ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E16-3

Is a Big Entrant a Threat to Incumbents?
The Role of Demand Substitutability inCompetition among the Big and the Smallby
Lijun Pan
Makoto Hanazono

Revised version as of August 2017

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Is a Big Entrant a Threat to Incumbents?
The Role of Demand Substitutability in Competition
among the Big and the Small
Lijun PANy
Nagoya University

Makoto HANAZONOz
Nagoya University

August 28, 2017

Abstract
We establish a model of market competition between large and small …rms andinvestigate the way in which demand substitutability a¤ects how the entry of big…rms impacts incumbents. We focus on the relative strength of two opposing e¤ectsof entry on large incumbent …rms’ demand: the direct substitution e¤ect amonglarge …rms (negative) and the indirect feedback e¤ect through the change in small…rms’ aggregated behavior (positive). If the substitutability between large andThis paper was previously circulated under the title “Competition among the Big and the Small withDi¤erent Product Substitution.”We would like to express our sincere gratitude to Ken-Ichi Shimomura,
Jacques-Francois Thisse, Toshihiro Matsumura, Takatoshi Tabuchi, and Daozhi Zeng for their detailed
discussion. We also thank Jay Pil Choi, Nisvan Erkal, Michal Fabinger, Daisuke Hirata, Mitsuru Igami,
Hiroaki Ino, Akifumi Ishihara, Kohei Kawamura, Jin Li, Noriaki Matsushima, Masashi Mizuno, Takeshi
Murooka, Hikaru Ogawa, Hiroshi Ohta, Dan Sasaki, Nicolas Schutz, and Yosuke Yasuda for their helpfulcomments. This paper was presented in the Royal Economic Society 2015 Conference, EARIE 2015,
Japanese Economic Association 2015 Fall Meeting, Singapore Economic Review 2015 Conference, Asian
Meeting of Econometric Society 2016, and Asia Paci…c Industrial Organization 2016 Conference. We
thank the participants at those conferences for their comments. We are also grateful to the seminar participants at Keio, Kwansei Gakuin, Tokyo, Tohoku, and Contract Theory Workshop for their comments.
The authors gratefully acknowledge the …nancial support of JSPS KAKENHI Grant No. 16H06854 andNo. 15H03346.
yCorresponding author: Lijun Pan, School of Economics, Nagoya University, Japan. Email:pan.lijun.nu@gmail.com
z
Makoto Hanazono, School of Economics, Nagoya University, Japan. Email: hanazono@soec.nagoyau.ac.jp

1

small …rms is su¢ ciently high, the indirect e¤ect dominates the direct e¤ect andlarge incumbents’ equilibrium prices and pro…ts increase. We show that welfaree¤ects are ambiguous, which calls for careful assessment when regulating large…rms’entry.
Keywords: big …rms, small …rms, substitutability, entry, market impacts
JEL classi…cation: D21, D43, L11, L13

1Introduction

The entry of a large …rm has substantial e¤ects on market competition and market
structure, and evidence suggests that the impact on large and small …rms di¤ers acrossmarkets. Igami (2011) studies the supermarket industry in Japan after deregulation(that is, the relaxation of the Large-Scale Retail Law) and shows that the entry of large
supermarkets has a negative impact on large (and medium) supermarkets and a neutralor positive impact on small supermarkets. However, IKEA’s entry into South Korea
in 2014 increased the pro…ts of large furniture makers while substantially lowering the
pro…ts of small furniture makers.1 Why do the impacts di¤er? What factors determine
these di¤erences? Does the entry of a large …rm enhance overall e¢ ciency?
To answer these questions, we consider the following model of competition amonglarge and small …rms and study the impact of large …rms’entry. A large …rm is modeled
as a multi-product …rm, and the number of products it o¤ers is a choice variable.2 A small…rm is modeled as a single-product …rm with free entry and exit. Following the modelof monopolistic competition, we treat each variety symmetrically. While the number oflarge …rms is …nite and exogenous, the variety of each large …rm is determined througholigopolistic competition and the measure of small …rms is determined by the free entry
condition. We consider a static game in which all decisions, including entry, variety
choices, and production, are simultaneous. Our model has the following two key features.
First, a large …rm can exert market power through variety choice and coordinated pricing
of its own varieties. Second, by adopting a quadratic quasi-linear utility function for the
representative consumer, demand in our model displays rich con…gurations of productsubstitutability within and across large and small …rms.3 We derive the condition underwhich a unique mixed market equilibrium exists (i.e., the coexistence of active large andsmall …rms) and investigate the impact of a large …rm’s entry on other …rms’behavior
and welfare.
Our main result shows how product substitutability a¤ects the impact of a large …rm’s
1

After IKEA’s entry into South Korea in 2014, large national furniture makers such as HANSSEM
and LIVART enjoyed an increase in revenue. After an IKEA store was established in Gwangmyeong, therevenue of LIVART’s Gwangmyeong branch increased by 27%, while HANSSEM’s Gwangmyeong store
saw a 10% rise in sales relative to the same period in the previous year. Small furniture makers, however,su¤ered from a revenue decrease of more than 70% on average. Source: Korea Bizwire. March 27, 2015,“Korea’s Large Furniture Makers Boost Revenues Thanks to IKEA,” http://koreabizwire.com/koreaslarge-furniture-makers-boost-revenues-thanks-to-ikea/32438
2
Bernard et al. (2010) show that multi-product …rms are nearly omnipresent in the U.S. manufacturingindustry. According to data from the period between 1979 and 1992, multi-product …rms account for
41% of all …rms but supply 91% of total output. In addition, 89% of multi-product …rms adjust theirproduct range every …ve years.
3
Our modeling approach draws closely on that of Parenti (2016), who also considers multi-productlarge …rms and single-product small …rms in a given market based on quasi-linear utility with quadratic
subutility. More broadly, a quasi-linear quadratic form of the utility function is also adopted by Singhand Vives (1984), Ottaviano and Thisse (1999), and Ottaviano et al. (2002). Unlike our paper, none ofthese papers considers di¤erent degrees of substitutability across …rms.

1

entry on incumbent large …rms. Indeed, we …nd a necessary and su¢ cient condition under which the product range, the price of each product, and the pro…t of each incumbent
all increase (Proposition 2). The key is to separate two e¤ects of a large …rm’s entry on
the demand for incumbent large …rms’products. The …rst e¤ect is the direct substitution e¤ect: the new large …rm’s products are substitutes for the incumbent large …rms’
products. This negatively a¤ects the demand for large …rms’products. The second e¤ect
is the indirect feedback e¤ect due to the change in the number of small …rms.4 A large…rm’s entry squeezes out some small …rms if the products are substitutes across large andsmall …rms, whereas such entry invites more small …rms to participate if the products
are complements. It is important to note that the resulting indirect feedback e¤ect is
non-negative on the demand for large …rms’products, regardless of whether large …rms’
and small …rms’products are substitutes or complements. If the degree of substitutability or complementarity between large …rms’and small …rms’products is relatively larger
than the substitutabilities within large …rms’products and within small …rms’products,
the indirect feedback e¤ect outweighs the direct substitution e¤ect, thereby resulting ina rise in demand for the large …rms’products, which characterizes the condition for an
increase in the product range, the price, and the pro…t of an incumbent large …rm.
Our …nding may explain the di¤erent impacts of a large …rm’s entry in the empirical evidence reported above. As Igami (2011) observes in his analysis of supermarkets
in Japan, large and medium incumbents seem to compete directly with large entrants,
while “small incumbents are insulated by product di¤erentiation and even bene…t fromthe positive demand externality (additional ‡ow of shoppers).”Thus, the substitutability
among large supermarkets is strong, while the substitutability between large and small
supermarkets is weakly positive (or even slightly negative).5 Therefore, large incumbents
may su¤er from intensi…ed competition from new large rivals because the feedback e¤ect
may not be strong enough to o¤set the substitution e¤ect. In contrast, small stores may
bene…t from the large entrants because the latter generate a positive demand externality(in addition to a negative but small substitution e¤ect) on the former. In the South
Korean furniture industry, large furniture makers have a superior ability to design furniture, which implies that the products of large furniture makers are more di¤erentiated4

Shimomura and Thisse (2012) are the …rst to identify this e¤ect but with a di¤erent term – the“market expansion e¤ect”–which is analogous to our “indirect feedback e¤ect”. Note that the “market
expansion e¤ect”originates from the exit of some small …rms in Shimomura and Thisse (2012), while the“indirect feedback e¤ect” in this paper stems from the exit of some small …rms when products are substitutes across large and small …rms and from the entry of small …rms when products are complementary
between large and small …rms.
5Large incumbents with a similar store size face direct competition with new large rivals. In contrast,
small retailers di¤erentiate themselves from large ones in terms of store size and serve di¤erent demands.
Therefore, the substitution e¤ect on small stores from large supermarkets is weak. Furthermore, newlarge retailers attract extra shopper tra¢ c, which raises the demand in the neighborhood and hence
creates pro…t opportunities for small stores. This process generates a positive demand externality forsmall supermarkets. If the positive demand externality dominates the substitution e¤ect, the existenceof large retailers would increase the demand for small ones, and thus, large and small stores are weakly
complementary.

2than those of small furniture makers and hence suggests that the substitutability amonglarge …rms is weaker than that between large and small …rms. Therefore, the feedbacke¤ect could outweigh the substitution e¤ect, resulting in a rise in demand for large …rms’
products, while the squeezing-out e¤ect on small …rms is strong.
The welfare e¤ects of a large …rm’s entry are ambiguous. The change in social welfare
depends on the replacement e¤ect, the variety e¤ect, and the quantity e¤ect (on producer
surplus). To …x ideas, consider the case in which large and small …rms’products are substitutes. Observe that some small …rms exit after the entry of a large …rm, while large…rms’(total) product range expands, and thus, a portion of small …rms’product range isreplaced by that of large …rms. Furthermore, the total variety range after entry may expand or shrink. We examine the associated changes in consumer welfare. For each variety
in the replacement range, the consumer welfare increases or decreases depending on therelative allocative ine¢ ciency between large and small …rms. Moreover, if the net variety
range expands (shrinks), consumer welfare improves (worsens) because consumers preferdiversi…ed consumption. For the associated change in producer surplus, the marginalchange in large …rms’aggregate pro…ts is always positive due to the expansion of their
total quantity.6 This argument shows that the welfare e¤ect depends on each e¤ect’s sign
and/or the relative strength. Our model thus implies that policy makers should carefully
assess the implications of a large …rm’s entry.
There are several strands of literature on markets with large and small …rms and theydi¤er in terms of how they capture large …rms’ market power. The …rst strand is theso-called dominant …rm model (e.g., Markham, 1951, Chen, 2003, and Gowrisankaran
and Holmes, 2004). The dominant …rm is large since it is the leader and the price-maker,
while the price-taking followers are small. Another strand employs the Stackelberg model(Etro, 2004, 2006, and Ino and Matsumura, 2012). In this model, the …rst mover is large
due to the commitment power in the market.
The third strand models a mixture of oligopolistic and monopolistic competition (Shimomura and Thisse, 2012; and Parenti, 2016). With respect to the modeling approach,
our paper is closely linked to these recent advances in studies of market structure in thepresence of large and small …rms. Shimomura and Thisse (2012) are the …rst to connect
oligopolistic and monopolistic competition, where oligopolistic …rms are large due to theirability to produce a large amount, while monopolistically competitive …rms are so smallthat they can produce only a negligible amount in the market. These authors establish ageneral equilibrium model in which large …rms account for (i) strategic interactions within
their group, (ii) the aggregate behavior of small …rms, and (iii) the endogenous income
generated by the pro…t distribution. This model is an important starting point for thepresent paper, where we introduce asymmetry in product substitution across oligopolistic
large …rms and small monopolistic competitors. In addition, this paper is also related toParenti (2016), who characterizes a similar mixed market structure in a representative
consumer model with linear quadratic preferences and examines the impact of trade lib6

Consumer welfare and producer surplus are also a¤ected by the price change due to a large …rm’s
entry. However, these e¤ects on social welfare are neutral.

