Theoretical Economics Letters, 2013, 3, 211-215
http://dx.doi.org/10.4236/tel.[phone] Published Online August 2013 (http://www.scirp.org/journal/tel)
Social Welfare under Quantity Competition and Price
Competition in a Mixed Duopoly with Network Effects:
An Analysis*
Yasuhiko Nakamura
College of Economics, Nihon University, Tokyo, Japan
Received May 6, 2013; revised June 6, 2013; accepted July 6, 2013
Copyright © 2013 Yasuhiko Nakamura. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In their recent work, Matsumura and Ogawa (2012) showed that in the context of a mixed duopoly, equilibrium social
welfare is higher in price-setting competition than in quantity-setting competition. We found that when the strength of
network effects is sufficiently high, the above result is totally reversed; thus, in a mixed duopoly, the presence of net-
work effects weakens the superiority of price-setting co mpetition with respect to equilibrium social welfare.
Keywords: Mixed Duopoly; Network Effect; Price Competition; Quantity Competition
1. Introduction
This study compares equilibrium social welfare between
price-setting competition and quantity-setting competi-
tion in a mixed duopoly with network effects. The net-
work effects that we consider in this paper were intro-
duced in Katz an d Shapiro [1] and applied in Hoernig [2 ],
Nakamura [3 ], and Nakamura [4]. These effects reflected
a simple mechanism where the surplus obtained by a
firm's client increases directly with the number of other
clients of this firm. In this study, we show that when the
strength of such network effects is sufficiently high, the
equilibrium social welfare is higher in a quantity-setting
competition than in a price-setting competition ; this find-
ing is strikingly different from those in th e existin g litera-
ture in this field including Ghosh and Mitra [5] and Ma-
tsumura and Ogawa [6].
Although we can consider the European automobile
industry as a representative example in a mixed oligopo-
listic industry, as described in Katz and Shapiro [1], net-
work effects as positive consumption externalities are
likely to arise in the automobile market1. In such an in-
dustry, foreign automobile firms’ sales may be less than
expected because of the perceptions that consumers hold
about the less experienced and smaller service networks
of new or less popular brands. Thus, in a mixed oligop-
oly, the presence of network effects as positive consump-
tion externalities sh ould be analyzed since they influ ence
the equilibrium market outcomes in both price-setting
and quantity-setting competitions.
By building a simple extention of the model from Ma-
tsumura and Ogawa [6] in line with Hoernig [2], Naka-
mura [3], and Nakamura [4], we find that a sufficiently
high level of network effects reverses the ranking order
of the equilibrium social welfare between price-setting
competition and quantity-setting competition. Thus, when
the strength of network effects is sufficiently high, the
equilibrium social welfare can be higher in quantity com-
petition than in price-setting competition, because the
relatively large total output levels in the former yield a
strictly positive influence on equilibrium social welfare
in accordance with the sufficiently high level of network
effects. Therefore, in a mixed market, the superiority of
price-setting competition is weakened if network effects
such as consumers’ expectations about each firm’s equi-
librium market share are sufficiently strong.
*We are grateful for the financial support by KAKENHI (25870113).
All remaining errors are our own.
1As one example, a Spanish public-owned automobile manufacturer,
SEAT competes with Volkswagen which is one of the most famous
German private automobile manufacturers. In addition, Renault which is
the most famous partia lly Fren ch public-owned fi rm compete s with many
p
rivate firms in Europe. Recentl
y
, Romanian private firm, Dacia merged
with Renault, and it became a s econd marque for t h e Renault gr o up . See
Barcena-Ruiz and Garzon [7] for other examples and detail discussions
of the European automobile industry as a mixed oligopolistic market.
2. Model
We formulate a mixed duopolistic model composed of
one public firm and one private firm, with an additional
term that reflects the network effects introduced in Katz
C
opyright © 2013 SciRes. TEL

