GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.289

Strategic Brand Proliferation: Monopoly vs. Duopoly

Kentaro Inomata

December 2012GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Strategic Brand Proliferation:
Monopoly vs. Duopoly∗
Kentaro Inomata†
December 24, 2012

Abstract
This paper investigates the possibility that market competition may
encourage a firm to proliferate its brand. To show this, using the multiproduct firm (MPF) model, we compare the incentive in monopoly withthe one in duopoly. We find that if firms compete in quantity and provide close substitutes, the incentive may be greater in duopoly than inmonopoly. This is because the MPF brands may act as complements
to each other in oligopoly, whereas they always act as substitutes inmonopoly. This result is contrary to that of Judd [RAND J. Econ 16(1985) 153].

JEL classiﬁcation: L13; D43
Keywords: brand proliferation, multi-product ﬁrm (MPF), product substitution, substitutes and complements, Cournot competition∗ I would like to thank Junichiro Ishida, Shingo Ishiguro and Noriaki Matsushima for detailed their comments and many suggestions. I am also indebted to Takanori Adachi, Satoshi
Fukuda, Jun-ichi Itaya, Akio Kawasaki, Keisuke Kawata, Hiroshi Kitamura, Akira Miyaoka,
Tadashi Morita, Tatsuhiko Nariu, Yoshihiro Yoshida and the seminar participants at Keio
University, Kyushu Sangyo University, and Osaka University.
† Graduate School of Economics, Osaka University. E-mail: kge001ik@mail2.econ.osakau.ac.jp1

1

Introduction

This paper investigates the possibility that in diﬀerentiated duopoly, the existence of rival ﬁrms may encourage one of the ﬁrms to proliferate its brand(providing a new diﬀerentiated good). As such, our analysis is motivated by thefollowing theoretical expectations. First, (i) a brand proliferating ﬁrm bears“a cost” in addition to the initial investment cost. In the context of industrialorganization and management (or business), this is called the cannibalizationeﬀect, which acts as an intra-ﬁrm competition. This is a factor that leads ﬁrms
to hesitate to proliferate their brands.1 However, (ii) in duopoly, the new goods
provided by ﬁrms may increase their own shares because of business-stealing
eﬀects. This eﬀect is positive for duopolists, but for monopolists. Thus, it isclear that brand proliferation by a duopolist plays a diﬀerent role than that bya monopolist.
The topic of strategic brand proliferation has been of great interest to researchers of industrial organization since Schmalensee (1978). For example,
an empirical study by Kadiyali et al. (1998) shows that brand proliferation
by a duopolist in the yogurt industry yields market expansion and therefore
both ﬁrms raise their demands. Similarly, Draganska and Jain (2005) extend
the argument of Kadiyali et al. (1998) and indicate that brand proliferation
by oligopolists in the yogurt industry may mitigate price competition. Theseﬁndings highlight the fact that brand proliferation acts as a strategic tool toyield the proliferating ﬁrm a proﬁt, although the cannibalization eﬀect hurts
the ﬁrm. In spite of such empirical arguments, it is not clear whether the incentive is greater in oligopoly (duopoly) than in monopoly. This paper provides atheoretical attempt to clarify this.
In previous works, brand proliferation has been mainly discussed in the context of entry deterrence. As implied by Schmalensee (1978), this is because
if the market size is constant, brand proliferation ﬁlls niches and makes entry
harder. Although brand proliferation as a tool of entry deterrence has beenbroadly studied, not many studies have compared the incentives between thecase in which market competition exists and the case where it does not. Judd’s(1985) study, one of the few on the subject, discusses the credibility of preemption by crowding the product spectrum. Using a spatial diﬀerentiated duopoly
model, Judd (1985) asserts that the implication by Schmalensee (1978) thatincumbents’ preemption by crowding the product spectrum may act as an entry
barrier is not persuasive. That is, an established ﬁrm cannot threaten the rival
entrant with proliferating its brands in advance, since providing multiple goods
is no longer beneﬁcial if the rival enters at least one of the markets. Thus, Judd(1985) implies that market competition discourages the ﬁrms from proliferating
1 The cannibalization eﬀect is the only characteristic of the MPF model that provides
multiple diﬀerentiated goods. Because of product substitutability between the ﬁrm’s own
goods, those goods compete with each other. Hence the demand for each good is likely to besmaller in the case of MPF than in the case of the single-product ﬁrm (SPF) model, which
provides only one variety even if there is no rival ﬁrm. Thus, a monopolist does not always
want to proliferate its brands. See also the studies on MPFs, including Lin (2004), Grossmann(2007), and Eckel and Neary (2010).