3

eralization on large and small …rms. Our model also employs a linear quadratic utility
function, but we go a step further by allowing for di¤erent substitutabilities of products
within and across large and small …rms. This approach is crucial for determining theinteractions between large and small …rms.7
This paper’s focus on the impact of a large …rm’s entry is shared by Shimomura
and Thisse (2012) and Ino and Matsumura (2012). Ino and Matsumura (2012) study ahomogeneous-good Stackelberg game with multiple leaders and free-entry followers and…nd that the impact of adding another Stackelberg leader is bene…cial to social welfare
since it eliminates some excessive entry by followers while increasing the total quantity
supplied. Shimomura and Thisse (2012) study a general equilibrium model featuring a
mixture of oligopolistic and monopolistic competition. They …nd that the entry of a large…rm increases the incumbent large …rms’pro…ts and raises consumer and social welfare.
The present paper di¤ers from Shimomura and Thisse (2012) in the following respects.
First, Shimomura and Thisse (2012) employ Cobb-Douglas CES nested utility, while weemploy a quasi-linear utility function with quadratic subutility.8 Therefore, this paper
does not account for the general equilibrium e¤ect on the di¤erentiated goods market
due to the endogenous income from large …rms’pro…ts. Second –and more importantly– Shimomura and Thisse (2012) assume the same substitution among the products oflarge and small …rms, but we consider richer substitutability among the products, whichis the essential element that distinguishes our model from theirs. Incorporating di¤erent
degrees of substitutability and complementarity, our study sheds new light on competition
among large and small …rms.9 The absence of the general equilibrium e¤ect in our modelallows us to highlight the role of di¤erent degrees of substitutability across large andsmall …rms with sharper results. Therefore, our framework provides a feasible way togenerate ‡exible patterns of production behavior and welfare changes.10
The remainder of the paper is organized as follows. We construct the model in Section
2. In Section 3, we analyze the equilibrium of the model and explore the impact of alarge …rm’s entry. Section 4 brie‡y discusses the robustness of the established results,
while Section 5 concludes.
7

Another strand of research di¤erentiates between large and small …rms from the perspective of …rmheterogeneity in cost, represented by Lahiri and Ono (1988) and Matsumura and Matsushima (2010).
8
For a detailed discussion of di¤erent classes of utility functions, see Parenti, Ushchev, and Thisse(2017).
9
There is an additional di¤erence between our model and Shimomura and Thisse’s (2012): in theirmodel, a large …rm is a single-product …rm supplying a non-negligbile quantity, while in our model, alarge …rm is a multi-product …rm: it chooses a variety range and the quantities of each chosen variety.
Since the equilibrium per-variety quantity level is the same, the variety range in our model e¤ectively
plays the same role as the large …rm’s quantity in Shimomura and Thisse (2012).
10
For example, social welfare may improve or worsen in our model, depending on the relative degree ofsubstitutability between large and small …rms, while it always improves in Shimomura and Thisse (2012)
and Ino and Matsumura (2012).

4

2

The Model

Consider a closed economy consisting of two sectors. Firms in sector 1 are perfectlycompetitive and produce a homogenous good under constant returns to scale. Sector 2
provides di¤erentiated goods that are produced by two types of …rms. The …rst type of…rm is large in size, and the number of these …rms is exogenous. The second type of …rm
is in…nitesimal and freely enters or exits the market. All the varieties in Sector 2 aredi¤erentiated so that no two …rms o¤er identical products.
The large and small …rms di¤er in four respects. First, each large …rm has a nonnegligible impact on the market and competes in an oligopolistic manner, whereas eachsmall …rm is negligible in the market and behaves as a monopolistic competitor. Second,
following the approach of Shimomura and Thisse (2012), we assume that the number oflarge …rms is exogenous, while the number of small …rms is endogenously determined byfree entry and exit. Third, each large …rm produces a range of varieties and strategicallychooses both the product range and the quantity of each variety, while each small …rm
produces only one variety of product. Fourth, the varieties are equally substitutablewithin the group of large …rms and that of small …rms, but the level of substitution
across these two types of …rms can be di¤erent.

2.1

Preferences and Demand

The utility of the representative consumer U is described by a quasi-linear utility function
with a quadratic subutility:#"Z
Z N
M Z !m
M Z !mN
X
X
22
m[q S (i)] di
qL (!)d!
[q mqS (i)di +U=
L (!)] d!
2 0
2 m=1 0
0
m=1 0
"M Z
"
#
#
2
Z N
Z N
2
M Z !mX !m
X
2
1
qS (i)diqS (i)diqLm (!)d!
qLm (!)d! +q 0 ;
3
2
2
0
0
m=1 0
m=1 0
(1)
where qS (i) is the quantity of small …rm i with i 2 [0; N ]. The output of each small …rm
is of zero measure, and the total mass of small …rms is N , describing the competitive
fringe. Large …rm m (m = 1; : : : ; M ) provides multiple varieties of products, with the
product range represented by ! m 0, which is chosen by large …rm m. The quantity of2,variety ! 2 [0; ! m ] is qLm (!). The total number of the incumbent large …rms is M
which is exogenously given. Here, we treat ! m continuously. The output of sector 1 is
q0 , which is treated as the numeraire.
Consumer preferences are characterized by …ve parameters, which are , , and j(j = 1; 2; 3). The intensity of preferences for the di¤erentiated good is measured by> 0, which determines the size of the di¤erentiated goods market, and > 0 implies theconsumer’s preference for diversi…ed product consumption. The substitutability among
varieties is characterized by j (j = 1; 2; 3). Speci…cally, the substitutability among the5varieties produced by small …rms and that among the varieties of large …rms are expressedby 1 and 2 , respectively, and the cross-substitutability between the varieties of large…rms and those of small …rms is expressed by 3 11 . The products are substitutes if j > 0
and complements if j < 0, and the products are closer substitutes (complements) whenj > 0 (j = 1; 2; 3) is higher. Finally, to ensure the concavity of the quadratic subutility,
we impose the following conditions:(i) =NP+ 1 > 0,(ii) =( m ! m ) + 2P> 0, and(iii) ( =N + 1 )[ =( m ! m ) + 2 ] > 23 .12
It should be noted here that the measure of small …rms, N , and the variety of eachlarge …rm, ! m , are determined in equilibrium. For the main analysis, we need greater‡exibility in the substitution parameters, j . To ensure this, we assume that there aresome large N < 1 and ! max < 1 such that 1 , 2 , and 3 satisfy (i) - (iii) for NNmax
and ! m !
for all m.
The representative consumer’s budget constraint is
Z N

pS (i)qS (i)di +0

M Z !mX

m=1

mpm
L (!)qL (!)d! + q0 = I,0where pS (i) and pm
L (!) are the prices of the small …rm i’s and large …rm m’s variety !,respectively. The price of the numeraire is normalized to 1. The representative consumer’s
income is I, which is exogenously given. The inverse demand functions facing small …rms
and large …rms are determined by the maximization of the consumer’s utility subject tothe budget constraint:13
pS (i) =qS (i)

1 QS

3 QL ;(2)pm
L (!) =

qLm (!)(3)
3 QS
2 QL ;
RN
R
PM!m m
where QSqS (i)di and QL
qL (!)d! are the total output of the small
m=1 00…rms and that of the large …rms, respectively.
11

A few existing works on multiproduct …rms allowed for substitutability between varieties withina …rm to di¤er from the substitutability between varieties across …rms (Allanson and Montagna, 2005;
Hottman et al., 2011; Bernard et al., 2016). Di¤erent from these works, this paper focuses on asymmetricsubstitutabilities across large and small …rms. We can generalize our model to allow for the asymmetricsubstitutability within and across multiproduct large …rms, and our results still hold qualitatively.
12
See Appendix A for details. Here, i (i = 1; 2; 3) can be positive or negative as long as the conditions
for the concavity of the utility function hold. We will discuss the di¤erent signs of i in Section 3.
13
We establish the strict concavity of the utility function to derive the inverse demand function fromthe variety-wise …rst-order conditions almost everywhere. For the optimization in an in…nite-dimensional
vector space, see Luenberger (1969, Lemma 1 in Section 8.7).

6

2.2

Firms

Both large and small …rms incur variable costs and …xed costs. All …rms incur a common
and constant marginal cost, which is normalized to zero, whereas the …xed cost may di¤er
across the two types of …rms.
Small Firms The pro…t of the small …rm is expressed by
S (i) = pS (i)qS (i)f,where S (i) is the pro…t of small …rm i, and f is the …xed cost of the small …rm.
Plugging pS (i) from equation (2) into the above pro…t function, S (i) can be rewrittenas[qS (i)]2 [ 1 QS + 3 QL ]qS (i) f;(4)
S (i) = qS (i)Each small …rm maximizes its pro…t with respect to its quantity qS (i), taking as given
the total output of small …rms, QS , and that of large …rms, QL .
The free entry and exit of small …rms pins the equilibrium pro…t of the small …rm tozero:[qS (i)]2 [ 1 QS + 3 QL ]qS (i) f = 0:
(5)
S (i) = qS (i)Large Firms The pro…t of the large …rm is
Z !m
m
m
m
(pm
L (!)qL (!)
L (! m ; qL ( )) =F )d!,

0m
where mL (! m ; qL ( )) is the pro…t of large …rm m, and F is the …xed production cost for
the large …rm to produce one variety.
mm
Substituting pm
L (! m ; qL ( ))L (!) from equation (3) into the above pro…t function,
can be rewritten as()Z!m
X Z !kkmm
qL (!)d!
qLm (!)d!
(6)
3 QS
2
L (! m ; qL ( )) =

Z !m

k6=m

[qLm (!)]2 d!

0

0

2

0

Z !m

2

qLm (!)d!