Y. NAKAMURA 212
and Shapiro [1] and applied by Hoernig [2], Nakamura
[3], and Nakamura [4]. We assume that firm 0 is a
welfare-maximizing public firm, whereas firm 1 is a pure
profit-maximizing private firm. Similar to Hoernig [2],
Nakamura [3], and Nakamura [4], firm faces a linear
demand of the following form: i
and 0,1;,
iiij
qanypbpi ij (1)
where and are demand parameters.
indicates the strength of network effects, and
i is consumers’ expectations on firm ’s equilibrium
mark et sh are. As explained in Hoernig [2], Nakamura [3 ],
and Nakamura [4], the above demand system can be
derived from the following quasi-linear concave utility
function of a representative consumer:
0a

1

0,1b

0,n
y i





22
001 01
0101 2
2
010101 01
2
,;,11
21
,,
1
aq qqq bqq
Uqqy ymbb
b
ybyq ybyq
nfy
b

 

y





where m denotes the income of the representative con-
sumer and represents some symmetric function
of expectations. In this paper, in the same manner as in
Hoernig [2], Nakamura [3], and Nakamura [4], we sup-
pose that

,f


22
01001 1
,221
2
f
yynybyyyb 2.
The marginal cost of production of both firms 0 and 1
is commonly assumed to be c. The profit function of firm
i is given by ii i
, where
i is given as (1)3. Consumer surplus is expressed as the
representative consumer’s utility as follows:

pcq 

,0,1;iji j
q

010100 11
,;,CS Uqqyypqpq,
whereas producer surplus is given by the sum of the
profits of both firms 0 and 1, 01

 . Finally, we sup-
pose that social welfare is defined as the sum of consum-
er surplus and producer surplus.
We consider “rational expectations” to be a subgame
perfect Nash equilibrium by imposing the rational ex-
pectations condition that 00 yq

and 11 à la Katz
and Shapiro [1], Hoernig [2], Nakamura [3], and Naka-
mura [4].
yq
3. Welfare Analysis
3.1. p-p Game
In this subsection, we discuss the p-p game where firms 0
and 1 simultaneously choose their price levels. By
considering y0 and y1 as given, public firm 0 maximizes
the social welfare with respect to 0 whereas private
firm 1 maximizes its profit with respect to . In this
case, the social welfare is given as follows:
p
1
p














 

0101
23
22 01
0
22
2222222
1100 01
2
2222
01 11
2
01
2
2101
2
,;,
21 122
1
21 21
21 2
21
2
21
21 21
21
21 1.
21
Wppyy
pbbcbpbp
bp
bb
abpbpnyn ybnyy
b
bn y ynyn y
b
ab bcnyy
b
bcbpnyy
b
1


 



 

 











Note that the term of the price level of firm 0, 0 is
independent of the term of consumers’ expectations
about the market share of both firms 0 and 1, 0 and
14. On the other hand, the profit of firm 1 is given as
follows:
p
y
y



 
1010110 11
,;, .ppyypcabpp ny
The respective reaction functions of firms 0 and 1, ,
are given as follows i
r


0, 1i:


001 1
,prpcbcbp (2)




11010 1
;2prp yacbpny. (3)
Note that the price level of firm 0 does not depend on
consumers’ expectations about the market share of both
firms 0 and 1 in its reaction function.
We obtain the rational expectations Nash equilibrium
outcomes by substituting the rational expectations assump-
tion that 00
yq

and 11 yq

into Equations (2) and (3).
The equilibrium market outcomes namely the output and
price levels of firms 0 and 1, profit of firm 0, producer
surplus and consumer surplus are given as follows:


 
01
2
01
22
11
,,
12
21
,,
22
pp pp
pp pp
abc abc
qq
nbn
ab cbnabc
ppc
bn bn
 


 

 
2This assumption in the form of

,f

 i mplies that the representative
consumer’s utility is the highest with respect to the consumption vector
of the goods produced by both the public firm and the private firm,
, when expectations are rational and correct.

01
,qq

3We assume that

1>abc
4Thus, as described below, in firm 0’s reaction function,its price level
does not depend on consumers’ expectations about the market share o
f
both firms 0 and 1, and .
0
y1 y
0 in order to ensure the non-negati-
vit
y
of all e
q
uilibrium outcomes.

