannot obtain itanalytically.14 Table 3 shows the numerical example with respect to the optimal degreeof privatization in case (J ) and we find that results similar to those in case ( f ) areobtained. From the facts, our results are partially different from those in Han andOgawa (2009), Lin (2007), Lin and Matsumura (2012), and Wang and Chen (2011)where there is no environmental problem: these works show that partial privatizationis always optimal for the government.

From the above discussion and our main results, we should pay attention to the para-meter d to determine the optimal degree of privatization when the private firms thatare partly or perfectly owned by foreign private investors are included as competitors

12 Here, we consider another popular model: mixed duopoly where one public firm and one domesticprivate firm produce a homogeneous product, and where the marginal cost is constant with that of the publicfirm being higher (c0(q0) = cq0 and c1(q1) = 0). When the private firm is domestic, we find that theoptimal degree of privatization depends on d though we cannot derive it analytically. When the private firmis foreign (case (C f )), we obtain the following optimal degree of privatization:

(1) when a > (2 + √2)c, αC f =

⎧⎪⎨⎪⎩

αC f if 0 ≤ d <c

a − c,

1 ifc

a − c≤ d <

a − 2c

a,

(2) when 2c < a ≤ (2 + √2)c, αC f = αC f ,

where

αC f = 3(a − c) − (2a − c)d

(3a − 2c)(1 − d).

We set new assumptions d < (a − 2c)/a in this case, instead of assuming (p1) d < 1 + c in Sect. 3.13 We use a model that is based on Cato and Matsumura (2012) and Wang and Chen (2011)14 The equilibrium output of each firm in the second stage is as follows:

q J0 = a(1 + c + (1 − γ )α − αd)

3 + 4c + c2 − (1 + γ + c)α + (1 + c)αd, q J

1 = a(1 + c − α + αd)

3 + 4c + c2 − (1 + γ + c)α + (1 + c)αd.

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Optimal degree of privatization

Table 3 Optimal degree of privatization in case (J ), α J

γ d

0 (%) 1/8 (%) 1/4 (%) 3/8 (%) 1/2 (%) 5/8 (%) 3/4 (%) 7/8 (%) 1 (%)

1/4 93.8 94.7 95.9 97.8 100 100 100 any 50.9

1/3 93.5 94.4 95.6 97.5 100 100 100 0 63.6

1/2 92.8 93.6 94.8 96.8 100 100 any 63.9 76.4

2/3 91.9 92.6 93.7 95.7 100 100 64.2 79.1 82.7

3/4 91.3 91.9 92.9 94.9 100 any 76.7 82.8 84.8

We set a = 100 and c = 6any implies that any α ∈ [0, 1] becomes the optimal degree of privatization

of the public firm. In particular, when the pollutant has a moderate degree of envi-ronmental damage, the optimal degree of privatization can change drastically; that is,full privatization or nationalization is optimal. Meanwhile, when all competitors ofthe public firm are private firms that are perfectly owned by domestic private investorsand the symmetry between the public and private firms except for in their objectivesholds, we need not care about the environmental problem to determine the optimaldegree of privatization.

This paper uses the simplest framework to consider the optimal degree of privati-zation in a mixed duopoly where the environmental problem exists. Therefore, severalextensions of this analysis are possible. One is that we can consider the case whereinthe government implements environmental policies. Earlier works examine the rela-tionships between the optimal degree of privatization and emission tax (Naito andOgawa 2009; Pal and Saha 2010; Wang et al. 2009). However, we can consider otherenvironmental policies, such as emission standard, emission quota, and tradable emis-sion permits. From the results of Kato (2006, 2010) and Naito and Ogawa (2009), themagnitude relationships of welfare among different environmental policies can varywith the degree of privatization. Thus, we need to examine how the differences amongenvironmental policies might affect the optimal degree of privatization. The otherextension is that we can consider the free-entry market, that is, the long-run equi-librium: full privatization (Ino and Matsumura 2010; Matsumura and Kanda 2005;Matsumura et al. 2009) and the optimal degree of privatization (Cato and Matsumura2012; Fujiwara 2007; Wang and Chen 2010). If we consider the free-entry market,the following question emerges: are similar results obtained even for our model? Toinvestigate this, we should pay attention to the timing of the game and the ownershippatterns of the private firm. These extensions would require much effort, and are leftfor future research.