F !m:

0

The large …rm maximizes its pro…t with respect to both its product range, ! m , and thequantity of each variety, qLm (!). Unlike a small …rm, the large …rm does not take thetotal output of large …rms, QL , as given. Instead, eachPlarge …rmas given the total
R !k takesmmk
output of other large …rms, QL , de…ned by QL
k6=m 0 qL (!)d!, in addition to
QS :

7

2.3

De…nition of Equilibrium

Since consumers are passive, an equilibrium state arises if no …rm wishes to unilaterally
deviate. Note that we consider market competition by the large …rms and small …rms,
in which all …rms behave simultaneously, including the entry decision of the small …rms.
Our solution concept is Nash equilibrium.
An equilibrium is characterized by the mass of small …rms, N , the output of eachsmall …rm, qS (i), 8i 2 N , the product range of each large …rm, ! m , 8m = 1; : : : ; M ,and the output of each variety for the large …rm, qLm (!), 8! 2 [0; ! m ], 8m = 1; : : : ; M ,such that each …rm maximizes the pro…ts given other …rms’behavior and no other small…rms can earn positive pro…ts due to free entry. An equilibrium is called a mixed market
equilibrium if QS > 0 and QL > 0.

2.4

WelfareThe social welfare comprises consumer welfare and producer surplus. Consumer welfare
is measured by
CW = U I;
and hence, the change in consumer welfare is the same as that in consumer utility.
Since small …rms earn zero pro…t, producer surplus is given by the sum of all large…rms’pro…ts:
M
X
m
PS =
L:
m=1

Then, social welfare is the sum of consumer welfare and producer surplus:
SW = U

I+

M
X

m
L:

(7)

m=1

3

Equilibrium Analysis

In this section, we derive the equilibrium results and conduct the comparative static
analysis to investigate the impact of the entry of a large …rm on the other …rms’behavior
and welfare.

3.1

Derivation of Mixed Market Equilibrium

Small Firms’ Pro…t Maximization and Entry A small …rm only accounts forthe impact of the market’s total production because its own impact on the market is
negligible. Thus, it does not internalize its externality into its production. The small

8

…rm maximizes its pro…t given by equation (4) with respect to its output qS (i). The…rst-order condition of a small …rm with respect to its output qS (i) is
2 qS (i)1 QS

3 QL = 0:

which implies that
1 QS

qS (i) = qS =

3 QL

:
2
All small …rms choose the same level of output. Hence, QS = N qS .
Using equation (2), the price of the small …rm can be expressed bypS (i) =

3 QL

2 +

1N

:

(8)

(9)

Accordingly, the equilibrium price of the small …rm decreases with the mass of small …rms
and the total output of large …rms.
Entry and exit are free for small …rms. Using equation (5) after plugging in (8) and(9), we can express the mass of small …rms (N ) as a function of the total output of large…rms (QL ):
#
"s
1
(
2 :
(10)
N (QL ) =3 QL )f
1
As shown by the above expression, the mass of small …rms decreases with the total outputof large …rms.
Substituting (10) into (8), the optimal quantity of each small …rm is
s
f
qS =
.
Owing to free entry and exit, the quantity produced by a small …rm is independent of the
behavior of large …rms. In other words, the aggregate behavior of small …rms responds
to the change in the market condition only by adjusting the competitive fringe. (See also
Lemma 1.)
Plugging qS into (9) yields the equilibrium price of small …rms:
p
pS =
f.

Large Firms’Pro…t Maximization and Variety Choice Unlike small …rms, large…rms impose a non-negligible impact on the market. Large …rm m maximizes its pro…tgiven by equation (6) with respect to its output. The …rst-order condition of large …rm
m with respect to the quantity of its !th variety, qLm (!), isZ !mmm
2 qL (!) 2 2
qLm (v)dv
= 0:
(11)
3 QS
2 QL
0

9

Equation (11) implies that qLm (!) = qLm for each ! 2 [0; ! m ]. Therefore, we haveZ !mqLm (v)dv = ! m qLm :0

Substituting the above symmetry property into equation (11), we solve the optimal quantity of each variety (qLm ) as a function of the expected aggregate output of small …rms
(QS ), the expected output of other large …rms (QL m ), and its own product range (! m ):
3 QS

qLm =

2( +

2 QL

m

(12)

:

2!m)

Everything else being equal, an increase in …rm m’s product range (larger ! m ) results
in a reduction in the quantity of each variety (smaller qLm ), implying the cannibalizatione¤ect.
Large …rm m maximizes its pro…t with respect to its product range. After substituting(12), the …rst-order condition of large …rm m with respect to its product range, ! m , isr2( +2!m) =

F

(

3 QS

2 QL

m

):

(13)

From equations (12) and (13), we obtain the optimal output per variety for the large
…rm:
s
F
qLm =
;
which is determined only by the …xed cost of large …rms and the demand parameters but
is independent of its product range and other …rms’behavior. (See also the discussionmafter Lemma
PM 1.) This equation also implies that qL = qL for each m = 1; :::; M . Thus,m
QL = k6=m ! k qL , and equation (13) can be expressed as2( +Substituting qL =2!m) =

r

F

(

3 QS

2

M
X

! k qL ):

(14)

k6=m

p
F= , the above equation can be rearranged intor
MX(
!k 2 ;2!m =3 QS )
2
F
k=1

(15)

which implies that ! m = ! for each m = 1; ::; M . Hence, we can express the productrange ! as a function of the expected aggregate output of small …rms QS :p
=F (2
3 QS )!(QS ) =:(16)
2 (M + 1)
10Mixed Market Equilibrium The above analysis indicates that …rms behave symmetp
rically within the group of large …rms
and
that
of
small
…rms.
That
is,
q
f=
(i)
=
q
=
S
S
p
m
for each i 2 [0; N ], qL = qL = F= for each m = 1; ::; M , and ! m = ! for eachm =p1; ::; M . Then, the total output of large …rms can be expressed byp
QL = M !qLm =M ! F= , and the aggregate output of small …rms is QS = N qS = N f = . Plugging
in these two expressions, (10) and (16) can be expressed as
s
"s
#
1
F
N =
(
) 2 ,
3M !
f
1
p
p
=F (
f= ) 23N! =.
2 (M + 1)
These two equations determine the equilibrium mass of small …rms N and the equilibrium
product range of a large …rm ! . A mixed market equilibrium exists if and only if N > 0
and ! > 0. Assuming this, the mass of small …rms and product range of large …rms inequilibrium aresppp[ 2 (M + 1)
2 [ 2 (M + 1) f
3M ]3M F ]N =,2
f
1 2+( 1 2
3 )M
r
p
p
p
( 1
2 ( 1 F
3)
3 f)
.
! =
2
F
1 2+( 1 23 )MPlugging N , ! , qS and qLm into equation (3), the price of the large …rm in equilibrium
is
p
p
p
p
[
(
)
2
(
2
1
3
1 F
3 f )]pL =F+.
2
1 2+( 1 23 )MSubstituting the equilibrium range of varieties ! , the output of each variety qLm and theequilibrium price of large …rms pL into equation (6), we obtain the equilibrium pro…t of
the large …rm:
p
p
p
( 1
2 ( 1 F3)
3 f) 2].
L = 2[
2
1 2+( 1 23 )MThe total output isQ =1
1 2+( 1 2

2M

3.2

p

[( 2

2
3 )M

f M( 1 +p
f +( 1

3)

2

p

3)

2 3) +

F ]g.

2(

p
2
f)

Conditions for a Unique Mixed Market Equilibrium

We focus on a market in which large and small …rms coexist. To ensure that the market
is mixed and stable in equilibrium, we need to guarantee that11

(i) large …rms earn positive pro…ts with a positive product range (that is, ! > 0);(ii) the mass of small …rms is positive (that is, N > 0); and(iii) the equilibrium is (locally) stable (that is, with slight perturbation from the mixed
market equilibrium, …rms’strategies would converge back to the equilibrium values)14 .
We identify the conditions for the existence and stability of a mixed market equilibrium through the aggregate reactions between large and small …rms. (A precise analysis
is available in Appendix B.1.)
The aggregate reaction of large …rms to the competitive fringe is
QL (QS ) = maxf0;

M(
2 (M + 1)

and that of the competitive fringe to large …rms is
QS (QL ) = maxf0;

1

(

1

p
2
F

p
f
2

3 QS )g;

3 QL )g:

(17)

(18)

Figure 1 depicts the aggregate reactions of large and small …rms under di¤erent conditions. When goods are substitutes across large and small …rms (that is, 3 > 0),
large and small …rms’ aggregate reactions behave like strategic substitutes. Figure 1a
depicts a stable mixed market equilibrium. First, the coexistence of large and small…rms in equilibrium requires that the two aggregate reaction functions intersect. Second, the (local) stability of the intersection requires that the slope of QL (QS ) is ‡atterthan the slope of QS (QL ) (that is, 3 M= 2 (M + 1) < 1 = 3 ). Even when the two aggregate reaction functions intersect, if the stability condition does not hold, and upona slight perturbation, the mixed market equilibrium is unstable, and …rms will deviate
from the intersection of QL (QS ) and QS (QL ) to the equilibrium with only small …rms
or the equilibrium with only large …rms (see Figure 1b). If the two aggregate reaction
functions do not intersect, there is no mixed market equilibrium even when the stability
condition is satis…ed. As shown in Figure 1c, for instance, there is only one equilibrium in which small …rms operate. This is the case when the …xed production cost ofa large …rm F is too high, meaning that large …rms are at a technological disadvantage
relative to small …rms. Similarly, there is only one market equilibrium in which large…rms operate if the …xed cost of a small …rm f is too high. Therefore, a unique mixed
market equilibrium requires that (i) QL (QS ) and QS (QL ) intersect and (ii) the slope of
QL (QS ) is ‡atter than the slope of QS (QL ). Given the second condition, the …rst condition implies that the intercept of QS (QLp
) should be smallerp
than the intercept of QL (QS )2 F )= 3 ), and the intercept
on the horizontal axis (that is, (2 f )= 1 < (of QS (Qp than the intercept of QL (QS ) on the vertical axis (that is,p L ) should be larger(
2 f )= 3 > (2 F )M=[ 2 (M + 1)]).
14

The local stability here is the pseudo-stability often seen in textbook discussions of Cournot models. In the dynamic adjustment, we assume non-cooperative behavior among large …rms and collective
behavior among small …rms. (See Appendix B.1 for details.) We would like to thank the anonymousreferees for their clarifying suggestions.

