Modern Economy, 2010, 1, 43-46
doi:10.4236/me.2010.11003 Published Online May 2010 (http://www.SciRP.org/journal/me)
Price-Setting Mixed Duopoly Models with
Complementary Goods
Kazuhiro Ohnishi
Institute for Basic Economic Science, Osaka, Japan
Received March 4, 2010; revised March 25, 2010; accepted April 5, 2010
Abstract
This paper considers domestic (resp. international) Bertrand mixed duopoly competition in which a state-
owned welfare-maximizing public firm and a domestic (resp. foreign) profit-maximizing private firm pro-
duce complementary goods. The main purpose of the paper is to present and to compare the equilibrium
outcomes of the two mixed duopoly models.
Keywords: Complementary Goods, Price Competition, Domestic Mixed Duopoly, International Mixed
Duopoly
1. Introduction
The analysis of mixed market models that incorporate
state-owned welfare-maximizing public firms has received
considerable attention in recent years and has been widely
performed by many researchers. However, most studies
consider quantity competition, such as [1-16].1
Some studies consider mixed markets with price com-
petition, such as [22-26]. These studies examine price-
setting mixed market models with domestic firms and do
not include foreign firms. Ohnishi [27] considers an in-
ternational mixed market in which a state-owned public
firm competes on price with a foreign private firm.
However, this study examines mixed duopoly competi-
tion in which public and foreign private firms produce
imperfectly substitutable goods. To the best of my know-
ledge, the analysis of international Bertrand mixed mar-
kets with public and foreign private firms producing
complementary goods has been ignored.
Therefore, we analyze the behavior of a state-owned
public firm and a foreign private firm in an international
price-setting model with complementary goods. We con-
sider both domestic and international mixed duopoly
models with complementary goods. The main purpose of
this paper is to present and to compare the equilibrium
outcomes of the two mixed duopoly models.
The remainder of the paper proceeds as follows. In
Section 2, we formulate a domestic Bertrand mixed du-
opoly model in which a state-owned public firm and a
domestic private firm produce complementary goods. We
show the slope of each firm’s reaction curve in the do-
mestic Bertrand mixed duopoly model with complemen-
tary goods. We present the equilibrium outcome of the
domestic Bertrand mixed duopoly model with comple-
mentary goods. Section 3 analyzes the equilibrium out-
come of an international Bertrand model in which a
state-owned public firm and a foreign private firm pro-
duce complementary goods. Section 4 compares the
equilibrium outcomes of the two mixed duopoly models.
Finally, Section 5 concludes the paper.
2. Domestic Mixed Duopoly with
Complementary Goods
In this section, we consider a market in which one state-
owned welfare-maximizing public firm and one domestic
profit-maximizing private firm produce complementary
goods. The basic structure is from Bárcena-Ruiz [25]. In
the remainder of this paper, subscripts S and D denote
the state-owned firm and the domestic private firm, re-
spectively. There is no possibility of entry or exit. On the
consumption side, there is a continuum of consumers of
the same type whose utility function is linear. The repre-
sentative consumer maximizes ,
where is the amount of good and is its price

SDSS DD
,Uq qpqpq
ii p
i
q


DS,i. The function is quadratic, strictly
concave and symmetric in and :

Uq
q

SD
,q
S D
q


SD
,Uq q





2
SSD
q qqq 
2
20a
SD
qD aq , where . The de-
1See, for example, [17-21] for excellent surveys.

K. OHNISHI 44
mand function is given by


23 2,S,D;
3
ij
i
app
qij

ij

D
. (1)
Each firm’s profit function is given by
 
S,D
ii i
pcq i   , (2)
where is the total cost for each unit of output. We
assume to assure that the production levels of
firms are positive. Domestic social welfare, which is the
sum of consumer surplus (CS) and profits, is given by
0c
ca
S
WCS

 . (3)
From (2) and (3), we derive the following reaction
functions in prices:
D
S
3
2
cp
R , (4)
S
D
32
4
acp
R . (5)
From (4) and (5), we can state the following:
Lemma 1. In the domestic Bertrand mixed market
with complementary goods, each firm’s reaction function
is downward sloping.
From (4) and (5), the equilibrium can be derived as
follows:
S
10 3
7
ca
p , (6)
D
6
7
ac
p , (7)

S2qa

c, (8)

D
8
7
ac
q . (9)
Furthermore, the profits and consumer surplus can be
expressed as follows:



S
6
7
acca


, (10)

2
D
48
49
ac


, (11)

2
37
49
ac
CS  . (12)
From (10), we see that the state-owned firm’s profit is
negative. Substituting (10), (11) and (12) into (3), social
welfare is obtained as