Y. NAKAMURA 213
 


2322
2
2
15 6232
.
21 12
pp
CS
abc bbnnb n
m
bnbn

 




 







2
02
22
2
2
1,
12
111,
12
pp
pp
ba bc
nbn
babcbb n
PS nbn



 


Furthermore, the payoffs of both firms 0 and 1 are
given as follows:
 


234 2
2
2
17 22474
,
21 12
pp abc bbbnnb n
Wm
bnbn

  




 


2
12
2
1
2
pp abc
bn


 
3.2. q-q Game social welfare with respect to q0 whereas private firm 1
maximizes its profit with respect to q1. In this case, the
social welfare is given as follows:
In this subsection, we discuss the q-q game where firms 0
and 1 simultaneously choose their output levels. By con-
sidering y0 and y1 as given, public firm 0 maximizes the










222
01 001101
0101 2
22
00100011101 1
2
212 21
,;, 21
222 22.
21
abqqqbqqq bcqq
Wqqy yb
nqybnq ynybnqynq ybnyyny
b




On the other hand, the profit of firm 1 is given as
follows:


0101
10101 12
1
,;, 1
abbqq bnyny
qqyy qc
b
 










11001
2001
,,
11
qrqyy
ab bcbqbnyny




2.
(5)
The respective reaction functions of firms 0 and 1,
are given as follows : i
r

0, 1i
 


2
001011 0
;, 1qrqyy bcbqny
1

We obtain the rational expectations Nash equilibrium
outcomes by substituting the rational expectations assump-
tion that 00 yq

and 11 yq

into Equations (4) and (5),
The equilibrium market outcomes including the output
and price levels of firms 0 and 1, profit of firm 0, and
consumer surplus are given as follows:
1abbny
(4)







2
01
2
2
11
1121
,,
21
211
qqqq qq
ba bc
babcbnn
qq
bnn
bnnn
 
 


 

 0
,pc






10
2
2
2
2
2
1,0
21
11
,
21
qq qq
qq
abc
pc bnn
ba bc
PS bnn

 







,
 
 
222
23
2
2
115221212123 .
212 11
qq babcbnnbnbnnn
CS m
bbnn n

 


 

 

Furthermore, the payoffs of both firms 0 and 1 are given as follows:

Y. NAKAMURA 214
  
 
22
24
2
2
174212124 ,
121 1
qq abc bbnnbnnn
Wm
bbnn n

 




 




2
2
12
2
11
.
21
qq ba bc
bnn









On comparing the equilibrium social welfare between price-setting competition and quantity competition, we obtain
the following result:
 




22
2422
2 2
22
14183483
0
22(1 )12
ppqq babcbnn n bnn
WW bnnn bn

  


 
 


2


  
242 2
24
442 10,1,0,1.
33
bbb b
nb
bb




Then, we obtain the proposition on the ranking order
of equilibrium social welfare between price-setting com-
petition and quantity-setting competition.
Proposition 1 When the strength of network effects in
a mixed duopoly is sufficiently high, that is,
 
242 224
>442133nbbbb b




b
1
, the
equilibrium social welfare is higher in quantity-setting
competition than in price-setting competition; otherwise
the opposite is the case.
In Figure 1, the difference in the equilibrium social
welfare between price-setting competition and quantity-
setting competition is described. Surprisingly, Proposi-
tion 1 implies that when the strength of network effects is
sufficiently high, the result on the ranking order of the
equilibrium social welfare between a price-setting com-
petition and a quantity-setting competition obtained in
Ghosh and Mitra [5] and Matsumura and Ogawa [6] is
totally reversed; hence, the equilibrium social welfare
can be higher in quantity-setting competition than in price-
setting competition. The intuition behind this result is
given by using the following lemma:
Lemma 1 For the rational expectations Nash equi-
librium in price and quantity competitions, the ranking
orders of the equilibrium market outcomes are given as
follows:
1) 001100 1 ,,,and
p
pqqppqqqqppppq
ppppqq qq 
 
