ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E07-3

Partial Privatization and Market-Opening
Policies: A Mixed Oligopoly ApproachbyLihua Han
Hikaru Ogawa

June 2007

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Partial Privatization and Market-Opening Policies:
A Mixed Oligopoly ApproachbyLihua Han *1
Hikaru Ogawa *2

Abstract
Using a simple mixed oligopoly model with partial privatization, we show thatthe relaxation of restrictions on foreign investment sets back privatization of public firm.
That is, as the share of foreign capital in each corporate joint venture increases in themixed oligopoly market, the government decreases the degree of privatization. By
contrast, given the share of foreign capital in each corporate joint venture, theprivatization is promoted by an increase in the number of firms operating in the market.
In sum, the two different strategies for market openings result in the opposite impacts
on the government’s incentive for privatization.

*1 Graduate Student, School of Economics, Nagoya University

E-mail: lihuah2003@hotmail.com*2 Associate Professor, School of Economics, Nagoya University

E-mail: Ogawa@soec.nagoya-u.ac.jp

1

Introduction

Over 15 years ago, De Fraja and Delbono (1989) developed a mixed oligopoly
model to show that the privatization of welfare-maximizing public firm may
improve social welfare1 . This result received much interest since it paradoxically argues that a welfare-maximizing public firm should not pursue social
welfare but it should seek private profits.
One of the important extensions in the literature is to account for the
possibility of partial privatization. This type of extension is motivated by
the observation that the government usually holds a part of the shares in privatized firms. In De Fraja and Delbono (1989), the government only chooses
whether to keep a full share of (public) firm or to fully privatize such (ex ante
public) firm. When this setting is extended by allowing partial privatization,interesting results are obtained in the literature. Assuming that a public firm
is ineﬃcient compared with a private firm, Fershtman (1990) shows that fullnationalization of public firm may reduce social welfare in the Cournot equilibrium. Matsumura (1998) formally shows that neither full privatization
nor full nationalization is optimal in a mixed market. Lee and Hwang (2003)
elaborate on the framework of Matsumura (1998) to show that partial privatization is a reasonable choice for government in a monopoly market as wellas in a mixed duopoly market. Furthermore, although the model is somewhat
diﬀerent from an original mixed oligopoly approach, Beladi and Chao (2006)
show that while an increase in partial privatization may lower the welfare in
the short term, it can make a positive contribution to social welfare in thelong term through capital inflow2 . These extensions make sense since the1

Excellent surveys on mixed oligopoly theories are provided in De Fraja and Delbono(1990) and Nett (1993).
2
There are more studies on partial privatization. George and La Manna (1996) extend
the Fershtman model to the game of Stackelberg competition, where the public firm is a
Stackelberg leader and a private firm the follower, and show that a change in objectivefunction of an ex ante public firm from social welfare to profit results in higher welfare.

1governments often keep a part of the shares of public firms from a practicalstandpoint3 .
The aim of this paper is similar to the studies mentioned above, and wefollow it in agreeing that the governments choose the degree of privatizationof public firm to maximize the social welfare. The feature of these studies isthat all of them are only concerned with the mixed market of domestic firms.
This is quite strange since it is often the case that the competitor of a public
firm is not only a domestic private firm but also a foreign firm, and a seriesof mixed market analyses indicate that the existence of foreign private firms
considerably changes the equilibrium outcome4 . The feature that diﬀerentiates the present paper from previous studies is that it incorporates foreign
investment into the model explicitly. By this extension, we are now able toclarify the eﬀects on the degree of privatization of market openings policies5 .
The market-openings to foreign investors and the privatization of public
firms have been among the most important items in reforming the economic
structure of most developing countries [Smith and Trebilcock (2001)]. A
typical example is the Chinese market, in which there has long been an effective public monopoly or public oligopolies in the market6 . The Chinese
government has started to privatize their public firms, while it applies strict
Tomaru (2006) proves that the optimal subsidy, output level, firms’ profit, and social
welfare are identical before and after the partial privatization of a public firm.
3
Although, in this paper, we use the traditional framework of mixed oligopoly, recent
studies on privatization develop a model with two countries each with public and private
firms competing in a single market. See Barcena-Ruiz and Garzon (2005), Dadpay andHeywood (2006), and Fujiwara (2006) for related studies.
4
For a mixed oligopoly model with foreign competitors, see Fjell and Pal (1996), Pal
and White (1998), Serizawa (2000), Fjell and Hevwood (2002), Matsumura (2003), Sun,
Zhang, and Li (2005), and Chang (2005).
5
To our knowledge, Chao and Yu (2006) incorporates the partial privatization andforeign competitors into the mixed oligopoly market. As diﬀerent from our study, theirattention is on the eﬀects of a change in the degree of privatization on the tariﬀ rate, sothat the degree of privatization is given exogenously.
6
Some other case examples on the foreign investment regulation are given in
Chakrabarti and Heywood (2004).

