Institute of Social Science, University of Tokyo
Discussion Paper Series No. F151

Competition and Privatization Policy:
The Relative Performance Approach
Toshihiro Matsumura
Institute of Social Science, University of Tokyoand
Makoto Okamura∗Economics Department, Hiroshima University
Discussion Paper Series No. F-151
August, 28, 2009

Abstract
We investigate the relationship between competition and privatization policies. Existing
works measured the toughness of competition by the number of ﬁrms and showed that the
optimal degree of privatization is higher when the number of ﬁrms is larger. We adopt therelative proﬁt approach where each ﬁrm maximizes the relative proﬁt rather than their own
proﬁt, and the resulting competition becomes tougher when ﬁrms care about the rivals’ proﬁt
more. We ﬁnd that the optimal degree of privatization is smaller when each ﬁrm cares about
the rival’s proﬁt more (so the market competition is tougher).

JEL classification numbers: H42, H44, L13, L32
Key words: partial privatization, relative proﬁt, mixed oligopoly, competition policy∗

Correspondence author: Makoto Okamura, Economics Department, Hiroshima University, 1-2-1, Kagamiyama,

Higashihiroshima-City, Hiroshima, 739-8525, Japan. E-mail: andrew-uf@w3.dion.ne.jp11

Introduction

Since the 1980s, we have observed a worldwide wave of privatization. Nevertheless, public andsemipublic (partially privatized) ﬁrms are still active. Many of them compete with private ﬁrms indeveloped, developing, and transitional economies.1 Recently, studies on mixed oligopoly involving
both private and public enterprises, have increasingly become popular.2 In the literature on mixed
oligopoly, the privatization of public enterprises is still an important research topic. In addition,in near future, the privatization of temporally nationalized ﬁrms in recent ﬁnancial crisis must
become important political and economic issues. In this paper we adopt partial privatization
model formulated by Matsumura (1998) and discus the optimal degree of privatization in mixed
oligopoly.3
The privatization of public ﬁrms often takes place in conjunction with reforms of the competition policy. Thus, the relationship between the privatization policy and the market competition
in a mixed oligopoly has attracted considerable attention among researchers. The existing works
on mixed oligopoly show that the privatization is more likely to improve welfare when the numberof private ﬁrms is large.4 However, since these works adopt the assumption of increasing marginal
1
2

See, e.g., Matsushima and Matsumura (2003).
This interest in mixed oligopolies is due to their importance in relation to European, Canadian, and Japanese

economies. Although they are less signiﬁcant in the United States, there are some examples of mixed oligopolies
such as the packaging and overnight-delivery industries. The analysis of mixed oligopoly dates back to at least
Merrill and Schneider (1966). Recently, the literature on mixed oligopoly has become richer and more diverse. See
Ishibashi and Matsumura (2006), Gil-Moltó and Poyago-Theotoky (2008) and Ishida and Matsushima (2009) andthe works cited by them for recent development in this ﬁeld.
3

Partial ownership in privatized ﬁrms is observed all over the world, and has become increasingly popular.

Partial privatization has been intensively discussed in the literature. For a general discussion, see Matsumura(1998). Also see Chang (2005), Chao and Yu (2006), and Fujiwara (2006) in the context of international trade,
Kato (2006) and Ohori (2006) for environmental problems, Tomaru (2006) for tax and subsidy policies, Matsumura
and Kanda (2005) and Fujiwara (2007) in free entry markets, Lu and Poddar (2007) for product diﬀerentiation, andBárcena-Ruiz and Garzón (2003) for merger problems.
4See, De Fraja and Delbono (1989), Fjell and Pal (1996), and Han and Ogawa (2008).

2

costs, the cost structure in the industry changes when the number of private ﬁrms changes.5 We
investigate the relationship between market competition and privatization policy by adopting a
diﬀerent approach: the relative proﬁt (performance) approach.
We now explain the relative proﬁt approach we adopted. We consider a model where ﬁrms
are concerned about relative proﬁts rather than their own proﬁts. Consider a symmetric n ﬁrm
private oligopoly where all ﬁrms are private. Firm i’s payoﬀ is πi − α( j=i πj ), where πi is ﬁrm
i’s proﬁts. Firms independently choose their outputs. Then, the equilibrium outcome converges
to the monopoly one when α converges to −1, it becomes the Cournot equilibrium when α = 0,
and it converges to the perfectly competitive one (Walrasian) when α is close to 1/(n-1). Thus,
we can interpret α as a parameter indicating the severity of competition, a larger α indicates amore competitive market.6 The relative performance approach enables us to treat the degree ofmarket competition as a continuous variable.7 We introduce this approach into a mixed oligopoly
setting and discuss the relationship between α and the optimal degree of privatization. We ﬁndthat optimal degree of privatization is decreasing in α, i.e., a tougher competition reduces the
optimal degree of privatization. This result makes sharp contrast to the existing result that anincrease in the number of ﬁrms increases the optimal degree of privatization.
We discuss the rationale for employing objective functions based on relative performance in5