12When 3 < 0, in aggregate, large …rms and small …rms behave like strategic complements (See Figure 1d). To ensure the existence of a unique market equilibrium inthis case, the slope of QL (QS ) should be ‡atter than the slope of QS (QL ) (that is,
3 M= 2 (M + 1) > 1 = 3 ).
[Figure 1 around here]
Summing up the above discussion, we formally establish the conditions in the following
proposition.
Proposition 1 A unique equilibrium exists, which is (locally) stable and features a mixed
market, if thepfollowing three conditions
hold:p2 f ) 3 ;p(i) (2 p F) 1 > ((ii) (
2 f )(M + 1) 2 > (2 F )M 3 ;2(iii) 1 2 + ( 1 23 )M > 0.
These conditions require that (a) the intensity of preferences for the di¤erentiatedgoods market, i.e., , is su¢ ciently high relative to the …xed costs of large and small…rms, i.e., F and f , and (b) the substitutability/complementarity across large and small…rms, i.e., j 3 j, is not too strong relative to the substitutabilities among large …rms, i.e.,
1 , and among small …rms, i.e., 2 .
Proof. See Appendix ??.
Conditions (i) and (iii) guarantee that large …rms operate with a positive productrange (! > 0), conditions (ii) and (iii) guarantee that the mass of small …rms is positive(N > 0), and condition (iii) is a necessary condition for the (local) stability of the mixed
market equilibrium.
These three conditions impose constraints on the parameters. No complementarity isallowed among the varieties of small …rms and among those of large …rms, that is, 1 > 0
and 2 > 0.15 In addition, the intensity of preferences
for thepdi¤erentiated goods market
pshould be su¢ ciently large (that is, > 2 f and > 2 F ).16 Conditions (ii) and(iii) imply that the range of parameters narrows with an increase in the number of large…rms M . In other words, the market condition for small …rms becomes more severe whenmore large …rms coexist.
We can express the three conditions parametrically by 1 = 3 and 2 = 3 , which represent the relative substitutability for the small …rm and that for the large …rm, respectively.
15

Notice that the concavity of a large …rm’s pro…t with respect to its product range requires 2 > 0.
To be precise, large …rm m’s pro…t after substituting equation (12) can be expressed as m
L = ! m [(
m 2
2
Q
Q
)
=4(
+
!
)
F
].
Thus,
>
0
guarantees
thatthesecond-orderderivative
of m
3 S
2 L
2 m
2
L
m 2
3
with respect to the product range ! m is negative, i.e.,
2 (3 QS
2 QL ) =2( + 2 ! m ) < 0.
Intuitively, 2 > 0 implies that the infra-marginal e¤ect of adding a new variety is negative.
Given that 2 > 0, if 1 < 0, condition (iii) of Proposition
1 is violated.
Thus, 1 > 0.
pp
16
Given that 1 > 0 and 2 > 0, if either < 2 f or < 2 F , then at leastthree
p one of the pconditions in Proposition 1 would be violated. For instance, when 3 > 0, if 2 f << 2 F,
condition (i) is violated. The results are analogous in other cases.

13

Figure 2 shows the areas where a unique mixed market equilibrium exists. The horizontal
axis and vertical axis are 1 = 3 and 2 = 3 , respectively. Note that, according to condition(iii), j 3 j should not be too high relative to 1 and 2 . If the varieties are highly substitutable across large and small …rms, then the mixed market equilibrium is unstable, andmarket competition typically causes one group to survive.17 If the varieties are highly
complementary between large and small …rms, then the total quantity of both groups of…rms would always expand. However, it is unrealistic to observe quantities and varieties
expand, exhausting all the income for products in a particular market.
In addition, if 1 = 2 = 3 > 0, condition (i) implies that f > F . That is,if the varieties are equally substitutable among all …rms, the existence of large …rms
requires that the …xed cost of a small …rm be larger than the large …rm’s per-variety …xed
production cost. If the large and small …rms share the same …xed production cost, thiscondition indicates that the small …rm’s entry cost should be positive to ensure that thelarge …rm enjoys economies of scope (Parenti, 2016). Even when the small …rm’s entry
cost is close to zero, the large …rm may also exist if it incurs a lower …xed cost to produce
a variety.
[Figure 2 around here]

3.3

The Impact of a Large Firm’s Entry

Now, we investigate the impact of a large …rm’s entry, that is, an exogenous increase in
the number of large …rms from M to M + 1. We have some preliminary results that will
be useful for subsequent analysis. Formally, based on qS and qL , we have the following
lemma:
Lemma 1 The entry of a large …rm has no impact on (i) the output and price level ofthe small …rm (that is, pS and qS ) or (ii) the output of each variety of the large …rm (thatis, qL ).
The …rst outcome is in line with the traditional monopolistic competition model.
Here, small …rms can be considered monopolistic competitors. The free entry and exitof small …rms shifts the demand curve such that there is only one equilibrium quantity
at which the marginal revenue is equal to the marginal cost and the price is equal to theaverage cost.
To see the second result, …rst note that pro…t maximization of a large …rm is accompanied by the following two conditions. First, for each variety, the marginal revenue from17

If 3 is high, according to equation (2), an increase in QL would substantially shift the demand
for a small …rm downward. Thus, small …rms would have much lower incentives to operate so that thecompetitive fringe signi…cantly shrinks, which further stimulates the expansion of large …rms. Eventually,
all small …rms would be driven out of the market, and only large …rms would operate. Analogously, there
is another equilibrium in which only small …rms operate.

14

output (price plus infra-marginal e¤ect) equals the marginal cost (M C), adjusted by theinternalization e¤ects on other varieties, that is,
pm
L

qLm
|{z}

= MC +

Infra-marginalm
2 ! m qL

| {z }:

Internalization

Second, the pro…t from adding one more variety is zero. The second condition implies
that the price equals the average cost, adjusted by the (average) cannibalization e¤ect,that is,m! m qLm:pm
L = M C + F=qL +| 2 {z}|{z}
Average Cost

Average Cannibalization

Note that the cannibalization e¤ect can be regarded as an aggregation of internalizatione¤ects, as each output of the marginal variety imposes a negative externality on its own
varieties, which implies that the average cannibalization must equal the internalization.
Combining the above two conditions, we obtain the following general condition: theinfra-marginal e¤ect equals the di¤erence between the marginal cost and the average cost.
Under linear demand and a constant marginal cost, we obtain a simpler condition: theinfra-marginal e¤ect ( qLm ) equals the average …xed cost (F=qLm ), which pins down theunique per-variety quantity.
3.3.1

Substitution between the Big and the Small ( j > 0, j = 1; 2; 3)

Based on the results in Lemma 1, we …rst investigate the impact of a large …rm’s entry on…rms’behavior when products are substitutes across large and small …rms. Proposition
2 establishes the results.
Proposition 2 The entry of a large …rm (that is, an increase in the number of large…rms, M ) will exert the following impact on …rms’behavior:(i) The competitive fringe shrinks (that is, N decreases);(ii) The price, product range, and pro…t of each large …rm increase if the substitutability across large and small …rms is su¢ ciently high relative to the substitutabilities within
the groups of large and small …rms, but otherwise, they all decline (formally, pL , ! and2
2
2
L increase if 1 2 < 3 , decrease if 1 2 > 3 , and remain the same if 1 2 = 3 ); and(iii) The total output rises if the substitutability among small …rms is higher than thesubstitutability across large and small …rms and declines otherwise (formally, Q increases
if 1 > 3 , decreases if 1 < 3 , and remains the same if 1 = 3 ).
Proof. See Appendix B.2.
We explain the economic intuition of the above results as follows. First, the entry ofa large …rm raises the total output of large …rms, which generates a negative impact on
the demand faced by small …rms by substituting some of the goods produced by small…rms. Consequently, some small …rms are squeezed out, and hence, the competitive fringe
shrinks. This argument explains Proposition 2-(i).
15Second, a large …rm’s entry may raise or reduce large …rms’prices and pro…ts. The
impact of a large …rm’s entry on incumbent large …rms depends primarily on two opposing e¤ects. On one hand, the entry of a large …rm intensi…es the competition amonglarge …rms through a negative substitution e¤ect. On the other hand, the shrinkage ofthe competitive fringe generates a positive impact on the demand faced by large …rms. If
the substitutability across large and small …rms is su¢ ciently high relative to the substitutability among large …rms and that among small …rms, the positive squeezing-out e¤ectoutweighs the negative substitution e¤ect, leading to an increase in the demand each large…rm faces. Therefore, each large …rm’s price, product range, and pro…t increase. If thesubstitutability across large and small …rms is relatively small, the negative substitutione¤ect dominates, resulting in a reduction in each large …rm’s price, product range, andpro…t.
For a precise illustration of the mechanism, we examine how the inverse demand ofeach variety shifts after a large …rm’s entry. Plugging the equilibrium values into theinverse demand, we have the following relationships:
pS =

qS

1 QS

3 QL ;

(19)

pL =

qL

2 QL

3 QS :(20)

Recall that in Lemma 1, qL , pS and qS are independent of the number of large or small…rms, which implies that the small …rms’inverse demand for each variety does not shift
after entry and thus that 1 QS + 3 QL remains constant. Hence, for an increase in large…rms’aggregate quantity, QL , small …rms’aggregate quantity is substituted by QS =( 3 = 1 ) QL . Now, we can see how the demand for a variety o¤ered by large …rms shifts
after entry. For an increase QL ; the substitution e¤ect is
2 QL , whereas the feedback
2e¤ect through the change in small …rms’aggregate quantity is
3 QS = ( 3 = 1 ) QL .
2
Thus, the total e¤ect is QL [ 231 2 ]= 1 : If
1 2 >3 , i.e., the substitution level
between large and small …rms is relatively small, then the negative substitution e¤ect
dominates the positive feedback e¤ect such that the large …rm’s price pL falls. Since therevenue from each variety decreases, the equilibrium product range of the large …rm !
must shrink, and consequently, the pro…t of each large …rm L decreases. The cases of18
2
2
1 2 < 3 and 1 2 = 3 are analogous.
Finally, the change in total output is determined by the comparison between the substitutability among small …rms ( 1 ) and the substitutability across large and small …rms
( 3 ). If the substitutability across large and small …rms is higher than the substitutability
among small …rms, that is, 3 > 1 , then the squeezing-out e¤ect is so strong that theincrease in the output of large …rms is insu¢ cient to o¤set the shrinkage of the competitive fringe. Thus, total output is reduced. Conversely, the squeezing-out e¤ect is modest,
meaning that the entry of a large …rm increases total output. Precisely, by equation (19),
it is straightforward that QS = ( 3 = 1 ) QL , and thus, the change in total output
18

When 1 2 = 23 , our result is consistent with that of Parenti (2016), who assumes that the productsubstitution is the same across large and small …rms, that is, 1 = 2 = 3 = .

16

is Q = QS + QL = ( 13 )= 1 QL , which is positive if 1 > 3 and negative
otherwise.
Our established model and …ndings are related to two major strands of the literature.
First, our model is closely linked to recent works on the coexistence of oligopolists andmonopolistic competitors. Representative works include Shimomura and Thisse (2012)
and Parenti (2016). Accounting for the endogenous income from the greater aggregate
pro…ts of large …rms, Shimomura and Thisse (2012) demonstrate that entry always increases the large incumbents’ quantities and pro…ts. Our result is complementary totheirs in that we focus on the interaction of demand substitutabilities but abstract fromthe income e¤ects on the di¤erentiated market using quasi-linear utility. Another related
paper is Parenti (2016), who analyzes the impact of trade liberalization when large andsmall …rms coexist in the same market. Based on a quasi-linear utility model with theassumption of symmetric substitutability among all varieties, he …nds that the entry of alarge …rm has no impact on large incumbents’behavior in a closed economy. While thisresult is consistent with our …ndings, we go a step further and introduce asymmetric substitutabilities across large and small …rms, showing how demand substitutability a¤ects
the impact of a large …rm’s entry.
Second, our result also sheds light on the literature on price-increasing competition.19
A key idea in this literature is that as the market becomes more competitive, the slope ofeach …rm’s demand curve can be steeper, especially in discrete choice models of productdi¤erentiation (Chen and Riordan (2008) call this the price sensitivity e¤ect). Our model
does not capture this e¤ect due to the assumption of representative consumers. We …ndthat large …rms’demand function may shift upward after the entry of a large …rm due tothe interaction between large and small …rms. The e¤ect of the exit of small …rms canbe large enough to dominate the negative e¤ects of a large …rm’s entry on the demand
for large incumbents.
Now let us consider how the entry of a large …rm in‡uences consumer welfare, producer
surplus and social welfare. We normalize the entry cost of the large entrant to zero.
Proposition 3 establishes the results.
Proposition 3 The entry of a large …rm generates the following impact on welfare:(i) Consumer welfare rises if
p

p
( 1 F

p
f ) + E( 1 2
3

2
3 )[

M
(M + 1)+]>0
D(M ) D(M + 1)

and falls otherwise;(ii) Producer surplus rises if(M + 1)D2 (M + 1)

M>0
D2 (M )

19

See Chen and Riordan (2007, 2008), Perlo¤ and Salop (1985), Perlo¤, Suslow, and Seguin (2005),
and Gabaix, Laibson, and Li (2005).