2rules on foreign capital to place a restraint on a remarkable increase in foreign investment7 . By the regulations on foreign investment, most of foreign
firms entering to the Chinese market have been forced to take the form ofjoint-stock companies. However, after China joins the WTO, the Chinese
government has to relax the limitation on foreign investment in stages. With
the background of this relaxation of foreign capital regulation, the present
concern is how the Chinese government changes its domestic industrial policy.
Specifically, the practician cares about whether the privatization advances oris set back with the opening of the domestic market to foreign investors.
Moreover, the relationship between market openings and privatization is not
only concerned in the developing countries but also in the developed countries, particularly in the EU. Concerns over the changes in domestic industrial policies have arisen as new major players become outward investors,
prompting a number of countries to review their privatization policies8 .
We explore the features of a mixed oligopoly model involving foreign
investment and privatization decision. Analysis shows that the degree of privatization of a public firm critically depends on the level of restrictions on themarket entry of foreign companies. As the restriction on the share of foreign
ownership of domestic firms is relaxed so that the share of foreign capitalin each joint venture increases, the government inhibits the privatization of
public firm. In this sense, the open market policy may lead governments
to pull back from their privatization programs. However, the open market
policy, measured by the increase in the number of firms operating in the7

In an automobile market, for instance, the controlling share of foreign capital is restricted to less than 50 percent, and the foreign enterprises can establish only one or two
joint-stock companies (China Automobile Industry Development Policy, State Council of
the People’s Republic of China, May 21, 2004).
8
With the background of increase in the degree of free-trade within the EU, BarcenaRuiz and Garzon (2005, p.502) state that “The competitive environment created with theimplementation of the Single Market has led Member States to take stock of the benefits
they obtain by holding on to state ownership in some companies”.

3market, accelerates the privatization. The two diﬀerent strategies for opening market result in the opposite impacts on the government’s incentive forprivatization.
This examination is important for several reasons. First, the present paper serves to extend the domestic mixed duopoly model with the partial privatization to an international mixed duopoly. Researchers who examine theeﬀects on the degree of privatization of other issues such as non-benevolent
government, political process, and uncertainty need a basic model. Second,
empirical researchers may wish to test how the market-opening policies affect the progress of privatization. This paper oﬀers a simple but fundamental
model to classify the eﬀects of diﬀerent market-opening policies on the privatization levels.
The remainder of the paper is organized as follows: Section 2 introduces
the model. Section 3 studies the market equilibrium and Section 4 concludes
the paper covering some possible extensions.

2

Model

We consider a mixed oligopoly market, where there is a single public firm
and n identical private firms. These firms operate in a homogeneous goodmarket with inverse demand given by p = a − Q, where p is market priceand Q ≡ q0 + q1 + ..., qn is the total quantity of the output in the market.
qj (j = 0, 1, .., n) is the output of firm j. Firm 0 is a (public) firm which

maximizes a certain objective function given later. Firm i = 1, ..., n is a
profit maximizing private firm, which is partially owned by foreign investors.
The cost function is identical for all firms and is given by Cj = f + 0.5qj2 .
We do not consider the entry problem, so f = 0 is assumed in the following
analysis.
The profit of firm j (j = 0, 1, .., n) is given by4πj = (a − Q)qj − 0.5qj2 .