2

their brands. This conclusion has been widely supported in the literature on
brand proliferation and entry deterrence.2
Although it seems that Judd’s (1985) suggestion is very persuasive, it overlooks one fact. In diﬀerentiated duopoly with single-product ﬁrms (SPFs), as
mentioned by Dixit (1979), an increase in the degree of product substitutability
decreases both proﬁts monotonically as long as there is no cost diﬀerence.3 It
operates positively on the incentive since the proﬁt diﬀerence between MPFs
and SPFs becomes greater if the proﬁt of MPF is constant. In contrast, in
diﬀerentiated duopoly with at least one MPF, it is not clear whether or not anincrease in competition decreases the proﬁt of the MPF. This is because thestrategic interaction between the two ﬁrms becomes complex. That is, if all
the degrees of product substitutability are not identical, such interactions bringabout asymmetric business-stealing eﬀects between the two ﬁrms. Hence, any
one of the ﬁrms may beneﬁt from an increase in competition. This point plays
an important role in brand proliferation in duopoly since it means the extensionof the proﬁt diﬀerence between MPFs and SPFs. Combining these two points,
we show that Judd’s (1985) suggestion is not valid in the case of MPF duopoly
with asymmetric product substitutabilities.
To show this, in the next section, we ﬁrst construct and analyze two basicmodels, a monopoly model and a duopoly model, and then compare the incentive
to proliferate the brand in monopoly with that in duopoly. The timings are
as follows. A monopolist (a duopolist), Firm 1, decides whether or not toproliferate its brand in Stage 1, then decides the quantity (competes in quantitywith Firm 2) in Stage 2. To compare the two cases and exclude the eﬀect of costadvantage, we assume that there is no asymmetry of marginal costs. In addition,
suppose that in the case of duopoly, there exist asymmetries in the degree of
product substitutability.4 That is, we divide one parameter that measures the
cross-price eﬀect into two diﬀerent parameters measuring the competition eﬀect
and the cannibalization eﬀect, respectively. The competition eﬀect indicates thestrategic interaction between the ﬁrms. Under the above settings, we ascertain
which incentive is stronger: monopoly or duopoly.
We ﬁrst show that the proﬁt of a duopolist may increase when the competition eﬀect becomes larger (Proposition 1). This result is similar to that of
Zanchettin (2006). Zanchettin’s (2006) study, an important study of industrialorganization, indicates the possibility that an increase in the degree of productsubstitutability between ﬁrms (competition eﬀect) increases the proﬁt of an efﬁcient ﬁrm. However, our result is diﬀerent from that of Zanchettin (2006) inthat our result does not need any asymmetry of the marginal costs, although
Zanchettin (2006) necessarily requires an asymmetry of the costs between two
2 For instance, Belleﬂamme and Peitz (2010, p. 417), one of the latest textbooks on industrial organization, states, “If the incumbent can withdraw its product at suﬃciently low
cost from a segment in which it faces a direct competitor, brand proliferation is not a credible
strategy for entry deterrence.” This is the essence of Judd (1985).
3 Zanchettin (2006) implies that in case of asymmetric costs, a more eﬃcient ﬁrm may
beneﬁt from an increase in competition eﬀect.
4 The asymmetry means the insight of consumers that “Goods A and B are not alike, but
are similar to C.”