Acknowledgments We are grateful to Susumu Cato, Akira Ogawa, Tsung-Hsiu Tsai, Leonard F.S. Wang,Yuexin Xie, and the two anonymous referees for their constructive suggestions and thoughtful comments. Wealso thank the participants for their helpful comments in the Non-linear Economic Theory Workshop at ChuoUniversity (November 2010), the International Workshop at the National University of Kaohsiung (June,2012), and the Autumn Meeting of the Japanese Economic Association (October, 2012). We acknowledgefor OECD and National Bureau of Statistics of China to allow us to reproduce their data. This researchis supported by a Grant-in-Aid for Young Scientists (B) from the Ministry of Education, Culture, Sports,Science and Technology, Japan (No. 23730247).

123

K. Kato

Appendix 1

Proof of Proposition 1 The first-order condition of the maximization problem of thegovernment in case (h) is

∂W h

∂α= a2(1 + c){(1 + c)2 − (1 + 3c + c2)α}(1 − 2d)2

{(1 + c)(3 + c + dα) − (2 + c)α}3 = 0. (18)

We can easily find that the denominator is positive. We focus on the numerator. Whend = 1/2, W h does not depend on α, and therefore, any α ∈ [0, 1] is optimal for thegovernment. When d �= 1/2, by solving the above equation in α, we can obtain thatthe optimal degree of partial privatization is αh . ��

Second-order condition of the maximization problem of the government

To determine whether αh is the maximizing value for W h , we calculate the second-order condition of the maximization problem for the government. Then, we obtain

∂2W h

∂α2 = − a2(1 + c)(1 − 2d)2gh(α)

{(1 + c)(3 + c + dα) − (2 + c)α}4 ≤ 0, (19)

where

gh(α) = (1 + c)(−3 + c + 3c2 + c3) + 2(2 + c)(1 + 3c + c2)α

+(3 − 2α)d + (9 − 8α)(1 + c)cd + (3 − 2α)c3d > 0, (20)

because of the assumption that c ≥ 1. Note that strict inequality holds when d �= 1/2.Therefore, the second-order condition is satisfied when d �= 1/2.

Appendix 2

Proof of Proposition 2 The first-order condition of the maximization problem of thegovernment in case ( f ) is

∂W f

∂α= 2a2(1 − d){(1 + c)(3 + 2c − 2(1 + c)d)−(3 + 6c + 2c2)(1 − d)α}

(1 + c)2(3 + c − α + dα)3 =0.

(21)

When d = 1, W f does not depend on α, and thus, any α ∈ [0, 1] is optimal for thegovernment. When d �= 1, we derive α f by solving the above equation with respectto α. Note that because both the sign and value of α f vary with the values of theparameters c and d, it is necessary to examine α f in detail.

First, we derive the condition where α f is positive. In order to obtain a positive α f ,the following conditions have to be satisfied:

123

Optimal degree of privatization

d < (>)3 + 2c

2(1 + c)and d < (>) 1. (22)

As (3 + 2c)/{2(1 + c)} > 1, we obtain

α f > 0 if

{d < 1,

d > 3+2c2(1+c) .

(23)

Next, we examine whether or not α f is less than 1. Calculating 1 − α f , we obtain

1 − α f = c − (1 + 2c)d

(3 + 6c + 2c2)(1 − d). (24)

When the above equation is positive, α f is less than 1. Thus, the conditions where α f

is less than 1 are given by

d < (>) 1 and d < (>)c

1 + 2c. (25)

As 1 > c/(1 + 2c), we obtain

α f < 1 if

{d < c

1+2c ,

d > 1.(26)

Summing up the above conditions, we can derive α f . ��

Second-order condition of the maximization problem of the government

To determine whether α f is the maximizing value for W f , we calculate the second-order condition of the maximization problem of the government. Then, we obtain

∂2W f

∂α2 = − 4a2(1 − d)2g f (α)

(1 + c)2(3 + c − α + dα)4 ≤ 0, (27)

where

g f (α) = c(3 + 3c + c2) + (3 + 6c + 2c2)α + 3(1 − α)(1 + 2c)d

+(3 − 2α)c2d > 0. (28)

Note that strict inequality holds when d �= 1. Therefore, the second-order conditionis satisfied when d �= 1.

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