(1)

Public firm 0 is owned by the government, so that its profit, π0 , will be
involved in the social welfare. Similarly, if private firm i (i = 1, .., n) is thedomestic firm, its profit, πi , should be involved in the social welfare. However,
πi will be eliminated from social welfare if firm i is owned by foreign investors.
In this paper, we define social welfare as
W = CS + π0 + α

X

πi ,

(2)

i

where CS ≡ 0.5Q2 is the consumer surplus. In (2), α (0 ≤ α ≤ 1) repre-

sents the extent of domestic ownership of a private firm. In this paper, therestrictions on foreign investment are represented by α. When α = 1, foreign
investments are not allowed so that private firms are owned by domestic residents/consumers. α = 0 corresponds to the case in which firm i is a foreign
enterprise which is completely owned by foreign investors. If α is positive(but not equal to one), private firms are considered as a joint ownership enterprise. In this case, it is natural to consider that 100α percent of firm i’s
profit should be attributed to domestic residents. In this paper, α is set to
be exogenous, and it is allowed to take a value less than 1. This assumption
follows Huizinga and Nielsen (1997), and is well-justified, when the interest is
the eﬀects on the equilibrium values of an exogenous change in α. In consideration of a foreign investment, we can examine how the abolition of control
on the foreign investment targeted by WTO changes the privatization policy.
The government owns a share of (1 − θ) ∈ [0, 1] of the public firm. The

manager of this firm will maximize the weighted average of social welfare andthe profit. Following Matsumura (1998), we define the objective function offirm 0 as

5V = θπ0 + (1 − θ)W .

(3)

Note that the manager of fully privatized firm (θ = 1) seeks the firm’s profit,
while the manager of a fully nationalized firm (θ = 0) maximizes social
welfare.

3

Equilibrium

The game is constructed by two-stage decision-making. The government
chooses the level of privatization, θ, to maximize (2) in the first-stage. Observing θ, the firms choose the quantity supplied in the second stage. The
private firm i maximizes (1) and the semi-public firm 0 maximizes (3). We
follow the concept of backward induction and solve the equilibrium from thesecond stage.

3.1

Second StageFor given θ, the standard Cournot-Nash equilibrium in the second stage canbe derived as

q0 = a[n(1 − α)(1 − θ) + 2]/∆(4)

qi = a(θ + 1)/∆,(5)

p = 2a(θ + 1)/∆,(6)where ∆ ≡ nαθ + 2θ + 4 + n(2 − α) > 0. Comparative statics results yield∂q0 /∂θ = a[2n2 (α − 1) + 2n(2α − 3) − 4]/∆2 < 0∂qi /∂θ = 2a(n + 2 − nα)/∆2 > 0∂p/∂θ = 4a(n + 2 − nα)/∆2 > 0
6

(7)
(8)
(9)

Hence, the privatization decreases firm 0’s output, and increases firm i’s
output and the market price. Using (4) and (5), the profits and consumer
surplus in the second stage can be obtained asa2 [n(1 − α)(1 − θ) + 2][4θ + 2 − n(1 − α)(1 − θ)],2∆2
3a2 (θ + 1)2
πi =
2∆2
2
a [nαθ + 2 + n(2 − α)]2
.
CS =
2∆2
π0 =

(10)
(11)
(12)

Notice that while firm i’s profit is always positive, the profit of firm 0 is
negative (positive) if 4θ + 2 − n(1 − α)(1 − θ) < (>)09 . Furthermore, from(10)-(12), we have

∂πi /∂θ = 6a2 (θ + 1)(n + 1 − nα)/∆3 > 0∂π0 /∂θ = 2a2 H/∆3 ,

(13)
(14)

where H ≡ n2 (n + 4)α2 + n(2n2 + 11n + 8)αθ + (n3 + 5n2 + 8n + 4) − n2 (n +4)α2 θ−n(2n2 +9n+8)α−(n3 +7n2 +10n+4)θ. (13) and (14) show that while

privatization serves to increase firm i’s profit, its eﬀect on the firm 0’s profit
is ambiguous. Furthermore, the following comparative statics result showsthat the privatization results in a negative eﬀect on the consumer surplus:∂CS/∂θ = 4a2 [nαθ + 2 + n(2 − α)](nα − n − 1)/∆3 < 0.

3.2

(15)

First StageIn the first stage, the government chooses the level of privatization of public
firm 0. Substituting (10)-(12) into (2), the objective function of the govern9

Lump-sum transfer from consumer to firm 0 may be conducted when π0 takes anegative value.