3

SPFs. Second, in contrast to Judd (1985), we suggest that there exists a case inwhich the incentive of a ﬁrm to provide another brand is greater in duopoly thanin monopoly if the competition eﬀect is suﬃciently large but the cannibalizationeﬀect is suﬃciently small (Proposition 2). The intuition behind the result is asfollows. A relatively high competition eﬀect aﬀects both ﬁrms negatively, but
a relatively small cannibalization eﬀect mitigates only the eﬀect on the proliferating ﬁrm, Firm 1. Accordingly, Firm 1 may beneﬁt from an increase in the
degree of the competition eﬀect. Namely, Firm 1 gains the power of successive
predation. Such a result is caused because the goods provided by an MPF may
not act as substitutes, but as complements in duopoly, although they always act
as substitutes in monopoly. This implies, for instance, the existence of brandloyalty, consumers’ cost of moving among multiple stores, or any other shopping
costs (Klemperer, 1992). That is, consumers loyal to Firm 1 prefer buying from
Firm 1 over Firm 2. In addition, if consumers’ cost of moving among multiple
stores is very high, they may want to buy similar goods at a single store rather
than at multiple stores. The consumers’ subjective costs are called shopping
costs which is a type of lump switching costs. If there exist shopping costs, thepower of predation by the proliferating ﬁrm may become substantially stronger.
Hence, we can see that Judd’s suggestion is not always true.
We extend the basic model to a generalized analysis after the main study.
That is, the asymmetry of the option of brand proliferation assumed in theprevious section will be relaxed. By this, we ﬁnd that there exist two asymmetricequilibria that hold the main proposition of this paper even if all ﬁrm conditions
are symmetric: when the ﬁxed cost for brand proliferation is relatively high, oneof the ﬁrms proliferates its brand and the other does not. This is simply becauseof the strategic substitute interaction between the ﬁrms.
There are several related studies in addition to the above papers. First,
in contrast to Judd (1985), Ishibashi (2003) suggests that preemption by amonopolist may operate as a tool for a credible entry barrier. Nevertheless,according to Ishibashi (2003), if an entrant can enter the market, the incumbent
no longer wants to proliferate its brands. Ishibashi (2003) also implies that
Judd’s (1985) suggestion holds if there exists at least one established rival ﬁrm.
However, this paper shows that an established ﬁrm may want to introduce a
new brand more aggressively if there is a rival in the industry. Second, Jing andZhang (2011) study endogenous product line selection and price promotions induopoly. Using a vertical diﬀerentiation model with MPFs, Jing and Zhang(2011) show that there is an equilibrium in which one of the ﬁrms oﬀers two
diﬀerent types of product, high-end and low-end, while the other oﬀers only
one type. The MPF with vertical diﬀerentiated goods beneﬁts from providing
multiple goods since the positive eﬀect of price discrimination dominates thenegative eﬀect of cannibalization. This is similar to our paper in that the positiveeﬀect of product line extension dominates the negative eﬀect of cannibalizationand yields a competitive advantage. However, Jing and Zhang (2011) analyze
a case of vertical diﬀerentiation in price competition, which stands in contrast
with our study of horizontal diﬀerentiated quantity competition.
Third, Klemperer and Padilla (1997) argue for brand proliferation as a tool4

of predation which indicates the possibility that in the case that there exist
consumers having shopping costs, an established ﬁrm beneﬁts from brand proliferation even if the new brand is independent of the other brands. Fourth,
adopting asymmetric parameters of product substitutability, Xing and Zhao
(2008) imply that asymmetries of the degrees of product substitutability among
three goods provided by an SPF and an MPF themselves bring about demand
shifts between the ﬁrms. This eﬀect is the most important feature of Xing andZhao (2008) and a similar eﬀect also occurs in our paper. However, Xing andZhao (2008) do not compare the monopoly case with the duopoly case.
Fifth, there is an investigation into whether competition makes ﬁrms proliferate brands in Hotelling’s spatial model. Tabuchi (2012) ﬁnds that in thecase of three or more ﬁrms, ﬁrms proliferate brands since the negative aspect ofbrand proliferation dominates the positive aspect that is the eﬀect of reducing
consumers’ cost of moving dominates the eﬀect of intensifying price competition,although it does not hold in duopoly. Thus, Tabuchi (2012) indicates the possibility that intensifying competition encourages ﬁrms to proliferate their brands.
This point is common to our paper. However, Tabuchi (2012) deﬁnitively differs from our paper, as we compare the case of no competitor with the case of acompetitor. Sixth, Raubitschek (1987) provides a fundamental study of brandproliferation by MPFs. This study, using a model of monopolistic competition
with MPFs, examines the relation between the number of ﬁrms and brands in anindustry, and concludes that the former is inversely proportional to the latter.
In contrast, we conclude that competition may encourage a ﬁrm to provide an
additional variety of a good. In other words, the number of goods in an industry
may be larger in oligopoly than in monopoly.
The remainder of this paper is as follows. In Section 2, we present the basicmodel and analyze the equilibrium in monopoly and duopoly. In Section 3, wecompare the ﬁrm’s incentives to provide another good in monopoly and duopoly.
In Section 4, we extend the basic model to a symmetric option case, and thengeneralize the main result. Finally, we conclude this paper.

