Modern Economy, 2011, 2, 45-48
doi:10.4236/me.2011.21007 Published Online February 2011 (http://www.SciRP.org/journal/me)
Partial Privatization in Price - Setting Mixed Duopolies
with Complementary Goods
Kazuhiro Ohnishi
Institute for Basic Economic Science, Osaka, Japan
Received October 7, 2010; revised October 22 , 20 10; accepted October 29, 2010
Abstract
We consider a domestic (resp. international) mixed duopoly model in which a domestic public firm and a
domestic (resp. foreign) private firm produce complementary goods. First, the domestic government chooses
the level of privatization to maximize domestic social welfare. Second, observing the level of privatization,
the firms simultaneously and independently choose prices. We present the equilibrium outcomes of the two
mixed duopoly models and shows that our result is in marked contrast to that of the price-setting mixed du-
opoly model with substitute goods.
Keywords: Partial Privatization, Price - Setting Model, Domestic Mixed Duopoly, International Mixed
Duopoly, Complementary Goods
1. Introduction
The theoretical analysis of partial privatization of state-
owned public firms has received significant attention in
recent years and has been extensively studied by many
economists, such as [1-17].
However, these studies analyze partial privatization in
mixed duopoly competition in which public and private
firms produce substitutable goods. To the best of my
knowledge, the analysis of partial privatization in Ber-
trand mixed markets with public and private firms pro-
ducing complementary goods has been ignored.
Therefore, we study partial privatization in price-set-
ting mixed duopoly competition with complementary
goods. We extend the analysis of Ohnishi [17], which
investigates a price-setting mixed duopoly model in-
volving a domestic public firm and a domestic private
firm to reassess the welfare effect of partial privatization.
Ohnishi demonstrates that partial privatization is not a
reasonable choice for the government that wishes to
maximize social welfare.
We consider both domestic and international mixed
duopoly models with complementary goods.1 We con-
sider the following situation. In the first stage, the do-
mestic government chooses the degree of privatization to
maximize domestic social welfare. In the second stage,
observing the degree of privatization, the firms simulta-
neously and independently choose prices.
The main purpose of this paper is to present the equi-
librium outcomes of the two mixed duopoly models and
to show that this result is in marked contrast to that of the
price - setting mixed duopoly model with substitute
goods.
The remainder of the paper is organized as follows. In
Section 2, we examine a domestic mixed duopoly model.
Section 3 examines an international mixed duopoly
model. Section 4 concludes the paper.
2. Domestic Mixed Duopoly
In this section, we consider a domestic mixed model with
two firms (firm P and firm D) and the government. These
firms produce complementary goods. There is no possi-
bility of entry or exit. On the consumption side, there is a
continuum of consumers of the same type whose utility
function is linear. Subscripts P and D denote firm P and
firm D, respectively. Following Bárcena-Ruiz [18], we
assume that the representative consumer maximizes


PDPP DD
,Uq qpqpq , where i q is the amount of
1As is well known, international mixed oligopolies are common in
developed and developing countries as well as in former communist
countries. Public firms compete against foreign private firms in many
industries, such as banking, life insurance, automobiles, airlines, steel,
shipbuilding and tobacco. For example, in the tobacco industries o
f
France, Italy, Russia, Spain, Austria, Turkey, China, Japan, etc, we can
find real-world examples in which public firms compete or competed
against foreign private f i r ms such as Philip Morris and R. J. Reynolds.

K. OHNISHI
46
good i and i p is its price

P,Di. The function

PD
,Uq q is quadratic, strictly concave and symmetric
in P q and D q:
 

22
PDP DPPD D
1
,2
2
Uq qaqqqbqqq  (1)
where a is a constant and


1, 0b is a measure of the
degree of complementarity among products. The demand
function is gi v e n by

2
1 (,P,D; )
1
ij
i
abpbp
qijij
b


 (2)
For simplicity, we assume that 1a and 0.5b

.
Each firm’s profit is

(P,D),
iiii
pcq i   (3)
where i
c is the marginal cost of firm i. Since the result
of this paper is not affected by i
c, we normalize it to
zero. Firm D aims to maximize its profit. Furthermore,
domestic social welfare, which is the sum of consumer
surplus and profits, is given by
PD
WCS

 . (4)
The objective function of firm P is given by

P1,VW
   (5)
where


0,1   is the level of privatization. That is, if
0

 firm P is purely public, whereas if 1


it is
purely private.
The game is constructed by the following two-stage
decision-making. In the first stage, the government
chooses the level of privatization,