If the marginal cost is constant and all ﬁrms have identical cost functions, the number of ﬁrms does not

aﬀect the cost structure in the industry. Note that most works on mixed oligopoly including above three papers
do not adopt this assumption. For the reason the mixed oligopoly literature does not adopt this assumption, see
Matsumura (1998).
6

For the related discussions, see Symeonidis (2000, 2008). Under the standard conditions in Cournot oligopoly,

the ratio between proﬁt margin (price minus marginal cost) and the price, known as Lerner index, is decreasing in
α. This index is intensively used in the empirical literature as a measure of market competition in product markets.
7

The conjectural variation approach is another approach that contains three models as special cases. However,the conjectural variation model assumes that ﬁrm 1’s output aﬀects that of ﬁrm 2 and vice versa. Needless to say,this assumption is inconsistent in any static model. The relative performance approach does not have this defect.
This is an advantage of this approach.

3

a general context. First, evaluations of managerial performances are often based on the relative
performance of managers as well as on their absolute performance. Outperforming managers
often obtain good positions in the management job markets. This provides the rationale forconsidering positive α in our model. Second, many laboratory (experimental) works pointed out
the spiteful behavior as well as reciprocal behavior or altruistic behavior, which is closely related
to the objective functions based on relative performance (both positive and negative α).8 Third,
relative performance, especially the positive α case, is quite important from the viewpoint of
evolutionary stability.9 Fourth, if private stockholders use α strategically, they adopt a positive
α.10 We believe that the relative performance approach is an important approach in the context
of mixed oligopoly, too.

2

The Model

Firms produce perfectly substitutable commodities for which the market demand function is given
by p = 1 − Q (price as a function of quantity). Firm i’s cost function is given by ci (qi ) = (c/2)(qi )2
where qi is the output quantity of ﬁrm i (i = 0, 1..., n) and c is a positive constant.11 We assume
that c > 1/2 so as to ensure the interior solution in the following game. Firm 0 is a semipublic(partially privatized) ﬁrm, and it competes against n private ﬁrms in the domestic product market.

Let πk (k = 0, 1, .., n) be ﬁrm k’s proﬁt. Let Vi := πi − α( j=i πj ) be ﬁrm i’s relative proﬁts.
89

See Coats and Neilson (2005), among others.
See Alchian (1950) and Vega-Redondo (1997). Vega-Redondo (1997) also shows that Cournot competition withrelative performance objectives yields the Bertrand outcome even in duopoly and that this outcome is evolutionary
stable.
10

See Kockesen et al. (2000). The payoﬀ function based on relative wage or relative wealth status has been

intensively discussed in the macroeconomics context, as well. Keynes (1936) discussed the rigidity of nominal wage
based on relative wage. See also Akerlof and Yellen (1988) and Corneo and Jeanne (1997, 1999). and Futagami andShibata (1998).
11This formulation of demand and cost is popular in the mixed oligopoly literature. See, among others, De Fraja

and Delbono (1989), Fjell and Pal (1996), Ogawa (2006), and Pal and White (1998).

4

Firm i’s (i = 1, 2, ..., n) payoﬀ is given by Vi and ﬁrm 0’s payoﬀ is given by U0 = θV0 + (1 − θ)W,
where W is the total social surplus (consumer surplus + the proﬁts of ﬁrms).12 θ ∈ [0, 1] represents
the degree of privatization.13 We assume that α ∈ [−1/n, 1/n). If α = 1/n, the equilibriumoutcome is Walrasian. We exclude the possibility of a negative price-cost margin (price minus
marginal cost) by assuming that α < 1/n.
In the ﬁrst stage, the government sells shares in ﬁrm 0 and chooses θ. In the second stage,
after observing θ, each ﬁrm i chooses qi (i = 0, 1, ..., n) independently (Cournot competition). Letn2i=0 qi be Q. The proﬁt of ﬁrm i is given by πi = p(Q)qi − ci (qi ) = (1 − Q)qi − cqi /2.

3

Equilibrium Analysis and Results

We adopt the subgame perfection as the equilibrium concept and solve the game by backwardinduction. First, we investigate the second stage competition. Let Q−i := j=i qj be the total
outputs of ﬁrms other than ﬁrm i. The ﬁrst-order conditions of ﬁrm 0 and ﬁrm i (i = 1, 2, ..., n)are
1 − (c + 1 + θ)q0 − (1 − αθ)Q−0 = 0,(1)

1 − (1 − α)Q−i − (c + 2)qi = 0 (i = 1, 2, ..., n).

(2)

Let superscript S be the second stage equilibrium. Solving the above equations yields
q0S =
12

c + 1 − α(n − 1) + αnθ,Hq1S = q2S = ..., = qnS =

c+α+θ
,
H

(3)

Another possible formulation of payoﬀs is V0 = π0 and Vi = πi − α(π1 +, ..., πi−1 + πi+1 +, ..., +πn ) wherei = 1, 2, ..., n (each private ﬁrm is concerned with the private rivals’ proﬁts only). Proposition holds in this setting
unless c is too small.
13

This is a standard formulation of the payoﬀ for a semipublic ﬁrm, if we replace Vi with πi . See Matsumura

(1998). See also Fujiwara (2007) for recent works adopting this approach. We assume that W is consumer surplus
plus proﬁts of ﬁrms rather than consumer surplus +ni=0 Vi . CEOs of ﬁrms might be concerned about the relative

performance of their ﬁrms because the good relative perform increases their future income, but we simply regard
this is just as income transfer and consider higher (lower) valuation for one CEO does not produce any net incomein the society.