2

The Basic ModelIn this section, we present a model to determine sure whether a ﬁrm’s incentive
to provide another diﬀerentiated good is stronger in duopoly than in monopoly.
First, we set up a two-stage monopoly model in which the monopolist, Firm 1,
chooses whether to become a multi-product ﬁrm (MPF) by paying ﬁxed cost
F or remain a single-product ﬁrm (SPF) in the ﬁrst stage, and then decides
the quantity level in the second stage. Second, we provide a diﬀerentiatedduopoly model where there exists a rival ﬁrm, Firm 2. Then, by comparing theformer with the latter, we derive the condition to accomplish the purpose of theanalysis. For simplicity, we assume that only Firm 1 has the option to provide
a new variety of good and become an MPF in this section (this assumption
will be relaxed in Section 4). This assumption can be interpreted as a situationin which Firm 2 faces some ﬁnancial restriction or has no technology to create5a new variety. Firm 1 provides goods A and B. Let qA and qB denote thequantities of good A and B respectively. Similarly, q2 indicates the quantity ofgood 2 provided by Firm 2. The inverse demand functions are as follows.5
pA =a − qA − γAB qB − γA2 q2 ,(1)

pB =a − qB − γAB qA − γB2 q2 ,p2 =a − q2 − γA2 qA − γB2 qB ,

(2)
(3)

where a is the intercept of the inverse demand for each demand. γij ∈ (0, 1)
indicates the degree of product substitutability between goods i and j for i, j ∈{A, B, 2}, i ̸= j. If there exists no rival, q2 = 0. Similarly, qB = 0 in the
single-product ﬁrm case.

2.1

Monopoly CaseFirst, this paper examines the monopoly case. After solving two proﬁt maximization problems of a monopolist (second stage), Firm 1 compares both proﬁts
and chooses the more proﬁtable one (ﬁrst stage).
2.1.1

Second Stage: Quantity Decision

In case of the single-product ﬁrm, the proﬁt function of Firm 1 isSπm(qA ) = (a − qA − c)qA ,(4)

where c > 0 is the constant marginal cost. The equilibrium quantity and proﬁtof single-product ﬁrm are
S
qm
=

a−c
,
2

S
πm
=(

a−c 2
) .
2

(5)

On the other hand, in case of the multi-product ﬁrm, the proﬁt function isM(qA , qB ) = (a − qA − γAB qB − c)qA + (a − qB − γAB qA − c)qB − F,πm

(6)where F is the ﬁxed cost for the entry of good B. γAB implies the degree of
product substitutability between goods A and B, which measures the cannibalization eﬀect. The MPF’s equilibrium quantities and proﬁt area−c,2(1 + γAB )

(7)

a−c 2
2
(
) − F.
1 + γAB2(8)

qA = qB =M
πm(γAB ) =By eq. (8), it is clear that the MPF’s quantities and proﬁt are the decreasing
function of γAB .
5 This type of inverse demand function is also used in Zhao and Xing (2006, 2008).

6

2.1.2First Stage: MPF or SPF?

In the ﬁrst stage, Firm 1 chooses whether to create an additional diﬀerentiated
good:

M
S

if
πm
(γAB ) − πm
< 0,
SP F
M
S
indiﬀerentif
πm (γAB ) − πm = 0,


M
S
MPF
if
πm
(γAB ) − πm
> 0.
These conditions can be rewritten as
if Fm (γAB ) < F,
SP F
indiﬀerent
if Fm (γAB ) = F,MPF
if Fm (γAB ) > F,(9)

AB a−c 2
where Fm (γAB ) ≡ 1−γ
1+γAB ( 2 ) . Fm (γAB ) also decreases in γAB .