17and falls otherwise; and(iii) Social welfare rises if
p

p
( 1 F

p
f) +
3

and falls otherwise,
where D(M ) = 1 2p+ ( 1 p2
and E = ( 1
2 ( 1 F3)
1.

1 2 E(

1
1+)>0
D(M ) D(M + 1)

23 )M
p > 0, D(M + 1) =3

1 2 +( 1 223 )(M + 1) > 0,f ) > 0 according to the conditions in Proposition

Proof. See Appendix B.3.
Proposition 3 shows that a large …rm’s entry will only conditionally raise consumer
welfare, producer surplus and social welfare. The change in consumer welfare is determined by three e¤ects. First, a portion of the small …rms’ product range is replacedby large …rms’output in the representative consumer’s consumption. This replacemente¤ect a¤ects consumer welfare because the quantity of one variety di¤ers across large andsmall …rms. Second, the entry of a large …rm a¤ects consumer welfare through the change
in the total variety range. The consumer bene…ts from an increase in the variety range.20
Third, the change in consumer welfare depends on the price charged by large …rms, which
is determined by the relative substitution levels. These three e¤ects determine the signof the change in consumer welfare.
Speci…cally, the impact of a large …rm’s entry on consumer welfare can be decomposedasq2
CW = ( S
| 2

q2
qL2
) VS + L V{z2} | 2 {z }Replacement E¤ect

Variety E¤ect1[QL (M ) + QL (M + 1)] pL :
2
{z
}
|

(21)

Price E¤ect

Here, VS is the (negative) change in the variety range of small …rms, and V is thechange in the total variety range. The …rst term characterizes the replacement e¤ect.
The utility obtained from each variety of the small …rm and that of the large …rm are
qS2 =2 and qL2 =2, respectively. Therefore, when the VS of small …rms’product range
is replaced by that of large …rms, the utility change is measured by the …rst term. The
second term characterizes the variety e¤ect. If the total variety range expands ( V ),the representative consumer’s utility gain is ( qL2 =2) V because (s)he consumes morevarieties from large …rms. The third term characterizes the price e¤ect arising from thesubstitutabilities among varieties. As shown in Proposition 2, the sign of this term is
ambiguous, depending on the relative substitution level across large and small …rms. If
the substitutability among large and small …rms is such that a large …rm’s entry raises
the price charged by large …rms (pL ), then the consumer su¤ers from a higher price.
20

See Ottaviano, Tabuchi and Thisse (2002) for a detailed discussion of the consumer’s “love forvariety.”18

Conversely, consumer welfare improves if the price charged by large …rms falls after alarge …rm’s entry. Moreover, owing to free entry and exit, the price charged by small…rms does not change with the number of large …rms, and thus, there is no price e¤ectof small …rms on consumer welfare. Therefore, the total impact on consumer welfare
depends on the comparison of the replacement e¤ect, the variety e¤ect, and the pricee¤ect.
The impact on producer surplus depends simply on the comparison between the pro…tchange due to the increase in the total quantity of large …rms (positive marginal e¤ect) andthe pro…t change due to the price change (infra-marginal e¤ect). A su¢ cient condition for
an increase in producer surplus is 1 2 < 23 , which means that the marginal and inframarginal e¤ects are both positive. Producer surplus decreases only if the infra-marginale¤ect is negative and outweighs the marginal e¤ect. Formally,1
P S = [pL (M ) + pL (M + 1)
2

1
2 qL ] QL + [QL (M ) + QL (M + 1)] pL .
2

Summing up the change in consumer welfare and that in producer surplus, the impact
on social welfare can be expressed byq2
SW = ( S
| 2

qL2
q2
1
) VS + L V + [pL (M ) + pL (M + 1)
{z2
} | 2 {z } |2{zReplacement E¤ect

Variety E¤ect

Quantity E¤ect

2 qL ] QL : (22)}Similar to CW , the …rst two terms here characterize the replacement e¤ect and varietye¤ect on consumer welfare. The third term, which is always positive, measures thequantity e¤ect (on producer surplus) arising from the increase in large …rms’total output.
Note that the price e¤ect on consumer welfare and the infra-marginal increase in producer
surplus o¤set one another. In consequence, the change in social welfare depends on thereplacement e¤ect and variety e¤ect on consumer welfare (the …rst two terms) and themarginal increase in producer surplus (the third term). Figure 3 provides a graphicaldepiction of these three e¤ects.
[Figure 3 around here]
We also analyze a few speci…c cases. First, we consider the case in which 1 = 2 = 3 ,i.e., the products are equal substitutes across large and small …rms. In this case, the large…rm’s price does not change, and thus, the price e¤ect on consumer welfare is zero. The
total output does not change, either. Moreover, by Proposition 1-(i), the condition for thecoexistence of large and small …rms implies that f > F . Therefore, we have CW < 0,
P S > 0, and SW > 0.21 Although the total consumption does not change, theproportion of large …rms’products increases in the consumption bundle, and therefore,
the representative consumer su¤ers from an increased average price. Producer surplus
21

Assuming the same product substitution across large and small …rms, Shimomura and Thisse (2012)
also …nd that a large entrant raises producer surplus and social welfare.

19

increases because of the expansion of the total output of large …rms. Social welfare
improves because the deterioration of consumer welfare is dominated by the increase inproducer surplus that results from the entry of a large …rm.22
Now, we consider the case in which the …xed cost of a small …rm is the same as
the per-variety …xed cost of a large …rm, i.e., f = F . In this case, the coexistence
condition in Proposition 1 implies that 1 > 3 , and thus, the total variety range (andtotal output) increases, while the replacement e¤ect is zero because per-variety output isthe same across large and small …rms. With the marginal increase in producer surplus,social welfare improves. A su¢ cient condition for the improvement of consumer welfareis 1 2 > 23 , which implies a decline in the large …rm’s price. However, a su¢ cientcondition for an increase in producer surplus is 1 2 < 23 , as shown above.
We summarize these …ndings in the following corollary:
Corollary 1 Let the parameters satisfy the conditions of Proposition 1 so that the market
is mixed in equilibrium. We consider the following special cases:(i) If the products are equally substitutable across large and small …rms ( 1 = 2 = 3 ),the entry of a large …rm results in a deterioration of consumer welfare, an increase inproducer surplus and improved social welfare;(ii) If the …xed cost of a small …rm is the same as the per-variety …xed cost of a large…rm (f = F ), the entry of a large …rm improves social welfare. A su¢ cient condition
for consumer welfare to increase is 1 2 > 23 , whereas a su¢ cient condition for producer
surplus to increase is 1 2 < 23 .
In the following example, we show the case in which the entry of a large …rm worsens
consumer welfare and social welfare.
Example 1 Consider a mixed market with 2 large …rms and a host of small …rms. Let
the size of the di¤erentiated goods market = 1, the preference for diversity = 1, thesubstitutability among small …rms’ products 1 = 0:4, the substitutability among large…rms’ products 2 = 0:625, the substitutability across small and large …rms’ products
3 = 0:5, a small …rm’s …xed cost f = 0:16, and a large …rm’s …xed production cost of one
variety F = 0:1444. In this case, 1 2 = 23 , and the conditions for the coexistence of large
and small …rms are satis…ed. The change in consumer welfare is 0:056, and the change
in social welfare is 0:032. Here, the consumer su¤ers from a negative replacement e¤ect
and a reduction of the total variety range, and these two negative e¤ects dominate themarginal increase in producer surplus.
3.3.2

Complementarity between the Big and the Small ( 1 > 0,
3 < 0)

2 > 0, andHaving examined the case in which all the products in the di¤erentiated goods market
are substitutes, we now consider the case in which the varieties produced by the large
22This case is also discussed in Parenti (2016).

20

…rms complement those produced by the small …rms; this complementarity between large
and small …rms implies 3 < 0. In this case, the results in the previous section are
mainly robust, except for the change in the number of small …rms and social welfare. We
establish the results in the following proposition.
Proposition 4 When large …rms’goods are complementary to those of small …rms (thatis, 3 < 0), the entry of a large …rm (that is, an increase in M ) generates the following
impact:(i) The impact on the behavior of large …rms is the same as in Proposition 2, but thecompetitive fringe expands (that is, N increases).
(ii) The impact on consumer welfare and producer surplus is also ambiguous, basedon the same conditions as in Proposition 3. Nevertheless, social welfare always increases.
First, the entry of a large …rm raises the total output of large …rms, which generates a
positive impact on the demand faced by small …rms, which provide goods that complement
large …rms’ goods. Consequently, more small …rms enter the market, and hence, thecompetitive fringe expands. The expansion of the competitive fringe thus generates a
positive e¤ect on the price charged by large …rms, according to equation (20). Thus, thechange in the price charged by the large …rm is determined by the comparison between
the negative substitution e¤ect
2 and the positive indirect e¤ect from the expansionof the competitive fringe, which is represented by 23 = 1 . Therefore, the impact on thelarge …rm is also determined by the same conditions as in Proposition 2.
Compared with the case in which all products are substitutes, it is more likely that
consumer welfare improves when the products are complementary between large and small…rms in that both the total variety range of large …rms and that of small …rms increase,i.e., the …rst two terms in expression (21) are positive. The only possible negative e¤ect
originates from the increases in the large …rms’price when 1 2 < 23 . In this case, thethird term of expression (21) is negative.
Because the large …rms’behavior is a¤ected in the same way as in the case in which
products are substitutes across large and small …rms, the impact on producer surplus
here follows the same intuition as in the previous section.
Finally, social welfare always increases here because there is no (negative) replacemente¤ect and variety expansion occurs in both large and small …rms.