2
43
49
ac
W . (13)
From (12) and (13), we see that though the state-owned
firm’s profit is negative, social welfare is higher than
consumer surplus.
3. International Mixed Duopoly with
Complementary Goods
In this section, we consider a market in which one state-
owned welfare-maximizing public firm and one foreign
profit-maximizing private firm produce complementary
goods. In the remainder of this paper, subscript F denotes
the foreign private firm. The utility, demand, and profit
functions are the same as those of the previous section.
Domestic social welfare, which is the sum of consumer
surplus and the state-owned firm’s profit, is given by
S
WCS


. (14)
From (2) and (14), we derive the following best re-
sponse:
S
Rc

, (15)
S
F
32
4
acp
R . (16)
From (15), we can state the following:
Lemma 2. In the international Bertrand mixed market
with complementary goods, the slope of the state-owned
firm’s reaction curve is zero.
It is shown in Ohnishi [27] that the slope of the
state-owned firm’s reaction curve is zero in international
Bertrand mixed duopoly competition with imperfect sub-
stitutable goods. From (15), we see that the result of in-
ternational Bertrand mixed duopoly competition with
complementary goods is similar to that of international
Bertrand mixed duopoly competition with imperfect sub-
stitutable goods.
From (15) and (16), the equilibrium can be derived as
follows:
S
pc

, (17)
F
3
4
ac
p , (18)


S
3
2
ac
q , (19)
F
qac

. (20)
From (17), we know that the state-owned firm pro-
duces an output such that price equals marginal cost.
From (19) and (20), we see that the state-owned firm’s
output is higher than the foreign private firm’s output.
Furthermore, the profits, consumer surplus, and social
welfare can be expressed as follows:
S0


, (21)

2
F
3
4
ac


, (22)

K. OHNISHI 45

2
7
8
ac
CS  , (23)

2
7
8
ac
W . (24)
We see that since the state-owned firm’s profit is zero,
social welfare is equal to consumer surplus.
4. Comparisons
In this section, we begin by presenting comparative stat-
ics results. In the remainder of this paper, superscripts H
and I denote the domestic and international mixed du-
opolies, respectively. We have the following results.
H
S0
p
a

,
H
D0
p
a

,
H
S0
q
a

,
H
D0
q
a

,
I
S0
p
a

,
I
F0
p
a

,
I
S0
q
a

,
I
F0
q
a

.
In each model, a rise in raises the private firm’s
price. However, in the domestic mixed duopoly model, a
rise in lowers the state-owned firm’s price, and in the
international mixed duopoly model, a rise in has no
influence on the state-owned firm’s price. Furthermore,
we have
a
a
a
H
S0
p
c

,
H
D0
p
c

,
H
S0
q
c

,
H
D0
q
c

,
I
S0
p
c

,
I
F0
p
c

,
I
S0
q
c

,
I
F0
q
c

.
A rise in raises the firms’ prices and decreases
their outputs. We see that the comparative statics results
are almost the same in both models.
c
We now compare the equilibrium outcomes of the two
models. The main result of this study is described by the
following proposition.
Proposition 1. In the equilibrium outcomes of the
domestic and international mixed duopoly models, 1)
H
SS
I

 ; 2) H
D
I
F

 ; 3) ; and 4) .
H
CS CSIHI WW
Proof. 1) From (21), we see that . Furthermore,
from (10) and ,
I
S0
 
0ca H
S

is negative.
2) From (11), (22) and 0ca

, both H
D

and I
F

are positive, and thus


2
H
D
22
I
F
48
49
1923 147
1964 196
ac
acac ac




2
The proofs of 3) and 4) are omitted, since they are the
same as the proof of 2). Q.E.D.
Proposition 1 states that consumer surplus is higher in
the international mixed duopoly equilibrium than in the
domestic mixed duopoly equilibrium.
5. Conclusions
We have first considered a domestic Bertrand model in
which a state-owned welfare-maximizing public firm and
a domestic profit-maximizing private firm produce com-
plementary goods. We have shown that each firm’s reac-
tion function is downward sloping in domestic Bertrand
mixed duopoly competition with complementary goods.
In addition, we have found that the state-owned firm’s
profit is negative in equilibrium.
Second, we have considered an international Bertrand
model in which a state-owned public firm and a foreign
private firm produce complementary goods. We have
then shown that the slope of the state-owned firm’s reac-
tion curve is zero, and so social welfare is equal to con-
sumer surplus.
Third, we have compared the equilibrium outcomes of
the two mixed models. We have demonstrated that
though the state-owned firm’s profit, the private firm’s
profit, and social welfare are higher in the domestic
mixed duopoly equilibrium than in the international
mixed duopoly equilibrium, consumer surplus associated
with the international mixed duopoly equilibrium ex-
ceeds consumer surplus associated with the domestic
mixed duopoly equilibrium.
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