2.2

Duopoly Case

Next, the case where there is a rival ﬁrm, Firm 2, is presented. For simplicity,
we assume that only Firm 1 has the option to become an MPF.
2.2.1

Second Stage: Cournot Competition

The proﬁt functions are
π1S = (a − qA − γA2 q2 − c)qA ,(10)π2S = (a − q2 − γA2 qA − c)q2 ,(11)

where γA2 denotes the degree of product substitutability between goods A, provided by Firm 1, and good 2, provided by Firm 2.6 γA2 measures the competitioneﬀect between the ﬁrms. The equilibrium quantities and proﬁts are
qA = q2 =a−c,
2 + γA2(12)

π1S = π2S = (

a−c 2
) .
2 + γA2(13)

Similarly, if Firm 1 becomes a multi-product ﬁrm, the proﬁt functions are
π1M =(a − qA − γAB qB − γA2 q2 − c)qA
+ (a − qB − γAB qA − γB2 q2 − c)qB − F,
π2M =(a − q2 − γA2 qA − γB2 qB − c)q2 ,

(14)
(15)

6 If γA2 = 0, these two ﬁrms are the diﬀerent market monopolists. In addition, if γA2 = 1,
the ﬁrms provide homogeneous goods.

7where γB2 indicates the degree of product substitutability between good B
and good 2. In the following, to focus on the relation between the degreeof the competition eﬀect (γA2 and γB2 ) and that of the cannibalization eﬀect(γAB ), we assume γA2 = γB2 and redeﬁne these parameters as γ12 . That is,
γA2 = γB2 ≡ γ12 .7 This redeﬁned parameter implies the degree of productsubstitutability between the goods of Firm 1 and those of Firm 2.
The equilibrium quantities and proﬁts are
M
M
= qB
=
qA

π1M =

2.2.2

(2 − γ12 )(a − c)
2 ),2(2 + 2γAB − γ12q2M =(1 + γAB )(2 − γ12 )2 (a − c)2
− F,
2 )2
2(2 + 2γAB − γ12(1 + γAB − γ12 )(a − c),2
2 + 2γAB − γ12

π2M =

(16)

(1 + γAB − γ12 )2 (a − c)2.
2 )2(2 + 2γAB − γ12(17)First Stage: MPF or SPF?

In the ﬁrst stage, Firm 1 in the diﬀerentiated duopoly chooses whether to enteran additional diﬀerentiated good:SMSP F if π1 (γ12 , γAB ) − π1 (γ12 ) < 0,M
indiﬀerent if π1 (γ12 , γAB ) − π1S (γ12 ) = 0,
M P F if π1M (γ12 , γAB ) − π1S (γ12 ) > 0.
These conditions can be rewritten asSP F if Fd (γ12 , γAB ) < F,indiﬀerent if Fd (γ12 , γAB ) = F,M P F if Fd (γ12 , γAB ) > F,
where
Fd (γ12 , γAB ) ≡

3

(18)

4
)
(1 − γAB )(8 + 8γAB − γ12(a − c)2 .
222(2 + 2γAB − γ12 ) (2 + γ12 )2

Does Quantity Competition encourage a Duopolist
to become an MPF?

In this section, we compare Firm 1’s incentive to provide another variety between the monopoly case and the duopoly case. Since the purpose of this paper
is to determine whether there exists a case where quantity competition encourages Firm 1 to provide another diﬀerentiated good, we combine two conditions(11) and (20). The condition where a duopolist, Firm 1 provides another diﬀerentiated good and becomes an MPF but monopolistic Firm 1 does not is
Fd (γ12 , γAB ) ≥ F ≥ Fm (γAB ).

(19)

7 In the sense that both goods A and B are produced by Firm 1 and only good 2 is producedby ﬁrm 2, γA2 and γB2 can be summarized in a competition eﬀect.