3.4

Implications

From the perspective of demand substitution, our …ndings may explain the distinct impacts of a large …rm’s entry across industries featuring the coexistence of large and small…rms. In the Japanese supermarket industry, large stores directly compete with one another, and thus, substitutability among large …rms is high. However, small stores are
insulated from such direct competition with large …rms and bene…t from the increasein consumer ‡ows from the opening of large stores; hence, the substitutability acrosslarge and small …rms is slightly negative. As implied by Proposition 4, in this case, the21feedback e¤ect may not be strong enough to o¤set the substitution e¤ect for large incumbents, which are expected to shrink their production scale and experience a decline inpro…ts. Moreover, because of the weak complementarity between large and small stores,
the entry of a large …rm increases the demand for small supermarkets. As a result, thecompetitive fringe may expand. Our theoretical analysis is consistent with the empirical evidence in Igami (2011), as discussed in the introduction.23 In the South Korean
furniture industry, large furniture makers di¤erentiate themselves from others through aunique design, while small furniture makers’products are less di¤erentiated. Thus, thesubstitutability between large …rms is weaker than that between large and small …rms.
Therefore, according to Proposition 2, the feedback e¤ect could outweigh the substitution e¤ect, resulting in a rise in demand for the large …rms’products and a signi…cantsqueezing-out e¤ect on small …rms.24
Our results also provide several policy implications for regulations on large …rm’s entry.
Many countries, such as France and the United Kingdom, enforce laws and regulations
to restrict the entry of large …rms. Our results partially support such restrictions interms of welfare because the entry of a large …rm may reduce consumer welfare or social
welfare in some cases; however, we also show that the entry of large …rms may improvewelfare under certain conditions. With di¤erent levels of substitution across large andsmall …rms, incumbent …rms react di¤erently to the entry of a large …rm. When products
are substitutes across large and small …rms ( 3 > 0), if the substitutability between large
and small …rms is su¢ ciently weak ( 23 < 1 2 ), a large …rm’s entry squeezes out a few
small …rms and intensi…es the competition among large …rms, and consumers bene…t fromreduced prices. Such entry behavior may be welcome in terms of the social bene…ts, withmodest costs paid by small …rms. However, if the cross-substitution e¤ect is su¢ ciently
strong ( 23 > 1 2 ), more small …rms will be squeezed out, and consumers are more
likely to su¤er due to the increased price. If the products of large and small …rms arecomplementary ( 3 < 0), such entry behavior actually expands small …rms’ business.
Owing to the ambiguity of the impact of a large …rm’s entry, our results suggest thatgovernments should implement more meticulous and ‡exible policies to address di¤erent
entry types.
23

Similar impacts on small local stores are observed after the entry of Wal-Mart in the United States.
Basker (2005) …nds that approximately four of two hundred small competitors closed within …ve years
after Wal-Mart’s entry, implying a minor e¤ect on the number of small stores. However, the e¤ect of
Wal-Mart’s entry on large competitors is more complex and subtle due to the di¤erence in their strategies
with respect to location choice and product mix, etc. (Basker, 2007). These asymmetric and endogenous
choices by large supermarkets are not captured in the present paper and may require further study. We
thank an anonymous referee for suggesting the connection between our model and the “Wal-Mart e¤ect.”24
The report on IKEA’s entry did not mention how the entry a¤ected the number of small furnituremakers but showed how small makers’ sales decreased. Given the sharp decline in their sales, somesmall …rms are very likely to shut down in the medium to long run. In addition, when small …rms areheterogenous in e¢ ciency, the least e¢ cient ones are predicted to exit …rst, while those remaining small…rms with higher e¢ ciency would experience revenue declines.

224Discussion

In the basic model, we have assumed that (i) large …rms are multiproduct …rms, (ii) there
is no vertical di¤erentiation across large and small …rms, and (iii) the marginal costs oflarge and small …rms are zero. In this section, we relax each of these three assumptions
and discuss the robustness of our results in the following extensions.
Single-product Large Firms In this subsection, we discuss the case when large…rms are single-product …rms, that is, ! = 1. our results are robust, and the change ineach large …rm’s output is qualitatively the same as the change in the large …rm’s variety
choice in our original model. Speci…cally, the impact of a large …rm’s entry on …rms’
behavior is the same as in Proposition 2. The welfare e¤ects are also ambiguous, withslight changes in the conditions. The conditions for the unique mixed market equilibrium
are also slightly modi…ed. We summarize the results in the following proposition.
Proposition 5 When both large and small …rms are single-product …rms,(i) There exists a unique mixed market equilibrium if the following three conditions
hold:2(i-1) 1 (2 + 2 ) + ( 1 20;3 )M >p(i-2) [2 + 2 (M + 1)p 3 M ] > 2 f [2 + 2 (M + 1)]; andp2f > [ 1 (2 + 2 ) + ( 1 2(i-3) ( 13) + 2 33 )M ] F=( + 2 ).
(ii) The impact of a large …rm’s entry on …rms’behavior is the same as in Propositions
2 and 4.
(iii) The entry of a large …rm generates the following
p impact on social welfare:(iii-1) Consumer welfare rises if ( 1 3 ) 2 3 f +( + 2 )( 1 2 23 )J[M=K(M )+(M + 1)=K(M + 1)] > 0 and falls otherwise;(iii-2) Producer surplus rises if (M + 1)=K 2 (M + 1) M=K 2 (M ) > 0 and falls
otherwise; andp(iii-3) Social welfare rises if( 1
f + 1 ( + 2 )(2 + 2 )J[1=K(M )+3)
2 3
1=K(M + 1)] > 2F K(M )K(M p+ 1)=J and falls otherwise,2
where J = ( 1f > 0, K(M ) = 1 (2 + 2 ) + ( 1 23) + 2 33 )M > 0, and2K(M + 1) = 1 (2 + 2 ) + ( 1 23 )(M + 1) > 0, according to the conditions in (i).
Proof. See Appendix B.4.
Vertical Di¤erentiation across Large and Small Firms In many industries,
large …rms and small …rms are vertically di¤erentiated. In this case, consumer preference23can be represented by
Z N
U = S
qS (i)di +

L

0

1

2
If

Z N

M Z !mX

qS (i)di

0

2

0

m=1
2

qLm (!)d!

2

2

"M Z
X !m
m=1

Z N

[qS (i)] di

2 m=1

0

qLm (!)d!

0

M Z !m
X

2

#2

3

Z N

qS (i)di0[qLm (!)]2 d!

0

"M Z
X !m
m=1

qLm (!)d! + q0 :0

L > (<) S , large …rms’products have a higher (lower) quality than small …rms’.

The (inverse) demand for small …rm i and that for large …rm m’s !th variety are
Z N
M Z !mX
qLm (!)d!;qS (i)dipS (i) = S
qS (i)
3
1
0

pL (!) =L

qLm (!)

2

M Z !m
X

m=1

0

m=1

qLm (!)d!

0

3

Z N

qS (i)di:0

In equilibrium, a small …rm’s quantity (qS ) and a large …rm’s per variety quantity (qL )are the same as in the basic model. The competitive fringe and a large …rm’s productrange are

N =
! =

s

r

p
p
2 [ 2 (M + 1) f
L 3M )
2
1 2+( 1 2
3 )M
p
p
p
2 ( 1 F
S 3)
3 f)
.
2
1 2+( 1 23 )M

( S 2 (M + 1)
f
( L 1
F

p
M
F]
3

,

We can obtain the conditions for a unique mixed market equilibrium following the same
approach that is adopted in the basic model. Moreover, it is readily veri…ed that thechange in …rms’ production behavior and welfare is almost the same as in the basicmodel. We summarize the results of this extension in the following proposition.
Proposition 6 When large and small …rms are vertically di¤erentiated, that is, S 6= L ,(i) There exists a unique mixed market equilibrium if the following three conditions
hold:
p
p
;
(i-1) ( L 2p F ) 1 > ( S 2 f ) 3p(i-2) ( S 2 f )(M + 1) 2 > ( L 2 F )M 3 ;2(i-3) 1 2 + ( 1 23 )M > 0.
(ii) The impact of a large …rm’s entry on …rms’behavior is the same as in Propositions
2 and 4.
(iii) The entry of a large …rm generates an ambiguous impact on welfare, depending ontheS 3)
pconditions
p samep as those in Proposition 3, except that E represents ( L 1
2 ( 1 F
3 f ), which is positive according to condition (i-1).
Proof. See the above discussion.
24#

Cost Asymmetry across Large and Small Firms In the basic model, we haveassumed zero marginal cost for both large and small …rms. In reality, however, marginal
cost is generally positive and may di¤er across large and small …rms. We consider two
possible extensions. In the …rst case, …rms incur a constant marginal cost. In the second
case, …rms incur an increasing variable cost.
In the …rst case, the variable costs of the large and small …rms are represented by cL qL
and cS qS , respectively. Under this setting, the aggregate reactions of large and small …rms
are
p
M
(
cL 2
F
3 QS )g;2 (M + 1)p
1
QS (QL ) = maxf0; (cS 2
F
3 QS )g:QL (QS ) = maxf0;

1

In equilibrium, a small …rm’s quantity (qS ) and a large …rm’s per variety quantity (qL )are the same as in the basic model. The competitive fringe and a large …rm’s productrange are slightly modi…ed:
s
p
p
p
(
cS ) 2 (M + 1) (cL ) 3 M 2 [ 2 (M + 1) f
3M F ],N =2
f
1 2+( 1 2
3 )M
r
p
p
p
(
cS ) 1 (cL ) 3 2 ( 1 F
3 f)
.
! =
2
F
1 2+( 1 23 )MThe conditions for a unique mixed market equilibrium and comparative analysis are
qualitatively the same as in the basic model. Speci…cally,
Proposition 7 When large and small …rms incur a constant marginal cost, that is, cL
for a large …rm and cS for a small …rm,(i) There exists a unique mixed market equilibrium if the following three conditions
hold:
p
p
cL 2 f ) 3p;(i-1) (cS 2 p F ) 1 > ((i-2) (cL 2 f )(M + 1) 2 > (cS 2 F )M 3 ;2(i-3) 1 2 + ( 1 23 )M > 0.
(ii) The impact of a large …rm’s entry on …rms’behavior is the same as in Propositions
2 and 4.
(iii) The entry of a large …rm generates an ambiguous impact on welfare, dependingon the same conditions
cS ) 1p as thosepp in Proposition 3, except that E represents ((cL ) 3 2 ( 1 F
3 f ), which is positive according to condition (i-1).
In the second case, we extend the basic model by assuming that variable costs arequadratic in quantity. Speci…cally, the variable costs of the large and small …rms can be
represented by tL qL2 and tS qS2 , respectively. Similar to the …rst case, our results are also
robust under this modi…ed setting.
25

In equilibrium, a small …rm’s quantity (qS ) and a large …rm’s per variety quantity(qL ) are respectivelyppqS = f =( + tS ), and qL = F=( + tL ).

The competitive fringe and a large …rm’s product range aresp
2[ 2 (M + 1) f ( + tS )+ tS [ 2 (M + 1)
3M ]
N =
2
f
1 2+( 1 2
3 )M
r
p
p
2( 1 F ( + tL )f ( + tS ))+ tL ( 1
3)
3
.
! =
2
F
1 2+( 1 2
3 )M

3M

p

F ( + tL )],In this case, the conditions for a unique mixed market equilibrium and comparative
analysis are also qualitatively the same as in the basic model. Speci…cally,
Proposition 8 When large and small …rms incur a variable cost that is quadratic inquantity, that is, tL qL2 for a large …rm and tS qS2 for a small …rm,(i) There exists a unique mixed market equilibrium if the following three conditions
hold:
p
p
2 ( + tp(i-1) (
2p( + tL )F ) 1 > (S )f ) 3 ;(i-2) (
2 ( + tS )f )(M + 1) 2 > (2 ( + tL )F )M 3 ;2(i-3) 1 2 + ( 1 23 )M > 0.
(ii) The impact of a large …rm’s entry on …rms’behavior is the same as in Propositions
2 and 4.
(iii) The entry of a large …rm generates an ambiguous impact on welfare, dependingon the3)pas those in Proposition 3, except that E represents ( 1p same conditions
2( 1 ( + tL )F( + tS )f ), which is positive according to condition (i-1).
3

To sum up, the entry of a large …rm will qualitatively exert the same impact achievedby Propositions 1, 2, 3, and 4 in all of the above four extensions.