5where H := (c + 1)(c + 1 + α) + cn(1 − α) + θ(c + n + 1 + α − α2 n).
Second, we consider the ﬁrst stage. The government chooses θ so as to maximize W . Let θ ∗be the optimal θ. The ﬁrst-order condition is1(1 − αn)2 (c + 1 + α)(−θJ + n(1 − α)c) = 0,J3

(4)

where J := (c + 1 + α)(c + n + 1 − α(n − 1)) − n(1 − α)(1 + α) > 0. From (4), we have:θ∗ =nc(1 − α)∈ (0, 1).
J

(5)

We present our result.
Proposition (i) θ ∗ is decreasing in α. (ii) θ ∗ is increasing in n.
Proof From (5), we have
nc 
2c
∂J 
∂θ ∗
= − 2 J + (1 − α)
= − 2 (c2 + 4c + (3 − α)(1 + α)) < 0
∂α
J
∂α
J
c(1 − α) ∂J  c(1 − α)(c + 1 − α)2
∂θ ∗
=
=
>0

J
−
n
∂n
J2
∂n
J2

Proposition (i) and (ii) present a sharp contrast. Proposition (ii) is similar to the existing result(an increase of the number of private ﬁrms is more likely to improve welfare).14 Suppose thatθ = 0. The marginal cost of public ﬁrm is equal to the equilibrium price, while that of private ﬁrm
is strictly lower than the price. Thus, in equilibrium the public ﬁrm’s marginal cost is higher thanthe private ﬁrm’s. An increase of θ reduces the public ﬁrm’s output and increases the other ﬁrms’
outputs. In other words, privatization induces the production substitution from the public ﬁrm
to the private ﬁrms. Since the marginal cost of private ﬁrm is lower than that of the public ﬁrm,
this production substitution economizes the total production costs, thereby improving welfare.
14

For example, De Fraja and Delbono (1989) discuss the case of α = 0 and show that there exists m such thatprivate oligopoly (θ = 1) yields a larger welfare than the mixed oligopoly θ = 0 if and only if n > m.

6

At the same time, the privatization reduces the total output, and hence consumer surplus. This
tradeoﬀ determines the optimal degree of privatization θ ∗ . An increase in the number of private
ﬁrms reduces the output of each private ﬁrm (and hence reduces the ﬁrm’s marginal cost). Thus,
an increase in the number of private ﬁrms strengthens the welfare-improving eﬀect of production
substitution mentioned above. This yields Proposition (ii).
On the contrary, given the number of private ﬁrms, an increase in α accelerates competition
and increases the output of each private ﬁrm (thereby increasing the ﬁrm’s marginal cost). Thus,
the welfare gain from production substitution from the public ﬁrm to the private ﬁrms is smallerwhen α is larger. In other words, if the competition is more severe in the market, the welfare gainfrom production substitution induced by the privatization is smaller. This yields Proposition (i).

4Concluding Remarks

In this paper we adopted a relative performance approach and investigate the relationship between
competition and privatization policy. We ﬁnd that an increase of the number of private ﬁrms
increases the optimal degree of privatization. However, given the number of ﬁrms, an increase of
the intensity of competition reduces the optimal degree of privatization. The literature on freeentry markets emphasized that measuring the intensity of competition by the number of ﬁrms
yields misleading implications.15 More intensive competition makes the market less proﬁtable.
As a result, tougher competition results in a smaller number of entering ﬁrms. If we incorporate
this eﬀects, our result that the optimal degree of privatization can be smaller under a tougher
competition is strengthen. In this paper we measure the intensity of competition by α (the weightof relative performance in the payoﬀ of ﬁrms). An increase in α directly reduces the optimal
degree of privatization θ ∗ . At the same time, an increase in α reduces the number of ﬁrms at freeentry market, and it indirectly reduces θ ∗ . Thus, we guess that at free entry market, a tougher
15

See Etro (2007).

7

competition is more likely to reduce the optimal degree of privatization.16
In this paper we assume that all private ﬁrms are domestic. However, public enterprises
often compete against both domestic and foreign private ﬁrms. The competition against foreignﬁrms often yields diﬀerent implications in mixed oligopoly. Incorporating this aspect and fully
investigate the relationship of three policies, privatization, competition, and international trade
policies remains for future researches.17

16

For the discussions of free entry market in mixed oligopoly, see Anderson et al (1997), Brandão and Castro(2007), Fujiwara (2007), and Matsumura and Kanda (2005).
17

For the discussion of foreign private ﬁrms, see Bárcena-Ruiz and Garzón (2005a,b), Corneo and Jeanne (1994),

Fjell and Heywood (2002), Heywood and Ye (2009), Inoue et al (2009), Matsumura (2003), and Ogawa and Sanjo(2007).

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11