8

This inequality implies the parameter condition that Firm 1’s incentive to provide diﬀerentiated good is larger in duopoly than in monopoly. Likewise, since
Fd (0, γAB ) = Fm (γAB ), the above condition can be rewritten as
Fd (γ12 , γAB ) ≥ F ≥ Fd (0, γAB ).

(20)

Hence, what we investigate hereafter is summarized as follows: whether the
existence of quantity competition enhances the proﬁtability of MPF.” To verify
this, we ﬁrst partially diﬀerentiate Fd (γ12 , γAB ) with respect to γ12 , then ﬁnd
the conditions of γ12 and γAB where the derivative is positive:∂Fd (γ12 , γAB )
>0
∂γ12

(21)

As a consequence, we obtain Proposition 1 and Corollary 1.
√
Proposition 1 (i) For γ12 ∈ (2 − 2(1 − γAB ), 1) and γAB ∈ (0, 21 ), the proﬁtof an MPF is monotonically increasing. In addition, (ii) the proﬁt diﬀerence
between MPF and SPF, Fd (γ12 , γAB ), is also monotonically increasing.
Proof of Proposition 1∂Fd (γ12 , γAB )
∂π1M
∂π1S
=
−
.
∂γ12
∂γ12
∂γ12

(22)

∂π S

1
Apparently, the second term of RHS (− ∂γ12
) is positive. Thus, the remaining
problem is only the sign of the ﬁrst term:∂π1M ∂π1M (q1 , q2 , γ12 , γAB ) ∂q2 (γ12 , γAB )
=
∂γ12
∂q2
∂γ12
M
∂π (q1 , q2 , γ12 , γAB )+ 1∂γ12∂q2 (γ12 , γAB )=2q1 (−γ12− q2 ),
∂γ12

(23)

(24)

where the ﬁrst term is positive (Business-stealing eﬀect) and the second term
is negative (price down eﬀect or market shrink eﬀect). If the former eﬀect∂π M

1dominates the latter, the sign of ∂γ12
becomes positive. It is a function of γ12and γAB :−2(γ12∂q2+ q2 ) > 0∂γ12⇔−γ12 (γ12 − 4) − 2(1 + γAB ) > 0.
(25)
Q.E.D.

The implications of Proposition 1 are as follows. (i) As the competition between
the ﬁrms becomes more intense, MPF yields more proﬁt. This result is similar to9that of Zanchettin (2006), who indicates that an intense competition may beneﬁt
an eﬃcient ﬁrm. However, our result holds even if ﬁrms’ costs are symmetric.
(ii) There exists the possibility that the existence of quantity competition gives
more beneﬁt to a duopolist that provides another diﬀerentiated good.
Corollary 1 The cross-derivative of the incentive in duopoly, Fd (γ12 , γAB ), is2
Fd (γ12 ,γAB )negative: ∂ ∂γ
< 0.
12 ∂γAB
Corollary 1 implies that the beneﬁt from a marginal decrease in the degree ofthe cannibalization eﬀect, γAB , is larger in duopoly than in monopoly. Accordingly, Proposition 1 and Corollary 1 yield a remarkable consequence: if thecannibalization eﬀect is suﬃciently low, a higher competition eﬀect makes the
MPF more proﬁtable. Based on these, Proposition 2 is derived.
Proposition 2 One of the duopolists wants to provide a diﬀerentiated good
more aggressively in duopoly than in monopoly if the competition eﬀect, γ12 , issuﬃciently large but the cannibalization eﬀect, γAB , is suﬃciently small.
Proof of Proposition 2 Take the limits γ12 → 1 and γAB → 0. Then, eq.
(22) is satisﬁed in the neighborhood of (γ12 , γAB ) = (1, 0) since Fd (γ12 , γAB )and Fm (γAB ) are continuous.
Q.E.D.
Further, Corollary 2 is derived from Proposition 2.
Corollary 2 The brands provided by an MPF competing with a rival that provides a close substitute act as complements as long as Proposition 2 holds: nevertheless, they act as substitutes in monopoly.
Proof of Corollary 2 Transform the inverse demand function of good A (B)into the demand function and diﬀerentiate it with respect to the price of good∂qi
B (A). Then, in duopoly, ∂p< 0, i, j ∈ {A, B}, i ̸= j, if Proposition 2 holds.
j
In contrast, in monopoly, the sign is always opposite.
Q.E.D.
The intuition behind Proposition 1 is as follows. If γ12 is suﬃciently large,
the driving out power becomes stronger. On the one hand, if γAB is smallenough, Firm 1’s goods A and B do not cannibalize each other too much. As
a result, these two eﬀects improve only Firm 1’s position, and yield it higher
proﬁt (and reduce Firm 2’s proﬁt). Such an eﬀect never occurs in monopoly.
This implies that the brands provided by the MPF do not act as substitutes,
but as complements to each other. This can explain the case in which a ﬁrm
that competes with its rivals intensively attempts to provide multiple similar
brands. For example, if Thomas can buy clothes by two apparel brands, 1 and
2, he may want to buy at one shop rather than at two. This is because he wants
to get a complete set of brand 1’s jeans. In this case, Thomas ﬁrst chooses the10