and
qq qqpppp
qq qq
q
,
2) ,
  
01 01
01 01
qq qqpp pp
qq qq
3) and
p
pqq qqpp
PSPSCSCS.
Note that the above equality of the price level of
private firm 1 is satisfied only when 5. 0n
From Lemma 1, even if network effects are introduced,
the ranking orders of equilibrium outcomes except for
the price level of firm 1 are the same as those obtained in
Ghosh and Mitra [5]. The introduction of network effects
widens the difference in the price level of firm 0 between
price-setting competition and quantity-setting competi-
tion, implying that such a low level of 0 creates a
strong downward pressure on firm 1’s price level. Thus,
when takes some sort of positive value, firm 0’s price
level is strictly higher in price-setting competition than in
quantity-setting competition. Therefore, in particular
when the strength of network effects is sufficiently low,
since the ranking order of the equilibrium market out-
comes seldom changes against the introduction of net-
work effects, the equilibrium social welfare is higher in
price competition than in quantity competition owing to
the large difference of the equilibrium producer surplus
re lative to that of the equilib ri um consumer surplus, which
is similar to Ghosh and Mitra [5] and Matsumura and
Ogawa [6].
qq
p
n
On the other hand, by following the formula of equi-
librium social welfare à la Ghosh and Mitra [5], and by
imposing the rational expectations condition that y0 = q0
and y1 = q1, we can represent the equilibrium social
welfare as follows:

  
010011010 1
,; ,,Wqqyqyqsqqdqq

  


  
   
01 01
2
01
2
01 01
11
where141,
141.
s
qq abcqqb
nq qb
dq qnqqb


 


 




As explained in Ghosh and Mitra [5] and Matsumura
and Ogawa [6], on the basis of the facts that


0s


and


0d

, the effect of on the equi-
librium social welfare outstrips the effect of

01
dq q



01
s
qq
when 0n

; consequently, the equilibrium social wel-
5When , the fact that such an equality holds is indicated in Ghosh0n
and Mitra [5].

Y. NAKAMURA 215
Figure 1. Comparison of the equilibrium social welfare be-
tween price-setting competition and quantity-setting com-
petition.
fare is higher in price-setting competition than in quan-
tity-setting competition if no network effects are con-
sidered. However, as becomes higher, in a quantity-
setting competition, the positive effect of
n


01
s
qq
owing to

on the equilibrium

01 01
qq qqpppp
qq qq




social welfare increases. On the other hand, the negative
effect of owing to
on the equilibrium social welfare decreases. Therefore,
the degree of the strength of network effects, n, reverses
the order of the equilibrium social welfare between price-
setting and quantity-setting competitions.

01
dq q

01 01
qq qqpppp
qq qq
4. Conclusions
This study compared the equilibrium social welfare be-
tween price-setting competition and quantity-competition
in a mixed duopolistic market with network effects. In
Ghosh and Mitra [5] and Matsumura and Ogawa [6],
where network effect is not considered, it was shown that
the equilibrium social welfare is always higher in price-
setting competition than in quantity competition6. How-
ever, in this paper, when the strength of network effects
is sufficiently high, we found that the ranking order of
the equilibrium social welfare between price-setting com-
petition and quantity-setting competition is totally revers-
ed. Thus, when network effects are explicitly considered,
the superiority of price competition with regard to social
welfare weakens in a mixed duopolistic market.
Furthermore, by following Matsumura and Ogawa [6],
where public firm 0 and private firm 1 can choose their
strategic variables (i.e., their price leve ls or outp ut lev e ls ),
even if the network effects are explicitly introduced, we
obtain the same results as those in their paper. Thus, in
the two-stage game wherein both firms 0 and 1 simul-
taneously choose either their price contracts or their
quantity contracts in the first stage, and accordingly en-
gage in market competition in the second stage, we find
that price-setting competition is a unique market compe-
tition structure led by dominant strategies such that both
the firms take the price contracts7.
REFERENCES
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6Matsumura and Ogawa [6] found that the equilibrium social welfare
becomes higher in price competition than in quantity competition re-
gardless of the relation of goods produced by both the firms, that is,
substitutable goods or complementary goods.
7Detailed discussions and proofs on the above are available upon re-
quest.