, to maximize do-
mestic social welfare. Observing

, the firms non-co-
operatively choose prices in the second stage. Through-
out this paper, the subgame perfect Nash equilibrium of
the price-setting game is examined.
We now examine the welfare effect of partial privati-
zation in the domestic mixed duopoly game. We obtain
the reaction functions in prices of the two firms:


D
P
332 ,
23
p
R



 (6)
P
D3
4
p
R  (7)
From (6) and (7), the equilibrium can be derived as
follows:




PD
PD
12 52
,
72 72
47 4452
,
37 237 2
pp
qq












Comparative static results yield P0dp d

,
D0dp d


, P0dq d


, and D0dq d

. Thus, the
privatization decreases each firm’s output and firm D’s
price, and increases firm P’s price.
Furthermore, the profits, consumer surplus, and social
welfare can be expressed as follows:





P2
41 274,
37 2






 (8)


2
D2
45 2,
37 2





 (9)



2
2
829 192,
37 2
CS




 (10)
In the first stage, the government chooses the level of
privatization. Substituting (8), (9) and (10) into (4), the
objective function of the government is obtained as



2
815 8.
72
W




 (11)
Social welfare W is illustrated as a function of

.
When 0


,


22249 2.449W, and when 1


,


26 252.24W
. In addition, when 0.1

, W



2 1322892.457. This can be stated in the following
proposition.
Proposition 1. In the domestic mixed market with
complementary goods, neither full nationalization nor
full privatization is a reasonable choice for the govern-
ment that wishes to maximize domestic social welfare;
that is, the optimal solution is partial privatization.
3. International Mixed Duopoly
In this section, we consider an international mixed du-
opoly model in which a state-owned public firm (firm P)
competes against a foreign private firm (firm F). Sub-
scripts P and F denote firm P and firm F, respectively. In
addition, an asterisk denotes the international mixed du-
opoly model.
Firm F aims to maximize its own profit. Furthermore,
domestic social welfare is given by
***
P.WCS

 (12)
The objective function of firm P is given by

** *
P1,VW
   (13)
where


0, 1   is the level of privatization. That is, if
0


firm P is purely public, whereas if 1


it is
purely private. The timing of this model is identical to
the domestic mixed duopoly model.
We examine the welfare effect of partial privatization
in the international mixed duopoly model. We obtain the

K. OHNISHI
47
reaction functions in pr i ces of the two firms:


*
F *
P
32 ,
23
p
R



 (14)
*
*P
F3.
4
p
R  (15)
From (14) and (15), the equilibrium can be derived as
follows:
 
 
**
PF
**
PF
3235 2
, ,
22 722 7
213745 2
, .
22 722 7
pp
qq














Comparative static results yield *
P0dp d
 ,
*
F0dp d
 , *
P0dq d
 , and *
F0dq d
 . Thus, the
privatization decreases each firm’s output and firm F’s
price, and increases firm P’s output.
Furthermore, the profits and consumer surplus can be
expressed as follows:
 

*
P2
6213 7,
22 7






 (16)


2
*
F2
12 52,
22 7





 (17)


2
*
2
2 36726340.
22 7
CS




 (18)
Substituting (16) and (18) into (12), the objective
function of the government is obtained as


2
*
2
18 55343.
22 7
W





 (19)
Social welfare *
W is illustrated as a function of

.
When 0

,

*211 2422.045W, and when 1


,

*12325 1.92W. In addition, when 0.2


,

*2526 106092.050W. This is stated in the fol-
lowing proposition.
Proposition 2. In the international mixed market with
complementary goods, neither full nationalization nor
full privatization is a reasonable choice for the govern-
ment that wishes to maximize domestic social welfare;
that is, the optimal solution is partial privatization.
4. Conclusions
We have studied partial privatization in price-setting
mixed duopolies with complementary goods. First, the
government chooses the level of privatization to maxi-
mize social welfare. Second, observing the level of pri-
vatization, the firms simultaneously and independently
choose prices. We have then presented the equilibrium
outcomes of the two mixed duopoly markets. We have
demonstrated that in each market, neither full nationali-
zation nor full privatization is a reasonable choice for the
government that wishes to maximize domestic social
welfare; that is, the optimal solution is partial privatiza-
tion. We have found that this result is in marked contrast
to that of the price-setting mixed duopoly model with
substitute goods.
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