shop, then purchases the jeans.8 Thus, the two types of jeans provided by brand
1 compete intensively with the similar one provided by brand 2. Of course, inthis case, the competitivenesses between A and 2 and between B and 2 are thesame.
In addition, the following result is also derived from Proposition 2.
Corollary 3 If the degree of the competition eﬀect equals that of the cannibalization eﬀect (γ12 = γAB ), there exists no case that a duopolist wants to add
another good more aggressively than a monopolist.
This indicates the importance of the separation of the measure of product substitutability into the measures of two eﬀects, the competition eﬀect and thecannibalization eﬀect.

4

Extension

In this section, we try to extend the basic model to a symmetric option case.

4.1

Symmetric option case

The assumption in which only one of the two ﬁrms has the option to provide
another good is relaxed in this subsection. That is, each ﬁrm has the option.
In this case, each ﬁrm chooses its strategy in the ﬁrst stage (Table 1).
XXX
XXXFirm 2
MPF
XXX
Firm 1X
MMMPF
π1 , π2M M
SPF
π1SM , π2M S

SPF
π1M S , π2SM
π1SS , π2SS

Table 1: MPF or SPF?
Note that the consequence of the asymmetric case is shown in the previous
section. Therefore, the remaining problem in this subsection is the result of
MPF vs. MPF (π1M M vs. π2M M ).
If both ﬁrms choose to provide good A with good B in the ﬁrst stage, theinverse demand functions that the two ﬁrms face in the second stage are givenby
p1A = a − q1A − γAB q1B − γ12 (q2A + q2B ),p1B = a − q1B − γAB q1A − γ12 (q2A + q2B ),

(26)
(27)

p2A = a − q2A − γAB q2B − γ12 (q1A + q1B ),p2B = a − q2B − γAB q2A − γ12 (q1A + q1B ).

(28)
(29)

8 Such an interpretation is common to the shopping costs mentioned in Klemperer (1992)
and Klemperer and Padilla (1997).

11

The proﬁt functions are
π1M M = (p1A − c)q1A + (p1B − c)q1B − F,(30)

π2M M = (p2A − c)q2A + (p2B − c)q2B − F.

(31)

The equilibrium quantity and proﬁt are, respectively,a−c,2(1 + γAB + γ12 )(1 + γAB )(a − c)2
π1M M = π2M M =− F.
2(1 + γ12 + γAB )2

q1A = q1B = q2A = q2B =

(32)
(33)

After that, in the ﬁrst stage, each ﬁrm chooses its strategy (MPF or SPF) given
the opponent’s strategy.
Then, we likewise compare the incentive in duopoly with a rival MPF to theone in monopoly. The condition is described by an inequality as follows.
FM M (γ12 , γAB ) ≥ F ≥ FM M (0, γAB )(34)