5

Conclusion

In this paper, we considered a market with large and small …rms that o¤er products that
may have di¤erent levels of substitution. In this market structure, we investigated theimpact of a large …rm’s entry on incumbent …rms’ behavior and welfare. As we notedat the beginning of the article, di¤erent industries featuring this mixed market structure
exhibit distinct impacts on large and small …rms. We focus on the relative strength of two
opposing e¤ects on the incumbent large …rms. The …rst e¤ect is the negative substitutione¤ect, and the second e¤ect is the positive squeezing-out (expanding) e¤ect due to theshrinkage (expansion) of small …rms when the products are substitutes (complements)
across big and small …rms. Which of these two e¤ects dominates hinges on the di¤erent
26substitutabilities between large and small …rms. The welfare e¤ect is also ambiguous,depending on the di¤erent levels of substitution and technology.
Many countries enforce laws to restrict the entry of large …rms’entry. However, ouranalysis indicates that the government should be cautious in adopting and enforcing suchlaws because such restrictions may not always be welfare-improving. If small …rms areable to di¤erentiate themselves from large …rms or are essentially complementary to large…rms, the entry of a large …rm into the local market may improve consumer welfare andsocial welfare. Our …nding suggests that policy makers should account for the speci…cdemand characteristics in di¤erent industries or markets when implementing zoning laws.
Despite the prevalence of markets with large and small …rms, only a few theoreticalstudies have been conducted on this topic thus far, such as Shimomura and Thisse (2012)
and Parenti (2016). This paper attempts to explore this market structure from theperspective of demand substitutability. Future research is advisable in several directions.
This paper assumes symmetric technology among large …rms and among small …rms, but
the heterogeneity of technology may also be present within large …rms or among small…rms. In addition, this paper employs quasi-linear utility with quadratic subutility to
express the representative consumer’s preference. Considering a more general framework
would be another interesting project. Finally, this paper investigates the change in social
welfare caused by a large …rm’s entry, but a comprehensive welfare analysis of this mixed
market structure –such as the optimal number of small …rms given the number of large…rms or the optimal number of large …rms – could be an important project for future
research.

27

Appendix
A

Concavity of the Quadratic Subutility Function

Before we show the concavity of the quadratic subutility function, several notations must
be introduced.25
Let qS : [0; N ] ! R and q0S : [0; N ] ! R be two square-integrable functions. Let
Z N
0qS (i)qS0 (i)di:hqS ; qS iS
0

Indeed, h ; iS is an inner product.
0m
For each m = 1;
; M , let qm
L : [0; ! m ] ! R and qL : [0; ! m0 ] ! R be square0
integrable functions. Let qL (q1L ;
; qM
(q01
; q0M
L ) and qL
L;L ). Let
M Z !mX
00hqL ; qL iL
qLm (!)qLm (!)d!:m=1

0

Also, h ; iL is an inner product.
Then, our utility function, that is, equation (1), can be written as
U = u(qS ; qL ) + q0
= (hqS ; 1iS + hqL ; 1iL )2(hqS ; qS iS + hqL ; qL iL )1
( 1 hqS ; 1iS 2 + 2 hqL ; 1iL 2 + 2 3 hqS ; 1iS hqL ; 1iL ) + q0 ;2
where 1 denotes the constant function, which is equal to 1 for each i 2 [0; N ], eachm = 1;; M , and each ! 2 [0; ! m ]. Recall that q0 is the consumption of the numeraire.
In the following, we show the concavity of the quadratic subutility function: if(i) =NP+ 1 > 0,(ii) =( m ! m ) + 2P> 0, and(iii) ( =N + 1 )[ =( m ! m ) + 2 ] > 23 ,then, for each 2 (0; 1), each q (qS ; qL ), and each q0 (q0S ; q0L ),u( q + (1

)q0 )

u(q)

(1

)u(q0 ) > 0.

By simple manipulation,u( q + (1)q0 )u(q) (1)u(q0 )(1)
f (k qS k2S + k qL k2L ) + 1 h qS ; 1iS 2=
2+ 2 h qL ; 1iL 2 + 2 3 h qS ; 1iS h qL ; 1iL g;
25

We thank an anonymous referee for suggesting a rigorous approach to show the concavity of theutility function on which this proof is based.

28

where qS qS q0S and qL qL q0L , and k kS and k kL are the norms associatedwith the inner products.
By the Cauchy-Schwarz inequality,
k qS k2S k1k2S

h qS ; 1iS 2 ;k qL k2L k1k2L

h qL ; 1iL 2 :Since
k1k2S = h1; 1iS =k1k2L = h1; 1iL =Z N

12 di = N;0M
X Z !m

m=1

then

2

1 di =

0

M
X

!m;

m=1

h qS ; 1iS 2 h qL ; 1iL 2
+ P
:
N
m !m

k qS k2S + k qL k2L
Therefore,u( q + (1)q0 )u(q) (1)u(q0 )
(1)
+ 1 h qS ; 1iS 2 +
2
N
P

m !m

+

h qL ; 1iL 2 + 2 3 h qS ; 1iS h qL ; 1iL :2Since 2 (0; 1), then (1) > 0. Note that conditions (i), (ii), and (iii) imply thatfor each h qS ; 1iS and each h qL ; 1iL ,N+

1h qS ; 1iS 2 +

P

m !m

+

2

h qL ; 1iL 2 + 2 3 h qS ; 1iS h qL ; 1iL > 0.

Thus, the concavity of the quadratic subutility function is shown.

B

ProofsB.1

Proof of Proposition 1

Q.E.D.

In the following, we conduct the dynamic adjustment process to identify the conditions
for the (local) stability of the mixed market equilibrium.
Large …rms choose their behavior non-cooperatively. Within the …rm, we assume
that each large …rm coordinates among the per-variety quantity qL and product range!. Thus, the per-variety quantity in equilibrium is by equations (12) and (13). Denote29

Qm! m qL as the total quantity of large …rm m (m = 1; :::; M ). We conduct the dynamicLadjustment of large …rms with respect to the aggregate behavior of each large …rm, that
is, Qm
L (m = 1; :::; M ).
To simplify the analysis, we assume collective behavior among small …rms, and the
dynamic adjustment of small …rms is p
in terms of the aggregate quantity, QS . A small…rms’s quantity in equilibrium is qS = f = by equations (8) and (10). The total output
of small …rms, QS , is adjusted by free entry and exit.
Gathering large and small …rms, we derive the stability conditions through the dynamic adjustment process with respect to the total quantity of small …rms, QS , and thetotal quantity of large …rm m, Qm
L (m = 1; :::; M ).
To be precise, we have the following dynamic adjustment system in terms of QS , andm
QL (= ! m qL ) (m = 1; :::; M ):Q_ S (QS ; Q1L ; :::; QM
L ) = d0 [Q_ 1L (QS ; Q1L ; :::; QM
L ) = d1 [

p
2
f

p
2
F

1 QS

3

M
X

Qm
L ];

m=1
1
2 (2QL +

3 QS

M
X

Qm
L )];

m=2

..
.
1
M
Q_ M
L (QS ; QL ; :::; QL ) = dM [

p
F
2

M
2 (2QL +

3 QS

M
X1

Qm
L )]:

m=1

where “ ”denotes the di¤erentiation with respect to time, i.e., Q_ S = dQS =dt, and Q_ m
L =mdQL =dt, (m = 1; :::; M ); and di (i = 0; 1;; M ) is the positive coe¢ cient measuring the
speed of dynamic adjustment. Without loss of generality, set di = 1, i = 0; 1;; M.
The Jacobian matrix of the above dynamic adjustment system is
10@ Q_ S =@QS @ Q_ S =@Q1L@ Q_ S =@QM
LC
B @ Q_ 1 =@Q@ Q_ 1L =@Q1L@ Q_ 1L =@QM
S
B
L
L C
J(QS ; Q1L ; :::; QM
)
=
C
B
..
..
..
L
..
A
@
.
.
.
.
M
M
1
M
M
_
_
_
@ QL =@QL@ QL =@QS @ QL =@QL
0
1
B
B
=B
@

1

..
.

3

3

3

2 2
..
.
2

3

...

2

..
.
2 2

C
C
C:
A

To guarantee the stability of the system, the Jacobian matrix should be negative
de…nite. That is, for any x (x0 ; x1 ; x2 ; :::; xM ) 6= (0; 0; :::; 0), xT Jx > 0.
| {z }
M +1

30

Let XPM

i=1 xi . Then,T

2
1 x0 + 2 2

x Jx =

MX

x2i + 2 3 x0

i=1

2
1 x0 + 2 2

=

M
X

M
X

xi + 2 2

i=1

By Cauchy-Schwarz inequality,

xi xj

i=1;i6=jx2i + 2 3 x0 X + 2 2

i=1

M
X

M
X

xi xj

i=1;i6=jPM22
i=1 xi = X =M . Thus,

M +1
2
2
1 x0 + 2 3 x0 X +2X

xT Jx =

M

x0 X

=

x0 X

=

A

1

3

3

2 (M + 1)=M

x0
X

x0
X

;

3
.
(M
+
1)=M
3
2
Note that J is negative de…nite if and only if A is positive de…nite, which requiresthat 1 > 0, 2 > 0, and 1 2 (M + 1)=M > 23 .
Therefore, the mixed market equilibrium is stable when 1 > 0, 2 > 0, and 1 2 (M +1) > 23 M .
Applying the stability condition, the denominator of N and ! is positive. Therefore,
N > 0 ifwhere A1(
and ! > 0 ifp2f )(M + 1) 2 > ((pF) 1 > (2

p2F )M 3 ;p2f ) 3:

Therefore, to ensure the existence of a unique equilibrium that is stable and features a26
mixed market,
p the following threep conditions should be satis…ed :(i) (2 p F) 1 > (2 f ) 3 ,p
2 F )M 3 , and(ii) (
2 f )(M + 1) 2 > (2(iii) 1 2 (M + 1) > 3 M .
Q.E.D.

B.2

Proof of Proposition 2.

ppp
2
Let D(M )
+
(
)M
,
and
E
(
)
2
(
1 2
1 2
1
3
1 F
3 f ). By
3assuming that a unique mixed market equilibrium arises before and after a large …rm’s
entry, D(M ) > 0, D(M + 1) > 0, and E > 0 (Proposition 1).
26

These three conditions imply that

1 > 0 and2 > 0.