7

ment in the first stage is obtained as
W =

a2 Γ
,
2∆2

(16)

where
Γ ≡ n(2n+7)αθ2 +6nαθ+2(n2 +2n+4)θ+(3n2 +8n+8)−n(2n+1)α−n(n+4)θ 2 .
The maximization of (16) with respect to θ yields∂W/∂θ = 2a2 Ω/∆3 ,(17)

where Ω ≡ n2 (n − 1)α2 + n(2n2 + 12n + 9)αθ + n2 (n + 1) − n2 (n + 5)α2 θ −n(2n2 − 1)α − (n3 + 7n2 + 10n + 4)θ. Setting ∂W/∂θ = 0, we obtain the

optimal degree of privatization as10θ=n(2n2 − 1)α − n2 (n − 1)α2 − n2 (n + 1).
n(2n2 + 12n + 9)α − n2 (n + 5)α2 − (n3 + 7n2 + 10n + 4)

(18)

From (18), we have the main results of this paper.

Proposition 1. Privatization is set back with the relaxing the regulation
on foreign investment; ∂θ/∂α > 0.
Proof. From (18), we have2n(n + 2)(nα − n − 1)2
∂θ
.
=
∂α
[n(2n2 + 12n + 9)α − n2 (n + 5)α2 − (n3 + 7n2 + 10n + 4)]2
For any n > 1 and 0 ≤ α ≤ 1, ∂θ/∂α > 0.
10

We can easily confirm that θ is in [0,1] for n > 1 and 0 ≤ α ≤ 1, and the second-ordercondition is satisfied, ∂ 2 W/∂θ 2 < 0.

8

Proposition 2. Privatization is promoted by an increase in the number offirms operating in the market; ∂θ/∂n > 0.
Proof. From (18), we have
Φ
∂θ
,
=
2
3
∂n
[(2α − α − 1)n + (12α − 5α2 − 7)n2 + (9α − 10)n − 4]2
where Φ ≡ 6(1 − α)4 n4 + 20(1 − α)3 n3 + 2(2α3 + 7α2 − 20α + 11)n2 +8(1 − α2 )n + 4α. For any n > 1 and 0 ≤ α ≤ 1, ∂θ/∂n > 0.

The above propositions show that the market-opening policy measured
by the increases in the share of foreign investment in private firm gives thegovernment less incentive to privatize its own public firms. This is simply
because if a private firm is domestically owned, its profit remains in the country. This provides the government with an incentive to privatize public firm.
By contrast, if foreign capitalists own private firms, private firm profits flowout to foreign countries, so that public firm decreases the degree of privatization so as not to yield profits to private firms and to keep their profits in the
country. The market-opening policy measured by the increase in the numberof private firms gives us an opposite result. It gives the government more
incentive to privatize its own public firms. The two diﬀerent strategies for
opening market result in the opposite impacts on the government’s incentive
for privatization.

4

Concluding Remarks

In this paper, we investigated the eﬀects of market-openings policies on the
degree of privatization in the mixed oligopoly market. A series of mixed
oligopoly analyses indicate that the existence of foreign investment considerably changes the equilibrium outcome. Nevertheless, market-openings strategies in the mixed market model with partial privatization have been nearly9neglected. By introducing a change in the levels of regulation on foreign
investment, we have shown alternative features of the degree of privatizationin mixed market.
We first show that the relaxation of restriction on foreign investment setsback privatization of public firm. Second, with the increase in the numberof private firms that are jointly owned by domestic and foreign investors,
the government increases the degree of privatization. In this sense, while
the market-opening policies called for by international agreement, such as
WTO, tend to set back the privatization of public firm, the increase in thenumber of firms operating in the market accelerates the privatization. The
two diﬀerent strategies for market-openings result in the opposite impacts on
the government’s incentive for privatization.
Finally, it should be noted that some of our assumptions could be relaxedwithout changing the main results of this paper. Specifically, although ourassumption of a quadratic cost function allows us to derive clear-cut results,
the assumption of constant marginal cost would only change the results in aquantitative sense unless public firm has less eﬃcient technology than private
firms. Furthermore, since our interest is the exogenous change in the regulation of foreign direct investment, we simply assume that the government
only controls the degree of privatization. However, the government may use
the level of foreign investment regulation as policy variables. The extensionof the present model, which is left for a future investigation, to the one thatconsiders the endogenous determination of FDI regulation and privatization
could well provide insightful implications on the optimal policies taken in themixed oligopoly market.

10References
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