where FM M (γ12 , γAB ) = π1M M − π1SM + F and FM M (0, γAB ) = Fm (γAB ). As aconsequence, we obtain the following result.
Proposition 3 If the rival ﬁrm also provides multiple goods (MPF vs. MPF),
then the incentive to provide another variety is always less in duopoly than inmonopoly.
Proof of Proposition 3 Suppose FM M (γ12 , γAB ) ≥ F ≥ FM M (0, γAB ). However, for γ12 , γAB ∈ [0, 1], the maximum value of FM M (γ12 , γAB )−FM M (0, γAB )is 0 if γ12 = 0 or γAB = 1. Therefore, ∀γ12 , γAB ∈ (0, 1), FM M (0, γAB ) >FM M (γ12 , γAB ). This contradicts the above inequality.
Q.E.D.
Then, we show that asymmetric equilibria can exist in the ﬁrst stage. In
the previous section, we suppose that only one of the ﬁrms has the option ofbrand proliferation. Nevertheless, in this subsection, because of the symmetryof the option, a symmetric equilibrium in which both ﬁrms proliferate theirbrands can be shown. However, at the same time, there can also exist two
asymmetric equilibria because of the ﬁxed costs. Thus, we now need to ascertain
the parameter domain of the ﬁxed cost that holds the asymmetric equilibria.
If it is non-empty, we can see the generality of the consequence of the previous
section.
As supposed in the previous section, we assume anew that FM S (γ12 , γAB ) ≥F where Fd (γ12 , γAB ) = FM S (γ12 , γAB ) = π1M S − π1SS + F . If so, the thing to
be shown is whether an inequality
π1M S ≥ π1SS ≥ π1SM ≥ π1M M(35)holds under FM S (γ12 , γAB ) ≥ F ≥ Fm (γAB ). The ﬁrst half of the inequality, π1M S ≥ π1SS , is the incentive to proliferate the brand given the opponent’s
12strategy for an SPF. Likewise, the second half of the inequality, π1SM ≥ π1M M ,implies the disincentive to provide another variety B given Firm 2’s strategy
for an MPF. Then, π1SS ≥ π1SM . We must now prove that the incentive
condition (FM S (γ12 , γAB ) ≥ F ) is consistent with the disincentive condition(F ≥ FM M (γ12 , γAB )). By Proposition 2 and Proposition 3, we obtain Proposition 4.
Proposition 4 If the competition eﬀect is suﬃciently large and the cannibalization eﬀect is suﬃciently small, the asymmetric equilibria where one of theﬁrms proliferates its brand and the other remains an SPF exist even if the rival
ﬁrm also has the option of brand proliferation.
Proof of Proposition 4 By Proposition 3, ∀γ12 , γAB ∈ (0, 1), Fm (γAB ) >FM M (γ12 , γAB ). Hence, FM S (γ12 , γAB ) ≥ F ≥ Fm (γAB ) > FM M (γ12 , γAB )∗∗holds in the neighborhood of (γ12, γAB
) = (1, 0).
Q.E.D.
In contrast, if the ﬁxed cost is relatively low, the disincentive condition
may not hold. In this case, the MPF strategy is the dominant strategy forboth players, and hence (MPF, MPF) is the unique Nash equilibrium. In otherwords, if the cost is relatively high, eq. (35) holds and quantity competition
may encourage one of the ﬁrms to provide an additional variety in the case of asymmetric option in addition to the asymmetric option in Section 3. Thus, fromthe above argument, we can conclude that our main proposition (Proposition
2) is persuasive if the ﬁxed costs are relatively high. Thus, our result is still
contrary to Judd (1985).

5

Concluding Remarks

In this paper, we investigate the incentive of a duopolist to provide anothervariety. Since the competition eﬀect and the cannibalization eﬀect have been
described by an identical parameter in most previous works, an increase in
the degree of product substitutability among the goods tends to mean that the
incentive to provide another variety and become an MPF is less in duopoly thanin monopoly. However, in this paper, it is seen that the eﬀects are separated.
As a result, we ﬁnd that if the competition eﬀect is suﬃciently high and thecannibalization eﬀect is suﬃciently low, the power of predation increases since
the two brands act as complements in duopoly but as substitutes in monopoly.
Thus, in this case, the existence of a rival ﬁrm encourages one of the ﬁrms
to provide another diﬀerentiated good and become an MPF. Such a result is
contrary to that of Judd (1985).
This paper constrains the game of brand proliferation to be one-shot. However, in reality, the strategic interaction is continuous: the rival SPF ought torespond to the strategy. Thus expansion to a sequential proliferation model isa subject for future research.

13

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