31

Deriving from the equilibrium values, the changes of the variables when the numberof large …rms increase from M to M + 1 are as follows:qS = 0; pS = 0; qL = 0,
s
2 3E
N =
< 0:
D(M )D(M + 1) f
And
pL =

2
2( 3

1 2 )E

D(M )D(M + 1)

,
r

( 23
1 2 )E,
D(M )D(M + 1) F
22
11
1 2 )E
2( 3
=
(
+
).
L
D(M )D(M + 1) D(M ) D(M + 1)

! =2
Thus, if 1 2 < 23 , then pL > 0, ! > 0, andpL < 0,
L > 0. If 1 2 > 3 , then2
! < 0, andpL = 0, ! = 0, andL < 0. If 1 2 = 3 , thenL = 0.
Recall that Q is the total quantity in the di¤erentiated goods market. Thus,

Q =Thus, if
Q.E.D.

B.3

3 , then1 >Q > 0. If

2( 1

3 )E

D(M )D(M + 1)

1 <3 , then.

Q < 0. If

1 =3 , then

Q = 0.

Proof of Proposition 3.

Consumer welfare, producer surplus, and social welfare can be expressed as
CW (M ) = Q=

2

2(N qS2 + M ! qL2 )(N qS2 + M ! qL2 ) +

1

1

2

QS2 +

2
2
2M E
P S(M ) = pL QL M F ! = 2,
D (M )SW (M ) = CW (M ) + P S(M ).

32

QS2
2

2

2

2

QL2 +

QL2

3 QS QL

3 QS QL ,

pS QS

pL QL

The impact of an increase from M to M + 1 on consumer welfare isq2
qS2(N (M + 1) N (M )) + L [(M + 1)! (M + 1) M ! (M )]22+ 1 (QS2 (M + 1) QS2 (M )) + 2 (QL2 (M + 1) QL2 (M ))22+ 3 (QS (M + 1)QL (M + 1) QS (M )QL (M ))q2
Q (M + 1) + QS (M )(qS2 qL2 ) VS + L V + S( 1 QS + 3 QL )=222
Q (M + 1) + QL (M )+ L
( 2 QL + 3 QS )2
qL2
Q (M + 1) + QS (M )22qL ) VS +
(qS
V + S
pS
=
2
2
2
Q (M + 1) + QL (M )
+ L
pL
2
qL2
QL (M + 1) + QL (M )22qL ) VS +
(qS
V
pL ,
=
2
2
2

CW =

where VS = N (M + 1) N (M ), and V = (N (M + 1) + (M + 1)! (M + 1))(N (M ) + M ! (M )).
By substituting the equilibrium values of the endogenous variables into the above
equation,
p
p
p
M +1
M
2E
2
f
( 1 F
f ) + E( 1 2
+
]g.
CW =
3
3 )[
2D(M )D(M + 1)
D(M ) D(M + 1)ppp2
CW > 0.
If( 1 F p 3 fp)+E( 1p2
3 )[M=D(M )+(M +1)=D(M +1)] > 0, then2
Otherwise, if p ( 1 pFf )+E( 1 23 p
3 )[M=D(M )+(M +1)=D(M +1)] < 0, then2
CW < 0. If( 1 F
3 f ) + E( 1 23 )[M=D(M ) + (M + 1)=D(M + 1)] = 0,then CW = 0.
The impact of an increase from M to M + 1 on producer surplus is
PS =

2
2E [

M +1
D2 (M + 1)

M].
D2 (M )

Thus if (M + 1)=D2 (M + 1) M=D2 (M ) > 0, then P S > 0. Otherwise, if (M +1)=D2 (M + 1) M=D2 (M ) < 0, then P S < 0. If (M + 1)=D2 (M + 1) M=D2 (M ) = 0,then P S = 0
The impact of an increase from M to M + 1 on social welfare is
p
p
p
1
1
2E
SW =
[
( 1 F
f ) + 1 2 E(+)] .
32D(M )D(M + 1)
D(M ) D(M + 1)pppThus, if
( p1 F p 3 f )p+ 1 2 E[1=D(M ) + 1=D(M + 1)] > 0, then SW > 0.
Otherwise,(p1 F
SW < 0.
3 f ) + 1 2 E[1=D(M ) + 1=D(M + 1)] < 0, then
p if
p
If
( 1 F
SW = 0. Q.E.D.
3 f ) + 1 2 E[1=D(M ) + 1=D(M + 1)] = 0, then
33

B.4

Proof of Proposition 5.

(i) When the large …rm supplies one variety, the computation shows that
s
f
;
qS =
( 1
qL =
1 (2 +
pL = ( +
s

N =

2)

3) + 2 3

p

2
3 )M

2) + ( 1 2( 1
1 (2 +[2 +f3) + 2 3

f23 )M2) + ( 1 2;p
2 f [2 +2
1 (2 + 2 ) + ( 1 23 )M

2 (M + 1)

f

;
p

3M ]

2 (M + 1)]:

To ensure the existence of a unique mixed market equilibrium, we should have the stabilityof the equilibrium, N > 0 and L > 0.
Following the same approach in Appendix (B.1), we have the following dynamic adjustment system in terms of QS , and qLm (m = 1; :::; M ):Q_ S (QS ; qL1 ; :::; qLM )p2
f= d2 [

q_L1 (QS ; qL1 ; :::; qLM )= d3 [

qL1

2

1 QS

3

M
X

f
];
qS

qLm

m=1
1
2 (2qL +

3 QS

M
X

qLm )];

m=2..
.
q_LM (QS ; qL1 ; :::; qLM ) = dM +2 [

2

qLM

M
2 (2qL +

3 QS

M
X1

qLm )]:

m=1

where “ ” denotes the di¤erentiation with respect to time, i.e., Q_ S = dQS =dt, and q_Lm =dqLm =dt, (m = 1; :::; M ); and di (i = 0; 1;; M ) is the positive coe¢ cient measuring the
speed of dynamic adjustment. Without loss of generality, set di = 1, i = 0; 1;; M.
The Jacobian matrix of the above dynamic adjustment system is01@ Q_ S =@QS @ Q_ S =@qL1@ Q_ S =@qLM
B @ q_1 =@Q@ q_L1 =@qL1@ q_L1 =@qLM C
S
BC
L
J(QS ; qL1 ; :::; qLM ) = B
C
..
..
.
.
..
..
@
A
.
.
M
M
1
M
M
@ q_L =@QS @ q_L =@qL
@ q_L =@qL
0
1
B
B
=B
@

1

..
.

3

3

3

2( +
..
.

3

2)

2

34

..

2

.

..
.
2( +

2)

C
C
C:
A

To guarantee the stability of the system, the Jacobian matrix should be negative de…nite.
That is, for any x (x0 ; x1 ; x2 ; :::; xM ) 6= (0; 0; :::; 0), xT Jx > 0.
| {z }
M +1
PM
Let X
i=1 xi . Then,T

2
1 x0 + 2(

x Jx =+

2)MX

x2i + 2 3 x0

i=1

2
1 x0 + 2(

=

+

2)

M
X

M
X

xi + 2 2

i=1

By Cauchy-Schwarz inequality,xT Jx =
=

PM

xi xj

i=1;i6=j2
1 x0 + 2 3 x0 X + (2

+

1

x0 X
x0 X

M
X

2
2
i=1 xi = X =M . Thus,

B

M +1
2
2 )X
M
3

2 +

3

=

xi xj

i=1;i6=jx2i + 2 3 x0 X + 2 2

i=1

M
X

x0
X

2 (M + 1)=M

x0
X

;

3
.
2
+
(M
+ 1)=M
3
2
Note that J is negative de…nite if and only if B is positive de…nite, which requires
2that 1 > 0 and 1 (2 + 2 ) + ( 1 23 )M > 0.
Therefore, the mixed market equilibrium is stable when 1 > 0 and 1 (2 + 2 ) +2( 1 2
3 )M > 0.
In addition, we should have N > 0, and L > 0; i.e., F < pL qL . Accordingly,pf [2 + 2 (M + 1)];[2 + 2 (M + 1)
M
]
>
2
3
s
p
F
2
:
f > [ 1 (2 + 2 ) + ( 1 2( 13) + 2 33 )M ]( + 2)

where B1p20(ii) Let J( 1
f , and K(M 0 )3) + 2 3
1 (2 + 2 ) + ( 1 23 )M . By (4-i),
J > 0, K(M ) > 0 and K(M + 1) > 0. From the equilibrium, the changes of the variables
when the number of large …rms increase from M to M + 1 are derived as follows:qS = 0;pS = 0;N =(2 + 2 ) 3 J
K(M )K(M + 1)sf

< 0 if35

3 > 0;

and > 0 if

3 < 0:

Moreover,( 23
1 2 )JqL =;K(M )K(M + 1)
( + 2 )( 23
1 2 )JpL =;K(M )K(M + 1)
2
( + 2 )( 23
11
1 2 )J[+]:
L =K(M )K(M + 1)
K(M ) K(M + 1)
which are positive if

1 2 <

2
3 , negative if

Q =which is positive if

1 >2
3 , and constant if

1 2 >

1 2 =

2
3 : Also(2 + 2 )( 13 )J;K(M )K(M + 1)

3 , negative if

3 ; and constant if

1 <

1 =

3:

(iii) The associated consumer welfare, producer surplus and social welfare are
CW (M ) = Q
P S(M ) =2

(N qS2 + M j1

j qL2 )

2

QS2

2

2

QL2

3 QS QLpS QS

pL QL ;

22M J;
K 2 (M )SW (M ) = CW (M ) + P S(M )F M:

The impact of an increase from M to M + 1 on consumer welfare is
J
M
M +1
2
[ + ( + 2 )( 1 2
+
)]:
3 )J(
2K(M )K(M + 1)
K(M ) K(M + 1)pwhere =
( 1 3 ) 2 3 f . CW is positive if +( + 2 )( 1 2 23 )J[M=K(M )+(M + 1)=K(M + 1)] > 0, and is negative otherwise.
The impact of a marginal increase of M on producer surplus is
CW =

PS =

( + 2 )J 2
M +1
[ 2
K(M )K(M + 1) K (M + 1)M]:
K 2 (M )which is positive if (M + 1)=K 2 (M + 1) M=K 2 (M ) > 0 > 0, and is negative otherwise.
The impact of a marginal increase of M on social welfare is
SW =

J
[ +
2K(M )K(M + 1)

1(

+

2 )(2

+

2 )J(

1
1
+
)]
K(M ) K(M + 1)

F.

which is positive if + 1 ( + 2 )(2 + 2 )J[1=K(M ) + 1=K(M + 1)] > 2F K(M )K(M +1)=J, and is negative otherwise.
Q.E.D.
36

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38

Figures

Figure 1a

Figure 1b

A Stable Mixed Market Equilibrium

An Unstable Mixed Market Equilibrium(Substitutes)(Substitutes)

Figure 1c

Figure 1d

No Mixed Market Equilibrium

A Stable Mixed Market Equilibrium(Substitutes)(Complements)

Figure 1 Existence and Stability of a Mixed Market Equilibrium

IFigure 2. Conditions for a Unique Mixed Market Equilibrium

II

𝑞2
2

2

𝑞
2

VL0

VS0
Before Entry

∆VL
VL1

2
○

1
○
𝑞2
2

𝑞2
2

3
○

VS1
After Entry

1 Replacement Effect○
2 Variety Effect○
3 Quantity Effect (on Producer Surplus)○

Figure 3 Social Welfare Change (Assume f > F)

III